lst:=[
# ::Package:: 

# ::Title:: 
#Rational Function Integration Problems Involving Trinomials


# ::Subsection::Closed:: 
#Integrands involving polynomials


# ::Subsubsection::Closed:: 
#Constants


[0, x, 1, 0],
[1, x, 1, x],
[5, x, 1, 5*x],
[-2, x, 1, -2*x],
[-3/2, x, 1, -3/2*x],
[Pi, x, 1, Pi*x],
[a, x, 1, a*x],
[3*a, x, 1, 3*a*x],
[Pi/sqrt(16 - E^2), x, 1, (Pi*x)/sqrt(16 - E^2)],


# ::Subsubsection::Closed:: 
#Monomials


[x, x, 1, x^2/2],
[x^2, x, 1, x^3/3],
[x^3, x, 1, x^4/4],
[x^100, x, 1, x^101/101],
[1/x, x, 1, log(x)],
[1/x^2, x, 1, -1/x],
[1/x^3, x, 1, -1/(2*x^2)],
[1/x^100, x, 1, -1/(99*x^99)],

[sqrt(x), x, 1, 2*x^(3/2)/3],
[x^(3/2), x, 1, 2*x^(5/2)/5],
[x^(5/2), x, 1, 2*x^(7/2)/7],
[1/sqrt(x), x, 1, 2*sqrt(x)],
[1/x^(3/2), x, 1, -2/sqrt(x)],
[1/x^(5/2), x, 1, -2/(3*x^(3/2))],

[x^(1/3), x, 1, (3/4)*x^(4/3)],
[x^(2/3), x, 1, (3/5)*x^(5/3)],

[x^n, x, 1, x^(1+n)/(1+n)],


# ::Subsubsection::Closed:: 
#Fully expanded polynomials


[a + b*x + c*x^2 + d*x^3, x, 1, a*x + (b*x^2)/2 + (c*x^3)/3 + (d*x^4)/4],
[-x^3 + x^4, x, 1, -(x^4/4) + x^5/5],
[-1 + x^5, x, 1, -x + x^6/6],
[7 + 4*x, x, 1, 7*x + 2*x^2],
[4*x + Pi*x^3, x, 1, 2*x^2 + (Pi*x^4)/4],
[2*x + 5*x^2, x, 1, x^2 + (5*x^3)/3],
[x^2/2 + x^3/3, x, 1, x^3/6 + x^4/12],
[3 - 5*x + 2*x^2, x, 1, 3*x - (5*x^2)/2 + (2*x^3)/3],
[-2*x + x^2 + x^3, x, 1, -x^2 + x^3/3 + x^4/4],
[1 - x^2 - 3*x^5, x, 1, x - x^3/3 - x^6/2],
[5 + 2*x + 3*x^2 + 4*x^3, x, 1, 5*x + x^2 + x^3 + x^4],


# ::Subsubsection::Closed:: 
#Fully expanded monomial sums


# Integrands of the form a*x^m + b*x^n + ... where m and n are rationals 
[a + b/x + c/x^2 + d/x^3, x, 1, -(d/(2*x^2)) - c/x + a*x + b*log(x)],
[x^(-5) + x + x^5, x, 1, -(1/(4*x^4)) + x^2/2 + x^6/6],
[x^(-3) + x^(-2) + x^(-1), x, 1, -(1/(2*x^2)) - 1/x + log(x)],
[-2/x^2 + 3/x, x, 1, 2/x + 3*log(x)],
[-1/(7*x^6) + x^6, x, 1, 1/(35*x^5) + x^7/7],
[1 + x^(-1) + x, x, 1, x + x^2/2 + log(x)],
[-3/x^3 + 4/x^2, x, 1, 3/(2*x^2) - 4/x],
[x^(-1) + 2*x + x^2, x, 1, x^2 + x^3/3 + log(x)],

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


# Integrands of the form a*x^m + b*x^n + ... where m and n are rationals 
[x^(5/6) - x^3, x, 1, (6*x^(11/6))/11 - x^4/4],
[33 + x^(1/33), x, 1, 33*x + (33*x^(34/33))/34],
[1/(2*sqrt(x)) + 2*sqrt(x), x, 1, sqrt(x) + (4*x^(3/2))/3],
[-x^(-2) + 10/x + 6*sqrt(x), x, 1, x^(-1) + 4*x^(3/2) + 10*log(x)],
[x^(-3/2) + x^(3/2), x, 1, -(2/sqrt(x)) + (2*x^(5/2))/5],
[-5*x^(3/2) + 7*x^(5/2), x, 1, -2*x^(5/2) + 2*x^(7/2)],
[2/sqrt(x) + sqrt(x) - x/2, x, 1, 4*sqrt(x) + (2*x^(3/2))/3 - x^2/4],
[-2/x + sqrt(x)/5 + x^(3/2), x, 1, (2*x^(3/2))/15 + (2*x^(5/2))/5 - 2*log(x)],


# ::Subsubsection::Closed:: 
#Products of polynomials and their derivative


# Integrands of the form x^(n-1)*(a+b*x^n)^m 
[x^(n - 1)*(a + b*x^n)^m, x, 2, (a + b*x^n)^(1 + m)/(b*(1 + m)*n)],
[x^2*(a + b*x^3)^m, x, 2, (a + b*x^3)^(1 + m)/(3*b*(1 + m))],

[x^(n - 1)*(a + b*x^n)^16, x, 2, (a + b*x^n)^17/(17*b*n)],
[x^2*(a + b*x^3)^16, x, 2, (a + b*x^3)^17/(51*b)],


# Integrands of the form (b+2*c*x)*(a+b*x+c*x^2)^m 
[(b + 2*c*x)*(a + b*x + c*x^2)^m, x, 1, (a + b*x + c*x^2)^(1 + m)/(1 + m)],
[(b + 2*c*x)*(b*x + c*x^2)^m, x, 1, (b*x + c*x^2)^(1 + m)/(1 + m)],
[x^m*(b + 2*c*x)*(b + c*x)^m, x, 1, (x^(1 + m)*(b + c*x)^(1 + m))/(1 + m)],

[(b + 2*c*x)*(a + b*x + c*x^2)^12, x, 1, (1/13)*(a + b*x + c*x^2)^13],
[(b + 2*c*x)*(b*x + c*x^2)^12, x, 1, (1/13)*x^13*(b + c*x)^13],
[x^12*(b + 2*c*x)*(b + c*x)^12, x, 1, (1/13)*x^13*(b + c*x)^13],


