lst:=[
# ::Package:: 

# ::Title:: 
#Algebraic Function Integration Problems Involving Linears


# ::Subsection::Closed:: 
#Integrands of the form (c (a+b x)^n)^(m/2)


# Integrands of the form (a+b*x)^n where n is a half-integer 
[(a + b*x)^(5/2), x, 1, (2*(a + b*x)^(7/2))/(7*b)],
[(a + b*x)^(3/2), x, 1, (2*(a + b*x)^(5/2))/(5*b)],
[sqrt(a + b*x), x, 1, (2*(a + b*x)^(3/2))/(3*b)],
[1/sqrt(a + b*x), x, 1, (2*sqrt(a + b*x))/b],
[1/(a + b*x)^(3/2), x, 1, -2/(b*sqrt(a + b*x))],
[1/(a + b*x)^(5/2), x, 1, -2/(3*b*(a + b*x)^(3/2))],


# Integrands of the form (c*(a+b*x)^2)^n where n is a half-integer 
[(c*(a + b*x)^2)^(5/2), x, 1, ((a + b*x)*(c*(a + b*x)^2)^(5/2))/(6*b)],
[(c*(a + b*x)^2)^(3/2), x, 1, ((a + b*x)*(c*(a + b*x)^2)^(3/2))/(4*b)],
[(c*(a + b*x)^2)^(1/2), x, 1, ((a + b*x)*sqrt(c*(a + b*x)^2))/(2*b)],
[1/(c*(a + b*x)^2)^(1/2), x, 1, ((a + b*x)*log(a + b*x))/(b*sqrt(c*(a + b*x)^2))],
[1/(c*(a + b*x)^2)^(3/2), x, 1, -((a + b*x)/(2*b*(c*(a + b*x)^2)^(3/2)))],
[1/(c*(a + b*x)^2)^(5/2), x, 1, -((a + b*x)/(4*b*(c*(a + b*x)^2)^(5/2)))],


# Integrands of the form (c*(a+b*x)^3)^n where n is a half-integer 
[(c*(a + b*x)^3)^(5/2), x, 1, (2*(a + b*x)*(c*(a + b*x)^3)^(5/2))/(17*b)],
[(c*(a + b*x)^3)^(3/2), x, 1, (2*(a + b*x)*(c*(a + b*x)^3)^(3/2))/(11*b)],
[(c*(a + b*x)^3)^(1/2), x, 1, (2*(a + b*x)*sqrt(c*(a + b*x)^3))/(5*b)],
[1/(c*(a + b*x)^3)^(1/2), x, 1, -((2*(a + b*x))/(b*sqrt(c*(a + b*x)^3)))],
[1/(c*(a + b*x)^3)^(3/2), x, 1, -((2*(a + b*x))/(7*b*(c*(a + b*x)^3)^(3/2)))],
[1/(c*(a + b*x)^3)^(5/2), x, 1, -((2*(a + b*x))/(13*b*(c*(a + b*x)^3)^(5/2)))],


# Integrands of the form (c/(a+b*x))^n where n is a half-integer 
[(c/(a + b*x))^(5/2), x, 1, -((2*(c/(a + b*x))^(5/2)*(a + b*x))/(3*b))],
[(c/(a + b*x))^(3/2), x, 1, -((2*(c/(a + b*x))^(3/2)*(a + b*x))/b)],
[(c/(a + b*x))^(1/2), x, 1, (2*sqrt(c/(a + b*x))*(a + b*x))/b],
[1/(c/(a + b*x))^(1/2), x, 1, (2*(a + b*x))/(3*b*sqrt(c/(a + b*x)))],
[1/(c/(a + b*x))^(3/2), x, 1, (2*(a + b*x))/(5*b*(c/(a + b*x))^(3/2))],
[1/(c/(a + b*x))^(5/2), x, 1, (2*(a + b*x))/(7*b*(c/(a + b*x))^(5/2))],


# Integrands of the form (c/(a+b*x)^2)^n where n is a half-integer 
[(c/(a + b*x)^2)^(5/2), x, 1, -(((c/(a + b*x)^2)^(5/2)*(a + b*x))/(4*b))],
[(c/(a + b*x)^2)^(3/2), x, 1, -(((c/(a + b*x)^2)^(3/2)*(a + b*x))/(2*b))],
[(c/(a + b*x)^2)^(1/2), x, 1, (sqrt(c/(a + b*x)^2)*(a + b*x)*log(a + b*x))/b],
[1/(c/(a + b*x)^2)^(1/2), x, 1, (a + b*x)/(2*b*sqrt(c/(a + b*x)^2))],
[1/(c/(a + b*x)^2)^(3/2), x, 1, (a + b*x)/(4*b*(c/(a + b*x)^2)^(3/2))],
[1/(c/(a + b*x)^2)^(5/2), x, 1, (a + b*x)/(6*b*(c/(a + b*x)^2)^(5/2))],


# Integrands of the form (c/(a+b*x)^3)^n where n is a half-integer 
[(c/(a + b*x)^3)^(5/2), x, 1, -((2*(c/(a + b*x)^3)^(5/2)*(a + b*x))/(13*b))],
[(c/(a + b*x)^3)^(3/2), x, 1, -((2*(c/(a + b*x)^3)^(3/2)*(a + b*x))/(7*b))],
[(c/(a + b*x)^3)^(1/2), x, 1, -((2*sqrt(c/(a + b*x)^3)*(a + b*x))/b)],
[1/(c/(a + b*x)^3)^(1/2), x, 1, (2*(a + b*x))/(5*b*sqrt(c/(a + b*x)^3))],
[1/(c/(a + b*x)^3)^(3/2), x, 1, (2*(a + b*x))/(11*b*(c/(a + b*x)^3)^(3/2))],
[1/(c/(a + b*x)^3)^(5/2), x, 1, (2*(a + b*x))/(17*b*(c/(a + b*x)^3)^(5/2))],


[sqrt((3 + 5*x)^2), x, 1, (1/10)*(3 + 5*x)*sqrt((3 + 5*x)^2)],
[sqrt((6 + 10*x)^2), x, 1, (1/5)*(3 + 5*x)*sqrt((3 + 5*x)^2)],

[1/sqrt((3 + 5*x)^2), x, 1, (3 + 5*x)*(log(3 + 5*x)/(5*sqrt((3 + 5*x)^2)))],
[1/sqrt((6 + 10*x)^2), x, 2, ((3 + 5*x)*log(3 + 5*x))/(10*sqrt((3 + 5*x)^2))],


[1/(sqrt(2 - 3*x)*sqrt(2 + 3*x)), x, 1, (1/3)*arcsin((3*x)/2)],
[1/sqrt((-(-2 + 3*x))*(2 + 3*x)), x, 2, (1/3)*arcsin((3*x)/2)],
[1/sqrt(4 - 9*x^2), x, 1, (1/3)*arcsin((3*x)/2)],

[1/(sqrt(-2 - 3*x)*sqrt(2 + 3*x)), x, 1, (sqrt(2 + 3*x)*log(2 + 3*x))/(3*sqrt(-2 - 3*x))],
[1/sqrt(-(2 + 3*x)^2), x, 1, ((2 + 3*x)*log(2 + 3*x))/(3*sqrt(-(2 + 3*x)^2))],


[1/(sqrt(3 - x)*sqrt(5 + x)), x, 1, -arcsin((1/4)*(-1 - x))],
[1/sqrt((-(-3 + x))*(5 + x)), x, 2, -arcsin((1/4)*(-1 - x))],
[1/sqrt(15 - 2*x - x^2), x, 1, -arcsin((1/4)*(-1 - x))],

[1/(sqrt(-3 - x)*sqrt(5 + x)), x, 1, arcsin(4 + x)],
[1/sqrt((-(3 + x))*(5 + x)), x, 2, arcsin(4 + x)],
[1/sqrt(-15 - 8*x - x^2), x, 1, arcsin(4 + x)],


# Integrands of the form (c*(a+b*x))^n where n is a half-integer 
[(c*(a + b*x))^(5/2), x, 2, (2*(c*(a + b*x))^(7/2))/(7*b*c)],
[(c*(a + b*x))^(3/2), x, 2, (2*(c*(a + b*x))^(5/2))/(5*b*c)],
[(c*(a + b*x))^(1/2), x, 2, (2*(c*(a + b*x))^(3/2))/(3*b*c)],
[1/(c*(a + b*x))^(1/2), x, 2, (2*sqrt(c*(a + b*x)))/(b*c)],
[1/(c*(a + b*x))^(3/2), x, 2, -(2/(b*c*sqrt(c*(a + b*x))))],
[1/(c*(a + b*x))^(5/2), x, 2, -(2/(3*b*c*(c*(a + b*x))^(3/2)))],


# Integrands of the form (c+d*(a+b*x))^n where n is a half-integer 
[(c + d*(a + b*x))^(5/2), x, 2, (2*(c + d*(a + b*x))^(7/2))/(7*b*d)],
[(c + d*(a + b*x))^(3/2), x, 2, (2*(c + d*(a + b*x))^(5/2))/(5*b*d)],
[sqrt(c + d*(a + b*x)), x, 2, (2*(c + d*(a + b*x))^(3/2))/(3*b*d)],
[1/sqrt(c + d*(a + b*x)), x, 2, (2*sqrt(c + d*(a + b*x)))/(b*d)],
[1/(c + d*(a + b*x))^(3/2), x, 2, -(2/(b*d*sqrt(c + d*(a + b*x))))],
[1/(c + d*(a + b*x))^(5/2), x, 2, -(2/(3*b*d*(c + d*(a + b*x))^(3/2)))],


# ::Subsection::Closed:: 
#Integrands of the form x^m (a+b x)^(n/2)


# Integrands of the form x^m*(a+b*x)^n where m is an integer and n is a half-integer 
[x*sqrt(a + b*x), x, 2, -((4*a*(a + b*x)^(3/2))/(15*b^2)) + (2*x*(a + b*x)^(3/2))/(5*b)],
[x*(a + b*x)^(3/2), x, 2, -((4*a*(a + b*x)^(5/2))/(35*b^2)) + (2*x*(a + b*x)^(5/2))/(7*b)],
[x*(a + b*x)^(5/2), x, 2, -((4*a*(a + b*x)^(7/2))/(63*b^2)) + (2*x*(a + b*x)^(7/2))/(9*b)],
[x/sqrt(a + b*x), x, 2, -((4*a*sqrt(a + b*x))/(3*b^2)) + (2*x*sqrt(a + b*x))/(3*b)],
[x/(a + b*x)^(3/2), x, 2, (4*a)/(b^2*sqrt(a + b*x)) + (2*x)/(b*sqrt(a + b*x))],
[x/(a + b*x)^(5/2), x, 2, -((4*a)/(3*b^2*(a + b*x)^(3/2))) - (2*x)/(b*(a + b*x)^(3/2))],

[x^2*sqrt(a + b*x), x, 3, (16*a^2*(a + b*x)^(3/2))/(105*b^3) - (8*a*x*(a + b*x)^(3/2))/(35*b^2) + (2*x^2*(a + b*x)^(3/2))/(7*b)],
[x^2*(a + b*x)^(3/2), x, 3, (16*a^2*(a + b*x)^(5/2))/(315*b^3) - (8*a*x*(a + b*x)^(5/2))/(63*b^2) + (2*x^2*(a + b*x)^(5/2))/(9*b)],
[x^2*(a + b*x)^(5/2), x, 3, (16*a^2*(a + b*x)^(7/2))/(693*b^3) - (8*a*x*(a + b*x)^(7/2))/(99*b^2) + (2*x^2*(a + b*x)^(7/2))/(11*b)],
[x^2/sqrt(a + b*x), x, 3, (16*a^2*sqrt(a + b*x))/(15*b^3) - (8*a*x*sqrt(a + b*x))/(15*b^2) + (2*x^2*sqrt(a + b*x))/(5*b)],
[x^2/(a + b*x)^(3/2), x, 3, -((16*a^2)/(3*b^3*sqrt(a + b*x))) - (8*a*x)/(3*b^2*sqrt(a + b*x)) + (2*x^2)/(3*b*sqrt(a + b*x))],
[x^2/(a + b*x)^(5/2), x, 3, (16*a^2)/(3*b^3*(a + b*x)^(3/2)) + (8*a*x)/(b^2*(a + b*x)^(3/2)) + (2*x^2)/(b*(a + b*x)^(3/2))],

[x^3*sqrt(a + b*x), x, 4, -((32*a^3*(a + b*x)^(3/2))/(315*b^4)) + (16*a^2*x*(a + b*x)^(3/2))/(105*b^3) - (4*a*x^2*(a + b*x)^(3/2))/(21*b^2) + (2*x^3*(a + b*x)^(3/2))/(9*b)],
[x^3*(a + b*x)^(3/2), x, 4, -((32*a^3*(a + b*x)^(5/2))/(1155*b^4)) + (16*a^2*x*(a + b*x)^(5/2))/(231*b^3) - (4*a*x^2*(a + b*x)^(5/2))/(33*b^2) + (2*x^3*(a + b*x)^(5/2))/(11*b)],
[x^3*(a + b*x)^(5/2), x, 4, -((32*a^3*(a + b*x)^(7/2))/(3003*b^4)) + (16*a^2*x*(a + b*x)^(7/2))/(429*b^3) - (12*a*x^2*(a + b*x)^(7/2))/(143*b^2) + (2*x^3*(a + b*x)^(7/2))/(13*b)],
[x^3/sqrt(a + b*x), x, 4, -((32*a^3*sqrt(a + b*x))/(35*b^4)) + (16*a^2*x*sqrt(a + b*x))/(35*b^3) - (12*a*x^2*sqrt(a + b*x))/(35*b^2) + (2*x^3*sqrt(a + b*x))/(7*b)],
[x^3/(a + b*x)^(3/2), x, 4, (32*a^3)/(5*b^4*sqrt(a + b*x)) + (16*a^2*x)/(5*b^3*sqrt(a + b*x)) - (4*a*x^2)/(5*b^2*sqrt(a + b*x)) + (2*x^3)/(5*b*sqrt(a + b*x))],
[x^3/(a + b*x)^(5/2), x, 4, -((32*a^3)/(3*b^4*(a + b*x)^(3/2))) - (16*a^2*x)/(b^3*(a + b*x)^(3/2)) - (4*a*x^2)/(b^2*(a + b*x)^(3/2)) + (2*x^3)/(3*b*(a + b*x)^(3/2))],

[sqrt(a + b*x)/x, x, 2, 2*sqrt(a + b*x) - 2*sqrt(a)*arctanh(sqrt(a + b*x)/sqrt(a))],
[(a + b*x)^(3/2)/x, x, 3, 2*a*sqrt(a + b*x) + (2/3)*(a + b*x)^(3/2) - 2*a^(3/2)*arctanh(sqrt(a + b*x)/sqrt(a))],
[(a + b*x)^(5/2)/x, x, 4, 2*a^2*sqrt(a + b*x) + (2/3)*a*(a + b*x)^(3/2) + (2/5)*(a + b*x)^(5/2) - 2*a^(5/2)*arctanh(sqrt(a + b*x)/sqrt(a))],
[1/(x*sqrt(a + b*x)), x, 1, (-2*arctanh(sqrt(a + b*x)/sqrt(a)))/sqrt(a)],
[1/(x*(a + b*x)^(3/2)), x, 2, 2/(a*sqrt(a + b*x)) - (2*arctanh(sqrt(a + b*x)/sqrt(a)))/a^(3/2)],
[1/(x*(a + b*x)^(5/2)), x, 3, 2/(3*a*(a + b*x)^(3/2)) + 2/(a^2*sqrt(a + b*x)) - (2*arctanh(sqrt(a + b*x)/sqrt(a)))/a^(5/2)],

[sqrt(a + b*x)/x^2, x, 2, -(sqrt(a + b*x)/x) - (b*arctanh(sqrt(a + b*x)/sqrt(a)))/sqrt(a)],
[(a + b*x)^(3/2)/x^2, x, 3, 3*b*sqrt(a + b*x) - (a + b*x)^(3/2)/x - 3*sqrt(a)*b*arctanh(sqrt(a + b*x)/sqrt(a))],
[(a + b*x)^(5/2)/x^2, x, 4, 5*a*b*sqrt(a + b*x) + (5/3)*b*(a + b*x)^(3/2) - (a + b*x)^(5/2)/x - 5*a^(3/2)*b*arctanh(sqrt(a + b*x)/sqrt(a))],
[1/(x^2*sqrt(a + b*x)), x, 2, -(sqrt(a + b*x)/(a*x)) + (b*arctanh(sqrt(a + b*x)/sqrt(a)))/a^(3/2)],
[1/(x^2*(a + b*x)^(3/2)), x, 3, 2/(a*x*sqrt(a + b*x)) - (3*sqrt(a + b*x))/(a^2*x) + (3*b*arctanh(sqrt(a + b*x)/sqrt(a)))/a^(5/2)],
[1/(x^2*(a + b*x)^(5/2)), x, 4, -((5*b)/(3*a^2*(a + b*x)^(3/2))) - 1/(a*x*(a + b*x)^(3/2)) - (5*b)/(a^3*sqrt(a + b*x)) + (5*b*arctanh(sqrt(a + b*x)/sqrt(a)))/a^(7/2)],

[sqrt(a + b*x)/x^3, x, 3, -(sqrt(a + b*x)/(2*x^2)) - (b*sqrt(a + b*x))/(4*a*x) + (b^2*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(3/2))],
[(a + b*x)^(3/2)/x^3, x, 3, -((3*b*sqrt(a + b*x))/(4*x)) - (a + b*x)^(3/2)/(2*x^2) - (3*b^2*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*sqrt(a))],
[(a + b*x)^(5/2)/x^3, x, 4, (15/4)*b^2*sqrt(a + b*x) - (5*b*(a + b*x)^(3/2))/(4*x) - (a + b*x)^(5/2)/(2*x^2) - (15/4)*sqrt(a)*b^2*arctanh(sqrt(a + b*x)/sqrt(a))],
[1/(x^3*sqrt(a + b*x)), x, 3, -(sqrt(a + b*x)/(2*a*x^2)) + (3*b*sqrt(a + b*x))/(4*a^2*x) - (3*b^2*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(5/2))],
[1/(x^3*(a + b*x)^(3/2)), x, 4, 2/(a*x^2*sqrt(a + b*x)) - (5*sqrt(a + b*x))/(2*a^2*x^2) + (15*b*sqrt(a + b*x))/(4*a^3*x) - (15*b^2*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(7/2))],
[1/(x^3*(a + b*x)^(5/2)), x, 5, 2/(3*a*x^2*(a + b*x)^(3/2)) + 14/(3*a^2*x^2*sqrt(a + b*x)) - (35*sqrt(a + b*x))/(6*a^3*x^2) + (35*b*sqrt(a + b*x))/(4*a^4*x) - (35*b^2*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(9/2))],


# Integrands of the form x^m*(-a+b*x)^n where m is an integer and n is a half-integer 
[sqrt(-a + b*x)/x, x, 2, 2*sqrt(-a + b*x) - 2*sqrt(a)*arctan(sqrt(-a + b*x)/sqrt(a))],
[(-a + b*x)^(3/2)/x, x, 3, -2*a*sqrt(-a + b*x) + (2/3)*(-a + b*x)^(3/2) + 2*a^(3/2)*arctan(sqrt(-a + b*x)/sqrt(a))],
[(-a + b*x)^(5/2)/x, x, 4, 2*a^2*sqrt(-a + b*x) - (2/3)*a*(-a + b*x)^(3/2) + (2/5)*(-a + b*x)^(5/2) - 2*a^(5/2)*arctan(sqrt(-a + b*x)/sqrt(a))],
[1/(x*sqrt(-a + b*x)), x, 1, (2*arctan(sqrt(-a + b*x)/sqrt(a)))/sqrt(a)],
[1/(x*(-a + b*x)^(3/2)), x, 2, -2/(a*sqrt(-a + b*x)) - (2*arctan(sqrt(-a + b*x)/sqrt(a)))/a^(3/2)],
[1/(x*(-a + b*x)^(5/2)), x, 3, -(2/(3*a*(-a + b*x)^(3/2))) + 2/(a^2*sqrt(-a + b*x)) + (2*arctan(sqrt(-a + b*x)/sqrt(a)))/a^(5/2)],

[sqrt(-a + b*x)/x^2, x, 2, -(sqrt(-a + b*x)/x) + (b*arctan(sqrt(-a + b*x)/sqrt(a)))/sqrt(a)],
[(-a + b*x)^(3/2)/x^2, x, 3, 3*b*sqrt(-a + b*x) - (-a + b*x)^(3/2)/x - 3*sqrt(a)*b*arctan(sqrt(-a + b*x)/sqrt(a))],
[(-a + b*x)^(5/2)/x^2, x, 4, -5*a*b*sqrt(-a + b*x) + (5/3)*b*(-a + b*x)^(3/2) - (-a + b*x)^(5/2)/x + 5*a^(3/2)*b*arctan(sqrt(-a + b*x)/sqrt(a))],
[1/(x^2*sqrt(-a + b*x)), x, 2, sqrt(-a + b*x)/(a*x) + (b*arctan(sqrt(-a + b*x)/sqrt(a)))/a^(3/2)],
[1/(x^2*(-a + b*x)^(3/2)), x, 3, -(2/(a*x*sqrt(-a + b*x))) - (3*sqrt(-a + b*x))/(a^2*x) - (3*b*arctan(sqrt(-a + b*x)/sqrt(a)))/a^(5/2)],
[1/(x^2*(-a + b*x)^(5/2)), x, 4, -((5*b)/(3*a^2*(-a + b*x)^(3/2))) + 1/(a*x*(-a + b*x)^(3/2)) + (5*b)/(a^3*sqrt(-a + b*x)) + (5*b*arctan(sqrt(-a + b*x)/sqrt(a)))/a^(7/2)],

[sqrt(-a + b*x)/x^3, x, 3, -(sqrt(-a + b*x)/(2*x^2)) + (b*sqrt(-a + b*x))/(4*a*x) + (b^2*arctan(sqrt(-a + b*x)/sqrt(a)))/(4*a^(3/2))],
[(-a + b*x)^(3/2)/x^3, x, 3, -((3*b*sqrt(-a + b*x))/(4*x)) - (-a + b*x)^(3/2)/(2*x^2) + (3*b^2*arctan(sqrt(-a + b*x)/sqrt(a)))/(4*sqrt(a))],
[(-a + b*x)^(5/2)/x^3, x, 4, (15/4)*b^2*sqrt(-a + b*x) - (5*b*(-a + b*x)^(3/2))/(4*x) - (-a + b*x)^(5/2)/(2*x^2) - (15/4)*sqrt(a)*b^2*arctan(sqrt(-a + b*x)/sqrt(a))],
[1/(x^3*sqrt(-a + b*x)), x, 3, sqrt(-a + b*x)/(2*a*x^2) + (3*b*sqrt(-a + b*x))/(4*a^2*x) + (3*b^2*arctan(sqrt(-a + b*x)/sqrt(a)))/(4*a^(5/2))],
[1/(x^3*(-a + b*x)^(3/2)), x, 4, -(2/(a*x^2*sqrt(-a + b*x))) - (5*sqrt(-a + b*x))/(2*a^2*x^2) - (15*b*sqrt(-a + b*x))/(4*a^3*x) - (15*b^2*arctan(sqrt(-a + b*x)/sqrt(a)))/(4*a^(7/2))],
[1/(x^3*(-a + b*x)^(5/2)), x, 5, -(2/(3*a*x^2*(-a + b*x)^(3/2))) + 14/(3*a^2*x^2*sqrt(-a + b*x)) + (35*sqrt(-a + b*x))/(6*a^3*x^2) + (35*b*sqrt(-a + b*x))/(4*a^4*x) + (35*b^2*arctan(sqrt(-a + b*x)/sqrt(a)))/(4*a^(9/2))],