# Products of symbolic powers of a polynomial and the polynomial's derivative 
[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^n, x, 2, (a + b*x + c*x^2 + d*x^3)^(1 + n)/(1 + n)],
[(b + 2*c*x + 3*d*x^2)*(b*x + c*x^2 + d*x^3)^n, x, 2, (b*x + c*x^2 + d*x^3)^(1 + n)/(1 + n)],
[(b + 2*c*x + 3*d*x^2)*x^n*(b + c*x + d*x^2)^n, x, 1, (x^(1 + n)*(b + c*x + d*x^2)^(1 + n))/(1 + n)],

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

[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^n, x, 2, (a + c*x^2 + d*x^3)^(1 + n)/(1 + n)],
[(2*c*x + 3*d*x^2)*(c*x^2 + d*x^3)^n, x, 2, (c*x^2 + d*x^3)^(1 + n)/(1 + n)],
[(2*c*x + 3*d*x^2)*x^n*(c*x + d*x^2)^n, x, 1, (x^(1 + n)*(c*x + d*x^2)^(1 + n))/(1 + n)],
[(2*c*x + 3*d*x^2)*x^(2*n)*(c + d*x)^n, x, 1, (x^(2 + 2*n)*(c + d*x)^(1 + n))/(1 + n)],

[x*(2*c + 3*d*x)*(a + c*x^2 + d*x^3)^n, x, 2, (a + c*x^2 + d*x^3)^(1 + n)/(1 + n)],
[x*(2*c + 3*d*x)*(c*x^2 + d*x^3)^n, x, 2, (c*x^2 + d*x^3)^(1 + n)/(1 + n)],
[x^(n+1)*(2*c + 3*d*x)*(c*x + d*x^2)^n, x, 1, (x^(1 + n)*(c*x + d*x^2)^(1 + n))/(1 + n)],
[x^(2*n+1)*(2*c + 3*d*x)*(c + d*x)^n, x, 1, (x^(2 + 2*n)*(c + d*x)^(1 + n))/(1 + n)],
[x^m*(a + b*x^n)^p*(a*(m + q + 1)*x^q + b*(m + q + n*(p + 1) + 1)*x^(n + q)), x, 1, x^(1 + m + q)*(a + b*x^n)^(1 + p)],


# Products of integer powers of a polynomial and the polynomial's derivative 
[(b + 2*c*x + 3*d*x^2)*(a + b*x + c*x^2 + d*x^3)^7, x, 2, (1/8)*(a + b*x + c*x^2 + d*x^3)^8],
[(b + 2*c*x + 3*d*x^2)*(b*x + c*x^2 + d*x^3)^7, x, 2, (1/8)*x^8*(b + c*x + d*x^2)^8],
[(b + 2*c*x + 3*d*x^2)*x^7*(b + c*x + d*x^2)^7, x, 1, (1/8)*x^8*(b + c*x + d*x^2)^8],

[(b + 3*d*x^2)*(a + b*x + d*x^3)^7, x, 2, (1/8)*(a + b*x + d*x^3)^8],
[(b + 3*d*x^2)*(b*x + d*x^3)^7, x, 2, (1/8)*x^8*(b + d*x^2)^8],
[(b + 3*d*x^2)*x^7*(b + d*x^2)^7, x, 1, (1/8)*x^8*(b + d*x^2)^8],

[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^7, x, 2, (1/8)*(a + c*x^2 + d*x^3)^8],
[(2*c*x + 3*d*x^2)*(c*x^2 + d*x^3)^7, x, 2, (1/8)*x^16*(c + d*x)^8],
[(2*c*x + 3*d*x^2)*x^7*(c*x + d*x^2)^7, x, 1, (1/8)*x^16*(c + d*x)^8],
[(2*c*x + 3*d*x^2)*x^14*(c + d*x)^7, x, 1, (x^(2 + 2*7)*(c + d*x)^8)/8],

[x*(2*c + 3*d*x)*(a + c*x^2 + d*x^3)^7, x, 2, (a + c*x^2 + d*x^3)^8/8],
[x*(2*c + 3*d*x)*(c*x^2 + d*x^3)^7, x, 2, (1/8)*x^16*(c + d*x)^8],
[x^8*(2*c + 3*d*x)*(c*x + d*x^2)^7, x, 1, (1/8)*x^16*(c + d*x)^8],
[x^15*(2*c + 3*d*x)*(c + d*x)^7, x, 1, (x^(2 + 2*7)*(c + d*x)^8)/8],

[(-4 + 4*x + x^2)*(5 - 12*x + 6*x^2 + x^3), x, 2, (5 - 12*x + 6*x^2 + x^3)^2/6],
[(2*x + x^3)*(1 + 4*x^2 + x^4), x, 2, (1/8)*(1 + 4*x^2 + x^4)^2],

[(1 + 2*x)*(x + x^2)^3*(-18 + 7*(x + x^2)^3)^2, x, 3, 81*x^4*(1 + x)^4 - 36*x^7*(1 + x)^7 + (49/10)*x^10*(1 + x)^10],
[x^3*(1 + x)^3*(1 + 2*x)*(-18 + 7*x^3*(1 + x)^3)^2, x, 2, 81*x^4*(1 + x)^4 - 36*x^7*(1 + x)^7 + (49/10)*x^10*(1 + x)^10, 81*x^4 + 324*x^5 + 486*x^6 + 288*x^7 - 171*x^8 - 756*x^9 - (12551*x^10)/10 - 1211*x^11 - (1071*x^12)/2 + 336*x^13 + 993*x^14 + (6174*x^15)/5 + 1029*x^16 + 588*x^17 + (441*x^18)/2 + 49*x^19 + (49*x^20)/10],


# ::Subsubsection::Closed:: 
#Disguised derivative divides examples


# Integrands of the form x^(12*m)*(a + b*x^(12*m+1))^12 
[x^12*(a + b*x^13)^12, x, 2, (a + b*x^13)^13/(169*b)],
[x^24*(a + b*x^25)^12, x, 2, (a + b*x^25)^13/(325*b)],
[x^36*(a + b*x^37)^12, x, 2, (a + b*x^37)^13/(481*b)],
[x^(12*m)*(a + b*x^(12*m + 1))^12, x, 2, (a + b*x^(1 + 12*m))^13/(13*b*(1 + 12*m))],

# Integrands of the form x^(12*(m-1))*(a*x + b*x^(12*m+2))^12 
[(a*x + b*x^14)^12, x, 3, (a + b*x^13)^13/(169*b)],
[x^12*(a*x + b*x^26)^12, x, 3, (a + b*x^25)^13/(325*b)],
[x^24*(a*x + b*x^38)^12, x, 3, (a + b*x^37)^13/(481*b)],
[x^(12*(m-1))*(a*x + b*x^(12*m + 2))^12, x, 3, (a + b*x^(1 + 12*m))^13/(13*b*(1 + 12*m))],

# Integrands of the form (a*x^m + b*x^(12*m+m+1))^12 
[(a*x + b*x^14)^12, x, 3, (a + b*x^13)^13/(169*b)],
[(a*x^2 + b*x^27)^12, x, 3, (a + b*x^25)^13/(325*b)],
[(a*x^3 + b*x^40)^12, x, 3, (a + b*x^37)^13/(481*b)],
[(a*x^m + b*x^(12*m + m + 1))^12, x, 3, (a + b*x^(1 + 12*m))^13/(13*b*(1 + 12*m))],


# Integrands of the form x^p*(a*x^n+b*x^(m*n+n+p+1))^m 
[x^p*(a*x^n + b*x^(12*n + n + p + 1))^12, x, 3, (a + b*x^(1 + 12*n + p))^13/(13*b*(1 + 12*n + p))],

[x^12*(a + b*x^13)^12, x, 2, (a + b*x^13)^13/(169*b)],
[x^12*(a*x + b*x^26)^12, x, 3, (a + b*x^25)^13/(325*b)],
[x^12*(a*x^2 + b*x^39)^12, x, 3, (a + b*x^37)^13/(481*b)],

[x^24*(a + b*x^25)^12, x, 2, (a + b*x^25)^13/(325*b)],
[x^24*(a*x + b*x^38)^12, x, 3, (a + b*x^37)^13/(481*b)],

[x^36*(a + b*x^37)^12, x, 2, (a + b*x^37)^13/(481*b)],


# ::Subsubsection::Closed:: 
#Miscellaneous polynomial expressions


[x^2*(2 + x)^5*(2 + 3*x), x, 1, (1/3)*x^3*(2 + x)^6],
[(5 - 2*x)^6*(2 + 3*x)^3*(-16 + 33*x), x, 1, (-(1/2))*(5 - 2*x)^7*(2 + 3*x)^4],


# Integrands of the form x^m*(a+b*x)^n*(c+d*x)^p 
[x*(a + b*x)*(c + d*x)^16, x, 3, ((b*c - 18*a*d)*(2*b*c + 17*a*d)*(c + d*x)^17)/(5814*b*d^3) - ((2*b*c + 17*a*d)*x*(c + d*x)^17)/(342*d^2) + ((a + b*x)^2*(c + d*x)^17)/(19*b*d)],
[x*(a + b*x)^2*(c + d*x)^16, x, 4, -(((b*c - a*d)^2*(3*b*c + 17*a*d)*(c + d*x)^17)/(58140*b*d^4)) + ((b*c - a*d)*(3*b*c + 17*a*d)*(a + b*x)*(c + d*x)^17)/(3420*b*d^3) - ((3*b*c + 17*a*d)*(a + b*x)^2*(c + d*x)^17)/(380*b*d^2) + ((a + b*x)^3*(c + d*x)^17)/(20*b*d)],
[x^2*(a + b*x)*(c + d*x)^16, x, 6, (a*c*(b*c - 18*a*d)*(c + d*x)^17)/(6120*b*d^3) - ((b*c - 18*a*d)*(b*c + 6*a*d)*(2*b*c + 17*a*d)*(c + d*x)^17)/(38760*b^2*d^4) - (a*c*x*(c + d*x)^17)/(360*d^2) + ((b*c + 6*a*d)*(2*b*c + 17*a*d)*x*(c + d*x)^17)/(2280*b*d^3) - (3*(b*c + 6*a*d)*(a + b*x)^2*(c + d*x)^17)/(380*b^2*d^2) + (x*(a + b*x)^2*(c + d*x)^17)/(20*b*d)],
# {x^2*(a + b*x)^2*(c + d*x)^16, x, 8, -((a*c*(b*c - a*d)^2*(c + d*x)^17)/(61047*b*d^4)) + ((b*c - a*d)^2*(2*b*c + 9*a*d)*(3*b*c + 17*a*d)*(c + d*x)^17)/(610470*b^2*d^5) + (a*c*(b*c - a*d)*(a + b*x)*(c + d*x)^17)/(3591*b*d^3) - ((b*c - a*d)*(2*b*c + 9*a*d)*(3*b*c + 17*a*d)*(a + b*x)*(c + d*x)^17)/(35910*b^2*d^4) - (a*c*(a + b*x)^2*(c + d*x)^17)/(399*b*d^2) + ((2*b*c + 9*a*d)*(3*b*c + 17*a*d)*(a + b*x)^2*(c + d*x)^17)/(3990*b^2*d^3) - ((2*b*c + 9*a*d)*(a + b*x)^3*(c + d*x)^17)/(210*b^2*d^2) + (x*(a + b*x)^3*(c + d*x)^17)/(21*b*d)} 


[(a + b*x)*(1 + (a*x + b*(x^2/2))^4), x, 2, a*x + (b*x^2)/2 + (1/160)*x^5*(2*a + b*x)^5],
[(a + b*x)*(1 + (c + a*x + b*(x^2/2))^4), x, 2, c + a*x + (b*x^2)/2 + (1/160)*(2*c + 2*a*x + b*x^2)^5],

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

[(a + c*x^2)*(1 + (a*x + c*(x^3/3))^5), x, 2, a*x + (c*x^3)/3 + (x^6*(3*a + c*x^2)^6)/4374],
[(a + c*x^2)*(1 + (d + a*x + c*(x^3/3))^5), x, 2, d + a*x + (c*x^3)/3 + (3*d + 3*a*x + c*x^3)^6/4374],

[(b*x + c*x^2)*(1 + (b*(x^2/2) + c*(x^3/3))^5), x, 2, (b*x^2)/2 + (c*x^3)/3 + (x^12*(3*b + 2*c*x)^6)/279936],
[(b*x + c*x^2)*(1 + (d + b*(x^2/2) + c*(x^3/3))^5), x, 2, d + (b*x^2)/2 + (c*x^3)/3 + (6*d + 3*b*x^2 + 2*c*x^3)^6/279936],