# ::Subsection::Closed:: 
#Integrands of the form x^(m/2) (a+b x)^n


# Integrands of the form x^m*(a+b*x)^n where m is a half-integer and n is an integer 
[sqrt(x)*(a + b*x), x, 2, (2/3)*a*x^(3/2) + (2/5)*b*x^(5/2)],
[sqrt(x)*(a + b*x)^2, x, 2, (2/3)*a^2*x^(3/2) + (4/5)*a*b*x^(5/2) + (2/7)*b^2*x^(7/2)],
[sqrt(x)*(a + b*x)^3, x, 2, (2/3)*a^3*x^(3/2) + (6/5)*a^2*b*x^(5/2) + (6/7)*a*b^2*x^(7/2) + (2/9)*b^3*x^(9/2)],
[sqrt(x)/(a + b*x), x, 2, (2*sqrt(x))/b - (2*sqrt(a)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/b^(3/2)],
[sqrt(x)/(a + b*x)^2, x, 2, -(sqrt(x)/(b*(a + b*x))) + arctan((sqrt(b)*sqrt(x))/sqrt(a))/(sqrt(a)*b^(3/2))],
[sqrt(x)/(a + b*x)^3, x, 3, -(sqrt(x)/(2*b*(a + b*x)^2)) + sqrt(x)/(4*a*b*(a + b*x)) + arctan((sqrt(b)*sqrt(x))/sqrt(a))/(4*a^(3/2)*b^(3/2))],

[x^(3/2)*(a + b*x), x, 2, (2/5)*a*x^(5/2) + (2/7)*b*x^(7/2)],
[x^(3/2)*(a + b*x)^2, x, 2, (2/5)*a^2*x^(5/2) + (4/7)*a*b*x^(7/2) + (2/9)*b^2*x^(9/2)],
[x^(3/2)*(a + b*x)^3, x, 2, (2/5)*a^3*x^(5/2) + (6/7)*a^2*b*x^(7/2) + (2/3)*a*b^2*x^(9/2) + (2/11)*b^3*x^(11/2)],
[x^(3/2)/(a + b*x), x, 3, -((2*a*sqrt(x))/b^2) + (2*x^(3/2))/(3*b) + (2*a^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/b^(5/2)],
[x^(3/2)/(a + b*x)^2, x, 3, (3*sqrt(x))/b^2 - x^(3/2)/(b*(a + b*x)) - (3*sqrt(a)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/b^(5/2)],
[x^(3/2)/(a + b*x)^3, x, 3, -(x^(3/2)/(2*b*(a + b*x)^2)) - (3*sqrt(x))/(4*b^2*(a + b*x)) + (3*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/(4*sqrt(a)*b^(5/2))],

[x^(5/2)*(a + b*x), x, 2, (2/7)*a*x^(7/2) + (2/9)*b*x^(9/2)],
[x^(5/2)*(a + b*x)^2, x, 2, (2/7)*a^2*x^(7/2) + (4/9)*a*b*x^(9/2) + (2/11)*b^2*x^(11/2)],
[x^(5/2)*(a + b*x)^3, x, 2, (2/7)*a^3*x^(7/2) + (2/3)*a^2*b*x^(9/2) + (6/11)*a*b^2*x^(11/2) + (2/13)*b^3*x^(13/2)],
[x^(5/2)/(a + b*x), x, 4, (2*a^2*sqrt(x))/b^3 - (2*a*x^(3/2))/(3*b^2) + (2*x^(5/2))/(5*b) - (2*a^(5/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/b^(7/2)],
[x^(5/2)/(a + b*x)^2, x, 4, -((5*a*sqrt(x))/b^3) + (5*x^(3/2))/(3*b^2) - x^(5/2)/(b*(a + b*x)) + (5*a^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/b^(7/2)],
[x^(5/2)/(a + b*x)^3, x, 4, (15*sqrt(x))/(4*b^3) - x^(5/2)/(2*b*(a + b*x)^2) - (5*x^(3/2))/(4*b^2*(a + b*x)) - (15*sqrt(a)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/(4*b^(7/2))],

[(a + b*x)/sqrt(x), x, 2, 2*a*sqrt(x) + (2/3)*b*x^(3/2)],
[(a + b*x)^2/sqrt(x), x, 2, 2*a^2*sqrt(x) + (4/3)*a*b*x^(3/2) + (2/5)*b^2*x^(5/2)],
[(a + b*x)^3/sqrt(x), x, 2, 2*a^3*sqrt(x) + 2*a^2*b*x^(3/2) + (6/5)*a*b^2*x^(5/2) + (2/7)*b^3*x^(7/2)],
[1/(sqrt(x)*(a + b*x)), x, 1, (2*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/(sqrt(a)*sqrt(b))],
[1/(sqrt(x)*(a + b*x)^2), x, 2, sqrt(x)/(a*(a + b*x)) + arctan((sqrt(b)*sqrt(x))/sqrt(a))/(a^(3/2)*sqrt(b))],
[1/(sqrt(x)*(a + b*x)^3), x, 3, sqrt(x)/(2*a*(a + b*x)^2) + (3*sqrt(x))/(4*a^2*(a + b*x)) + (3*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/(4*a^(5/2)*sqrt(b))],

[(a + b*x)/x^(3/2), x, 2, -((2*a)/sqrt(x)) + 2*b*sqrt(x)],
[(a + b*x)^2/x^(3/2), x, 2, -((2*a^2)/sqrt(x)) + 4*a*b*sqrt(x) + (2/3)*b^2*x^(3/2)],
[(a + b*x)^3/x^(3/2), x, 2, -((2*a^3)/sqrt(x)) + 6*a^2*b*sqrt(x) + 2*a*b^2*x^(3/2) + (2/5)*b^3*x^(5/2)],
[1/(x^(3/2)*(a + b*x)), x, 2, -2/(a*sqrt(x)) - (2*sqrt(b)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/a^(3/2)],
[1/(x^(3/2)*(a + b*x)^2), x, 3, -(2/(a*sqrt(x)*(a + b*x))) - (3*b*sqrt(x))/(a^2*(a + b*x)) - (3*sqrt(b)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/a^(5/2)],
[1/(x^(3/2)*(a + b*x)^3), x, 4, -(2/(a*sqrt(x)*(a + b*x)^2)) - (5*b*sqrt(x))/(2*a^2*(a + b*x)^2) - (15*b*sqrt(x))/(4*a^3*(a + b*x)) - (15*sqrt(b)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/(4*a^(7/2))],

[(a + b*x)/x^(5/2), x, 2, -((2*a)/(3*x^(3/2))) - (2*b)/sqrt(x)],
[(a + b*x)^2/x^(5/2), x, 2, -((2*a^2)/(3*x^(3/2))) - (4*a*b)/sqrt(x) + 2*b^2*sqrt(x)],
[(a + b*x)^3/x^(5/2), x, 2, -((2*a^3)/(3*x^(3/2))) - (6*a^2*b)/sqrt(x) + 6*a*b^2*sqrt(x) + (2/3)*b^3*x^(3/2)],
[1/(x^(5/2)*(a + b*x)), x, 3, -(2/(3*a*x^(3/2))) + (2*b)/(a^2*sqrt(x)) + (2*b^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/a^(5/2)],
[1/(x^(5/2)*(a + b*x)^2), x, 4, -(5/(3*a^2*x^(3/2))) + (5*b)/(a^3*sqrt(x)) + 1/(a*x^(3/2)*(a + b*x)) + (5*b^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/a^(7/2)],
[1/(x^(5/2)*(a + b*x)^3), x, 5, -(2/(3*a*x^(3/2)*(a + b*x)^2)) + (14*b)/(3*a^2*sqrt(x)*(a + b*x)^2) + (35*b^2*sqrt(x))/(6*a^3*(a + b*x)^2) + (35*b^2*sqrt(x))/(4*a^4*(a + b*x)) + (35*b^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a)))/(4*a^(9/2))],


# Integrands of the form x^m*(-a+b*x)^n where m is a half-integer and n is an integer 
[sqrt(x)*(-a + b*x), x, 2, (-(2/3))*a*x^(3/2) + (2/5)*b*x^(5/2)],
[sqrt(x)*(-a + b*x)^2, x, 2, (2/3)*a^2*x^(3/2) - (4/5)*a*b*x^(5/2) + (2/7)*b^2*x^(7/2)],
[sqrt(x)*(-a + b*x)^3, x, 2, (-(2/3))*a^3*x^(3/2) + (6/5)*a^2*b*x^(5/2) - (6/7)*a*b^2*x^(7/2) + (2/9)*b^3*x^(9/2)],
[sqrt(x)/(-a + b*x), x, 2, (2*sqrt(x))/b - (2*sqrt(a)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/b^(3/2)],
[sqrt(x)/(-a + b*x)^2, x, 2, sqrt(x)/(b*(a - b*x)) - arctanh((sqrt(b)*sqrt(x))/sqrt(a))/(sqrt(a)*b^(3/2))],
[sqrt(x)/(-a + b*x)^3, x, 3, -(sqrt(x)/(2*b*(a - b*x)^2)) + sqrt(x)/(4*a*b*(a - b*x)) + arctanh((sqrt(b)*sqrt(x))/sqrt(a))/(4*a^(3/2)*b^(3/2))],

[x^(3/2)*(-a + b*x), x, 2, (-(2/5))*a*x^(5/2) + (2/7)*b*x^(7/2)],
[x^(3/2)*(-a + b*x)^2, x, 2, (2/5)*a^2*x^(5/2) - (4/7)*a*b*x^(7/2) + (2/9)*b^2*x^(9/2)],
[x^(3/2)*(-a + b*x)^3, x, 2, (-(2/5))*a^3*x^(5/2) + (6/7)*a^2*b*x^(7/2) - (2/3)*a*b^2*x^(9/2) + (2/11)*b^3*x^(11/2)],
[x^(3/2)/(-a + b*x), x, 3, (2*a*sqrt(x))/b^2 + (2*x^(3/2))/(3*b) - (2*a^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/b^(5/2)],
[x^(3/2)/(-a + b*x)^2, x, 3, (3*sqrt(x))/b^2 + x^(3/2)/(b*(a - b*x)) - (3*sqrt(a)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/b^(5/2)],
[x^(3/2)/(-a + b*x)^3, x, 3, -(x^(3/2)/(2*b*(a - b*x)^2)) + (3*sqrt(x))/(4*b^2*(a - b*x)) - (3*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/(4*sqrt(a)*b^(5/2))],

[x^(5/2)*(-a + b*x), x, 2, (-(2/7))*a*x^(7/2) + (2/9)*b*x^(9/2)],
[x^(5/2)*(-a + b*x)^2, x, 2, (2/7)*a^2*x^(7/2) - (4/9)*a*b*x^(9/2) + (2/11)*b^2*x^(11/2)],
[x^(5/2)*(-a + b*x)^3, x, 2, (-(2/7))*a^3*x^(7/2) + (2/3)*a^2*b*x^(9/2) - (6/11)*a*b^2*x^(11/2) + (2/13)*b^3*x^(13/2)],
[x^(5/2)/(-a + b*x), x, 4, (2*a^2*sqrt(x))/b^3 + (2*a*x^(3/2))/(3*b^2) + (2*x^(5/2))/(5*b) - (2*a^(5/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/b^(7/2)],
[x^(5/2)/(-a + b*x)^2, x, 4, (5*a*sqrt(x))/b^3 + (5*x^(3/2))/(3*b^2) + x^(5/2)/(b*(a - b*x)) - (5*a^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/b^(7/2)],
[x^(5/2)/(-a + b*x)^3, x, 4, (15*sqrt(x))/(4*b^3) - x^(5/2)/(2*b*(a - b*x)^2) + (5*x^(3/2))/(4*b^2*(a - b*x)) - (15*sqrt(a)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/(4*b^(7/2))],

[(-a + b*x)/sqrt(x), x, 2, -2*a*sqrt(x) + (2/3)*b*x^(3/2)],
[(-a + b*x)^2/sqrt(x), x, 2, 2*a^2*sqrt(x) - (4/3)*a*b*x^(3/2) + (2/5)*b^2*x^(5/2)],
[(-a + b*x)^3/sqrt(x), x, 2, -2*a^3*sqrt(x) + 2*a^2*b*x^(3/2) - (6/5)*a*b^2*x^(5/2) + (2/7)*b^3*x^(7/2)],
[1/(sqrt(x)*(-a + b*x)), x, 1, (-2*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/(sqrt(a)*sqrt(b))],
[1/(sqrt(x)*(-a + b*x)^2), x, 2, sqrt(x)/(a*(a - b*x)) + arctanh((sqrt(b)*sqrt(x))/sqrt(a))/(a^(3/2)*sqrt(b))],
[1/(sqrt(x)*(-a + b*x)^3), x, 3, -(sqrt(x)/(2*a*(a - b*x)^2)) - (3*sqrt(x))/(4*a^2*(a - b*x)) - (3*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/(4*a^(5/2)*sqrt(b))],

[(-a + b*x)/x^(3/2), x, 2, (2*a)/sqrt(x) + 2*b*sqrt(x)],
[(-a + b*x)^2/x^(3/2), x, 2, -((2*a^2)/sqrt(x)) - 4*a*b*sqrt(x) + (2/3)*b^2*x^(3/2)],
[(-a + b*x)^3/x^(3/2), x, 2, (2*a^3)/sqrt(x) + 6*a^2*b*sqrt(x) - 2*a*b^2*x^(3/2) + (2/5)*b^3*x^(5/2)],
[1/(x^(3/2)*(-a + b*x)), x, 2, 2/(a*sqrt(x)) - (2*sqrt(b)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/a^(3/2)],
[1/(x^(3/2)*(-a + b*x)^2), x, 3, -(2/(a*sqrt(x)*(a - b*x))) + (3*b*sqrt(x))/(a^2*(a - b*x)) + (3*sqrt(b)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/a^(5/2)],
[1/(x^(3/2)*(-a + b*x)^3), x, 4, 2/(a*sqrt(x)*(a - b*x)^2) - (5*b*sqrt(x))/(2*a^2*(a - b*x)^2) - (15*b*sqrt(x))/(4*a^3*(a - b*x)) - (15*sqrt(b)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/(4*a^(7/2))],

[(-a + b*x)/x^(5/2), x, 2, (2*a)/(3*x^(3/2)) - (2*b)/sqrt(x)],
[(-a + b*x)^2/x^(5/2), x, 2, -((2*a^2)/(3*x^(3/2))) + (4*a*b)/sqrt(x) + 2*b^2*sqrt(x)],
[(-a + b*x)^3/x^(5/2), x, 2, (2*a^3)/(3*x^(3/2)) - (6*a^2*b)/sqrt(x) - 6*a*b^2*sqrt(x) + (2/3)*b^3*x^(3/2)],
[1/(x^(5/2)*(-a + b*x)), x, 3, 2/(3*a*x^(3/2)) + (2*b)/(a^2*sqrt(x)) - (2*b^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/a^(5/2)],
[1/(x^(5/2)*(-a + b*x)^2), x, 4, -(5/(3*a^2*x^(3/2))) - (5*b)/(a^3*sqrt(x)) + 1/(a*x^(3/2)*(a - b*x)) + (5*b^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/a^(7/2)],
[1/(x^(5/2)*(-a + b*x)^3), x, 5, 2/(3*a*x^(3/2)*(a - b*x)^2) + (14*b)/(3*a^2*sqrt(x)*(a - b*x)^2) - (35*b^2*sqrt(x))/(6*a^3*(a - b*x)^2) - (35*b^2*sqrt(x))/(4*a^4*(a - b*x)) - (35*b^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a)))/(4*a^(9/2))],


# ::Subsection::Closed:: 
#Integrands of the form x^(m/2) (a+b x)^(n/2)


# Integrands of the form x^m*(2+b*x)^n where m and n are half-integers 
[sqrt(x)*sqrt(2 + b*x), x, 3, (sqrt(x)*sqrt(2 + b*x))/(2*b) + (1/2)*x^(3/2)*sqrt(2 + b*x) - arcsinh((sqrt(b)*sqrt(x))/sqrt(2))/b^(3/2)],
[sqrt(x)*(2 + b*x)^(3/2), x, 4, -((sqrt(x)*sqrt(2 + b*x))/(2*b)) - (sqrt(x)*(2 + b*x)^(3/2))/(6*b) + (sqrt(x)*(2 + b*x)^(5/2))/(3*b) - arcsinh((sqrt(b)*sqrt(x))/sqrt(2))/b^(3/2)],
[sqrt(x)*(2 + b*x)^(5/2), x, 5, -((5*sqrt(x)*sqrt(2 + b*x))/(8*b)) - (5*sqrt(x)*(2 + b*x)^(3/2))/(24*b) - (sqrt(x)*(2 + b*x)^(5/2))/(12*b) + (sqrt(x)*(2 + b*x)^(7/2))/(4*b) - (5*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(3/2))],
[sqrt(x)/sqrt(2 + b*x), x, 2, (sqrt(x)*sqrt(2 + b*x))/b - (2*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(3/2)],
[sqrt(x)/(2 + b*x)^(3/2), x, 2, (-2*sqrt(x))/(b*sqrt(2 + b*x)) + (2*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(3/2)],
[sqrt(x)/(2 + b*x)^(5/2), x, 1, x^(3/2)/(3*(2 + b*x)^(3/2))],

[x^(3/2)*sqrt(2 + b*x), x, 4, -((sqrt(x)*sqrt(2 + b*x))/(2*b^2)) + (x^(3/2)*sqrt(2 + b*x))/(6*b) + (1/3)*x^(5/2)*sqrt(2 + b*x) + arcsinh((sqrt(b)*sqrt(x))/sqrt(2))/b^(5/2)],
[x^(3/2)*(2 + b*x)^(3/2), x, 5, -((3*sqrt(x)*sqrt(2 + b*x))/(8*b^2)) + (x^(3/2)*sqrt(2 + b*x))/(8*b) + (1/4)*x^(5/2)*sqrt(2 + b*x) + (1/4)*x^(5/2)*(2 + b*x)^(3/2) + (3*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(5/2))],
[x^(3/2)*(2 + b*x)^(5/2), x, 6, (3*sqrt(x)*sqrt(2 + b*x))/(8*b^2) + (sqrt(x)*(2 + b*x)^(3/2))/(8*b^2) + (sqrt(x)*(2 + b*x)^(5/2))/(20*b^2) - (3*sqrt(x)*(2 + b*x)^(7/2))/(20*b^2) + (x^(3/2)*(2 + b*x)^(7/2))/(5*b) + (3*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(5/2))],
[x^(3/2)/sqrt(2 + b*x), x, 3, -((3*sqrt(x)*sqrt(2 + b*x))/(2*b^2)) + (x^(3/2)*sqrt(2 + b*x))/(2*b) + (3*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(5/2)],
[x^(3/2)/(2 + b*x)^(3/2), x, 3, -((2*x^(3/2))/(b*sqrt(2 + b*x))) + (3*sqrt(x)*sqrt(2 + b*x))/b^2 - (6*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(5/2)],
[x^(3/2)/(2 + b*x)^(5/2), x, 3, -((2*x^(3/2))/(3*b*(2 + b*x)^(3/2))) - (2*sqrt(x))/(b^2*sqrt(2 + b*x)) + (2*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(5/2)],

[x^(5/2)*sqrt(2 + b*x), x, 5, (5*sqrt(x)*sqrt(2 + b*x))/(8*b^3) - (5*x^(3/2)*sqrt(2 + b*x))/(24*b^2) + (x^(5/2)*sqrt(2 + b*x))/(12*b) + (1/4)*x^(7/2)*sqrt(2 + b*x) - (5*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(7/2))],
[x^(5/2)*(2 + b*x)^(3/2), x, 6, (3*sqrt(x)*sqrt(2 + b*x))/(8*b^3) - (x^(3/2)*sqrt(2 + b*x))/(8*b^2) + (x^(5/2)*sqrt(2 + b*x))/(20*b) + (3/20)*x^(7/2)*sqrt(2 + b*x) + (1/5)*x^(7/2)*(2 + b*x)^(3/2) - (3*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(7/2))],
[x^(5/2)*(2 + b*x)^(5/2), x, 7, (5*sqrt(x)*sqrt(2 + b*x))/(16*b^3) - (5*x^(3/2)*sqrt(2 + b*x))/(48*b^2) + (x^(5/2)*sqrt(2 + b*x))/(24*b) + (1/8)*x^(7/2)*sqrt(2 + b*x) + (1/6)*x^(7/2)*(2 + b*x)^(3/2) + (1/6)*x^(7/2)*(2 + b*x)^(5/2) - (5*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/(8*b^(7/2))],
[x^(5/2)/sqrt(2 + b*x), x, 4, (5*sqrt(x)*sqrt(2 + b*x))/(2*b^3) - (5*x^(3/2)*sqrt(2 + b*x))/(6*b^2) + (x^(5/2)*sqrt(2 + b*x))/(3*b) - (5*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(7/2)],
[x^(5/2)/(2 + b*x)^(3/2), x, 4, -((2*x^(5/2))/(b*sqrt(2 + b*x))) - (15*sqrt(x)*sqrt(2 + b*x))/(2*b^3) + (5*x^(3/2)*sqrt(2 + b*x))/(2*b^2) + (15*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(7/2)],
[x^(5/2)/(2 + b*x)^(5/2), x, 4, -((2*x^(5/2))/(3*b*(2 + b*x)^(3/2))) - (10*x^(3/2))/(3*b^2*sqrt(2 + b*x)) + (5*sqrt(x)*sqrt(2 + b*x))/b^3 - (10*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/b^(7/2)],

[sqrt(2 + b*x)/sqrt(x), x, 2, sqrt(x)*sqrt(2 + b*x) + (2*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[(2 + b*x)^(3/2)/sqrt(x), x, 3, (3/2)*sqrt(x)*sqrt(2 + b*x) + (1/2)*sqrt(x)*(2 + b*x)^(3/2) + (3*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[(2 + b*x)^(5/2)/sqrt(x), x, 4, (5/2)*sqrt(x)*sqrt(2 + b*x) + (5/6)*sqrt(x)*(2 + b*x)^(3/2) + (1/3)*sqrt(x)*(2 + b*x)^(5/2) + (5*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[1/(sqrt(x)*sqrt(2 + b*x)), x, 1, (2*arcsinh((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[1/(sqrt(x)*(2 + b*x)^(3/2)), x, 1, sqrt(x)/sqrt(2 + b*x)],
[1/(sqrt(x)*(2 + b*x)^(5/2)), x, 2, sqrt(x)/(3*(2 + b*x)^(3/2)) + sqrt(x)/(3*sqrt(2 + b*x))],

[sqrt(2 + b*x)/x^(3/2), x, 2, (-2*sqrt(2 + b*x))/sqrt(x) + 2*sqrt(b)*arcsinh((sqrt(b)*sqrt(x))/sqrt(2))],
[(2 + b*x)^(3/2)/x^(3/2), x, 3, 3*b*sqrt(x)*sqrt(2 + b*x) - (2*(2 + b*x)^(3/2))/sqrt(x) + 6*sqrt(b)*arcsinh((sqrt(b)*sqrt(x))/sqrt(2))],
[(2 + b*x)^(5/2)/x^(3/2), x, 4, (15/2)*b*sqrt(x)*sqrt(2 + b*x) + (5/2)*b*sqrt(x)*(2 + b*x)^(3/2) - (2*(2 + b*x)^(5/2))/sqrt(x) + 15*sqrt(b)*arcsinh((sqrt(b)*sqrt(x))/sqrt(2))],
[1/(x^(3/2)*sqrt(2 + b*x)), x, 1, -(sqrt(2 + b*x)/sqrt(x))],
[1/(x^(3/2)*(2 + b*x)^(3/2)), x, 2, 1/(sqrt(x)*sqrt(2 + b*x)) - sqrt(2 + b*x)/sqrt(x)],
[1/(x^(3/2)*(2 + b*x)^(5/2)), x, 3, -(1/(sqrt(x)*(2 + b*x)^(3/2))) - (2*b*sqrt(x))/(3*(2 + b*x)^(3/2)) - (2*b*sqrt(x))/(3*sqrt(2 + b*x))],

[sqrt(2 + b*x)/x^(5/2), x, 1, -(2 + b*x)^(3/2)/(3*x^(3/2))],
[(2 + b*x)^(3/2)/x^(5/2), x, 3, -((2*b*sqrt(2 + b*x))/sqrt(x)) - (2*(2 + b*x)^(3/2))/(3*x^(3/2)) + 2*b^(3/2)*arcsinh((sqrt(b)*sqrt(x))/sqrt(2))],
[(2 + b*x)^(5/2)/x^(5/2), x, 4, 5*b^2*sqrt(x)*sqrt(2 + b*x) - (10*b*(2 + b*x)^(3/2))/(3*sqrt(x)) - (2*(2 + b*x)^(5/2))/(3*x^(3/2)) + 10*b^(3/2)*arcsinh((sqrt(b)*sqrt(x))/sqrt(2))],
[1/(x^(5/2)*sqrt(2 + b*x)), x, 2, -(sqrt(2 + b*x)/(3*x^(3/2))) + (b*sqrt(2 + b*x))/(3*sqrt(x))],
[1/(x^(5/2)*(2 + b*x)^(3/2)), x, 3, 1/(x^(3/2)*sqrt(2 + b*x)) - (2*sqrt(2 + b*x))/(3*x^(3/2)) + (2*b*sqrt(2 + b*x))/(3*sqrt(x))],
[1/(x^(5/2)*(2 + b*x)^(5/2)), x, 4, 1/(3*x^(3/2)*(2 + b*x)^(3/2)) + 1/(x^(3/2)*sqrt(2 + b*x)) - (2*sqrt(2 + b*x))/(3*x^(3/2)) + (2*b*sqrt(2 + b*x))/(3*sqrt(x))],