[(a + b*x + c*x^2)*(1 + (a*x + b*(x^2/2) + c*(x^3/3))^5), x, 2, a*x + (b*x^2)/2 + (c*x^3)/3 + (x^6*(6*a + 3*b*x + 2*c*x^2)^6)/279936],
[(a + b*x + c*x^2)*(1 + (d + a*x + b*(x^2/2) + c*(x^3/3))^5), x, 2, d + a*x + (b*x^2)/2 + (c*x^3)/3 + (6*d + 6*a*x + 3*b*x^2 + 2*c*x^3)^6/279936],

[(a + c*x^2)*(1 + (a*x + c*(x^3/3))^n), x, 2, a*x + (c*x^3)/3 + (3^(-1 - n)*(3*a*x + c*x^3)^(1 + n))/(1 + n)],
[(b*x + c*x^2)*(1 + (b*(x^2/2) + c*(x^3/3))^n), x, 2, (b*x^2)/2 + (c*x^3)/3 + (6^(-1 - n)*(3*b*x^2 + 2*c*x^3)^(1 + n))/(1 + n)],
[(a + b*x + c*x^2)*(1 + (a*x + b*(x^2/2) + c*(x^3/3))^n), x, 2, a*x + (b*x^2)/2 + (c*x^3)/3 + (6^(-1 - n)*(6*a*x + 3*b*x^2 + 2*c*x^3)^(1 + n))/(1 + n)],


# Integrands of the form f[x^m*(a+b*x^n)^p] where m==-n*p 
# Test to ensure NormalForm does not result in infinite recursion 
[f((a + b*x)/x), x, 1, Int(f(b + a/x), x)],
[f((a + b*x^2)/x^2), x, 1, Int(f(b + a/x^2), x)],
[f(x/(a + b*x)), x, 1, Int(f(1/(b + a/x)), x)],
[f(x^2/(a + b*x^2)), x, 1, Int(f(1/(b + a/x^2)), x)],
[f(x^2/(a + b*x)^2), x, 2, -subst(Int(f(1/(b + a*x)^2)/x^2, x), x, 1/x)],
[f(x^4/(a + b*x^2)^2), x, 1, Int(f(1/(b + a/x^2)^2), x)],


[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14), x, 2, (1/21)*(2 + 3*x)^7 + (1/42)*(2 + 3*x)^14 + (1/63)*(2 + 3*x)^21],
[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14)^2, x, 3, (1/21)*(2 + 3*x)^7 + (1/21)*(2 + 3*x)^14 + (1/21)*(2 + 3*x)^21 + (1/42)*(2 + 3*x)^28 + (1/105)*(2 + 3*x)^35],


# Miscellaneous polynomial integrands 
[c*(a + b*x), x, 1, (c*(a + b*x)^2)/(2*b)],
[((c + d)*(a + b*x))/e, x, 1, ((c + d)*(a + b*x)^2)/(2*b*e)],
[(2 + x)*(3 + x), x, 2, 6*x + (5*x^2)/2 + x^3/3],
[(-4*x + 3*x^3)^6, x, 3, (4096*x^7)/7 - 2048*x^9 + (34560*x^11)/11 - (34560*x^13)/13 + 1296*x^15 - (5832*x^17)/17 + (729*x^19)/19],
[x^8*(a + b*x^7), x, 2, (a*x^9)/9 + (b*x^16)/16],
[x^2*(a + b*x^2)^3, x, 2, (a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^7)/7 + (b^3*x^9)/9],
[x*(a + b*x + c*x^2), x, 2, (a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4],
[x^6*(-4 + 3*x^2), x, 2, (-4*x^7)/7 + x^9/3],


# ::Subsection::Closed:: 
#Integrands involving powers of quadratic trinomials


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


# Integrands of the form x^m*(a+b*x+c*x^2) where m is an integer 
[x^4*(a + b*x + c*x^2), x, 2, (a*x^5)/5 + (b*x^6)/6 + (c*x^7)/7],
[x^3*(a + b*x + c*x^2), x, 2, (a*x^4)/4 + (b*x^5)/5 + (c*x^6)/6],
[x^2*(a + b*x + c*x^2), x, 2, (a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5],
[x*(a + b*x + c*x^2), x, 2, (a*x^2)/2 + (b*x^3)/3 + (c*x^4)/4],
[a + b*x + c*x^2, x, 1, a*x + (b*x^2)/2 + (c*x^3)/3],
[a + x*(b + c*x), x, 3, a*x + (b*x^2)/2 + (c*x^3)/3],
[(a + b*x + c*x^2)/x, x, 2, b*x + (c*x^2)/2 + a*log(x)],
[(a + b*x + c*x^2)/x^2, x, 2, -(a/x) + c*x + b*log(x)],
[(a + b*x + c*x^2)/x^3, x, 2, -(a/(2*x^2)) - b/x + c*log(x)],
[(a + b*x + c*x^2)/x^4, x, 2, -(a/(3*x^3)) - b/(2*x^2) - c/x],