# Integrands of the form x^m*(a+b*x)^n where m and n are half-integers 
[sqrt(x)*sqrt(a + b*x), x, 3, (a*sqrt(x)*sqrt(a + b*x))/(4*b) + (1/2)*x^(3/2)*sqrt(a + b*x) - (a^2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(4*b^(3/2))],
[sqrt(x)*(a + b*x)^(3/2), x, 4, -((a^2*sqrt(x)*sqrt(a + b*x))/(8*b)) - (a*sqrt(x)*(a + b*x)^(3/2))/(12*b) + (sqrt(x)*(a + b*x)^(5/2))/(3*b) - (a^3*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(8*b^(3/2))],
[sqrt(x)*(a + b*x)^(5/2), x, 5, -((5*a^3*sqrt(x)*sqrt(a + b*x))/(64*b)) - (5*a^2*sqrt(x)*(a + b*x)^(3/2))/(96*b) - (a*sqrt(x)*(a + b*x)^(5/2))/(24*b) + (sqrt(x)*(a + b*x)^(7/2))/(4*b) - (5*a^4*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(64*b^(3/2))],
[sqrt(x)/sqrt(a + b*x), x, 2, (sqrt(x)*sqrt(a + b*x))/b - (a*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/b^(3/2)],
[sqrt(x)/(a + b*x)^(3/2), x, 2, (-2*sqrt(x))/(b*sqrt(a + b*x)) + (2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/b^(3/2)],
[sqrt(x)/(a + b*x)^(5/2), x, 1, (2*x^(3/2))/(3*a*(a + b*x)^(3/2))],

[x^(3/2)*sqrt(a + b*x), x, 4, -((a^2*sqrt(x)*sqrt(a + b*x))/(8*b^2)) + (a*x^(3/2)*sqrt(a + b*x))/(12*b) + (1/3)*x^(5/2)*sqrt(a + b*x) + (a^3*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(8*b^(5/2))],
[x^(3/2)*(a + b*x)^(3/2), x, 5, -((3*a^3*sqrt(x)*sqrt(a + b*x))/(64*b^2)) + (a^2*x^(3/2)*sqrt(a + b*x))/(32*b) + (1/8)*a*x^(5/2)*sqrt(a + b*x) + (1/4)*x^(5/2)*(a + b*x)^(3/2) + (3*a^4*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(64*b^(5/2))],
[x^(3/2)*(a + b*x)^(5/2), x, 6, (3*a^4*sqrt(x)*sqrt(a + b*x))/(128*b^2) + (a^3*sqrt(x)*(a + b*x)^(3/2))/(64*b^2) + (a^2*sqrt(x)*(a + b*x)^(5/2))/(80*b^2) - (3*a*sqrt(x)*(a + b*x)^(7/2))/(40*b^2) + (x^(3/2)*(a + b*x)^(7/2))/(5*b) + (3*a^5*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(128*b^(5/2))],
[x^(3/2)/sqrt(a + b*x), x, 3, -((3*a*sqrt(x)*sqrt(a + b*x))/(4*b^2)) + (x^(3/2)*sqrt(a + b*x))/(2*b) + (3*a^2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(4*b^(5/2))],
[x^(3/2)/(a + b*x)^(3/2), x, 3, -((2*x^(3/2))/(b*sqrt(a + b*x))) + (3*sqrt(x)*sqrt(a + b*x))/b^2 - (3*a*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/b^(5/2)],
[x^(3/2)/(a + b*x)^(5/2), x, 3, -((2*x^(3/2))/(3*b*(a + b*x)^(3/2))) - (2*sqrt(x))/(b^2*sqrt(a + b*x)) + (2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/b^(5/2)],

[x^(5/2)*sqrt(a + b*x), x, 5, (5*a^3*sqrt(x)*sqrt(a + b*x))/(64*b^3) - (5*a^2*x^(3/2)*sqrt(a + b*x))/(96*b^2) + (a*x^(5/2)*sqrt(a + b*x))/(24*b) + (1/4)*x^(7/2)*sqrt(a + b*x) - (5*a^4*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(64*b^(7/2))],
[x^(5/2)*(a + b*x)^(3/2), x, 6, (3*a^4*sqrt(x)*sqrt(a + b*x))/(128*b^3) - (a^3*x^(3/2)*sqrt(a + b*x))/(64*b^2) + (a^2*x^(5/2)*sqrt(a + b*x))/(80*b) + (3/40)*a*x^(7/2)*sqrt(a + b*x) + (1/5)*x^(7/2)*(a + b*x)^(3/2) - (3*a^5*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(128*b^(7/2))],
[x^(5/2)*(a + b*x)^(5/2), x, 7, (5*a^5*sqrt(x)*sqrt(a + b*x))/(512*b^3) - (5*a^4*x^(3/2)*sqrt(a + b*x))/(768*b^2) + (a^3*x^(5/2)*sqrt(a + b*x))/(192*b) + (1/32)*a^2*x^(7/2)*sqrt(a + b*x) + (1/12)*a*x^(7/2)*(a + b*x)^(3/2) + (1/6)*x^(7/2)*(a + b*x)^(5/2) - (5*a^6*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(512*b^(7/2))],
[x^(5/2)/sqrt(a + b*x), x, 4, (5*a^2*sqrt(x)*sqrt(a + b*x))/(8*b^3) - (5*a*x^(3/2)*sqrt(a + b*x))/(12*b^2) + (x^(5/2)*sqrt(a + b*x))/(3*b) - (5*a^3*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(8*b^(7/2))],
[x^(5/2)/(a + b*x)^(3/2), x, 4, -((2*x^(5/2))/(b*sqrt(a + b*x))) - (15*a*sqrt(x)*sqrt(a + b*x))/(4*b^3) + (5*x^(3/2)*sqrt(a + b*x))/(2*b^2) + (15*a^2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(4*b^(7/2))],
[x^(5/2)/(a + b*x)^(5/2), x, 4, -((2*x^(5/2))/(3*b*(a + b*x)^(3/2))) - (10*x^(3/2))/(3*b^2*sqrt(a + b*x)) + (5*sqrt(x)*sqrt(a + b*x))/b^3 - (5*a*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/b^(7/2)],

[sqrt(a + b*x)/sqrt(x), x, 2, sqrt(x)*sqrt(a + b*x) + (a*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/sqrt(b)],
[(a + b*x)^(3/2)/sqrt(x), x, 3, (3/4)*a*sqrt(x)*sqrt(a + b*x) + (1/2)*sqrt(x)*(a + b*x)^(3/2) + (3*a^2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(4*sqrt(b))],
[(a + b*x)^(5/2)/sqrt(x), x, 4, (5/8)*a^2*sqrt(x)*sqrt(a + b*x) + (5/12)*a*sqrt(x)*(a + b*x)^(3/2) + (1/3)*sqrt(x)*(a + b*x)^(5/2) + (5*a^3*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/(8*sqrt(b))],
[1/(sqrt(x)*sqrt(a + b*x)), x, 1, (2*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x)))/sqrt(b)],
[1/(sqrt(x)*(a + b*x)^(3/2)), x, 1, (2*sqrt(x))/(a*sqrt(a + b*x))],
[1/(sqrt(x)*(a + b*x)^(5/2)), x, 2, (2*sqrt(x))/(3*a*(a + b*x)^(3/2)) + (4*sqrt(x))/(3*a^2*sqrt(a + b*x))],

[sqrt(a + b*x)/x^(3/2), x, 2, (-2*sqrt(a + b*x))/sqrt(x) + 2*sqrt(b)*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x))],
[(a + b*x)^(3/2)/x^(3/2), x, 3, 3*b*sqrt(x)*sqrt(a + b*x) - (2*(a + b*x)^(3/2))/sqrt(x) + 3*a*sqrt(b)*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x))],
[(a + b*x)^(5/2)/x^(3/2), x, 4, (15/4)*a*b*sqrt(x)*sqrt(a + b*x) + (5/2)*b*sqrt(x)*(a + b*x)^(3/2) - (2*(a + b*x)^(5/2))/sqrt(x) + (15/4)*a^2*sqrt(b)*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x))],
[1/(x^(3/2)*sqrt(a + b*x)), x, 1, (-2*sqrt(a + b*x))/(a*sqrt(x))],
[1/(x^(3/2)*(a + b*x)^(3/2)), x, 2, 2/(a*sqrt(x)*sqrt(a + b*x)) - (4*sqrt(a + b*x))/(a^2*sqrt(x))],
[1/(x^(3/2)*(a + b*x)^(5/2)), x, 3, -(2/(a*sqrt(x)*(a + b*x)^(3/2))) - (8*b*sqrt(x))/(3*a^2*(a + b*x)^(3/2)) - (16*b*sqrt(x))/(3*a^3*sqrt(a + b*x))],

[sqrt(a + b*x)/x^(5/2), x, 1, (-2*(a + b*x)^(3/2))/(3*a*x^(3/2))],
[(a + b*x)^(3/2)/x^(5/2), x, 3, -((2*b*sqrt(a + b*x))/sqrt(x)) - (2*(a + b*x)^(3/2))/(3*x^(3/2)) + 2*b^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x))],
[(a + b*x)^(5/2)/x^(5/2), x, 4, 5*b^2*sqrt(x)*sqrt(a + b*x) - (10*b*(a + b*x)^(3/2))/(3*sqrt(x)) - (2*(a + b*x)^(5/2))/(3*x^(3/2)) + 5*a*b^(3/2)*arctanh((sqrt(b)*sqrt(x))/sqrt(a + b*x))],
[1/(x^(5/2)*sqrt(a + b*x)), x, 2, -((2*sqrt(a + b*x))/(3*a*x^(3/2))) + (4*b*sqrt(a + b*x))/(3*a^2*sqrt(x))],
[1/(x^(5/2)*(a + b*x)^(3/2)), x, 3, 2/(a*x^(3/2)*sqrt(a + b*x)) - (8*sqrt(a + b*x))/(3*a^2*x^(3/2)) + (16*b*sqrt(a + b*x))/(3*a^3*sqrt(x))],
[1/(x^(5/2)*(a + b*x)^(5/2)), x, 4, 2/(3*a*x^(3/2)*(a + b*x)^(3/2)) + 4/(a^2*x^(3/2)*sqrt(a + b*x)) - (16*sqrt(a + b*x))/(3*a^3*x^(3/2)) + (32*b*sqrt(a + b*x))/(3*a^4*sqrt(x))],


# Integrands of the form x^m*(2-b*x)^n where m and n are half-integers 
[sqrt(x)*sqrt(2 - b*x), x, 3, -((sqrt(x)*sqrt(2 - b*x))/(2*b)) + (1/2)*x^(3/2)*sqrt(2 - b*x) + arcsin((sqrt(b)*sqrt(x))/sqrt(2))/b^(3/2)],
[sqrt(x)*(2 - b*x)^(3/2), x, 4, (sqrt(x)*sqrt(2 - b*x))/(2*b) + (sqrt(x)*(2 - b*x)^(3/2))/(6*b) - (sqrt(x)*(2 - b*x)^(5/2))/(3*b) + arcsin((sqrt(b)*sqrt(x))/sqrt(2))/b^(3/2)],
[sqrt(x)*(2 - b*x)^(5/2), x, 5, (5*sqrt(x)*sqrt(2 - b*x))/(8*b) + (5*sqrt(x)*(2 - b*x)^(3/2))/(24*b) + (sqrt(x)*(2 - b*x)^(5/2))/(12*b) - (sqrt(x)*(2 - b*x)^(7/2))/(4*b) + (5*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(3/2))],
[sqrt(x)/sqrt(1 - x), x, 2, (-sqrt(1 - x))*sqrt(x) + arcsin(sqrt(x))],
[sqrt(x)/sqrt(2 - b*x), x, 2, -((sqrt(x)*sqrt(2 - b*x))/b) + (2*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(3/2)],
[sqrt(x)/(2 - b*x)^(3/2), x, 2, (2*sqrt(x))/(b*sqrt(2 - b*x)) - (2*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(3/2)],
[sqrt(x)/(2 - b*x)^(5/2), x, 1, x^(3/2)/(3*(2 - b*x)^(3/2))],

[x^(3/2)*sqrt(2 - b*x), x, 4, -((sqrt(x)*sqrt(2 - b*x))/(2*b^2)) - (x^(3/2)*sqrt(2 - b*x))/(6*b) + (1/3)*x^(5/2)*sqrt(2 - b*x) + arcsin((sqrt(b)*sqrt(x))/sqrt(2))/b^(5/2)],
[x^(3/2)*(2 - b*x)^(3/2), x, 5, -((3*sqrt(x)*sqrt(2 - b*x))/(8*b^2)) - (x^(3/2)*sqrt(2 - b*x))/(8*b) + (1/4)*x^(5/2)*sqrt(2 - b*x) + (1/4)*x^(5/2)*(2 - b*x)^(3/2) + (3*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(5/2))],
[x^(3/2)*(2 - b*x)^(5/2), x, 6, (3*sqrt(x)*sqrt(2 - b*x))/(8*b^2) + (sqrt(x)*(2 - b*x)^(3/2))/(8*b^2) + (sqrt(x)*(2 - b*x)^(5/2))/(20*b^2) - (3*sqrt(x)*(2 - b*x)^(7/2))/(20*b^2) - (x^(3/2)*(2 - b*x)^(7/2))/(5*b) + (3*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(5/2))],
[x^(3/2)/sqrt(2 - b*x), x, 3, -((3*sqrt(x)*sqrt(2 - b*x))/(2*b^2)) - (x^(3/2)*sqrt(2 - b*x))/(2*b) + (3*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(5/2)],
[x^(3/2)/(2 - b*x)^(3/2), x, 3, (2*x^(3/2))/(b*sqrt(2 - b*x)) + (3*sqrt(x)*sqrt(2 - b*x))/b^2 - (6*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(5/2)],
[x^(3/2)/(2 - b*x)^(5/2), x, 3, (2*x^(3/2))/(3*b*(2 - b*x)^(3/2)) - (2*sqrt(x))/(b^2*sqrt(2 - b*x)) + (2*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(5/2)],

[x^(5/2)*sqrt(2 - b*x), x, 5, -((5*sqrt(x)*sqrt(2 - b*x))/(8*b^3)) - (5*x^(3/2)*sqrt(2 - b*x))/(24*b^2) - (x^(5/2)*sqrt(2 - b*x))/(12*b) + (1/4)*x^(7/2)*sqrt(2 - b*x) + (5*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(7/2))],
[x^(5/2)*(2 - b*x)^(3/2), x, 6, -((3*sqrt(x)*sqrt(2 - b*x))/(8*b^3)) - (x^(3/2)*sqrt(2 - b*x))/(8*b^2) - (x^(5/2)*sqrt(2 - b*x))/(20*b) + (3/20)*x^(7/2)*sqrt(2 - b*x) + (1/5)*x^(7/2)*(2 - b*x)^(3/2) + (3*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/(4*b^(7/2))],
[x^(5/2)*(2 - b*x)^(5/2), x, 7, -((5*sqrt(x)*sqrt(2 - b*x))/(16*b^3)) - (5*x^(3/2)*sqrt(2 - b*x))/(48*b^2) - (x^(5/2)*sqrt(2 - b*x))/(24*b) + (1/8)*x^(7/2)*sqrt(2 - b*x) + (1/6)*x^(7/2)*(2 - b*x)^(3/2) + (1/6)*x^(7/2)*(2 - b*x)^(5/2) + (5*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/(8*b^(7/2))],
[x^(5/2)/sqrt(2 - b*x), x, 4, -((5*sqrt(x)*sqrt(2 - b*x))/(2*b^3)) - (5*x^(3/2)*sqrt(2 - b*x))/(6*b^2) - (x^(5/2)*sqrt(2 - b*x))/(3*b) + (5*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(7/2)],
[x^(5/2)/(2 - b*x)^(3/2), x, 4, (2*x^(5/2))/(b*sqrt(2 - b*x)) + (15*sqrt(x)*sqrt(2 - b*x))/(2*b^3) + (5*x^(3/2)*sqrt(2 - b*x))/(2*b^2) - (15*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(7/2)],
[x^(5/2)/(2 - b*x)^(5/2), x, 4, (2*x^(5/2))/(3*b*(2 - b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*sqrt(2 - b*x)) - (5*sqrt(x)*sqrt(2 - b*x))/b^3 + (10*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/b^(7/2)],

[sqrt(2 - b*x)/sqrt(x), x, 2, sqrt(x)*sqrt(2 - b*x) + (2*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[(2 - b*x)^(3/2)/sqrt(x), x, 3, (3/2)*sqrt(x)*sqrt(2 - b*x) + (1/2)*sqrt(x)*(2 - b*x)^(3/2) + (3*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[(2 - b*x)^(5/2)/sqrt(x), x, 4, (5/2)*sqrt(x)*sqrt(2 - b*x) + (5/6)*sqrt(x)*(2 - b*x)^(3/2) + (1/3)*sqrt(x)*(2 - b*x)^(5/2) + (5*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[1/(sqrt(x)*sqrt(2 - b*x)), x, 1, (2*arcsin((sqrt(b)*sqrt(x))/sqrt(2)))/sqrt(b)],
[1/(sqrt(x)*(2 - b*x)^(3/2)), x, 1, sqrt(x)/sqrt(2 - b*x)],
[1/(sqrt(x)*(2 - b*x)^(5/2)), x, 2, sqrt(x)/(3*(2 - b*x)^(3/2)) + sqrt(x)/(3*sqrt(2 - b*x))],

[sqrt(2 - b*x)/x^(3/2), x, 2, (-2*sqrt(2 - b*x))/sqrt(x) - 2*sqrt(b)*arcsin((sqrt(b)*sqrt(x))/sqrt(2))],
[(2 - b*x)^(3/2)/x^(3/2), x, 3, -3*b*sqrt(x)*sqrt(2 - b*x) - (2*(2 - b*x)^(3/2))/sqrt(x) - 6*sqrt(b)*arcsin((sqrt(b)*sqrt(x))/sqrt(2))],
[(2 - b*x)^(5/2)/x^(3/2), x, 4, (-(15/2))*b*sqrt(x)*sqrt(2 - b*x) - (5/2)*b*sqrt(x)*(2 - b*x)^(3/2) - (2*(2 - b*x)^(5/2))/sqrt(x) - 15*sqrt(b)*arcsin((sqrt(b)*sqrt(x))/sqrt(2))],
[1/(x^(3/2)*sqrt(2 - b*x)), x, 1, -(sqrt(2 - b*x)/sqrt(x))],
[1/(x^(3/2)*(2 - b*x)^(3/2)), x, 2, 1/(sqrt(x)*sqrt(2 - b*x)) - sqrt(2 - b*x)/sqrt(x)],
[1/(x^(3/2)*(2 - b*x)^(5/2)), x, 3, -(1/(sqrt(x)*(2 - b*x)^(3/2))) + (2*b*sqrt(x))/(3*(2 - b*x)^(3/2)) + (2*b*sqrt(x))/(3*sqrt(2 - b*x))],

[sqrt(2 - b*x)/x^(5/2), x, 1, -(2 - b*x)^(3/2)/(3*x^(3/2))],
[(2 - b*x)^(3/2)/x^(5/2), x, 3, (2*b*sqrt(2 - b*x))/sqrt(x) - (2*(2 - b*x)^(3/2))/(3*x^(3/2)) + 2*b^(3/2)*arcsin((sqrt(b)*sqrt(x))/sqrt(2))],
[(2 - b*x)^(5/2)/x^(5/2), x, 4, 5*b^2*sqrt(x)*sqrt(2 - b*x) + (10*b*(2 - b*x)^(3/2))/(3*sqrt(x)) - (2*(2 - b*x)^(5/2))/(3*x^(3/2)) + 10*b^(3/2)*arcsin((sqrt(b)*sqrt(x))/sqrt(2))],
[1/(x^(5/2)*sqrt(2 - b*x)), x, 2, -(sqrt(2 - b*x)/(3*x^(3/2))) - (b*sqrt(2 - b*x))/(3*sqrt(x))],
[1/(x^(5/2)*(2 - b*x)^(3/2)), x, 3, 1/(x^(3/2)*sqrt(2 - b*x)) - (2*sqrt(2 - b*x))/(3*x^(3/2)) - (2*b*sqrt(2 - b*x))/(3*sqrt(x))],
[1/(x^(5/2)*(2 - b*x)^(5/2)), x, 4, 1/(3*x^(3/2)*(2 - b*x)^(3/2)) + 1/(x^(3/2)*sqrt(2 - b*x)) - (2*sqrt(2 - b*x))/(3*x^(3/2)) - (2*b*sqrt(2 - b*x))/(3*sqrt(x))],


# Integrands of the form x^m*(a-b*x)^n where m and n are half-integers 
[sqrt(x)*sqrt(a - b*x), x, 3, -((a*sqrt(x)*sqrt(a - b*x))/(4*b)) + (1/2)*x^(3/2)*sqrt(a - b*x) + (a^2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(4*b^(3/2))],
[sqrt(x)*(a - b*x)^(3/2), x, 4, (a^2*sqrt(x)*sqrt(a - b*x))/(8*b) + (a*sqrt(x)*(a - b*x)^(3/2))/(12*b) - (sqrt(x)*(a - b*x)^(5/2))/(3*b) + (a^3*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(8*b^(3/2))],
[sqrt(x)*(a - b*x)^(5/2), x, 5, (5*a^3*sqrt(x)*sqrt(a - b*x))/(64*b) + (5*a^2*sqrt(x)*(a - b*x)^(3/2))/(96*b) + (a*sqrt(x)*(a - b*x)^(5/2))/(24*b) - (sqrt(x)*(a - b*x)^(7/2))/(4*b) + (5*a^4*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(64*b^(3/2))],
[sqrt(x)/sqrt(a - b*x), x, 2, -((sqrt(x)*sqrt(a - b*x))/b) + (a*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/b^(3/2)],
[sqrt(x)/(a - b*x)^(3/2), x, 2, (2*sqrt(x))/(b*sqrt(a - b*x)) - (2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/b^(3/2)],
[sqrt(x)/(a - b*x)^(5/2), x, 1, (2*x^(3/2))/(3*a*(a - b*x)^(3/2))],

[x^(3/2)*sqrt(a - b*x), x, 4, -((a^2*sqrt(x)*sqrt(a - b*x))/(8*b^2)) - (a*x^(3/2)*sqrt(a - b*x))/(12*b) + (1/3)*x^(5/2)*sqrt(a - b*x) + (a^3*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(8*b^(5/2))],
[x^(3/2)*(a - b*x)^(3/2), x, 5, -((3*a^3*sqrt(x)*sqrt(a - b*x))/(64*b^2)) - (a^2*x^(3/2)*sqrt(a - b*x))/(32*b) + (1/8)*a*x^(5/2)*sqrt(a - b*x) + (1/4)*x^(5/2)*(a - b*x)^(3/2) + (3*a^4*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(64*b^(5/2))],
[x^(3/2)*(a - b*x)^(5/2), x, 6, (3*a^4*sqrt(x)*sqrt(a - b*x))/(128*b^2) + (a^3*sqrt(x)*(a - b*x)^(3/2))/(64*b^2) + (a^2*sqrt(x)*(a - b*x)^(5/2))/(80*b^2) - (3*a*sqrt(x)*(a - b*x)^(7/2))/(40*b^2) - (x^(3/2)*(a - b*x)^(7/2))/(5*b) + (3*a^5*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(128*b^(5/2))],
[x^(3/2)/sqrt(a - b*x), x, 3, -((3*a*sqrt(x)*sqrt(a - b*x))/(4*b^2)) - (x^(3/2)*sqrt(a - b*x))/(2*b) + (3*a^2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(4*b^(5/2))],
[x^(3/2)/(a - b*x)^(3/2), x, 3, (2*x^(3/2))/(b*sqrt(a - b*x)) + (3*sqrt(x)*sqrt(a - b*x))/b^2 - (3*a*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/b^(5/2)],
[x^(3/2)/(a - b*x)^(5/2), x, 3, (2*x^(3/2))/(3*b*(a - b*x)^(3/2)) - (2*sqrt(x))/(b^2*sqrt(a - b*x)) + (2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/b^(5/2)],

[x^(5/2)*sqrt(a - b*x), x, 5, -((5*a^3*sqrt(x)*sqrt(a - b*x))/(64*b^3)) - (5*a^2*x^(3/2)*sqrt(a - b*x))/(96*b^2) - (a*x^(5/2)*sqrt(a - b*x))/(24*b) + (1/4)*x^(7/2)*sqrt(a - b*x) + (5*a^4*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(64*b^(7/2))],
[x^(5/2)*(a - b*x)^(3/2), x, 6, -((3*a^4*sqrt(x)*sqrt(a - b*x))/(128*b^3)) - (a^3*x^(3/2)*sqrt(a - b*x))/(64*b^2) - (a^2*x^(5/2)*sqrt(a - b*x))/(80*b) + (3/40)*a*x^(7/2)*sqrt(a - b*x) + (1/5)*x^(7/2)*(a - b*x)^(3/2) + (3*a^5*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(128*b^(7/2))],
[x^(5/2)*(a - b*x)^(5/2), x, 7, -((5*a^5*sqrt(x)*sqrt(a - b*x))/(512*b^3)) - (5*a^4*x^(3/2)*sqrt(a - b*x))/(768*b^2) - (a^3*x^(5/2)*sqrt(a - b*x))/(192*b) + (1/32)*a^2*x^(7/2)*sqrt(a - b*x) + (1/12)*a*x^(7/2)*(a - b*x)^(3/2) + (1/6)*x^(7/2)*(a - b*x)^(5/2) + (5*a^6*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(512*b^(7/2))],
[x^(5/2)/sqrt(a - b*x), x, 4, -((5*a^2*sqrt(x)*sqrt(a - b*x))/(8*b^3)) - (5*a*x^(3/2)*sqrt(a - b*x))/(12*b^2) - (x^(5/2)*sqrt(a - b*x))/(3*b) + (5*a^3*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(8*b^(7/2))],
[x^(5/2)/(a - b*x)^(3/2), x, 4, (2*x^(5/2))/(b*sqrt(a - b*x)) + (15*a*sqrt(x)*sqrt(a - b*x))/(4*b^3) + (5*x^(3/2)*sqrt(a - b*x))/(2*b^2) - (15*a^2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(4*b^(7/2))],
[x^(5/2)/(a - b*x)^(5/2), x, 4, (2*x^(5/2))/(3*b*(a - b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*sqrt(a - b*x)) - (5*sqrt(x)*sqrt(a - b*x))/b^3 + (5*a*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/b^(7/2)],

[sqrt(a - b*x)/sqrt(x), x, 2, sqrt(x)*sqrt(a - b*x) + (a*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/sqrt(b)],
[(a - b*x)^(3/2)/sqrt(x), x, 3, (3/4)*a*sqrt(x)*sqrt(a - b*x) + (1/2)*sqrt(x)*(a - b*x)^(3/2) + (3*a^2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(4*sqrt(b))],
[(a - b*x)^(5/2)/sqrt(x), x, 4, (5/8)*a^2*sqrt(x)*sqrt(a - b*x) + (5/12)*a*sqrt(x)*(a - b*x)^(3/2) + (1/3)*sqrt(x)*(a - b*x)^(5/2) + (5*a^3*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/(8*sqrt(b))],
[1/(sqrt(x)*sqrt(a - b*x)), x, 1, (2*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x)))/sqrt(b)],
[1/(sqrt(x)*sqrt(1 - b*x)), x, 1, (2*arcsin(sqrt(b)*sqrt(x)))/sqrt(b)],
[1/(sqrt(x)*sqrt(1 - x)), x, 1, 2*arcsin(sqrt(x))],
[1/(sqrt(x)*(a - b*x)^(3/2)), x, 1, (2*sqrt(x))/(a*sqrt(a - b*x))],
[1/(sqrt(x)*(a - b*x)^(5/2)), x, 2, (2*sqrt(x))/(3*a*(a - b*x)^(3/2)) + (4*sqrt(x))/(3*a^2*sqrt(a - b*x))],

[sqrt(a - b*x)/x^(3/2), x, 2, (-2*sqrt(a - b*x))/sqrt(x) - 2*sqrt(b)*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x))],
[(a - b*x)^(3/2)/x^(3/2), x, 3, -3*b*sqrt(x)*sqrt(a - b*x) - (2*(a - b*x)^(3/2))/sqrt(x) - 3*a*sqrt(b)*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x))],
[(a - b*x)^(5/2)/x^(3/2), x, 4, (-(15/4))*a*b*sqrt(x)*sqrt(a - b*x) - (5/2)*b*sqrt(x)*(a - b*x)^(3/2) - (2*(a - b*x)^(5/2))/sqrt(x) - (15/4)*a^2*sqrt(b)*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x))],
[1/(x^(3/2)*sqrt(a - b*x)), x, 1, (-2*sqrt(a - b*x))/(a*sqrt(x))],
[1/(x^(3/2)*(a - b*x)^(3/2)), x, 2, 2/(a*sqrt(x)*sqrt(a - b*x)) - (4*sqrt(a - b*x))/(a^2*sqrt(x))],
[1/(x^(3/2)*(a - b*x)^(5/2)), x, 3, -(2/(a*sqrt(x)*(a - b*x)^(3/2))) + (8*b*sqrt(x))/(3*a^2*(a - b*x)^(3/2)) + (16*b*sqrt(x))/(3*a^3*sqrt(a - b*x))],

[sqrt(a - b*x)/x^(5/2), x, 1, (-2*(a - b*x)^(3/2))/(3*a*x^(3/2))],
[(a - b*x)^(3/2)/x^(5/2), x, 3, (2*b*sqrt(a - b*x))/sqrt(x) - (2*(a - b*x)^(3/2))/(3*x^(3/2)) + 2*b^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x))],
[(a - b*x)^(5/2)/x^(5/2), x, 4, 5*b^2*sqrt(x)*sqrt(a - b*x) + (10*b*(a - b*x)^(3/2))/(3*sqrt(x)) - (2*(a - b*x)^(5/2))/(3*x^(3/2)) + 5*a*b^(3/2)*arctan((sqrt(b)*sqrt(x))/sqrt(a - b*x))],
[1/(x^(5/2)*sqrt(a - b*x)), x, 2, -((2*sqrt(a - b*x))/(3*a*x^(3/2))) - (4*b*sqrt(a - b*x))/(3*a^2*sqrt(x))],
[1/(x^(5/2)*(a - b*x)^(3/2)), x, 3, 2/(a*x^(3/2)*sqrt(a - b*x)) - (8*sqrt(a - b*x))/(3*a^2*x^(3/2)) - (16*b*sqrt(a - b*x))/(3*a^3*sqrt(x))],
[1/(x^(5/2)*(a - b*x)^(5/2)), x, 4, 2/(3*a*x^(3/2)*(a - b*x)^(3/2)) + 4/(a^2*x^(3/2)*sqrt(a - b*x)) - (16*sqrt(a - b*x))/(3*a^3*x^(3/2)) - (32*b*sqrt(a - b*x))/(3*a^4*sqrt(x))],


# ::Subsection::Closed:: 
#Integrands of the form (a+b x)^m (c+d x)^(n/2)


# Integrands of the form (a+b*x)^5*(a*c+b*c*x)^n where n is a half-integer 
[(a + b*x)^5*(a*c + b*c*x)^(3/2), x, 2, (2*(a*c + b*c*x)^(15/2))/(15*b*c^6)],
[(a + b*x)^5*(a*c + b*c*x)^(1/2), x, 2, (2*(a*c + b*c*x)^(13/2))/(13*b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(1/2), x, 2, (2*(a*c + b*c*x)^(11/2))/(11*b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(3/2), x, 2, (2*(a*c + b*c*x)^(9/2))/(9*b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(5/2), x, 2, (2*(a*c + b*c*x)^(7/2))/(7*b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(7/2), x, 2, (2*(a*c + b*c*x)^(5/2))/(5*b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(9/2), x, 2, (2*(a*c + b*c*x)^(3/2))/(3*b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(11/2), x, 2, (2*sqrt(a*c + b*c*x))/(b*c^6)],
[(a + b*x)^5/(a*c + b*c*x)^(13/2), x, 2, -(2/(b*c^6*sqrt(a*c + b*c*x)))],


# ::Subsection::Closed:: 
#Integrands of the form (a+b x)^(m/2) (c+d x)^(n/2)


# Integrands of the form (1-x)^m*(1+x)^m == (1-x^2)^m where m is an integer 
[(1 - x)^(3/2)*(1 + x)^(3/2), x, 4, (3/8)*x*sqrt(1 - x^2) + (1/4)*x*(1 - x^2)^(3/2) + (3*arcsin(x))/8],
[sqrt(1 - x)*sqrt(1 + x), x, 3, (1/2)*x*sqrt(1 - x)*sqrt(1 + x) + arcsin(x)/2, (1/2)*x*sqrt(1 - x^2) + arcsin(x)/2],
[1/(sqrt(1 - x)*sqrt(1 + x)), x, 1, arcsin(x)],
[1/((1 - x)^(3/2)*(1 + x)^(3/2)), x, 2, x/(sqrt(1 - x)*sqrt(1 + x)), x/sqrt(1 - x^2)],


# Integrands of the form (3-x)^m*(-2+x)^m == (-6+5*x-x^2)^m where m is a half-integer 
[(3 - x)^(3/2)*(-2 + x)^(3/2), x, 4, (-(3/64))*(5 - 2*x)*sqrt(-6 + 5*x - x^2) - (1/8)*(5 - 2*x)*(-6 + 5*x - x^2)^(3/2) - (3/128)*arcsin(5 - 2*x)],
[sqrt(3 - x)*sqrt(-2 + x), x, 3, (-(1/4))*(5 - 2*x)*sqrt(-6 + 5*x - x^2) - (1/8)*arcsin(5 - 2*x)],
[1/(sqrt(3 - x)*sqrt(-2 + x)), x, 1, -arcsin(5 - 2*x)],
[1/((3 - x)^(3/2)*(-2 + x)^(3/2)), x, 2, -((2*(5 - 2*x))/sqrt(-6 + 5*x - x^2))],


# ::Subsection::Closed:: 
#Integrands of the form x^m (a+b x)^(n/2) (c+d x)^(p/2)


# Note that the derivative of (a+b*x)^n/(-a-b*x)^n is zero. 
[x^2*sqrt(a + b*x)/sqrt(-a - b*x), x, 2, (x^3*sqrt(a + b*x))/(3*sqrt(-a - b*x))],
[x*sqrt(a + b*x)/sqrt(-a - b*x), x, 2, (x^2*sqrt(a + b*x))/(2*sqrt(-a - b*x))],
[sqrt(a + b*x)/sqrt(-a - b*x), x, 2, (x*sqrt(a + b*x))/sqrt(-a - b*x)],
[sqrt(a + b*x)/(x*sqrt(-a - b*x)), x, 2, (sqrt(a + b*x)*log(x))/sqrt(-a - b*x)],
[sqrt(a + b*x)/(x^2*sqrt(-a - b*x)), x, 2, -(sqrt(a + b*x)/(x*sqrt(-a - b*x)))],
[sqrt(a + b*x)/(x^3*sqrt(-a - b*x)), x, 2, -(sqrt(a + b*x)/(2*x^2*sqrt(-a - b*x)))],
[sqrt(a + b*x)/(x^m*sqrt(-a - b*x)), x, 2, (x^(1 - m)*sqrt(a + b*x))/((1 - m)*sqrt(-a - b*x))],

[x^2*((a + b*x)^n/(-a - b*x)^n), x, 2, ((1/3)*x^3*(a + b*x)^n)/(-a - b*x)^n],
[x*((a + b*x)^n/(-a - b*x)^n), x, 2, ((1/2)*x^2*(a + b*x)^n)/(-a - b*x)^n],
[((a + b*x)^n/(-a - b*x)^n), x, 2, (x*(a + b*x)^n)/(-a - b*x)^n],
[((a + b*x)^n/(x*(-a - b*x)^n)), x, 2, ((a + b*x)^n*log(x))/(-a - b*x)^n],
[((a + b*x)^n/(x^2*(-a - b*x)^n)), x, 2, -((a + b*x)^n/((-a - b*x)^n*x))],
[((a + b*x)^n/(x^3*(-a - b*x)^n)), x, 2, -((a + b*x)^n/((-a - b*x)^n*(2*x^2)))],
[((a + b*x)^n/(x^m*(-a - b*x)^n)), x, 2, (x^(1 - m)*(a + b*x)^n)/((-a - b*x)^n*(1 - m))],


# Integrands of the form x^m/(Sqrt[a+b*x]*(c+d*x)^n) where m is an integer and n>0 is a half-integer 
[x^3/(sqrt(a + b*x)*sqrt(c + d*x)), x, 7, -((2*a*c*sqrt(a + b*x)*sqrt(c + d*x))/(3*b^2*d^2)) + (5*(b*c + a*d)^2*sqrt(a + b*x)*sqrt(c + d*x))/(8*b^3*d^3) - (5*(b*c + a*d)*x*sqrt(a + b*x)*sqrt(c + d*x))/(12*b^2*d^2) + (x^2*sqrt(a + b*x)*sqrt(c + d*x))/(3*b*d) + (3*a*c*(b*c + a*d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(2*b^(5/2)*d^(5/2)) - (5*(b*c + a*d)^3*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(8*b^(7/2)*d^(7/2))],
[x^2/(sqrt(a + b*x)*sqrt(c + d*x)), x, 4, -((3*(b*c + a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^2*d^2)) + (x*sqrt(a + b*x)*sqrt(c + d*x))/(2*b*d) - (a*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(3/2)) + (3*(b*c + a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(5/2)*d^(5/2))],
[x/(sqrt(a + b*x)*sqrt(c + d*x)), x, 2, (sqrt(a + b*x)*sqrt(c + d*x))/(b*d) - ((b*c + a*d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(3/2))],
[1/(sqrt(a + b*x)*sqrt(c + d*x)), x, 1, (2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*sqrt(d))],
[1/(x*sqrt(a + b*x)*sqrt(c + d*x)), x, 1, -((2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*sqrt(c)))],
[1/(x^2*sqrt(a + b*x)*sqrt(c + d*x)), x, 2, -((sqrt(a + b*x)*sqrt(c + d*x))/(a*c*x)) + ((b*c + a*d)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(3/2))],
[1/(x^3*sqrt(a + b*x)*sqrt(c + d*x)), x, 4, -((sqrt(a + b*x)*sqrt(c + d*x))/(2*a*c*x^2)) + (3*(b*c + a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(4*a^2*c^2*x) + (b*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(3/2)) - (3*(b*c + a*d)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(4*a^(5/2)*c^(5/2))],

[x^3/(sqrt(a + b*x)*(c + d*x)^(3/2)), x, 9, -((2*c^3*sqrt(a + b*x))/(d^3*(b*c - a*d)*sqrt(c + d*x))) - (7*c*sqrt(a + b*x)*sqrt(c + d*x))/(4*b*d^3) - (3*a*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^2*d^2) + (x*sqrt(a + b*x)*sqrt(c + d*x))/(2*b*d^2) + (3*c^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(7/2)) + (3*(b*c + a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(5/2)*d^(7/2))],
[x^2/(sqrt(a + b*x)*(c + d*x)^(3/2)), x, 5, (2*c^2*sqrt(a + b*x))/(d^2*(b*c - a*d)*sqrt(c + d*x)) + (sqrt(a + b*x)*sqrt(c + d*x))/(b*d^2) - (3*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(5/2)) - (a*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(3/2))],
[x/(sqrt(a + b*x)*(c + d*x)^(3/2)), x, 3, -((2*c*sqrt(a + b*x))/(d*(b*c - a*d)*sqrt(c + d*x))) + (2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(3/2))],
[1/(sqrt(a + b*x)*(c + d*x)^(3/2)), x, 1, (2*sqrt(a + b*x))/((b*c - a*d)*sqrt(c + d*x))],
[1/(x*sqrt(a + b*x)*(c + d*x)^(3/2)), x, 3, -((2*d*sqrt(a + b*x))/(c*(b*c - a*d)*sqrt(c + d*x))) - (2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(3/2))],
[1/(x^2*sqrt(a + b*x)*(c + d*x)^(3/2)), x, 5, (2*d^2*sqrt(a + b*x))/(c^2*(b*c - a*d)*sqrt(c + d*x)) - (sqrt(a + b*x)*sqrt(c + d*x))/(a*c^2*x) + (b*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(3/2)) + (3*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(5/2))],
[1/(x^3*sqrt(a + b*x)*(c + d*x)^(3/2)), x, 9, -((2*d^3*sqrt(a + b*x))/(c^3*(b*c - a*d)*sqrt(c + d*x))) - (sqrt(a + b*x)*sqrt(c + d*x))/(2*a*c^2*x^2) + (3*b*sqrt(a + b*x)*sqrt(c + d*x))/(4*a^2*c^2*x) + (7*d*sqrt(a + b*x)*sqrt(c + d*x))/(4*a*c^3*x) - (3*d^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(7/2)) - (3*(b*c + a*d)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(4*a^(5/2)*c^(7/2))],

[x^5/(sqrt(a + b*x)*(c + d*x)^(5/2)), x, 22, -((2*a^5*sqrt(a + b*x))/(3*b^5*(b*c - a*d)*(c + d*x)^(3/2))) + (20*a^3*(a + b*x)^(3/2))/(3*b^5*d*(c + d*x)^(3/2)) + (10*a^4*(a + b*x)^(3/2))/(3*b^5*(b*c - a*d)*(c + d*x)^(3/2)) - (20*a^2*(a + b*x)^(5/2))/(3*b^5*d*(c + d*x)^(3/2)) + (10*a*(a + b*x)^(7/2))/(3*b^5*d*(c + d*x)^(3/2)) - (2*(a + b*x)^(9/2))/(3*b^5*d*(c + d*x)^(3/2)) + (20*a^3*sqrt(a + b*x))/(b^4*d^2*sqrt(c + d*x)) - (4*a^5*sqrt(a + b*x))/(3*b^4*(b*c - a*d)^2*sqrt(c + d*x)) - (100*a^2*(a + b*x)^(3/2))/(3*b^4*d^2*sqrt(c + d*x)) + (70*a*(a + b*x)^(5/2))/(3*b^4*d^2*sqrt(c + d*x)) - (6*(a + b*x)^(7/2))/(b^4*d^2*sqrt(c + d*x)) + (175*a*c*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^2*d^4) + (25*a^2*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^3*d^3) + (105*(b*c - a*d)^2*sqrt(a + b*x)*sqrt(c + d*x))/(8*b^3*d^5) - (35*c*(a + b*x)^(3/2)*sqrt(c + d*x))/(4*b^2*d^4) - (245*a*(a + b*x)^(3/2)*sqrt(c + d*x))/(12*b^3*d^3) + (7*(a + b*x)^(5/2)*sqrt(c + d*x))/(b^3*d^3) - (50*a^2*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(5/2)*d^(7/2)) + (30*a^3*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(7/2)*d^(5/2)) - (175*a*(b*c - a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(7/2)*d^(9/2)) - (105*(b*c - a*d)^3*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(8*b^(7/2)*d^(11/2))],
[x^4/(sqrt(a + b*x)*(c + d*x)^(5/2)), x, 16, (2*a^4*sqrt(a + b*x))/(3*b^4*(b*c - a*d)*(c + d*x)^(3/2)) - (4*a^2*(a + b*x)^(3/2))/(b^4*d*(c + d*x)^(3/2)) - (8*a^3*(a + b*x)^(3/2))/(3*b^4*(b*c - a*d)*(c + d*x)^(3/2)) + (8*a*(a + b*x)^(5/2))/(3*b^4*d*(c + d*x)^(3/2)) - (2*(a + b*x)^(7/2))/(3*b^4*d*(c + d*x)^(3/2)) - (12*a^2*sqrt(a + b*x))/(b^3*d^2*sqrt(c + d*x)) + (4*a^4*sqrt(a + b*x))/(3*b^3*(b*c - a*d)^2*sqrt(c + d*x)) + (40*a*(a + b*x)^(3/2))/(3*b^3*d^2*sqrt(c + d*x)) - (14*(a + b*x)^(5/2))/(3*b^3*d^2*sqrt(c + d*x)) - (35*c*sqrt(a + b*x)*sqrt(c + d*x))/(4*b*d^4) - (45*a*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^2*d^3) + (35*(a + b*x)^(3/2)*sqrt(c + d*x))/(6*b^2*d^3) + (20*a*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(7/2)) - (8*a^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(5/2)*d^(5/2)) + (35*(b*c - a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(5/2)*d^(9/2))],
[x^3/(sqrt(a + b*x)*(c + d*x)^(5/2)), x, 7, -((2*c^3*sqrt(a + b*x))/(3*d^3*(b*c - a*d)*(c + d*x)^(3/2))) - (4*b*c^3*sqrt(a + b*x))/(3*d^3*(b*c - a*d)^2*sqrt(c + d*x)) + (6*c^2*sqrt(a + b*x))/(d^3*(b*c - a*d)*sqrt(c + d*x)) + (sqrt(a + b*x)*sqrt(c + d*x))/(b*d^3) - (5*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(7/2)) - (a*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(5/2))],
[x^2/(sqrt(a + b*x)*(c + d*x)^(5/2)), x, 5, (2*c^2*sqrt(a + b*x))/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) + (4*b*c^2*sqrt(a + b*x))/(3*d^2*(b*c - a*d)^2*sqrt(c + d*x)) - (4*c*sqrt(a + b*x))/(d^2*(b*c - a*d)*sqrt(c + d*x)) + (2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(5/2))],
[x/(sqrt(a + b*x)*(c + d*x)^(5/2)), x, 5, -((2*c*sqrt(a + b*x))/(3*d*(b*c - a*d)*(c + d*x)^(3/2))) + (2*(b*c - 3*a*d)*sqrt(a + b*x))/(3*d*(b*c - a*d)^2*sqrt(c + d*x))],
[1/(sqrt(a + b*x)*(c + d*x)^(5/2)), x, 2, (2*sqrt(a + b*x))/(3*(b*c - a*d)*(c + d*x)^(3/2)) + (4*b*sqrt(a + b*x))/(3*(b*c - a*d)^2*sqrt(c + d*x))],
[1/(x*sqrt(a + b*x)*(c + d*x)^(5/2)), x, 6, -((2*d*sqrt(a + b*x))/(3*c*(b*c - a*d)*(c + d*x)^(3/2))) - (4*b*d*sqrt(a + b*x))/(3*c*(b*c - a*d)^2*sqrt(c + d*x)) - (2*d*sqrt(a + b*x))/(c^2*(b*c - a*d)*sqrt(c + d*x)) - (2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(5/2))],
[1/(x^2*sqrt(a + b*x)*(c + d*x)^(5/2)), x, 7, (2*d^2*sqrt(a + b*x))/(3*c^2*(b*c - a*d)*(c + d*x)^(3/2)) + (4*b*d^2*sqrt(a + b*x))/(3*c^2*(b*c - a*d)^2*sqrt(c + d*x)) + (4*d^2*sqrt(a + b*x))/(c^3*(b*c - a*d)*sqrt(c + d*x)) - (sqrt(a + b*x)*sqrt(c + d*x))/(a*c^3*x) + (b*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(5/2)) + (5*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(7/2))],
[1/(x^3*sqrt(a + b*x)*(c + d*x)^(5/2)), x, 11, -((2*d^3*sqrt(a + b*x))/(3*c^3*(b*c - a*d)*(c + d*x)^(3/2))) - (4*b*d^3*sqrt(a + b*x))/(3*c^3*(b*c - a*d)^2*sqrt(c + d*x)) - (6*d^3*sqrt(a + b*x))/(c^4*(b*c - a*d)*sqrt(c + d*x)) - (sqrt(a + b*x)*sqrt(c + d*x))/(2*a*c^3*x^2) + (3*b*sqrt(a + b*x)*sqrt(c + d*x))/(4*a^2*c^3*x) + (11*d*sqrt(a + b*x)*sqrt(c + d*x))/(4*a*c^4*x) - (b*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(7/2)) - (8*d^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(9/2)) - (3*(b*c + a*d)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(4*a^(5/2)*c^(9/2))],