# Integrands of the form x^m*(a+b*x+c*x^2)^2 where m is an integer 
[x^4*(a + b*x + c*x^2)^2, x, 2, (a^2*x^5)/5 + (1/3)*a*b*x^6 + (1/7)*(b^2 + 2*a*c)*x^7 + (1/4)*b*c*x^8 + (c^2*x^9)/9],
[x^3*(a + b*x + c*x^2)^2, x, 2, (a^2*x^4)/4 + (2/5)*a*b*x^5 + (1/6)*(b^2 + 2*a*c)*x^6 + (2/7)*b*c*x^7 + (c^2*x^8)/8],
[x^2*(a + b*x + c*x^2)^2, x, 2, (a^2*x^3)/3 + (1/2)*a*b*x^4 + (1/5)*(b^2 + 2*a*c)*x^5 + (1/3)*b*c*x^6 + (c^2*x^7)/7],
[x*(a + b*x + c*x^2)^2, x, 2, (a^2*x^2)/2 + (2/3)*a*b*x^3 + (1/4)*(b^2 + 2*a*c)*x^4 + (2/5)*b*c*x^5 + (c^2*x^6)/6],
[(a + b*x + c*x^2)^2, x, 2, a^2*x + a*b*x^2 + (1/3)*(b^2 + 2*a*c)*x^3 + (1/2)*b*c*x^4 + (c^2*x^5)/5],
[(a + b*x + c*x^2)^2/x, x, 2, 2*a*b*x + (1/2)*(b^2 + 2*a*c)*x^2 + (2/3)*b*c*x^3 + (c^2*x^4)/4 + a^2*log(x)],
[(a + b*x + c*x^2)^2/x^2, x, 2, -(a^2/x) + (b^2 + 2*a*c)*x + b*c*x^2 + (c^2*x^3)/3 + 2*a*b*log(x)],
[(a + b*x + c*x^2)^2/x^3, x, 2, -(a^2/(2*x^2)) - (2*a*b)/x + 2*b*c*x + (c^2*x^2)/2 + (b^2 + 2*a*c)*log(x)],
[(a + b*x + c*x^2)^2/x^4, x, 2, -(a^2/(3*x^3)) - (a*b)/x^2 - (b^2 + 2*a*c)/x + c^2*x + 2*b*c*log(x)],
[(a + b*x + c*x^2)^2/x^5, x, 2, -(a^2/(4*x^4)) - (2*a*b)/(3*x^3) - (b^2 + 2*a*c)/(2*x^2) - (2*b*c)/x + c^2*log(x)],
[(a + b*x + c*x^2)^2/x^6, x, 2, -(a^2/(5*x^5)) - (a*b)/(2*x^4) - (b^2 + 2*a*c)/(3*x^3) - (b*c)/x^2 - c^2/x],
[(a + b*x + c*x^2)^2/x^7, x, 2, -(a^2/(6*x^6)) - (2*a*b)/(5*x^5) - (b^2 + 2*a*c)/(4*x^4) - (2*b*c)/(3*x^3) - c^2/(2*x^2)],


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


# Integrands of the form x^m/(a+b*x+c*x^2)^2 where m is an integer 
[x^4/(a + b*x + c*x^2)^2, x, 7, x/c^2 + (b*(b^2 - 2*a*c))/(2*c^4*(a + b*x + c*x^2)) - ((2*a^2*c^2 + b^2*(b^2 - 4*a*c))*(b + 2*c*x))/(2*c^4*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (4*a^2*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c*(b^2 - 4*a*c)^(3/2)) - (4*(b^2/2 - a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^3*sqrt(b^2 - 4*a*c)) - (b*log(a + b*x + c*x^2))/c^3],
[x^3/(a + b*x + c*x^2)^2, x, 7, -((b^2 - a*c)/(2*c^3*(a + b*x + c*x^2))) + (b*(b^2 - 3*a*c)*(b + 2*c*x))/(2*c^3*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (3*b*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^2*sqrt(b^2 - 4*a*c)) - (2*b*(b^2 - 3*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^2*(b^2 - 4*a*c)^(3/2)) + log(a + b*x + c*x^2)/(2*c^2)],
[x^2/(a + b*x + c*x^2)^2, x, 3, -(x/(c*(a + b*x + c*x^2))) - (a*(b + 2*c*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (4*a*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[x/(a + b*x + c*x^2)^2, x, 3, (2*a + b*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*b*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[1/(a + b*x + c*x^2)^2, x, 2, -((b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[1/(x*(a + b*x + c*x^2)^2), x, 8, 1/(2*a*(a + b*x + c*x^2)) + (b*(b + 2*c*x))/(2*a*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*b*c*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a*(b^2 - 4*a*c)^(3/2)) + (b*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^2*sqrt(b^2 - 4*a*c)) + log(x)/a^2 - log(a + b*x + c*x^2)/(2*a^2)],
[1/(x^2*(a + b*x + c*x^2)^2), x, 8, -(1/(a^2*x)) - b/(2*a^2*(a + b*x + c*x^2)) + (((-b^2)*c + 2*c*(b^2 - a*c))*(b + 2*c*x))/(2*a^2*c*(-b^2 + 4*a*c)*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)) - (2*b*log(x))/a^3 + (b*log(a + b*x + c*x^2))/a^3, -(1/(a^2*x)) - b/(2*a^2*(a + b*x + c*x^2)) - ((b^2 - 2*a*c)*(b + 2*c*x))/(2*a^2*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (2*c*(b^2 - 2*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(3/2)) - (2*(b^2 - a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*sqrt(b^2 - 4*a*c)) - (2*b*log(x))/a^3 + (b*log(a + b*x + c*x^2))/a^3],
[1/(x^3*(a + b*x + c*x^2)^2), x, 8, -(1/(2*a^2*x^2)) + (2*b)/(a^3*x) - ((-b^2)*c + a*c^2)/(2*a^3*c*(a + b*x + c*x^2)) + ((2*c*(-b^3 + 2*a*b*c) - b*((-b^2)*c + a*c^2))*(b + 2*c*x))/(2*a^3*c*(-b^2 + 4*a*c)*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*log(x))/a^4 + ((-3*b^2*c + 2*a*c^2)*log(a + b*x + c*x^2))/(2*a^4*c), -(1/(2*a^2*x^2)) + (2*b)/(a^3*x) + (b^2 - a*c)/(2*a^3*(a + b*x + c*x^2)) + (b*(b^2 - 3*a*c)*(b + 2*c*x))/(2*a^3*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*b*c*(b^2 - 3*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)) + (3*b*(b^2 - 2*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*sqrt(b^2 - 4*a*c)) + ((3*b^2 - 2*a*c)*log(x))/a^4 - ((3*b^2 - 2*a*c)*log(a + b*x + c*x^2))/(2*a^4)],
[1/(x^4*(a + b*x + c*x^2)^2), x, 8, -(1/(3*a^2*x^3)) + b/(a^3*x^2) - (3*b^2 - 2*a*c)/(a^4*x) - (b*(b^2 - 2*a*c))/(2*a^4*(a + b*x + c*x^2)) - ((2*a^2*c^2 + b^2*(b^2 - 4*a*c))*(b + 2*c*x))/(2*a^4*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (4*(b^4 + a^2*c^2)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^5*sqrt(b^2 - 4*a*c)) + (2*c*(2*a^2*c^2 + 7*b^2*(b^2 - 4*a*c))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*(b^2 - 4*a*c)^(3/2)) - (2*b*(2*b^2 - 3*a*c)*log(x))/a^5 + (b*(2*b^2 - 3*a*c)*log(a + b*x + c*x^2))/a^5],

# Integrands of the form x^m/(a+b*x+c*x^2)^3 where m is an integer 
[x^4/(a + b*x + c*x^2)^3, x, 7, -((a*x)/(c^2*(a + b*x + c*x^2)^2)) - (b*x^2)/(2*c^2*(a + b*x + c*x^2)^2) - x^3/(c*(a + b*x + c*x^2)^2) - (a^2*(b + 2*c*x))/(2*c^2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*a^2*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (12*a^2*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[x^3/(a + b*x + c*x^2)^3, x, 5, -(x^2/(2*c*(a + b*x + c*x^2)^2)) + (a*(2*a + b*x))/(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*a*b*(b + 2*c*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (6*a*b*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[x^2/(a + b*x + c*x^2)^3, x, 5, b/(12*c^2*(a + b*x + c*x^2)^2) - x/(3*c*(a + b*x + c*x^2)^2) - ((b^2 + 2*a*c)*(b + 2*c*x))/(12*c^2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((2*a + b^2/c)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (2*(b^2 + 2*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[x/(a + b*x + c*x^2)^3, x, 4, (2*a + b*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*b*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (6*b*c*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[1/(a + b*x + c*x^2)^3, x, 3, -((b + 2*c*x)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2)) + (3*c*(b + 2*c*x))/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (12*c^2*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[1/(x*(a + b*x + c*x^2)^3), x, 12, 1/(4*a*(a + b*x + c*x^2)^2) + (b*(b + 2*c*x))/(4*a*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + 1/(2*a^2*(a + b*x + c*x^2)) - (3*b*c*(b + 2*c*x))/(2*a*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (b*(b + 2*c*x))/(2*a^2*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (6*b*c^2*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a*(b^2 - 4*a*c)^(5/2)) - (2*b*c*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(3/2)) + (b*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*sqrt(b^2 - 4*a*c)) + log(x)/a^3 - log(a + b*x + c*x^2)/(2*a^3)],
[1/(x^2*(a + b*x + c*x^2)^3), x, 12, -(1/(a^3*x)) - b/(4*a^2*(a + b*x + c*x^2)^2) - ((b^2 - 2*a*c)*(b + 2*c*x))/(4*a^2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - b/(a^3*(a + b*x + c*x^2)) + (3*c*(b^2 - 2*a*c)*(b + 2*c*x))/(2*a^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((b^2 - a*c)*(b + 2*c*x))/(a^3*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x + c*x^2))/(2*a^4), -(1/(a^3*x)) - b/(4*a^2*(a + b*x + c*x^2)^2) - ((b^2 - 2*a*c)*(b + 2*c*x))/(4*a^2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - b/(a^3*(a + b*x + c*x^2)) + (3*c*(b^2 - 2*a*c)*(b + 2*c*x))/(2*a^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((b^2 - a*c)*(b + 2*c*x))/(a^3*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (6*c^2*(b^2 - 2*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(5/2)) - ((3*b^2 - 2*a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*sqrt(b^2 - 4*a*c)) + (4*c*(b^2 - a*c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x + c*x^2))/(2*a^4)],
# {1/(x^3*(a + b*x + c*x^2)^3), x, 12, -(1/(2*a^3*x^2)) + (3*b)/(a^4*x) + (b^2 - a*c)/(4*a^3*(a + b*x + c*x^2)^2) + (b*(b^2 - 3*a*c)*(b + 2*c*x))/(4*a^3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*b^2 - 2*a*c)/(2*a^4*(a + b*x + c*x^2)) - (3*b*c*(b^2 - 3*a*c)*(b + 2*c*x))/(2*a^3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (3*b*(b^2 - 2*a*c)*(b + 2*c*x))/(2*a^4*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (6*b*c^2*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^3*(b^2 - 4*a*c)^(5/2)) + (3*b*(2*b^2 - 3*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*Sqrt[b^2 - 4*a*c]) - (6*b*c*(b^2 - 2*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*(b^2 - 4*a*c)^(3/2)) + (3*(2*b^2 - a*c)*Log[x])/a^5 - (3*(2*b^2 - a*c)*Log[a + b*x + c*x^2])/(2*a^5)} 
# {1/(x^4*(a + b*x + c*x^2)^3), x, 0, 0} 


# Integrands of the form x^m/(a+b*x+c*x^2)^4 where m is an integer 
[x^3/(c + d*x + e*x^2)^4, x, 7, -((d^2 + 5*c*e)/(60*e^3*(c + d*x + e*x^2)^3)) + (d*x)/(20*e^2*(c + d*x + e*x^2)^3) - x^2/(4*e*(c + d*x + e*x^2)^3) + (d*(d^2 + 6*c*e)*(d + 2*e*x))/(60*e^3*(d^2 - 4*c*e)*(c + d*x + e*x^2)^3) - (d*(d^2 + 6*c*e)*(d + 2*e*x))/(12*e^2*(d^2 - 4*c*e)^2*(c + d*x + e*x^2)^2) + (d*(6*c + d^2/e)*(d + 2*e*x))/(2*(d^2 - 4*c*e)^3*(c + d*x + e*x^2)) - (2*d*(d^2 + 6*c*e)*arctanh((d + 2*e*x)/sqrt(d^2 - 4*c*e)))/(d^2 - 4*c*e)^(7/2)],


# Integrands of the form 1/(a+b*x+c*x^2) where b^2-4*a*c==0 
[(4 + 12*x + 9*x^2)^(-1), x, 2, -(1/(3*(2 + 3*x)))],
[(4 - 12*x + 9*x^2)^(-1), x, 2, 1/(3*(2 - 3*x))],
[(a^2 + 2*a*b*x + b^2*x^2)^(-1), x, 2, -(1/(b*(a + b*x)))],
[(a^2 - 2*a*b*x + b^2*x^2)^(-1), x, 2, 1/(b*(a - b*x))],