# Integrands of the form x^m/((a+b*x)^(3/2)*(c+d*x)^n) where m is an integer and n>0 is a half-integer 
[x^3/((a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 9, (4*a*c^2)/(b*d^2*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) - (3*(b*c + a*d)^2)/(4*b^3*d^3*sqrt(a + b*x)*sqrt(c + d*x)) - (3*(b*c + a*d)^3)/(4*b^3*d^3*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) + (3*(b*c + a*d)*x)/(2*b^2*d^2*sqrt(a + b*x)*sqrt(c + d*x)) + x^2/(b*d*sqrt(a + b*x)*sqrt(c + d*x)) - (4*a*c*(b*c + a*d)*sqrt(c + d*x))/(b*d^2*(b*c - a*d)^2*sqrt(a + b*x)) + (3*(b*c + a*d)^3*sqrt(c + d*x))/(2*b^2*d^3*(b*c - a*d)^2*sqrt(a + b*x)) - (3*(b*c + a*d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(5/2)*d^(5/2))],
[x^2/((a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 5, (b*c + a*d)/(2*b^2*d^2*sqrt(a + b*x)*sqrt(c + d*x)) + (b*c + a*d)^2/(2*b^2*d^2*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) - x/(b*d*sqrt(a + b*x)*sqrt(c + d*x)) - ((b*c + a*d)^2*sqrt(c + d*x))/(b*d^2*(b*c - a*d)^2*sqrt(a + b*x)) + (2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(3/2))],
[x/((a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 3, -((2*c)/(d*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x))) + (2*(b*c + a*d)*sqrt(c + d*x))/(d*(b*c - a*d)^2*sqrt(a + b*x))],
[1/((a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 2, 2/((b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) - (4*b*sqrt(c + d*x))/((b*c - a*d)^2*sqrt(a + b*x))],
[1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 4, -((2*d)/(c*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x))) + (2*b*(b*c + a*d)*sqrt(c + d*x))/(a*c*(b*c - a*d)^2*sqrt(a + b*x)) - (2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(3/2))],
[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 7, -((4*b*d)/(a*c*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x))) - (3*(b*c + a*d))/(2*a^2*c^2*sqrt(a + b*x)*sqrt(c + d*x)) + (3*(b*c + a*d)^2)/(2*a^2*c^2*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) - 1/(a*c*x*sqrt(a + b*x)*sqrt(c + d*x)) + (8*b^2*d*sqrt(c + d*x))/(a*c*(b*c - a*d)^2*sqrt(a + b*x)) - (3*b*(b*c + a*d)^2*sqrt(c + d*x))/(a^2*c^2*(b*c - a*d)^2*sqrt(a + b*x)) + (3*(b*c + a*d)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(5/2)*c^(5/2))],
[1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)), x, 12, -((3*b*d)/(2*a^2*c^2*sqrt(a + b*x)*sqrt(c + d*x))) + (13*b*d*(b*c + a*d))/(2*a^2*c^2*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) + (15*(b*c + a*d)^2)/(8*a^3*c^3*sqrt(a + b*x)*sqrt(c + d*x)) - (15*(b*c + a*d)^3)/(8*a^3*c^3*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) - 1/(2*a*c*x^2*sqrt(a + b*x)*sqrt(c + d*x)) + (5*(b*c + a*d))/(4*a^2*c^2*x*sqrt(a + b*x)*sqrt(c + d*x)) - (13*b^2*d*(b*c + a*d)*sqrt(c + d*x))/(a^2*c^2*(b*c - a*d)^2*sqrt(a + b*x)) + (15*b*(b*c + a*d)^3*sqrt(c + d*x))/(4*a^3*c^3*(b*c - a*d)^2*sqrt(a + b*x)) + (3*b*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(5/2)*c^(5/2)) - (15*(b*c + a*d)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(4*a^(7/2)*c^(7/2))],

[x^5/((a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 19, (2*a^5)/(b^5*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2)) + (8*a^5*d*sqrt(a + b*x))/(3*b^5*(b*c - a*d)^2*(c + d*x)^(3/2)) + (10*a^4*sqrt(a + b*x))/(3*b^5*(b*c - a*d)*(c + d*x)^(3/2)) - (20*a^2*c*(a + b*x)^(3/2))/(3*b^4*d*(b*c - a*d)*(c + d*x)^(3/2)) + (10*a*(a + b*x)^(5/2))/(3*b^5*d*(c + d*x)^(3/2)) - (2*(a + b*x)^(7/2))/(3*b^5*d*(c + d*x)^(3/2)) - (20*a^2*sqrt(a + b*x))/(b^4*d^2*sqrt(c + d*x)) + (16*a^5*d*sqrt(a + b*x))/(3*b^4*(b*c - a*d)^3*sqrt(c + d*x)) + (20*a^4*sqrt(a + b*x))/(3*b^4*(b*c - a*d)^2*sqrt(c + d*x)) + (50*a*(a + b*x)^(3/2))/(3*b^4*d^2*sqrt(c + d*x)) - (14*(a + b*x)^(5/2))/(3*b^4*d^2*sqrt(c + d*x)) - (35*c*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^2*d^4) - (65*a*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^3*d^3) + (35*(a + b*x)^(3/2)*sqrt(c + d*x))/(6*b^3*d^3) + (25*a*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(5/2)*d^(7/2)) - (5*a^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(7/2)*d^(5/2)) + (35*(b*c - a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(7/2)*d^(9/2))],
[x^4/((a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 14, -((2*a^4)/(b^4*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2))) - (8*a^4*d*sqrt(a + b*x))/(3*b^4*(b*c - a*d)^2*(c + d*x)^(3/2)) - (8*a^3*sqrt(a + b*x))/(3*b^4*(b*c - a*d)*(c + d*x)^(3/2)) + (8*a*(a + b*x)^(3/2))/(3*b^4*d*(c + d*x)^(3/2)) + (4*a^2*(a + b*x)^(3/2))/(b^4*(b*c - a*d)*(c + d*x)^(3/2)) - (2*(a + b*x)^(5/2))/(3*b^4*d*(c + d*x)^(3/2)) + (8*a*sqrt(a + b*x))/(b^3*d^2*sqrt(c + d*x)) - (16*a^4*d*sqrt(a + b*x))/(3*b^3*(b*c - a*d)^3*sqrt(c + d*x)) - (16*a^3*sqrt(a + b*x))/(3*b^3*(b*c - a*d)^2*sqrt(c + d*x)) - (10*(a + b*x)^(3/2))/(3*b^3*d^2*sqrt(c + d*x)) + (5*sqrt(a + b*x)*sqrt(c + d*x))/(b^2*d^3) - (5*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(7/2)) - (3*a*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(5/2)*d^(5/2))],
[x^3/((a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 10, (2*a^3)/(b^3*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2)) + (8*a^3*d*sqrt(a + b*x))/(3*b^3*(b*c - a*d)^2*(c + d*x)^(3/2)) + (2*a^2*sqrt(a + b*x))/(b^3*(b*c - a*d)*(c + d*x)^(3/2)) - (2*(a + b*x)^(3/2))/(3*b^3*d*(c + d*x)^(3/2)) - (2*a*(a + b*x)^(3/2))/(b^3*(b*c - a*d)*(c + d*x)^(3/2)) - (2*sqrt(a + b*x))/(b^2*d^2*sqrt(c + d*x)) + (16*a^3*d*sqrt(a + b*x))/(3*b^2*(b*c - a*d)^3*sqrt(c + d*x)) + (4*a^2*sqrt(a + b*x))/(b^2*(b*c - a*d)^2*sqrt(c + d*x)) + (2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(b^(3/2)*d^(5/2))],
[x^2/((a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 7, -((2*a^2)/(b^2*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2))) - (8*a^2*d*sqrt(a + b*x))/(3*b^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (4*a*sqrt(a + b*x))/(3*b^2*(b*c - a*d)*(c + d*x)^(3/2)) + (2*(a + b*x)^(3/2))/(3*b^2*(b*c - a*d)*(c + d*x)^(3/2)) - (16*a^2*d*sqrt(a + b*x))/(3*b*(b*c - a*d)^3*sqrt(c + d*x)) - (8*a*sqrt(a + b*x))/(3*b*(b*c - a*d)^2*sqrt(c + d*x))],
[x/((a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 6, (2*a)/(b*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2)) + (8*a*d*sqrt(a + b*x))/(3*b*(b*c - a*d)^2*(c + d*x)^(3/2)) + (2*sqrt(a + b*x))/(3*b*(b*c - a*d)*(c + d*x)^(3/2)) + (16*a*d*sqrt(a + b*x))/(3*(b*c - a*d)^3*sqrt(c + d*x)) + (4*sqrt(a + b*x))/(3*(b*c - a*d)^2*sqrt(c + d*x))],
[1/((a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 3, -(2/((b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2))) - (8*d*sqrt(a + b*x))/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) - (16*b*d*sqrt(a + b*x))/(3*(b*c - a*d)^3*sqrt(c + d*x))],
[1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 10, (2*b)/(a*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2)) + (8*b*d*sqrt(a + b*x))/(3*a*(b*c - a*d)^2*(c + d*x)^(3/2)) - (2*d*sqrt(a + b*x))/(3*a*c*(b*c - a*d)*(c + d*x)^(3/2)) + (16*b^2*d*sqrt(a + b*x))/(3*a*(b*c - a*d)^3*sqrt(c + d*x)) - (4*b*d*sqrt(a + b*x))/(3*a*c*(b*c - a*d)^2*sqrt(c + d*x)) - (2*d*sqrt(a + b*x))/(a*c^2*(b*c - a*d)*sqrt(c + d*x)) - (2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(5/2))],
[1/(x^2*(a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 15, -((2*d^2)/(c^2*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(3/2))) - (8*d^3*sqrt(a + b*x))/(3*c^2*(b*c - a*d)^2*(c + d*x)^(3/2)) - (3*b)/(2*a^2*c^2*sqrt(a + b*x)*sqrt(c + d*x)) - (5*d)/(2*a*c^3*sqrt(a + b*x)*sqrt(c + d*x)) - (3*b*d)/(a*c^2*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) + d^2/(c^3*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) + (3*(b*c + a*d)^2)/(2*a^2*c^3*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) - 1/(a*c^2*x*sqrt(a + b*x)*sqrt(c + d*x)) - (16*b*d^3*sqrt(a + b*x))/(3*c^2*(b*c - a*d)^3*sqrt(c + d*x)) + (6*b^2*d*sqrt(c + d*x))/(a*c^2*(b*c - a*d)^2*sqrt(a + b*x)) - (2*b*d^2*sqrt(c + d*x))/(c^3*(b*c - a*d)^2*sqrt(a + b*x)) - (3*b*(b*c + a*d)^2*sqrt(c + d*x))/(a^2*c^3*(b*c - a*d)^2*sqrt(a + b*x)) + (3*b*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(5/2)*c^(5/2)) + (5*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(3/2)*c^(7/2))],
# {1/(x^3*(a + b*x)^(3/2)*(c + d*x)^(5/2)), x, 0, 0} 


# Integrands of the form x^m/((a+b*x)^(5/2)*(c+d*x)^n) where m is an integer and n>0 is a half-integer 
[x^3/((a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 16, -((2*a*c)/(3*b^2*d^2*(a + b*x)^(3/2)*(c + d*x)^(3/2))) - (a*c*(b*c + a*d))/(2*b^2*d^2*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*c + a*d)^2/(24*b^3*d^3*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*c + a*d)^3/(24*b^3*d^3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - ((b*c + a*d)*x)/(4*b^2*d^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - x^2/(b*d*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (3*a*c*(b*c + a*d))/(b*d^2*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) + (b*c + a*d)^3/(4*b^2*d^3*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) + (4*a*c*(b*c + a*d)*sqrt(c + d*x))/(d^2*(b*c - a*d)^3*(a + b*x)^(3/2)) - ((b*c + a*d)^3*sqrt(c + d*x))/(3*b*d^3*(b*c - a*d)^3*(a + b*x)^(3/2)) - (8*a*c*(b*c + a*d)*sqrt(c + d*x))/(d*(b*c - a*d)^4*sqrt(a + b*x)) + (2*(b*c + a*d)^3*sqrt(c + d*x))/(3*b*d^2*(b*c - a*d)^4*sqrt(a + b*x))],
[x^2/((a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 10, (a*c)/(3*b*d*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*c + a*d)/(12*b^2*d^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (b*c + a*d)^2/(12*b^2*d^2*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - x/(2*b*d*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*c)/(d*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) + (b*c + a*d)^2/(2*b*d^2*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) - (8*a*b*c*sqrt(c + d*x))/(3*d*(b*c - a*d)^3*(a + b*x)^(3/2)) - (2*(b*c + a*d)^2*sqrt(c + d*x))/(3*d^2*(b*c - a*d)^3*(a + b*x)^(3/2)) + (16*a*b*c*sqrt(c + d*x))/(3*(b*c - a*d)^4*sqrt(a + b*x)) + (4*(b*c + a*d)^2*sqrt(c + d*x))/(3*d*(b*c - a*d)^4*sqrt(a + b*x))],
[x/((a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 5, -((2*c)/(3*d*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))) - (2*(b*c + a*d))/(d*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) + (8*b*(b*c + a*d)*sqrt(c + d*x))/(3*d*(b*c - a*d)^3*(a + b*x)^(3/2)) - (16*b*(b*c + a*d)*sqrt(c + d*x))/(3*(b*c - a*d)^4*sqrt(a + b*x))],
[1/((a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 4, 2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (4*b)/((b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) - (16*b^2*sqrt(c + d*x))/(3*(b*c - a*d)^3*(a + b*x)^(3/2)) + (32*b^2*d*sqrt(c + d*x))/(3*(b*c - a*d)^4*sqrt(a + b*x))],
[1/(x*(a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 9, -((2*d)/(3*c*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))) - (2*b*(b*c + a*d))/(a*c*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) - (2*d)/(a*c^2*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) + (8*b^2*(b*c + a*d)*sqrt(c + d*x))/(3*a*c*(b*c - a*d)^3*(a + b*x)^(3/2)) - (16*b^2*d*(b*c + a*d)*sqrt(c + d*x))/(3*a*c*(b*c - a*d)^4*sqrt(a + b*x)) + (2*b*(b*c + a*d)*sqrt(c + d*x))/(a^2*c^2*(b*c - a*d)^2*sqrt(a + b*x)) - (2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(5/2)*c^(5/2))],
[1/(x^2*(a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 14, -((8*b*d)/(3*a*c*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))) - (5*(b*c + a*d))/(6*a^2*c^2*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (5*(b*c + a*d)^2)/(6*a^2*c^2*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - 1/(a*c*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (16*b^2*d)/(a*c*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) + (5*b*(b*c + a*d)^2)/(a^2*c^2*(b*c - a*d)^2*(a + b*x)^(3/2)*sqrt(c + d*x)) - (5*(b*c + a*d))/(2*a^3*c^3*sqrt(a + b*x)*sqrt(c + d*x)) + (5*(b*c + a*d)^2)/(2*a^3*c^3*(b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x)) + (64*b^3*d*sqrt(c + d*x))/(3*a*c*(b*c - a*d)^3*(a + b*x)^(3/2)) - (20*b^2*(b*c + a*d)^2*sqrt(c + d*x))/(3*a^2*c^2*(b*c - a*d)^3*(a + b*x)^(3/2)) - (128*b^3*d^2*sqrt(c + d*x))/(3*a*c*(b*c - a*d)^4*sqrt(a + b*x)) + (40*b^2*d*(b*c + a*d)^2*sqrt(c + d*x))/(3*a^2*c^2*(b*c - a*d)^4*sqrt(a + b*x)) - (5*b*(b*c + a*d)^2*sqrt(c + d*x))/(a^3*c^3*(b*c - a*d)^2*sqrt(a + b*x)) + (5*(b*c + a*d)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(a^(7/2)*c^(7/2))],
# {1/(x^3*(a + b*x)^(5/2)*(c + d*x)^(5/2)), x, 0, 0} 


# Integrands of the form x^m*Sqrt[a+b*x]*(c+d*x)^(5/2) where m is an integer 
# {x^4*Sqrt[a + b*x]*(c + d*x)^(5/2), x, 36, -((5*c^4*(b*c - a*d)^3*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^3*d^5)) + (7*c^3*(b*c - a*d)^4*Sqrt[a + b*x]*Sqrt[c + d*x])/(32*b^4*d^5) - (63*c^2*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(256*b^5*d^5) + (33*c*(b*c - a*d)^6*Sqrt[a + b*x]*Sqrt[c + d*x])/(256*b^6*d^5) - (429*(b*c - a*d)^7*Sqrt[a + b*x]*Sqrt[c + d*x])/(16384*b^7*d^5) - (5*c^4*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^2*d^5) + (7*c^3*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(3/2))/(48*b^3*d^5) - (21*c^2*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(3/2))/(128*b^4*d^5) + (11*c*(b*c - a*d)^5*Sqrt[a + b*x]*(c + d*x)^(3/2))/(128*b^5*d^5) - (143*(b*c - a*d)^6*Sqrt[a + b*x]*(c + d*x)^(3/2))/(8192*b^6*d^5) - (c^4*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*b*d^5) + (7*c^3*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(5/2))/(60*b^2*d^5) - (21*c^2*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(5/2))/(160*b^3*d^5) + (11*c*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(5/2))/(160*b^4*d^5) - (143*(b*c - a*d)^5*Sqrt[a + b*x]*(c + d*x)^(5/2))/(10240*b^5*d^5) + (c^4*Sqrt[a + b*x]*(c + d*x)^(7/2))/(4*d^5) + (c^3*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(10*b*d^5) - (9*c^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(7/2))/(80*b^2*d^5) + (33*c*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(7/2))/(560*b^3*d^5) - (429*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(7/2))/(35840*b^4*d^5) - (4*c^3*Sqrt[a + b*x]*(c + d*x)^(9/2))/(5*d^5) - (c^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(9/2))/(10*b*d^5) + (11*c*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(9/2))/(210*b^2*d^5) - (143*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(9/2))/(13440*b^3*d^5) + (c^2*Sqrt[a + b*x]*(c + d*x)^(11/2))/d^5 + (c*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(11/2))/(21*b*d^5) - (13*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(11/2))/(1344*b^2*d^5) - (4*c*Sqrt[a + b*x]*(c + d*x)^(13/2))/(7*d^5) - ((b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(13/2))/(112*b*d^5) + (Sqrt[a + b*x]*(c + d*x)^(15/2))/(8*d^5) - (5*c^4*(b*c - a*d)^4*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(7/2)*d^(11/2)) + (7*c^3*(b*c - a*d)^5*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(32*b^(9/2)*d^(11/2)) - (63*c^2*(b*c - a*d)^6*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(256*b^(11/2)*d^(11/2)) + (33*c*(b*c - a*d)^7*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(256*b^(13/2)*d^(11/2)) - (429*(b*c - a*d)^8*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(16384*b^(15/2)*d^(11/2))} 
[x^3*sqrt(a + b*x)*(c + d*x)^(5/2), x, 27, (5*c^3*(b*c - a*d)^3*sqrt(a + b*x)*sqrt(c + d*x))/(64*b^3*d^4) - (21*c^2*(b*c - a*d)^4*sqrt(a + b*x)*sqrt(c + d*x))/(128*b^4*d^4) + (63*c*(b*c - a*d)^5*sqrt(a + b*x)*sqrt(c + d*x))/(512*b^5*d^4) - (33*(b*c - a*d)^6*sqrt(a + b*x)*sqrt(c + d*x))/(1024*b^6*d^4) + (5*c^3*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(3/2))/(96*b^2*d^4) - (7*c^2*(b*c - a*d)^3*sqrt(a + b*x)*(c + d*x)^(3/2))/(64*b^3*d^4) + (21*c*(b*c - a*d)^4*sqrt(a + b*x)*(c + d*x)^(3/2))/(256*b^4*d^4) - (11*(b*c - a*d)^5*sqrt(a + b*x)*(c + d*x)^(3/2))/(512*b^5*d^4) + (c^3*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(5/2))/(24*b*d^4) - (7*c^2*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(5/2))/(80*b^2*d^4) + (21*c*(b*c - a*d)^3*sqrt(a + b*x)*(c + d*x)^(5/2))/(320*b^3*d^4) - (11*(b*c - a*d)^4*sqrt(a + b*x)*(c + d*x)^(5/2))/(640*b^4*d^4) - (13*c^3*sqrt(a + b*x)*(c + d*x)^(7/2))/(40*d^4) + (3*a*c^2*sqrt(a + b*x)*(c + d*x)^(7/2))/(40*b*d^3) + (9*c*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(7/2))/(160*b^2*d^4) - (33*(b*c - a*d)^3*sqrt(a + b*x)*(c + d*x)^(7/2))/(2240*b^3*d^4) + (13*c^2*sqrt(a + b*x)*(c + d*x)^(9/2))/(20*d^4) - (a*c*sqrt(a + b*x)*(c + d*x)^(9/2))/(20*b*d^3) - (11*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(9/2))/(840*b^2*d^4) - (43*c*sqrt(a + b*x)*(c + d*x)^(11/2))/(84*d^4) + (a*sqrt(a + b*x)*(c + d*x)^(11/2))/(84*b*d^3) + (sqrt(a + b*x)*(c + d*x)^(13/2))/(7*d^4) + (5*c^3*(b*c - a*d)^4*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(64*b^(7/2)*d^(9/2)) - (21*c^2*(b*c - a*d)^5*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(128*b^(9/2)*d^(9/2)) + (63*c*(b*c - a*d)^6*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(512*b^(11/2)*d^(9/2)) - (33*(b*c - a*d)^7*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(1024*b^(13/2)*d^(9/2))],
[x^2*sqrt(a + b*x)*(c + d*x)^(5/2), x, 19, -((5*c^2*(b*c - a*d)^3*sqrt(a + b*x)*sqrt(c + d*x))/(64*b^3*d^3)) + (7*c*(b*c - a*d)^4*sqrt(a + b*x)*sqrt(c + d*x))/(64*b^4*d^3) - (21*(b*c - a*d)^5*sqrt(a + b*x)*sqrt(c + d*x))/(512*b^5*d^3) - (5*c^2*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(3/2))/(96*b^2*d^3) + (7*c*(b*c - a*d)^3*sqrt(a + b*x)*(c + d*x)^(3/2))/(96*b^3*d^3) - (7*(b*c - a*d)^4*sqrt(a + b*x)*(c + d*x)^(3/2))/(256*b^4*d^3) - (c^2*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(5/2))/(24*b*d^3) + (7*c*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(5/2))/(120*b^2*d^3) - (7*(b*c - a*d)^3*sqrt(a + b*x)*(c + d*x)^(5/2))/(320*b^3*d^3) + (3*c^2*sqrt(a + b*x)*(c + d*x)^(7/2))/(10*d^3) - (a*c*sqrt(a + b*x)*(c + d*x)^(7/2))/(20*b*d^2) - (3*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(7/2))/(160*b^2*d^3) - (5*c*sqrt(a + b*x)*(c + d*x)^(9/2))/(12*d^3) + (a*sqrt(a + b*x)*(c + d*x)^(9/2))/(60*b*d^2) + (sqrt(a + b*x)*(c + d*x)^(11/2))/(6*d^3) - (5*c^2*(b*c - a*d)^4*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(64*b^(7/2)*d^(7/2)) + (7*c*(b*c - a*d)^5*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(64*b^(9/2)*d^(7/2)) - (21*(b*c - a*d)^6*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(512*b^(11/2)*d^(7/2))],
[x*sqrt(a + b*x)*(c + d*x)^(5/2), x, 12, (5*c*(b*c - a*d)^3*sqrt(a + b*x)*sqrt(c + d*x))/(64*b^3*d^2) - (7*(b*c - a*d)^4*sqrt(a + b*x)*sqrt(c + d*x))/(128*b^4*d^2) + (5*c*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(3/2))/(96*b^2*d^2) - (7*(b*c - a*d)^3*sqrt(a + b*x)*(c + d*x)^(3/2))/(192*b^3*d^2) + (c*(b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(5/2))/(24*b*d^2) - (7*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(5/2))/(240*b^2*d^2) - (11*c*sqrt(a + b*x)*(c + d*x)^(7/2))/(40*d^2) + (a*sqrt(a + b*x)*(c + d*x)^(7/2))/(40*b*d) + (sqrt(a + b*x)*(c + d*x)^(9/2))/(5*d^2) + (5*c*(b*c - a*d)^4*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(64*b^(7/2)*d^(5/2)) - (7*(b*c - a*d)^5*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(128*b^(9/2)*d^(5/2))],
[sqrt(a + b*x)*(c + d*x)^(5/2), x, 5, -((5*(b*c - a*d)^3*sqrt(a + b*x)*sqrt(c + d*x))/(64*b^3*d)) - (5*(b*c - a*d)^2*sqrt(a + b*x)*(c + d*x)^(3/2))/(96*b^2*d) - ((b*c - a*d)*sqrt(a + b*x)*(c + d*x)^(5/2))/(24*b*d) + (sqrt(a + b*x)*(c + d*x)^(7/2))/(4*d) - (5*(b*c - a*d)^4*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(64*b^(7/2)*d^(3/2))],
[sqrt(a + b*x)*(c + d*x)^(5/2)/x, x, 15, (7*a*c*d*sqrt(a + b*x)*sqrt(c + d*x))/(4*b) - (3*a^2*d^2*sqrt(a + b*x)*sqrt(c + d*x))/(4*b^2) + (5*(b*c - a*d)^2*sqrt(a + b*x)*sqrt(c + d*x))/(8*b^2) + (5/12)*c*sqrt(a + b*x)*(c + d*x)^(3/2) + (a*d*sqrt(a + b*x)*(c + d*x)^(3/2))/(12*b) + (1/3)*sqrt(a + b*x)*(c + d*x)^(5/2) - 2*sqrt(a)*c^(5/2)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))) + (3*a*c^2*sqrt(d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/sqrt(b) - (a^2*c*d^(3/2)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/b^(3/2) + (3*a*sqrt(d)*(b*c - a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(5/2)) + (5*(b*c - a*d)^3*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(8*b^(5/2)*sqrt(d))],
# {Sqrt[a + b*x]*(c + d*x)^(5/2)/x^2, x, 19, 0} 


# Integrands of the form x^m*Sqrt[1+x]/(1-x)^(5/2) where m is an integer 
[x^3*sqrt(1 + x)/(1 - x)^(5/2), x, 14, -((6*sqrt(1 + x))/sqrt(1 - x)) - 3*sqrt(1 - x)*sqrt(1 + x) + (1 + x)^(3/2)/(3*(1 - x)^(3/2)) - (1/2)*x*sqrt(1 - x^2) + (11*arcsin(x))/2, -((6*sqrt(1 + x))/sqrt(1 - x)) - (5/2)*sqrt(1 - x)*sqrt(1 + x) + (5*(1 + x)^(3/2))/(3*(1 - x)^(3/2)) + (10*(1 + x)^(3/2))/sqrt(1 - x) - (35/6)*sqrt(1 - x)*(1 + x)^(3/2) - (2*(1 + x)^(5/2))/(1 - x)^(3/2) - (14*(1 + x)^(5/2))/(3*sqrt(1 - x)) + (2*(1 + x)^(7/2))/(3*(1 - x)^(3/2)) + (11*arcsin(x))/2],
[x^2*sqrt(1 + x)/(1 - x)^(5/2), x, 9, -((4*sqrt(1 + x))/sqrt(1 - x)) - sqrt(1 - x)*sqrt(1 + x) + (1 + x)^(3/2)/(3*(1 - x)^(3/2)) + 3*arcsin(x), (4*sqrt(1 + x))/sqrt(1 - x) - 5*sqrt(1 - x)*sqrt(1 + x) - (1 + x)^(3/2)/(1 - x)^(3/2) - (10*(1 + x)^(3/2))/(3*sqrt(1 - x)) + (2*(1 + x)^(5/2))/(3*(1 - x)^(3/2)) + 3*arcsin(x)],
[x*sqrt(1 + x)/(1 - x)^(5/2), x, 5, -((2*sqrt(1 + x))/sqrt(1 - x)) + (1 + x)^(3/2)/(3*(1 - x)^(3/2)) + arcsin(x)],
[sqrt(1 + x)/(1 - x)^(5/2), x, 1, (1 + x)^(3/2)/(3*(1 - x)^(3/2))],
[sqrt(1 + x)/(-1 + x)^(5/2), x, 1, -((1 + x)^(3/2)/(3*(-1 + x)^(3/2)))],
[sqrt(1 + x)/(x*(1 - x)^(5/2)), x, 9, (2*sqrt(1 + x))/(3*(1 - x)^(3/2)) + (5*sqrt(1 + x))/(3*sqrt(1 - x)) - 2*arctanh(sqrt(1 - x)/sqrt(1 + x))],
[sqrt(1 + x)/(x^2*(1 - x)^(5/2)), x, 12, (4*sqrt(1 + x))/sqrt(1 - x) + (1 + x)^(3/2)/(3*(1 - x)^(3/2)) - ((1 - x)^(3/2)*(1 + x)^(3/2))/x - x*sqrt(1 - x^2) - 6*arctanh(sqrt(1 - x)/sqrt(1 + x))],
[sqrt(1 + x)/(x^3*(1 - x)^(5/2)), x, 15, (6*sqrt(1 + x))/sqrt(1 - x) - (1/2)*sqrt(1 - x)*sqrt(1 + x) + (1 + x)^(3/2)/(3*(1 - x)^(3/2)) - ((1 - x)^(3/2)*(1 + x)^(3/2))/(2*x^2) - (3*(1 - x)^(3/2)*(1 + x)^(3/2))/x - 3*x*sqrt(1 - x^2) - 11*arctanh(sqrt(1 - x)/sqrt(1 + x))],


# Integrands of the form x^m*(a+b*x)^n*(c+d*x)^p where m is an integer, and n and p are half-integers 
[x^3*sqrt(a + b*x)/sqrt(c + d*x), x, 12, (55*a*c^2*sqrt(a + b*x)*sqrt(c + d*x))/(48*b*d^3) + (23*a^2*c*sqrt(a + b*x)*sqrt(c + d*x))/(48*b^2*d^2) - (5*(7*b*c - a*d)*(b*c + a*d)^2*sqrt(a + b*x)*sqrt(c + d*x))/(64*b^3*d^4) - (3*a*c*x*sqrt(a + b*x)*sqrt(c + d*x))/(8*b*d^2) + (5*(7*b*c - a*d)*(b*c + a*d)*x*sqrt(a + b*x)*sqrt(c + d*x))/(96*b^2*d^3) - ((7*b*c - a*d)*x^2*sqrt(a + b*x)*sqrt(c + d*x))/(24*b*d^2) + (x^3*sqrt(a + b*x)*sqrt(c + d*x))/(4*d) + (3*a^2*c^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(3/2)*d^(5/2)) - (3*a*c*(7*b*c - a*d)*(b*c + a*d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(16*b^(5/2)*d^(7/2)) - (9*a*c*(b*c + a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(16*b^(5/2)*d^(7/2)) + (5*(7*b*c - a*d)*(b*c + a*d)^3*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(64*b^(7/2)*d^(9/2))],
[x^2*sqrt(a + b*x)/sqrt(c + d*x), x, 7, -((2*a*c*sqrt(a + b*x)*sqrt(c + d*x))/(3*b*d^2)) + ((5*b*c - a*d)*(b*c + a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(8*b^2*d^3) - ((5*b*c - a*d)*x*sqrt(a + b*x)*sqrt(c + d*x))/(12*b*d^2) + (x^2*sqrt(a + b*x)*sqrt(c + d*x))/(3*d) + (3*a*c^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(2*sqrt(b)*d^(5/2)) + (a^2*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(2*b^(3/2)*d^(3/2)) - ((5*b*c - a*d)*(b*c + a*d)^2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(8*b^(5/2)*d^(7/2))],
[x*sqrt(a + b*x)/sqrt(c + d*x), x, 4, -(((3*b*c - a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(4*b*d^2)) + (x*sqrt(a + b*x)*sqrt(c + d*x))/(2*d) - (a*c*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(3/2)) + ((3*b*c - a*d)*(b*c + a*d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(4*b^(3/2)*d^(5/2))],
[sqrt(a + b*x)/sqrt(c + d*x), x, 2, (sqrt(a + b*x)*sqrt(c + d*x))/d - ((b*c - a*d)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*d^(3/2))],
[sqrt(a + b*x)/(x*sqrt(c + d*x)), x, 3, -((2*sqrt(a)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/sqrt(c)) + (2*sqrt(b)*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/sqrt(d)],
[sqrt(a + b*x)/(x^2*sqrt(c + d*x)), x, 2, -((sqrt(a + b*x)*sqrt(c + d*x))/(c*x)) - ((b*c - a*d)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(3/2))],
[sqrt(a + b*x)/(x^3*sqrt(c + d*x)), x, 4, -((sqrt(a + b*x)*sqrt(c + d*x))/(2*c*x^2)) - ((b*c - 3*a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(4*a*c^2*x) + (b*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(sqrt(a)*c^(3/2)) + ((b*c - 3*a*d)*(b*c + a*d)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(4*a^(3/2)*c^(5/2))],
[sqrt(a + b*x)/(x^4*sqrt(c + d*x)), x, 7, -((sqrt(a + b*x)*sqrt(c + d*x))/(3*c*x^3)) - ((b*c - 5*a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(12*a*c^2*x^2) + (2*b*d*sqrt(a + b*x)*sqrt(c + d*x))/(3*a*c^2*x) + ((b*c - 5*a*d)*(b*c + a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(8*a^2*c^3*x) - (b^2*d*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(2*a^(3/2)*c^(3/2)) - (3*b*d^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(2*sqrt(a)*c^(5/2)) - ((b*c - 5*a*d)*(b*c + a*d)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(8*a^(5/2)*c^(7/2))],
[sqrt(a + b*x)/(x^5*sqrt(c + d*x)), x, 12, -((sqrt(a + b*x)*sqrt(c + d*x))/(4*c*x^4)) - ((b*c - 7*a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(24*a*c^2*x^3) + (3*b*d*sqrt(a + b*x)*sqrt(c + d*x))/(8*a*c^2*x^2) + (5*(b*c - 7*a*d)*(b*c + a*d)*sqrt(a + b*x)*sqrt(c + d*x))/(96*a^2*c^3*x^2) - (23*b^2*d*sqrt(a + b*x)*sqrt(c + d*x))/(48*a^2*c^2*x) - (55*b*d^2*sqrt(a + b*x)*sqrt(c + d*x))/(48*a*c^3*x) - (5*(b*c - 7*a*d)*(b*c + a*d)^2*sqrt(a + b*x)*sqrt(c + d*x))/(64*a^3*c^4*x) - (3*b^2*d^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(4*a^(3/2)*c^(5/2)) - (3*b*d*(b*c - 7*a*d)*(b*c + a*d)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(16*a^(5/2)*c^(7/2)) + (9*b*d*(b*c + a*d)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(16*a^(5/2)*c^(7/2)) + (5*(b*c - 7*a*d)*(b*c + a*d)^3*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + d*x))))/(64*a^(7/2)*c^(9/2))],

[x^2/(sqrt(-1 + x)*sqrt(1 + x)), x, 2, (1/2)*sqrt(-1 + x)*x*sqrt(1 + x) + arccosh(x)/2],
[x/(sqrt(-1 + x)*sqrt(1 + x)), x, 1, sqrt(-1 + x)*sqrt(1 + x)],
[1/(sqrt(-1 + x)*sqrt(1 + x)), x, 1, arccosh(x)],
[1/(x*sqrt(-1 + x)*sqrt(1 + x)), x, 1, 2*arctan(sqrt(-1 + x)/sqrt(1 + x))],
[1/(x^2*sqrt(-1 + x)*sqrt(1 + x)), x, 1, (sqrt(-1 + x)*sqrt(1 + x))/x],
[(-1 + 2*x^2)/(sqrt(-1 + x)*sqrt(1 + x)), x, 5, sqrt(-1 + x)*x*sqrt(1 + x)],

[x^2*sqrt(-1 + x)*sqrt(1 + x), x, 4, (-(1/8))*sqrt(-1 + x)*x*sqrt(1 + x) + (1/4)*sqrt(-1 + x)*x^3*sqrt(1 + x) - arccosh(x)/8, (1/8)*sqrt(-1 + x)*sqrt(1 + x) + (1/8)*(-1 + x)^(3/2)*sqrt(1 + x) + (1/4)*(-1 + x)^(3/2)*x*(1 + x)^(3/2) - arccosh(x)/8],
[x*sqrt(-1 + x)*sqrt(1 + x), x, 1, ((-1 + x)^(3/2)*(1 + x)^(3/2))/3],
[sqrt(-1 + x)*sqrt(1 + x), x, 3, (1/2)*sqrt(-1 + x)*x*sqrt(1 + x) - arccosh(x)/2, (1/2)*sqrt(-1 + x)*sqrt(1 + x) + (1/2)*(-1 + x)^(3/2)*sqrt(1 + x) - arccosh(x)/2],
[sqrt(-1 + x)*sqrt(1 + x)/x, x, 2, sqrt(-1 + x)*sqrt(1 + x) - 2*arctan(sqrt(-1 + x)/sqrt(1 + x))],
[sqrt(-1 + x)*sqrt(1 + x)/x^2, x, 4, -((sqrt(-1 + x)*sqrt(1 + x))/x) + arccosh(x), (-sqrt(-1 + x))*sqrt(1 + x) - (-1 + x)^(3/2)*sqrt(1 + x) + ((-1 + x)^(3/2)*(1 + x)^(3/2))/x + arccosh(x)],

[1/(sqrt(1 + 2*x)*sqrt(3 + 2*x)), x, 1, arcsinh(sqrt(1 + 2*x)/sqrt(2))],
[1/(x*sqrt(-2 + 3*x)*sqrt(3 + 5*x)), x, 1, sqrt(2/3)*arctan((sqrt(3/2)*sqrt(-2 + 3*x))/sqrt(3 + 5*x))],

[1/(x*(1 + x)^(3/2)*(-1 + x)^(3/2)), x, 2, -(1/(sqrt(-1 + x)*sqrt(1 + x))) - 2*arctan(sqrt(-1 + x)/sqrt(1 + x))],

[sqrt(-1 + x)/(x^2*sqrt(1 + x)), x, 2, -((sqrt(-1 + x)*sqrt(1 + x))/x) + 2*arctan(sqrt(-1 + x)/sqrt(1 + x))],
# Simplified form of above integrand: 
[sqrt((-1 + x)/(1 + x))/x^2, x, 3, -((sqrt(-((1 - x)/(1 + x)))*(1 + x))/x) + 2*arctan(sqrt(-((1 - x)/(1 + x))))],

[x*sqrt(1 - x)*sqrt(1 + x), x, 1, (-(1/3))*(1 - x)^(3/2)*(1 + x)^(3/2)],
# Checks to ensure that expansion occurs before substitution for fractional powers of linears: 
[x*(1 + sqrt(1 - x)*sqrt(1 + x)), x, 4, x^2/2 - (1/3)*(1 - x)^(3/2)*(1 + x)^(3/2)],
[x*(1 + 1/(sqrt(2 + x)*sqrt(3 + x))), x, 5, x^2/2 + sqrt(2 + x)*sqrt(3 + x) - 5*arcsinh(sqrt(2 + x))],

[x^3*(2 + 3*x)^(3/2)*(1 + 4*x)^(1/2), x, 23, (213575*sqrt(2 + 3*x)*sqrt(1 + 4*x))/42467328 + (42715*(2 + 3*x)^(3/2)*sqrt(1 + 4*x))/15925248 - (37613*(2 + 3*x)^(5/2)*sqrt(1 + 4*x))/995328 + (539*(2 + 3*x)^(7/2)*sqrt(1 + 4*x))/13824 - (293*(2 + 3*x)^(9/2)*sqrt(1 + 4*x))/19440 + (1/486)*(2 + 3*x)^(11/2)*sqrt(1 + 4*x) + (1067875*arcsinh(sqrt(3/5)*sqrt(1 + 4*x)))/(84934656*sqrt(3))],


[1/(sqrt(-1 + a + b*x)*sqrt(1 + a + b*x)), x, 1, (2/b)*arcsinh(sqrt(-1 + a + b*x)/sqrt(2))],


# ::Subsection::Closed:: 
#Integrands of the form (a+b x)^(m/4) (c+d x)^(n/4)


# Integrands of the form (c+d*x)^(n+1/4)/(a+b*x)^(1/4) where n is an integer 
[(c + d*x)^(5/4)/(a + b*x)^(1/4), x, 6, (5*(b*c - a*d)*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(8*b^2) + ((a + b*x)^(3/4)*(c + d*x)^(5/4))/(2*b) - (5*(b*c - a*d)^2*arctan((d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))))/(16*b^(9/4)*d^(3/4)) + (5*(b*c - a*d)^2*arctanh((d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))))/(16*b^(9/4)*d^(3/4))],
[(c + d*x)^(1/4)/(a + b*x)^(1/4), x, 5, ((a + b*x)^(3/4)*(c + d*x)^(1/4))/b - ((b*c - a*d)*arctan((d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))))/(2*b^(5/4)*d^(3/4)) + ((b*c - a*d)*arctanh((d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))))/(2*b^(5/4)*d^(3/4))],
[1/((a + b*x)^(1/4)*(c + d*x)^(3/4)), x, 4, -((2*arctan((d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))))/(b^(1/4)*d^(3/4))) + (2*arctanh((d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))))/(b^(1/4)*d^(3/4))],
[1/((a + b*x)^(1/4)*(c + d*x)^(7/4)), x, 1, (4*(a + b*x)^(3/4))/(3*(b*c - a*d)*(c + d*x)^(3/4))],
[1/((a + b*x)^(1/4)*(c + d*x)^(11/4)), x, 2, (4*(a + b*x)^(3/4))/(7*(b*c - a*d)*(c + d*x)^(7/4)) + (16*b*(a + b*x)^(3/4))/(21*(b*c - a*d)^2*(c + d*x)^(3/4))],


# ::Subsection::Closed:: 
#Integrands of the form x^m (a+b x)^n


# Integrands of the form x^m*(a+b*x)^n where m is an integer and n is symbolic 
[x^3*(a + b*x)^n, x, 4, -((6*a^3*(a + b*x)^(1 + n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))) + (6*a^2*x*(a + b*x)^(1 + n))/(b^3*(2 + n)*(3 + n)*(4 + n)) - (3*a*x^2*(a + b*x)^(1 + n))/(b^2*(3 + n)*(4 + n)) + (x^3*(a + b*x)^(1 + n))/(b*(4 + n))],
[x^2*(a + b*x)^n, x, 3, (2*a^2*(a + b*x)^(1 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (2*a*x*(a + b*x)^(1 + n))/(b^2*(2 + n)*(3 + n)) + (x^2*(a + b*x)^(1 + n))/(b*(3 + n))],
[x*(a + b*x)^n, x, 2, -((a*(a + b*x)^(1 + n))/(b^2*(1 + n)*(2 + n))) + (x*(a + b*x)^(1 + n))/(b*(2 + n))],
[(a + b*x)^n, x, 1, (a + b*x)^(1 + n)/(b*(1 + n))],
[(a + b*x)^n/x, x, 0, Int((a + b*x)^n/x, x)],
[(a + b*x)^n/x^2, x, 1, -((a + b*x)^(1 + n)/(a*x)) + (b*n*Int((a + b*x)^n/x, x))/a],
[(a + b*x)^n/x^3, x, 2, -((a + b*x)^(1 + n)/(2*a*x^2)) + (b*(1 - n)*(a + b*x)^(1 + n))/(2*a^2*x) - (b^2*(1 - n)*n*Int((a + b*x)^n/x, x))/(2*a^2)],


# Integrands of the form x^m*(a+b*x)^n where m is symbolic and n is an integer 
[x^m*(a + b*x)^3, x, 2, (a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(2 + m))/(2 + m) + (3*a*b^2*x^(3 + m))/(3 + m) + (b^3*x^(4 + m))/(4 + m)],
[x^m*(a + b*x)^2, x, 2, (a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(2 + m))/(2 + m) + (b^2*x^(3 + m))/(3 + m)],
[x^m*(a + b*x), x, 2, (a*x^(1 + m))/(1 + m) + (b*x^(2 + m))/(2 + m)],
[x^m/(a + b*x), x, 0, Int(x^m/(a + b*x), x)],
[x^m/(a + b*x)^2, x, 1, x^(1 + m)/(a*(a + b*x)) - (m*Int(x^m/(a + b*x), x))/a],
[x^m/(a + b*x)^3, x, 2, x^(1 + m)/(2*a*(a + b*x)^2) + ((1 - m)*x^(1 + m))/(2*a^2*(a + b*x)) - ((1 - m)*m*Int(x^m/(a + b*x), x))/(2*a^2)],


# Integrands of the form x^m*(a+b*x)^n where m and n are symbolic 
[x^(-4 + n)/(a + b*x)^n, x, 3, -((x^(-3 + n)*(a + b*x)^(1 - n))/(a*(1 - n))) + (2*x^(-3 + n)*(a + b*x)^(2 - n))/(a^2*(1 - n)*(2 - n)) - (2*x^(-3 + n)*(a + b*x)^(3 - n))/(a^3*(1 - n)*(2 - n)*(3 - n))],
[x^(-3 + n)/(a + b*x)^n, x, 2, -((x^(-2 + n)*(a + b*x)^(1 - n))/(a*(1 - n))) + (x^(-2 + n)*(a + b*x)^(2 - n))/(a^2*(1 - n)*(2 - n))],
[x^(-2 + n)/(a + b*x)^n, x, 1, -((x^(-1 + n)*(a + b*x)^(1 - n))/(a*(1 - n)))],
[x^(-1 + n)/(a + b*x)^n, x, 1, (x^n*(a + b*x)^(1 - n))/(a*n) - (b*Int(x^n/(a + b*x)^n, x))/(a*n)],
[x^n/(a + b*x)^n, x, 0, Int(x^n/(a + b*x)^n, x)],
[x^(1 + n)/(a + b*x)^n, x, 1, (x^(1 + n)*(a + b*x)^(1 - n))/(2*b) - (a*(1 + n)*Int(x^n/(a + b*x)^n, x))/(2*b)],
[x^(2 + n)/(a + b*x)^n, x, 2, -((a*(2 + n)*x^(1 + n)*(a + b*x)^(1 - n))/(6*b^2)) + (x^(2 + n)*(a + b*x)^(1 - n))/(3*b) + (a^2*(1 + n)*(2 + n)*Int(x^n/(a + b*x)^n, x))/(6*b^2)],


# ::Subsection::Closed:: 
#Integrands involving sums of fractional powers of linear binomials


# Integrands of the form (a+b*Sqrt[x])^n where n is a positive integer 
[a + b*sqrt(x), x, 1, a*x + (2/3)*b*x^(3/2)],
[(a + b*sqrt(x))^2, x, 3, a^2*x + (4/3)*a*b*x^(3/2) + (b^2*x^2)/2],
[(a + b*sqrt(x))^3, x, 3, -((a*(a + b*sqrt(x))^4)/(10*b^2)) + (2*(a + b*sqrt(x))^4*sqrt(x))/(5*b)],
[(a + b*sqrt(x))^4, x, 3, -((a*(a + b*sqrt(x))^5)/(15*b^2)) + ((a + b*sqrt(x))^5*sqrt(x))/(3*b)],
[(a + b*sqrt(x))^5, x, 3, -((a*(a + b*sqrt(x))^6)/(21*b^2)) + (2*(a + b*sqrt(x))^6*sqrt(x))/(7*b)],
[(a + b*sqrt(x))^6, x, 3, -((a*(a + b*sqrt(x))^7)/(28*b^2)) + ((a + b*sqrt(x))^7*sqrt(x))/(4*b)],
[(a + b*sqrt(x))^7, x, 3, -((a*(a + b*sqrt(x))^8)/(36*b^2)) + (2*(a + b*sqrt(x))^8*sqrt(x))/(9*b)],
[(a + b*sqrt(x))^8, x, 3, -((a*(a + b*sqrt(x))^9)/(45*b^2)) + ((a + b*sqrt(x))^9*sqrt(x))/(5*b)],