# Integrands of the form 1/(a+b*x+c*x^2) 
[(2 + 4*x + 3*x^2)^(-1), x, 1, arctan((2 + 3*x)/sqrt(2))/sqrt(2)],
[(2 + 4*x - 3*x^2)^(-1), x, 1, -(arctanh((2 - 3*x)/sqrt(10))/sqrt(10))],
[(2 + 5*x + 3*x^2)^(-1), x, 1, -2*arctanh(5 + 6*x)],
[(2 + 5*x - 3*x^2)^(-1), x, 1, (-(2/7))*arctanh(5/7 - (6*x)/7)],
[(1 + Pi*x + 2*x^2)^(-1), x, 1, (-2*arctanh((Pi + 4*x)/sqrt(-8 + Pi^2)))/sqrt(-8 + Pi^2)],
[(1 + Pi*x - 2*x^2)^(-1), x, 1, (-2*arctanh((Pi - 4*x)/sqrt(8 + Pi^2)))/sqrt(8 + Pi^2)],
[(1 + Pi*x + 3*x^2)^(-1), x, 1, (2*arctan((Pi + 6*x)/sqrt(12 - Pi^2)))/sqrt(12 - Pi^2)],
[(1 + Pi*x - 3*x^2)^(-1), x, 1, (-2*arctanh((Pi - 6*x)/sqrt(12 + Pi^2)))/sqrt(12 + Pi^2)],
[(a + b*x + c*x^2)^(-1), x, 1, (-2*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/sqrt(b^2 - 4*a*c)],
[(a + c*x + b*x^2)^(-1), x, 1, (2*arctan((c + 2*b*x)/sqrt(4*a*b - c^2)))/sqrt(4*a*b - c^2)],
[(b + 2*a*x + b*x^2)^(-1), x, 1, -(arctanh((a + b*x)/sqrt(a^2 - b^2))/sqrt(a^2 - b^2))],
[(b + 2*a*x - b*x^2)^(-1), x, 1, -(arctanh((a - b*x)/sqrt(a^2 + b^2))/sqrt(a^2 + b^2))],


# Integrands of the form 1/(a+b*x+c*x^2)^2 
[(2 + 4*x + 3*x^2)^(-2), x, 2, (2 + 3*x)/(4*(2 + 4*x + 3*x^2)) + (3*arctan((2 + 3*x)/sqrt(2)))/(4*sqrt(2))],
[(2 + 4*x - 3*x^2)^(-2), x, 2, -((2 - 3*x)/(20*(2 + 4*x - 3*x^2))) - (3*arctanh((2 - 3*x)/sqrt(10)))/(20*sqrt(10))],
[(2 + 5*x + 3*x^2)^(-2), x, 2, -((5 + 6*x)/(2 + 5*x + 3*x^2)) + 12*arctanh(5 + 6*x)],
[(2 + 5*x - 3*x^2)^(-2), x, 2, -((5 - 6*x)/(49*(2 + 5*x - 3*x^2))) - (12/343)*arctanh(5/7 - (6*x)/7)],
[(a + b*x + c*x^2)^(-2), x, 2, -((b + 2*c*x)/((b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*c*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[(a + c*x + b*x^2)^(-2), x, 2, (c + 2*b*x)/((4*a*b - c^2)*(a + c*x + b*x^2)) + (4*b*arctan((c + 2*b*x)/sqrt(4*a*b - c^2)))/(4*a*b - c^2)^(3/2)],
[(b + 2*a*x + b*x^2)^(-2), x, 2, -((a + b*x)/(2*(a^2 - b^2)*(b + 2*a*x + b*x^2))) + (b*arctanh((a + b*x)/sqrt(a^2 - b^2)))/(2*(a^2 - b^2)^(3/2))],
[(b + 2*a*x - b*x^2)^(-2), x, 2, -((a - b*x)/(2*(a^2 + b^2)*(b + 2*a*x - b*x^2))) - (b*arctanh((a - b*x)/sqrt(a^2 + b^2)))/(2*(a^2 + b^2)^(3/2))],


# Integrands of the form x^m*(a+b*x+c*x^2)^n where m and n are integers 
[x/(4 + 4*x + x^2), x, 5, 2/(2 + x) + log(2 + x)],
[x/(5 + 2*x + x^2), x, 2, (-(1/2))*arctan(1/2 + x/2) + (1/2)*log(5 + 2*x + x^2)],
[x/(6 - 5*x + x^2), x, 4, -2*log(2 - x) + 3*log(3 - x)],
[x/(2 + 2*x + x^2)^2, x, 3, -(1/(2 + 2*x + x^2)) - x/(2*(2 + 2*x + x^2)) - (1/2)*arctan(1 + x)],
[x/(1 + x + x^2)^3, x, 4, -(1/(3*(1 + x + x^2)^2)) - x/(6*(1 + x + x^2)^2) - (1 + 2*x)/(6*(1 + x + x^2)) - (2*arctan((1 + 2*x)/sqrt(3)))/(3*sqrt(3))],

[x^2/(1 + x + x^2), x, 4, x - arctan((1 + 2*x)/sqrt(3))/sqrt(3) - log(1 + x + x^2)/2],
[x^2/(2 - 3*x + x^2), x, 4, x - log(1 - x) + 4*log(2 - x)],
[x^2/(-6 + x + x^2), x, 4, x + (4/5)*log(2 - x) - (9/5)*log(3 + x)],
[x^2/(2 + 2*x + x^2)^2, x, 3, (2 + 2*x + x^2)^(-1) + arctan(1 + x)],

[x^3/(2 - 3*x + x^2), x, 5, 3*x + x^2/2 - log(1 - x) + 8*log(2 - x)],
[x^3/(1 + 2*x + x^2), x, 6, -2*x + x^2/2 + 1/(1 + x) + 3*log(1 + x)],
[x^3/(1 - 2*x + x^2), x, 6, 1/(1 - x) + 2*x + x^2/2 + 3*log(1 - x)],

[x^4/(4 + 4*x + x^2), x, 6, 12*x - 2*x^2 + x^3/3 - 16/(2 + x) - 32*log(2 + x)],

[(1 + x + x^2)/x, x, 2, x + x^2/2 + log(x)],
[(9 + 6*x + x^2)/x^2, x, 2, -9/x + x + 6*log(x)],
[(3 - 4*x + x^2)/x^3, x, 2, -(3/(2*x^2)) + 4/x + log(x)],
[(1 + 2*x + x^2)/x^4, x, 2, -(1/(3*x^3)) - 1/x^2 - 1/x],

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


[1/((a/b)^(2/n) + x^2 - 2*(a/b)^(1/n)*x*cos((Pi - 2*k*Pi)/n)), x, 1, (arctan(((x - (a/b)^(1/n)*cos(((1 - 2*k)*Pi)/n))*csc(((1 - 2*k)*Pi)/n))/(a/b)^(n^(-1)))*csc(((1 - 2*k)*Pi)/n))/(a/b)^(n^(-1))],


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


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


# Integrands of the form (d+e*x)^m/(a+b*x+c*x^2) where m is an integer 
[(d + e*x)^3/(a + b*x + c*x^2), x, 5, (e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^2)/(2*c) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^3*sqrt(b^2 - 4*a*c)) + (e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2 - a*c*e^2)*log(a + b*x + c*x^2))/(2*c^3)],
[(d + e*x)^2/(a + b*x + c*x^2), x, 4, (e^2*x)/c - ((b^2*e^2 + 2*c*(c*d^2 - e*(b*d + a*e)))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^2*sqrt(b^2 - 4*a*c)) + (e*(2*c*d - b*e)*log(a + b*x + c*x^2))/(2*c^2)],
[(d + e*x)/(a + b*x + c*x^2), x, 2, -(((2*d - (b*e)/c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/sqrt(b^2 - 4*a*c)) + (e*log(a + b*x + c*x^2))/(2*c)],
[1/((d + e*x)*(a + b*x + c*x^2)), x, 5, -(((2*c*d - b*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(sqrt(b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e)))) + (e*log(d + e*x))/(c*d^2 - e*(b*d - a*e)) - (e*log(a + b*x + c*x^2))/(2*(c*d^2 - e*(b*d - a*e)))],
[1/((d + e*x)^2*(a + b*x + c*x^2)), x, 6, -(e/((c*d^2 - e*(b*d - a*e))*(d + e*x))) - ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(sqrt(b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e))^2) + (e*(2*c*d - b*e)*log(d + e*x))/(c*d^2 - e*(b*d - a*e))^2 - (e*(2*c*d - b*e)*log(a + b*x + c*x^2))/(2*(c*d^2 - e*(b*d - a*e))^2)],
# {1/((d + e*x)^3*(a + b*x + c*x^2)), x, 7, -(e/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2)) - (e*(2*c*d - b*e))/((c*d^2 - e*(b*d - a*e))^2*(d + e*x)) + ((3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + b^3*e^3 - 3*a*b*c*e^3 - 2*c^2*d*(c*d^2 - 3*a*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - e*(b*d - a*e))^3) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[d + e*x])/(c*d^2 - e*(b*d - a*e))^3 - (e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2 - a*c*e^2)*Log[a + b*x + c*x^2])/(2*(c*d^2 - e*(b*d - a*e))^3)} 


# Integrands of the form (d+e*x)^m/(a+b*x+c*x^2)^2 where m is an integer 
# {(d + e*x)^3/(a + b*x + c*x^2)^2, x, 6, -(((2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(b + 2*c*x))/(2*c^2*(b^2 - 4*a*c)*(a + b*x + c*x^2))) - (e*(2*c*d - b*e)*(d + e*x))/(2*c^2*(a + b*x + c*x^2)) - (e*(d + e*x)^2)/(2*c*(a + b*x + c*x^2)) - (e^2*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + (2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*(b^2 - 4*a*c)^(3/2)) + (e^3*Log[a + b*x + c*x^2])/(2*c^2)} 
[(d + e*x)^2/(a + b*x + c*x^2)^2, x, 3, -(((c*d^2 - e*(b*d - a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2))) - (e*(d + e*x))/(c*(a + b*x + c*x^2)) + (4*(c*d^2 - e*(b*d - a*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[(d + e*x)/(a + b*x + c*x^2)^2, x, 3, -(e/(2*c*(a + b*x + c*x^2))) - ((2*c*d - b*e)*(b + 2*c*x))/(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) + (2*(2*c*d - b*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[1/((d + e*x)*(a + b*x + c*x^2)^2), x, 8, e/(2*(c*d^2 - e*(b*d - a*e))*(a + b*x + c*x^2)) - ((2*c*d - b*e)*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e))*(a + b*x + c*x^2)) - (e^2*(2*c*d - b*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(sqrt(b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e))^2) + (2*c*(2*c*d - b*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(3/2)*(c*d^2 - e*(b*d - a*e))) + (e^3*log(d + e*x))/(c*d^2 - e*(b*d - a*e))^2 - (e^3*log(a + b*x + c*x^2))/(2*(c*d^2 - e*(b*d - a*e))^2)],
# {1/((d + e*x)^2*(a + b*x + c*x^2)^2), x, 9, -(e^3/((c*d^2 - e*(b*d - a*e))^2*(d + e*x))) + (e*(2*c*d - b*e))/(2*(c*d^2 - e*(b*d - a*e))^2*(a + b*x + c*x^2)) + ((2*b*c*d*e - b^2*e^2 - 2*c*(c*d^2 - a*e^2))*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e))^2*(a + b*x + c*x^2)) - (2*e^2*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2 - a*c*e^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - e*(b*d - a*e))^3) - (2*c*(2*b*c*d*e - b^2*e^2 - 2*c*(c*d^2 - a*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - e*(b*d - a*e))^2) + (2*e^3*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - e*(b*d - a*e))^3 - (e^3*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(c*d^2 - e*(b*d - a*e))^3} 
# {1/((d + e*x)^3*(a + b*x + c*x^2)^2), x, 10, -(e^3/(2*(c*d^2 - e*(b*d - a*e))^2*(d + e*x)^2)) - (2*e^3*(2*c*d - b*e))/((c*d^2 - e*(b*d - a*e))^3*(d + e*x)) + (e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2 - a*c*e^2))/(2*(c*d^2 - e*(b*d - a*e))^3*(a + b*x + c*x^2)) + ((3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + b^3*e^3 - 3*a*b*c*e^3 - 2*c^2*d*(c*d^2 - 3*a*e^2))*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(c*d^2 - e*(b*d - a*e))^3*(a + b*x + c*x^2)) - (2*c*(3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + b^3*e^3 - 3*a*b*c*e^3 - 2*c^2*d*(c*d^2 - 3*a*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - e*(b*d - a*e))^3) + (3*e^2*(6*b*c^2*d^2*e - 4*b^2*c*d*e^2 + b^3*e^3 - 2*a*b*c*e^3 - 4*c^2*d*(c*d^2 - a*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])/(c*d^2 - e*(b*d - a*e))^4 - (e^3*(10*c^2*d^2 - 10*b*c*d*e + 3*b^2*e^2 - 2*a*c*e^2)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)} 


# Integrands of