# Integrands of the form x^m/(Sqrt[a+b*x]+Sqrt[c+b*x])^n where m and n are integers 
[x^2/(sqrt(a + b*x) + sqrt(c + b*x)), x, 9, (16*a^2*(a + b*x)^(3/2))/(105*b^3*(a - c)) - (8*a*x*(a + b*x)^(3/2))/(35*b^2*(a - c)) + (2*x^2*(a + b*x)^(3/2))/(7*b*(a - c)) - (16*c^2*(c + b*x)^(3/2))/(105*b^3*(a - c)) + (8*c*x*(c + b*x)^(3/2))/(35*b^2*(a - c)) - (2*x^2*(c + b*x)^(3/2))/(7*b*(a - c))],
[x/(sqrt(a + b*x) + sqrt(c + b*x)), x, 7, -((4*a*(a + b*x)^(3/2))/(15*b^2*(a - c))) + (2*x*(a + b*x)^(3/2))/(5*b*(a - c)) + (4*c*(c + b*x)^(3/2))/(15*b^2*(a - c)) - (2*x*(c + b*x)^(3/2))/(5*b*(a - c))],
[1/(sqrt(a + b*x) + sqrt(c + b*x)), x, 4, (2*(a + b*x)^(3/2))/(3*b*(a - c)) - (2*(c + b*x)^(3/2))/(3*b*(a - c))],
[1/(x*(sqrt(a + b*x) + sqrt(c + b*x))), x, 7, (2*sqrt(a + b*x))/(a - c) - (2*sqrt(c + b*x))/(a - c) - (2*sqrt(a)*arctanh(sqrt(a + b*x)/sqrt(a)))/(a - c) + (2*sqrt(c)*arctanh(sqrt(c + b*x)/sqrt(c)))/(a - c)],
[1/(x^2*(sqrt(a + b*x) + sqrt(c + b*x))), x, 7, -(sqrt(a + b*x)/((a - c)*x)) + sqrt(c + b*x)/((a - c)*x) - (b*arctanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(a - c)) + (b*arctanh(sqrt(c + b*x)/sqrt(c)))/((a - c)*sqrt(c))],

[x^2/(sqrt(a + b*x) + sqrt(c + b*x))^2, x, 12, ((a + c)*x^3)/(3*(a - c)^2) + (b*x^4)/(2*(a - c)^2) - (a*c*sqrt(a + b*x)*sqrt(c + b*x))/(8*b^3*(a - c)) + (5*(a + c)^2*sqrt(a + b*x)*sqrt(c + b*x))/(32*b^3*(a - c)) + (a*c*(a + b*x)^(3/2)*sqrt(c + b*x))/(4*b^3*(a - c)^2) - (5*(a + c)^2*(a + b*x)^(3/2)*sqrt(c + b*x))/(16*b^3*(a - c)^2) + (5*(a + c)*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(12*b^3*(a - c)^2) - (x*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(2*b^2*(a - c)^2) - (a*c*arctanh(sqrt(a + b*x)/sqrt(c + b*x)))/(8*b^3) + (5*(a + c)^2*arctanh(sqrt(a + b*x)/sqrt(c + b*x)))/(32*b^3)],
[x/(sqrt(a + b*x) + sqrt(c + b*x))^2, x, 8, ((a + c)*x^2)/(2*(a - c)^2) + (2*b*x^3)/(3*(a - c)^2) - ((a + c)*sqrt(a + b*x)*sqrt(c + b*x))/(4*b^2*(a - c)) + ((a + c)*(a + b*x)^(3/2)*sqrt(c + b*x))/(2*b^2*(a - c)^2) - (2*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(3*b^2*(a - c)^2) - ((a + c)*arctanh(sqrt(a + b*x)/sqrt(c + b*x)))/(4*b^2)],
[1/(sqrt(a + b*x) + sqrt(c + b*x))^2, x, -7, (a - c)^2/(8*b*(sqrt(a + b*x) + sqrt(c + b*x))^4) + arctanh(sqrt(a + b*x)/sqrt(c + b*x))/(2*b)],
[1/(x*(sqrt(a + b*x) + sqrt(c + b*x))^2), x, 7, (2*b*x)/(a - c)^2 - (2*sqrt(a + b*x)*sqrt(c + b*x))/(a - c)^2 - (2*(a + c)*arctanh(sqrt(a + b*x)/sqrt(c + b*x)))/(a - c)^2 + (4*sqrt(a)*sqrt(c)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + b*x))))/(a - c)^2 + ((a + c)*log(x))/(a - c)^2],
[1/(x^2*(sqrt(a + b*x) + sqrt(c + b*x))^2), x, 11, -((a + c)/((a - c)^2*x)) + (b*sqrt(a + b*x)*sqrt(c + b*x))/(a*(a - c)*c) - (b*(a + c)*sqrt(a + b*x)*sqrt(c + b*x))/(a*(a - c)^2*c) - (2*b*(a + b*x)^(3/2)*sqrt(c + b*x))/(a*(a - c)^2*c) + (2*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(a*(a - c)^2*c*x) + (b*arctanh(sqrt(a + b*x)/sqrt(c + b*x)))/(a*c) - (b*(a + c)^2*arctanh(sqrt(a + b*x)/sqrt(c + b*x)))/(a*(a - c)^2*c) + (2*b*(a + c)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + b*x))))/(sqrt(a)*(a - c)^2*sqrt(c)) + (2*b*log(x))/(a - c)^2],

[x^2/(sqrt(a + b*x) + sqrt(c + b*x))^3, x, 17, -((16*a^3*(a + b*x)^(3/2))/(63*b^3*(a - c)^3)) + (16*a^2*c*(a + b*x)^(3/2))/(35*b^3*(a - c)^3) + (8*a^2*x*(a + b*x)^(3/2))/(21*b^2*(a - c)^3) - (24*a*c*x*(a + b*x)^(3/2))/(35*b^2*(a - c)^3) - (10*a*x^2*(a + b*x)^(3/2))/(21*b*(a - c)^3) + (6*c*x^2*(a + b*x)^(3/2))/(7*b*(a - c)^3) + (8*x^3*(a + b*x)^(3/2))/(9*(a - c)^3) - (16*a*c^2*(c + b*x)^(3/2))/(35*b^3*(a - c)^3) + (16*c^3*(c + b*x)^(3/2))/(63*b^3*(a - c)^3) + (24*a*c*x*(c + b*x)^(3/2))/(35*b^2*(a - c)^3) - (8*c^2*x*(c + b*x)^(3/2))/(21*b^2*(a - c)^3) - (6*a*x^2*(c + b*x)^(3/2))/(7*b*(a - c)^3) + (10*c*x^2*(c + b*x)^(3/2))/(21*b*(a - c)^3) - (8*x^3*(c + b*x)^(3/2))/(9*(a - c)^3)],
[x/(sqrt(a + b*x) + sqrt(c + b*x))^3, x, 13, (12*a^2*(a + b*x)^(3/2))/(35*b^2*(a - c)^3) - (4*a*c*(a + b*x)^(3/2))/(5*b^2*(a - c)^3) - (18*a*x*(a + b*x)^(3/2))/(35*b*(a - c)^3) + (6*c*x*(a + b*x)^(3/2))/(5*b*(a - c)^3) + (8*x^2*(a + b*x)^(3/2))/(7*(a - c)^3) + (4*a*c*(c + b*x)^(3/2))/(5*b^2*(a - c)^3) - (12*c^2*(c + b*x)^(3/2))/(35*b^2*(a - c)^3) - (6*a*x*(c + b*x)^(3/2))/(5*b*(a - c)^3) + (18*c*x*(c + b*x)^(3/2))/(35*b*(a - c)^3) - (8*x^2*(c + b*x)^(3/2))/(7*(a - c)^3)],
[1/(sqrt(a + b*x) + sqrt(c + b*x))^3, x, -9, (a - c)^2/(10*b*(sqrt(a + b*x) + sqrt(c + b*x))^5) - 1/(2*b*(sqrt(a + b*x) + sqrt(c + b*x)))],
[1/(x*(sqrt(a + b*x) + sqrt(c + b*x))^3), x, 9, (2*(a + 3*c)*sqrt(a + b*x))/(a - c)^3 + (8*(a + b*x)^(3/2))/(3*(a - c)^3) - (2*(3*a + c)*sqrt(c + b*x))/(a - c)^3 - (8*(c + b*x)^(3/2))/(3*(a - c)^3) - (2*sqrt(a)*(a + 3*c)*arctanh(sqrt(a + b*x)/sqrt(a)))/(a - c)^3 + (2*sqrt(c)*(3*a + c)*arctanh(sqrt(c + b*x)/sqrt(c)))/(a - c)^3],
[1/(x^2*(sqrt(a + b*x) + sqrt(c + b*x))^3), x, 11, (8*b*sqrt(a + b*x))/(a - c)^3 - ((a + 3*c)*sqrt(a + b*x))/((a - c)^3*x) - (8*b*sqrt(c + b*x))/(a - c)^3 + ((3*a + c)*sqrt(c + b*x))/((a - c)^3*x) - (9*sqrt(a)*b*arctanh(sqrt(a + b*x)/sqrt(a)))/(a - c)^3 - (3*b*c*arctanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(a - c)^3) + (3*a*b*arctanh(sqrt(c + b*x)/sqrt(c)))/((a - c)^3*sqrt(c)) + (9*b*sqrt(c)*arctanh(sqrt(c + b*x)/sqrt(c)))/(a - c)^3],


# Integrands of the form x^m/(Sqrt[a+b*x]+Sqrt[a+c*x])^n where m and n are integers 
[x^2/(sqrt(a + b*x) + sqrt(a + c*x)), x, 7, -((4*a*(a + b*x)^(3/2))/(15*b^2*(b - c))) + (2*x*(a + b*x)^(3/2))/(5*b*(b - c)) + (4*a*(a + c*x)^(3/2))/(15*(b - c)*c^2) - (2*x*(a + c*x)^(3/2))/(5*(b - c)*c)],
[x/(sqrt(a + b*x) + sqrt(a + c*x)), x, 4, (2*(a + b*x)^(3/2))/(3*b*(b - c)) - (2*(a + c*x)^(3/2))/(3*(b - c)*c)],
[1/(sqrt(a + b*x) + sqrt(a + c*x)), x, 7, (2*sqrt(a + b*x))/(b - c) - (2*sqrt(a + c*x))/(b - c) - (2*sqrt(a)*arctanh(sqrt(a + b*x)/sqrt(a)))/(b - c) + (2*sqrt(a)*arctanh(sqrt(a + c*x)/sqrt(a)))/(b - c)],
[1/(x*(sqrt(a + b*x) + sqrt(a + c*x))), x, 7, -(sqrt(a + b*x)/((b - c)*x)) + sqrt(a + c*x)/((b - c)*x) - (b*arctanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(b - c)) + (c*arctanh(sqrt(a + c*x)/sqrt(a)))/(sqrt(a)*(b - c))],
[1/(x^2*(sqrt(a + b*x) + sqrt(a + c*x))), x, 9, -(sqrt(a + b*x)/(2*(b - c)*x^2)) - (b*sqrt(a + b*x))/(4*a*(b - c)*x) + sqrt(a + c*x)/(2*(b - c)*x^2) + (c*sqrt(a + c*x))/(4*a*(b - c)*x) + (b^2*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(3/2)*(b - c)) - (c^2*arctanh(sqrt(a + c*x)/sqrt(a)))/(4*a^(3/2)*(b - c))],

[x^2/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 7, (2*a*x)/(b - c)^2 + ((b + c)*x^2)/(2*(b - c)^2) - (a*sqrt(a + b*x)*sqrt(a + c*x))/(2*b*(b - c)*c) - ((a + b*x)^(3/2)*sqrt(a + c*x))/(b*(b - c)^2) + (a^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(2*b^(3/2)*c^(3/2))],
[x/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 7, ((b + c)*x)/(b - c)^2 - (2*sqrt(a + b*x)*sqrt(a + c*x))/(b - c)^2 + (4*a*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(b - c)^2 - (2*a*(b + c)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(sqrt(b)*(b - c)^2*sqrt(c)) + (2*a*log(x))/(b - c)^2],
[1/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 11, (2*a)/((sqrt(b) + sqrt(c))^2*(a + sqrt(b)*sqrt(c)*x + sqrt(a + b*x)*sqrt(a + c*x))) + (2*(b + c)*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(b - c)^2 - (4*sqrt(b)*sqrt(c)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(b - c)^2 + ((b + c)*log(x))/(b - c)^2, -((2*a)/((b - c)^2*x)) - (sqrt(a + b*x)*sqrt(a + c*x))/(a*(b - c)) - ((b + c)*sqrt(a + b*x)*sqrt(a + c*x))/(a*(b - c)^2) - (2*c*(a + b*x)^(3/2)*sqrt(a + c*x))/(a^2*(b - c)^2) + (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(a^2*(b - c)^2*x) + (2*(b + c)*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(b - c)^2 + arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x)))/(sqrt(b)*sqrt(c)) - ((b + c)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(sqrt(b)*(b - c)^2*sqrt(c)) + ((b + c)*log(x))/(b - c)^2],
[1/(x*(sqrt(a + b*x) + sqrt(a + c*x))^2), x, 15, -(a/((b - c)^2*x^2)) - (b + c)/((b - c)^2*x) - (b*c*sqrt(a + b*x)*sqrt(a + c*x))/(a^2*(b - c)^2) + ((b + c)*sqrt(a + b*x)*sqrt(a + c*x))/(4*a^2*(b - c)) + ((b + c)^2*sqrt(a + b*x)*sqrt(a + c*x))/(4*a^2*(b - c)^2) + (c*(b + c)*(a + b*x)^(3/2)*sqrt(a + c*x))/(2*a^3*(b - c)^2) + ((a + b*x)^(3/2)*(a + c*x)^(3/2))/(a^2*(b - c)^2*x^2) - ((b + c)*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(2*a^3*(b - c)^2*x) + (2*b*c*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(a*(b - c)^2) - ((b + c)^2*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(2*a*(b - c)^2) - ((b + c)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(4*a*sqrt(b)*sqrt(c)) - (sqrt(b)*sqrt(c)*(b + c)*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(a*(b - c)^2) + ((b + c)^3*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(4*a*sqrt(b)*(b - c)^2*sqrt(c))],
[1/(x^2*(sqrt(a + b*x) + sqrt(a + c*x))^2), x, 16, -((2*a)/(3*(b - c)^2*x^3)) - (b + c)/(2*(b - c)^2*x^2) + (b*c*(b + c)*sqrt(a + b*x)*sqrt(a + c*x))/(2*a^3*(b - c)^2) - ((b + c)^2*sqrt(a + b*x)*sqrt(a + c*x))/(8*a^3*(b - c)) - ((b + c)^3*sqrt(a + b*x)*sqrt(a + c*x))/(8*a^3*(b - c)^2) - (c*(b + c)^2*(a + b*x)^(3/2)*sqrt(a + c*x))/(4*a^4*(b - c)^2) + (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*a^2*(b - c)^2*x^3) - ((b + c)*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(2*a^3*(b - c)^2*x^2) + ((b + c)^2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(4*a^4*(b - c)^2*x) - (b*c*(b + c)*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(a^2*(b - c)^2) + ((b + c)^3*arctanh(sqrt(a + b*x)/sqrt(a + c*x)))/(4*a^2*(b - c)^2) + ((b + c)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(8*a^2*sqrt(b)*sqrt(c)) + (sqrt(b)*sqrt(c)*(b + c)^2*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(2*a^2*(b - c)^2) - ((b + c)^4*arctanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(8*a^2*sqrt(b)*(b - c)^2*sqrt(c))],

[x^2/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 9, (8*a*sqrt(a + b*x))/(b - c)^3 + (2*(b + 3*c)*(a + b*x)^(3/2))/(3*b*(b - c)^3) - (8*a*sqrt(a + c*x))/(b - c)^3 - (2*(3*b + c)*(a + c*x)^(3/2))/(3*(b - c)^3*c) - (8*a^(3/2)*arctanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 + (8*a^(3/2)*arctanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3],
[x/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 11, (2*(b + 3*c)*sqrt(a + b*x))/(b - c)^3 - (4*a*sqrt(a + b*x))/((b - c)^3*x) - (2*(3*b + c)*sqrt(a + c*x))/(b - c)^3 + (4*a*sqrt(a + c*x))/((b - c)^3*x) - (6*sqrt(a)*b*arctanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 - (6*sqrt(a)*c*arctanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 + (6*sqrt(a)*b*arctanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3 + (6*sqrt(a)*c*arctanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3],
[1/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 13, -((2*a*sqrt(a + b*x))/((b - c)^3*x^2)) - (2*b*sqrt(a + b*x))/((b - c)^3*x) - (3*c*sqrt(a + b*x))/((b - c)^3*x) + (2*a*sqrt(a + c*x))/((b - c)^3*x^2) + (3*b*sqrt(a + c*x))/((b - c)^3*x) + (2*c*sqrt(a + c*x))/((b - c)^3*x) - (3*b*c*arctanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(b - c)^3) + (3*b*c*arctanh(sqrt(a + c*x)/sqrt(a)))/(sqrt(a)*(b - c)^3)],
[1/(x*(sqrt(a + b*x) + sqrt(a + c*x))^3), x, 17, -((4*a*sqrt(a + b*x))/(3*(b - c)^3*x^3)) - (5*b*sqrt(a + b*x))/(6*(b - c)^3*x^2) - (3*c*sqrt(a + b*x))/(2*(b - c)^3*x^2) + (b^2*sqrt(a + b*x))/(4*a*(b - c)^3*x) - (3*b*c*sqrt(a + b*x))/(4*a*(b - c)^3*x) + (4*a*sqrt(a + c*x))/(3*(b - c)^3*x^3) + (3*b*sqrt(a + c*x))/(2*(b - c)^3*x^2) + (5*c*sqrt(a + c*x))/(6*(b - c)^3*x^2) + (3*b*c*sqrt(a + c*x))/(4*a*(b - c)^3*x) - (c^2*sqrt(a + c*x))/(4*a*(b - c)^3*x) - (b^3*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(3/2)*(b - c)^3) + (3*b^2*c*arctanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(3/2)*(b - c)^3) - (3*b*c^2*arctanh(sqrt(a + c*x)/sqrt(a)))/(4*a^(3/2)*(b - c)^3) + (c^3*arctanh(sqrt(a + c*x)/sqrt(a)))/(4*a^(3/2)*(b - c)^3)],
[1/(x^2*(sqrt(a + b*x) + sqrt(a + c*x))^3), x, 21, -((a*sqrt(a + b*x))/((b - c)^3*x^4)) - (b*sqrt(a + b*x))/(2*(b - c)^3*x^3) - (c*sqrt(a + b*x))/((b - c)^3*x^3) + (b^2*sqrt(a + b*x))/(8*a*(b - c)^3*x^2) - (b*c*sqrt(a + b*x))/(4*a*(b - c)^3*x^2) - (3*b^3*sqrt(a + b*x))/(16*a^2*(b - c)^3*x) + (3*b^2*c*sqrt(a + b*x))/(8*a^2*(b - c)^3*x) + (a*sqrt(a + c*x))/((b - c)^3*x^4) + (b*sqrt(a + c*x))/((b - c)^3*x^3) + (c*sqrt(a + c*x))/(2*(b - c)^3*x^3) + (b*c*sqrt(a + c*x))/(4*a*(b - c)^3*x^2) - (c^2*sqrt(a + c*x))/(8*a*(b - c)^3*x^2) - (3*b*c^2*sqrt(a + c*x))/(8*a^2*(b - c)^3*x) + (3*c^3*sqrt(a + c*x))/(16*a^2*(b - c)^3*x) + (3*b^4*arctanh(sqrt(a + b*x)/sqrt(a)))/(16*a^(5/2)*(b - c)^3) - (3*b^3*c*arctanh(sqrt(a + b*x)/sqrt(a)))/(8*a^(5/2)*(b - c)^3) + (3*b*c^3*arctanh(sqrt(a + c*x)/sqrt(a)))/(8*a^(5/2)*(b - c)^3) - (3*c^4*arctanh(sqrt(a + c*x)/sqrt(a)))/(16*a^(5/2)*(b - c)^3)],


# ::Subsection::Closed:: 
#Miscellaneous integrands involving fractional powers of linears


[sqrt(-x)*(sqrt(-x) + x), x, 3, (2/5)*(-x)^(5/2) - x^2/2],
[sqrt(d*x)*(a + b*x + c*x^2), x, 3, (2*a*(d*x)^(3/2))/(3*d) + (2*b*(d*x)^(5/2))/(5*d^2) + (2*c*(d*x)^(7/2))/(7*d^3)],


# Integrands of the form x^m*(a+b*x)^n where m is a half-integer and n is symbolic 
[(a + b*x)^n/sqrt(x), x, 1, 2*subst(Int((a + b*x^2)^n, x), x, sqrt(x))],
[sqrt(x)*(a + b*x)^n, x, 2, (2*sqrt(x)*(a + b*x)^(1 + n))/(b*(3 + 2*n)) - (2*a*subst(Int((a + b*x^2)^n, x), x, sqrt(x)))/(b*(3 + 2*n))],
[(a + b*x)^n/x^(3/2), x, 2, -((2*(a + b*x)^(1 + n))/(a*sqrt(x))) + (2*b*(1 + 2*n)*subst(Int((a + b*x^2)^n, x), x, sqrt(x)))/a],
[x^(3/2)*(a + b*x)^n, x, 3, -((6*a*sqrt(x)*(a + b*x)^(1 + n))/(b^2*(3 + 2*n)*(5 + 2*n))) + (2*x^(3/2)*(a + b*x)^(1 + n))/(b*(5 + 2*n)) + (6*a^2*subst(Int((a + b*x^2)^n, x), x, sqrt(x)))/(b^2*(15 + 16*n + 4*n^2))],
[(a + b*x)^n/x^(5/2), x, 3, -((2*(a + b*x)^(1 + n))/(3*a*x^(3/2))) + (2*b*(1 - 2*n)*(a + b*x)^(1 + n))/(3*a^2*sqrt(x)) - (2*b^2*(1 - 4*n^2)*subst(Int((a + b*x^2)^n, x), x, sqrt(x)))/(3*a^2)],


# Integrands of the form x^m*(a+b*x)^n where m is symbolic and n is a half-integer 
[x^m/sqrt(a + b*x), x, 1, (2*subst(Int((-(a/b) + x^2/b)^m, x), x, sqrt(a + b*x)))/b],
[x^m*sqrt(a + b*x), x, 2, (2*x^(1 + m)*sqrt(a + b*x))/(3 + 2*m) + (2*a*subst(Int((-(a/b) + x^2/b)^m, x), x, sqrt(a + b*x)))/(b*(3 + 2*m))],
[x^m/(a + b*x)^(3/2), x, 2, (2*x^(1 + m))/(a*sqrt(a + b*x)) - (2*(1 + 2*m)*subst(Int((-(a/b) + x^2/b)^m, x), x, sqrt(a + b*x)))/(a*b)],
[x^m*(a + b*x)^(3/2), x, 3, (6*a*x^(1 + m)*sqrt(a + b*x))/((3 + 2*m)*(5 + 2*m)) + (2*x^(1 + m)*(a + b*x)^(3/2))/(5 + 2*m) + (6*a^2*subst(Int((-(a/b) + x^2/b)^m, x), x, sqrt(a + b*x)))/(b*(15 + 16*m + 4*m^2))],
[x^m/(a + b*x)^(5/2), x, 3, (2*x^(1 + m))/(3*a*(a + b*x)^(3/2)) + (2*(1 - 2*m)*x^(1 + m))/(3*a^2*sqrt(a + b*x)) - (2*(1 - 4*m^2)*subst(Int((-(a/b) + x^2/b)^m, x), x, sqrt(a + b*x)))/(3*a^2*b)],


# Integrands of the form 1/(Sqrt[a+b*x]*Sqrt[c+d*x]) 
[1/(sqrt(4+b*x)*sqrt(-4+b*x)), x, 1, arccosh((b*x)/4)/b],

[1/(sqrt((-b+b*c)/d+b*x)*sqrt(c+d*x)), x, 1, (2*arcsinh((sqrt(d)*sqrt(-((b*(1 - c))/d) + b*x))/sqrt(b)))/(sqrt(b)*sqrt(d))],
[1/(sqrt(x)*sqrt(-3 + 2*x)), x, 1, sqrt(2)*arcsinh(sqrt(-3 + 2*x)/sqrt(3))],
[1/(sqrt(2+3*x)*sqrt(-3+2*x)), x, 1, sqrt(2/3)*arcsinh(sqrt(3/13)*sqrt(-3 + 2*x))],
[1/(sqrt(a+b*x)*sqrt(c+d*x)), x, 1, (2*arctanh((sqrt(d)*sqrt(a + b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*sqrt(d))],

[1/(sqrt((b-b*c)/d+b*x)*sqrt(c-d*x)), x, 1, (2*arcsin((sqrt(d)*sqrt((b*(1 - c))/d + b*x))/sqrt(b)))/(sqrt(b)*sqrt(d))],
[1/(sqrt(4 - x)*sqrt(x)), x, 1, 2*arcsin(sqrt(x)/2)],
[1/(sqrt(x)*sqrt(3 - 2*x)), x, 1, sqrt(2)*arcsin(sqrt(2/3)*sqrt(x))],
[1/(sqrt(3+5*x)*sqrt(3-2*x)), x, 1, sqrt(2/5)*arcsin(sqrt(2/21)*sqrt(3 + 5*x))],
[1/(sqrt(a-b*x)*sqrt(c+d*x)), x, 1, -((2*arctan((sqrt(d)*sqrt(a - b*x))/(sqrt(b)*sqrt(c + d*x))))/(sqrt(b)*sqrt(d)))],


[1/(a + b*sqrt(c + d*x)), x, 4, (2*sqrt(c + d*x))/(b*d) - (2*a*log(a + b*sqrt(c + d*x)))/(b^2*d)],
[x/(a + b*sqrt(c + d*x)), x, 5, (2*(a^2 - b^2*c)*sqrt(c + d*x))/(b^3*d^2) - (a*(c + d*x))/(b^2*d^2) + (2*(c + d*x)^(3/2))/(3*b*d^2) - (2*a*(a^2 - b^2*c)*log(a + b*sqrt(c + d*x)))/(b^4*d^2)],
[x^2/(a + b*sqrt(c + d*x)), x, 5, (2*(a^2 - b^2*c)^2*sqrt(c + d*x))/(b^5*d^3) - (a*(a^2 - 2*b^2*c)*(c + d*x))/(b^4*d^3) + (2*(a^2 - 2*b^2*c)*(c + d*x)^(3/2))/(3*b^3*d^3) - (a*(c + d*x)^2)/(2*b^2*d^3) + (2*(c + d*x)^(5/2))/(5*b*d^3) - (2*a*(a^2 - b^2*c)^2*log(a + b*sqrt(c + d*x)))/(b^6*d^3)],


[(1 + sqrt(x))/(-1 + sqrt(x)), x, 5, 4*sqrt(x) + x + 4*log(1 - sqrt(x))],
[(1 - sqrt(2 + 3*x))/(1 + sqrt(2 + 3*x)), x, 5, (1/3)*(-2 - 3*x) + (4/3)*sqrt(2 + 3*x) - (4/3)*log(1 + sqrt(2 + 3*x))],
[(-1 + sqrt(a + b*x))/(1 + sqrt(a + b*x)), x, 5, -((4*sqrt(a + b*x))/b) + (a + b*x)/b + (4*log(1 + sqrt(a + b*x)))/b],


# Integrands of the form (c+d*x)^m/(a+b*x^2) where m is a half-integer 
[sqrt(x)/(1 + x^2), x, 6, -(arctan(1 - sqrt(2)*sqrt(x))/sqrt(2)) + arctan(1 + sqrt(2)*sqrt(x))/sqrt(2) + log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) - log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))],
[sqrt(x)/(a + b*x^2), x, 4, arctan((b^(1/4)*sqrt(x))/(-a)^(1/4))/((-a)^(1/4)*b^(3/4)) - arctanh((b^(1/4)*sqrt(x))/(-a)^(1/4))/((-a)^(1/4)*b^(3/4))],
[sqrt(x)/(1 - x^2), x, 4, -arctan(sqrt(x)) + arctanh(sqrt(x))],
[sqrt(x)/(a - b*x^2), x, 4, -(arctan((b^(1/4)*sqrt(x))/a^(1/4))/(a^(1/4)*b^(3/4))) + arctanh((b^(1/4)*sqrt(x))/a^(1/4))/(a^(1/4)*b^(3/4))],

[sqrt(2 + 3*x)/(1 + x^2), x, 4, (-(1/2))*sqrt(4 + 2*sqrt(13))*arctan((sqrt(-4 + 2*sqrt(13))*sqrt(2 + 3*x))/(2 - sqrt(13) + 3*x)) - (1/2)*sqrt(-4 + 2*sqrt(13))*arctanh((sqrt(4 + 2*sqrt(13))*sqrt(2 + 3*x))/(2 + sqrt(13) + 3*x))],
[sqrt(2 + 3*x)/(1 - x^2), x, 4, -arctan(sqrt(2 + 3*x)) + sqrt(5)*arctanh(sqrt(2 + 3*x)/sqrt(5))],
[sqrt(2 + 3*x)/(a + b*x^2), x, 4, (sqrt(3*sqrt(-a)*sqrt(b) - 2*b)*arctan((sqrt(b)*sqrt(2 + 3*x))/sqrt(3*sqrt(-a)*sqrt(b) - 2*b)))/(sqrt(-a)*b) - (sqrt(3*sqrt(-a)*sqrt(b) + 2*b)*arctanh((sqrt(b)*sqrt(2 + 3*x))/sqrt(3*sqrt(-a)*sqrt(b) + 2*b)))/(sqrt(-a)*b)],
[sqrt(2 + 3*x)/(a - b*x^2), x, 4, -((sqrt(3*sqrt(a)*sqrt(b) - 2*b)*arctan((sqrt(b)*sqrt(2 + 3*x))/sqrt(3*sqrt(a)*sqrt(b) - 2*b)))/(sqrt(a)*b)) + (sqrt(3*sqrt(a)*sqrt(b) + 2*b)*arctanh((sqrt(b)*sqrt(2 + 3*x))/sqrt(3*sqrt(a)*sqrt(b) + 2*b)))/(sqrt(a)*b)],

[sqrt(c + d*x)/(1 + x^2), x, 4, (d*sqrt(c - sqrt(-d^2))*arctanh(sqrt(c + d*x)/sqrt(c - sqrt(-d^2))))/sqrt(-d^2) - (d*sqrt(c + sqrt(-d^2))*arctanh(sqrt(c + d*x)/sqrt(c + sqrt(-d^2))))/sqrt(-d^2)],
[sqrt(c + d*x)/(1 - x^2), x, 4, (-sqrt(c - d))*arctanh(sqrt(c + d*x)/sqrt(c - d)) + sqrt(c + d)*arctanh(sqrt(c + d*x)/sqrt(c + d))],
[sqrt(c + d*x)/(a + b*x^2), x, 4, (sqrt(b*c - sqrt(-a)*sqrt(b)*d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(-a)*sqrt(b)*d)))/(sqrt(-a)*b) - (sqrt(b*c + sqrt(-a)*sqrt(b)*d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(-a)*sqrt(b)*d)))/(sqrt(-a)*b)],
[sqrt(c + d*x)/(a - b*x^2), x, 4, -((sqrt(b*c - sqrt(a)*sqrt(b)*d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(a)*sqrt(b)*d)))/(sqrt(a)*b)) + (sqrt(b*c + sqrt(a)*sqrt(b)*d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(a)*sqrt(b)*d)))/(sqrt(a)*b)],

[1/(sqrt(c + d*x)*(a + b*x^2)), x, 4, arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(-a)*sqrt(b)*d))/(sqrt(-a)*sqrt(b*c - sqrt(-a)*sqrt(b)*d)) - arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(-a)*sqrt(b)*d))/(sqrt(-a)*sqrt(b*c + sqrt(-a)*sqrt(b)*d))],
[1/(sqrt(c + d*x)*(a - b*x^2)), x, 4, -(arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(a)*sqrt(b)*d))/(sqrt(a)*sqrt(b*c - sqrt(a)*sqrt(b)*d))) + arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(a)*sqrt(b)*d))/(sqrt(a)*sqrt(b*c + sqrt(a)*sqrt(b)*d))],
[(c + d*x)^(3/2)/(a - b*x^2), x, 5, -((2*d*sqrt(c + d*x))/b) + ((2*sqrt(b)*c*d - (b*c^2 + a*d^2)/sqrt(a))*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(a)*sqrt(b)*d)))/(b*sqrt(b*c - sqrt(a)*sqrt(b)*d)) + ((2*sqrt(b)*c*d + (b*c^2 + a*d^2)/sqrt(a))*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(a)*sqrt(b)*d)))/(b*sqrt(b*c + sqrt(a)*sqrt(b)*d))],
[1/((c + d*x)^(3/2)*(a + b*x^2)), x, 5, -((2*d)/((b*c^2 + a*d^2)*sqrt(c + d*x))) + (sqrt(b)*((sqrt(b)*c)/sqrt(-a) + d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(-a)*sqrt(b)*d)))/(sqrt(b*c - sqrt(-a)*sqrt(b)*d)*(b*c^2 + a*d^2)) - (sqrt(b)*((sqrt(b)*c)/sqrt(-a) - d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(-a)*sqrt(b)*d)))/(sqrt(b*c + sqrt(-a)*sqrt(b)*d)*(b*c^2 + a*d^2))],
[1/((c + d*x)^(3/2)*(a - b*x^2)), x, 5, (2*d)/((b*c^2 - a*d^2)*sqrt(c + d*x)) - (sqrt(b)*((sqrt(b)*c)/sqrt(a) + d)*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c - sqrt(a)*sqrt(b)*d)))/(sqrt(b*c - sqrt(a)*sqrt(b)*d)*(b*c^2 - a*d^2)) - (sqrt(b)*(1 - (sqrt(b)*c)/(sqrt(a)*d))*d*arctanh((sqrt(b)*sqrt(c + d*x))/sqrt(b*c + sqrt(a)*sqrt(b)*d)))/(sqrt(b*c + sqrt(a)*sqrt(b)*d)*(b*c^2 - a*d^2))],


[sqrt(d + e*x)/(a + b*x + c*x^2), x, 4, -((sqrt(2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*arctanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(c)*sqrt(b^2 - 4*a*c))) + (sqrt(2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*arctanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(c)*sqrt(b^2 - 4*a*c))],
[1/(sqrt(d + e*x)*(a + b*x + c*x^2)), x, 4, -((2*sqrt(2)*sqrt(c)*arctanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))) + (2*sqrt(2)*sqrt(c)*arctanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))],
[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)), x, 5, -((2*e)/((c*d^2 - b*d*e + a*e^2)*sqrt(d + e*x))) + (sqrt(2)*sqrt(c)*(e - (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*arctanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2)) + (sqrt(2)*sqrt(c)*(e + (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*arctanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(c*d^2 - b*d*e + a*e^2))],


# Miscellaneous integrands involving algebraic functions of linear binomials 
[1/(sqrt(x)*(1 + x)), x, 1, 2*arctan(sqrt(x))],
[1/((-1 - x)*sqrt(x)), x, 1, -2*arctan(sqrt(x))],
[1/(sqrt(x)*(4 + 9*x)), x, 1, arctan((3*sqrt(x))/2)/3],
[1/(sqrt(x)*(9 + x)), x, 1, (2*arctan(sqrt(x)/3))/3],
[1/((-2 + x)*sqrt(2 + x)), x, 1, -arctanh(sqrt(2 + x)/2)],
[1/(sqrt(x)*(1 + x^2)), x, 6, -(arctan(1 - sqrt(2)*sqrt(x))/sqrt(2)) + arctan(1 + sqrt(2)*sqrt(x))/sqrt(2) - log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) + log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))],

[1/(sqrt(x) + sqrt(1 + x)), x, 4, -((2*x^(3/2))/3) + (2/3)*(1 + x)^(3/2)],
[1/(sqrt(-1 + x) + sqrt(1 + x)), x, 4, (-(1/3))*(-1 + x)^(3/2) + (1/3)*(1 + x)^(3/2)],
[1/(1 + x - sqrt(2 + x)), x, 3, (2*arctanh((1 - 2*sqrt(2 + x))/sqrt(5)))/sqrt(5) + log(-1 - x + sqrt(2 + x))],
[1/(x - sqrt(1 + x)), x, 3, (2*arctanh((1 - 2*sqrt(1 + x))/sqrt(5)))/sqrt(5) + log(-x + sqrt(1 + x))],
[1/(x - sqrt(2 + x)), x, 5, (4/3)*log(2 - sqrt(2 + x)) + (2/3)*log(1 + sqrt(2 + x))],
[1/(x - sqrt(1 - x)), x, 3, (2*arctanh((1 + 2*sqrt(1 - x))/sqrt(5)))/sqrt(5) + log(-sqrt(1 - x) + x)],
[1/(sqrt(-1 + x) + sqrt(x)), x, 4, (-(2/3))*(-1 + x)^(3/2) + (2*x^(3/2))/3],
[1/(4 + sqrt(4 - x) - x), x, 2, -2*log(1 + sqrt(4 - x))],

[(sqrt(1 - x) + sqrt(1 + x))*sqrt(1 - x), x, 5, x - x^2/2 + (1/2)*x*sqrt(1 - x^2) + arcsin(x)/2],
[(sqrt(1 - x) + sqrt(1 + x))^2, x, 5, 2*x + x*sqrt(1 - x^2) + arcsin(x)],
[(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x)), x, 7, -2*x - x*sqrt(1 - x^2) - arcsin(x)],
[(sqrt(1 - x) + sqrt(1 + x))^2/x, x, 5, 2*sqrt(1 - x)*sqrt(1 + x) - 4*arctanh(sqrt(1 - x)/sqrt(1 + x)) + 2*log(x)],
[(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x))/x, x, 7, -2*sqrt(1 - x)*sqrt(1 + x) + 4*arctanh(sqrt(1 - x)/sqrt(1 + x)) - 2*log(x)],
[(sqrt(1 - x) + sqrt(1 + x))/(-sqrt(1 - x) + sqrt(1 + x)), x, 8, sqrt(1 - x)*sqrt(1 + x) - 2*arctanh(sqrt(1 - x)/sqrt(1 + x)) + log(x)],
[(-sqrt(-1 + x) + sqrt(1 + x))/(sqrt(-1 + x) + sqrt(1 + x)), x, 7, x^2/2 - (1/2)*sqrt(-1 + x)*x*sqrt(1 + x) + arccosh(x)/2, x^2/2 - (1/2)*sqrt(-1 + x)*sqrt(1 + x) - (1/2)*(-1 + x)^(3/2)*sqrt(1 + x) + arccosh(x)/2],

[(2*sqrt(3 - x) + 3/sqrt(1 + x))^2/x, x, 7, -4*x + 12*arcsin((1 - x)/2) - 24*sqrt(3)*arctanh(sqrt(3 - x)/(sqrt(3)*sqrt(1 + x))) - 18*arctanh(1 + 2*x) + 12*log(x)],

[sqrt(1 - x)/(1 + sqrt(x)), x, 6, -2*sqrt(1 - x) + sqrt(1 - x)*sqrt(x) - arcsin(sqrt(x))],
[sqrt(1 - x)/(1 - sqrt(x)), x, 6, -2*sqrt(1 - x) - sqrt(1 - x)*sqrt(x) + arcsin(sqrt(x))],
[x/((1 + x)*sqrt(2 + x)), x, 4, 2*sqrt(2 + x) + 2*arctanh(sqrt(2 + x))],
[x/(sqrt(2 + 5*x)*(-2 + 9*x)), x, 4, (2/45)*sqrt(2 + 5*x) - (2*arctanh((3*sqrt(2 + 5*x))/(2*sqrt(7))))/(27*sqrt(7))],
[((-2 + x)*(1 + x))/sqrt(x), x, 3, -4*sqrt(x) - (2*x^(3/2))/3 + (2*x^(5/2))/5],
[(-1 + x^2)/sqrt(x), x, 2, -2*sqrt(x) + (2*x^(5/2))/5],
[sqrt(x)*(-2 + x^2), x, 2, -((4*x^(3/2))/3) + (2*x^(7/2))/7],
[sqrt(x)/(1 + x^3), x, 2, (2*arctan(x^(3/2)))/3],
[(1 + x)/sqrt(1 - x), x, 2, -((10*sqrt(1 - x))/3) - (2/3)*sqrt(1 - x)*x],
[(-x + 3*x^2)/sqrt(1 + x), x, 2, 8*sqrt(1 + x) - (14/3)*(1 + x)^(3/2) + (6/5)*(1 + x)^(5/2)],
[(x - 2*x^3)/sqrt(2 + 3*x), x, 2, (-(4/81))*sqrt(2 + 3*x) - (10/81)*(2 + 3*x)^(3/2) + (8/135)*(2 + 3*x)^(5/2) - (4/567)*(2 + 3*x)^(7/2)],
[sqrt(x)/(9 + x), x, 2, 2*sqrt(x) - 6*arctan(sqrt(x)/3)],
[(1 + x)/sqrt(x), x, 2, 2*sqrt(x) + (2*x^(3/2))/3],
[sqrt(x)/(1 + x), x, 2, 2*sqrt(x) - 2*arctan(sqrt(x))],
[(2 + x)*sqrt(2 + 3*x), x, 2, (52/135)*(2 + 3*x)^(3/2) + (2/15)*x*(2 + 3*x)^(3/2)],
[(7 + x)/sqrt(5 - x), x, 2, -((62*sqrt(5 - x))/3) - (2/3)*sqrt(5 - x)*x],
[(2 + 3*x)/sqrt(-9 + x), x, 2, 40*sqrt(-9 + x) + 2*sqrt(-9 + x)*x],
[sqrt(-1 + x)/(1 + x)^2, x, 2, -(sqrt(-1 + x)/(1 + x)) + arctan(sqrt(-1 + x)/sqrt(2))/sqrt(2)],
[sqrt(-1 + x)/(1 + x)^3, x, 3, -(sqrt(-1 + x)/(2*(1 + x)^2)) + sqrt(-1 + x)/(8*(1 + x)) + arctan(sqrt(-1 + x)/sqrt(2))/(8*sqrt(2))],
# {Sqrt[-1 + x]/(1 + x^2)^3, x, 1/(256 Sqrt[-1+Sqrt[2]] (1+x^2)^2) (8 Sqrt[-1+Sqrt[2]] Sqrt[-1+x] (-1+x (19+x (-1+11 x)))+2 (-36+25 Sqrt[2]) (1+x^2)^2 ArcTan[(Sqrt[2 (-1+Sqrt[2])]-2 Sqrt[-1+x])/Sqrt[2 (1+Sqrt[2])]]-2 (-36+25 Sqrt[2]) (1+x^2)^2 ArcTan[(Sqrt[2 (-1+Sqrt[2])]+2 Sqrt[-1+x])/Sqrt[2 (1+Sqrt[2])]]+(14+11 Sqrt[2]) (1+x^2)^2 Log[-1+Sqrt[2]-Sqrt[2 (-1+Sqrt[2])] Sqrt[-1+x]+x]-(14+11 Sqrt[2]) (1+x^2)^2 Log[-1+Sqrt[2]+Sqrt[2 (-1+Sqrt[2])] Sqrt[-1+x]+x])} 
[sqrt(1 + x^2)/(-1 + x^2), x, 3, arcsinh(x) - sqrt(2)*arctanh((sqrt(2)*x)/sqrt(1 + x^2))],
[(sqrt(1 + x)*(1 + x^3))/(1 + x^2), x, 7, -2*sqrt(1 + x) - (2/3)*(1 + x)^(3/2) + (2/5)*(1 + x)^(5/2) - sqrt(1 + sqrt(2))*arctan((sqrt(-2 + 2*sqrt(2))*sqrt(1 + x))/(1 - sqrt(2) + x)) + sqrt(-1 + sqrt(2))*arctanh((sqrt(2 + 2*sqrt(2))*sqrt(1 + x))/(1 + sqrt(2) + x))],
[-1/sqrt(-1 + x) + 1/sqrt(x), x, 3, -2*sqrt(-1 + x) + 2*sqrt(x)],
[sqrt(-1 + x)/(1 + x), x, 2, 2*sqrt(-1 + x) - 2*sqrt(2)*arctan(sqrt(-1 + x)/sqrt(2))],
[sqrt(-2 + x)/(2 + x), x, 2, 2*sqrt(-2 + x) - 4*arctan(sqrt(-2 + x)/2)],

[1/(x*(a + b*x)^(1/3)), x, 5, (sqrt(3)*arctan((a^(1/3) + 2*(a + b*x)^(1/3))/(sqrt(3)*a^(1/3))))/a^(1/3) + log(a^(1/3) - (a + b*x)^(1/3))/a^(1/3) - log(a^(2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3))/(2*a^(1/3))],
[(1 + x)^(1/3)/x, x, 6, 3*(1 + x)^(1/3) - sqrt(3)*arctan((1 + 2*(1 + x)^(1/3))/sqrt(3)) + log(1 - (1 + x)^(1/3)) - (1/2)*log(1 + (1 + x)^(1/3) + (1 + x)^(2/3))],
[(a + b*x)^(-1/3), x, 1, (3*(a + b*x)^(2/3))/(2*b)],
[(-1 + x)^(1/3)/(1 + x)^(1/3), x, 6, (-1 + x)^(1/3)*(1 + x)^(2/3) - (2*arctan((1 + (2*(-1 + x)^(1/3))/(1 + x)^(1/3))/sqrt(3)))/sqrt(3) + (2/3)*log(1 - (-1 + x)^(1/3)/(1 + x)^(1/3)) - (1/3)*log(1 + (-1 + x)^(2/3)/(1 + x)^(2/3) + (-1 + x)^(1/3)/(1 + x)^(1/3))],
[x/(1 + x)^(1/3), x, 2, (-(9/10))*(1 + x)^(2/3) + (3/5)*x*(1 + x)^(2/3)],
[(1 - x)^(1/3)/(1 + x), x, 6, 3*(1 - x)^(1/3) - 2^(1/3)*sqrt(3)*arctan((1 + 2^(2/3)*(1 - x)^(1/3))/sqrt(3)) + 2^(1/3)*log(2^(1/3) - (1 - x)^(1/3)) - log(2^(2/3) + 2^(1/3)*(1 - x)^(1/3) + (1 - x)^(2/3))/2^(2/3)],
[x*(c + x)^(1/3), x, 2, (-(9/28))*c*(c + x)^(4/3) + (3/7)*x*(c + x)^(4/3)],
[(3 - 2*x)^(1/3)*(7 + x), x, 2, (-(321/112))*(3 - 2*x)^(4/3) - (3/14)*(3 - 2*x)^(4/3)*x],

[(1 - x)^(2/3)*x^3, x, 4, -((243*(1 - x)^(5/3))/3080) - (81/616)*(1 - x)^(5/3)*x - (27/154)*(1 - x)^(5/3)*x^2 - (3/14)*(1 - x)^(5/3)*x^3],
[x*(1 + 2*x)^(2/3), x, 2, (-(9/160))*(1 + 2*x)^(5/3) + (3/16)*x*(1 + 2*x)^(5/3)],
[x^(2/3)/(1 + x), x, 6, (3*x^(2/3))/2 + sqrt(3)*arctan((1 - 2*x^(1/3))/sqrt(3)) + log(1 + x^(1/3)) - (1/2)*log(1 - x^(1/3) + x^(2/3))],
[(1 - x)^(1/4)/(1 + x), x, 5, 4*(1 - x)^(1/4) - 2*2^(1/4)*arctan((1 - x)^(1/4)/2^(1/4)) - 2*2^(1/4)*arctanh((1 - x)^(1/4)/2^(1/4))],
[x/(3 + x)^(1/5), x, 2, (-(25/12))*(3 + x)^(4/5) + (5/9)*x*(3 + x)^(4/5)],
[(-1 + x)^(1/6)*x, x, 2, (36/91)*(-1 + x)^(7/6) + (6/13)*(-1 + x)^(7/6)*x],

[(2*x - x^2)^(3/2)/(2 - 2*x), x, 3, (-(1/2))*sqrt(2*x - x^2) - (1/6)*(2*x - x^2)^(3/2) + (1/2)*arctanh(sqrt(2*x - x^2))],
[(2*x - x^2)^(1/2)/(2 - 2*x), x, 2, (-(1/2))*sqrt(2*x - x^2) + (1/2)*arctanh(sqrt(2*x - x^2))],
[1/((2 - 2*x)*(2*x - x^2)^(1/2)), x, 1, (1/2)*arctanh(sqrt(2*x - x^2))],
[1/((2 - 2*x)*(2*x - x^2)^(3/2)), x, 2, -(1/(2*sqrt(2*x - x^2))) + (1/2)*arctanh(sqrt(2*x - x^2))],
[1/((2 - 2*x)*(2*x - x^2)^(5/2)), x, 3, -(1/(6*(2*x - x^2)^(3/2))) - 1/(2*sqrt(2*x - x^2)) + (1/2)*arctanh(sqrt(2*x - x^2))],

[(x + sqrt(2*x - x^2))/(2 - 2*x), x, 7, -(x/2) - (1/2)*sqrt(2*x - x^2) + (1/2)*arctanh(sqrt(2*x - x^2)) - (1/2)*log(1 - x)],

[sqrt(x)/(sqrt(2 - x) - sqrt(x)), x, -37, -(x/2) - (1/2)*sqrt((2 - x)*x) + arctanh(sqrt(2 - x)/sqrt(x)) - (1/2)*log(1 - x)]
]:
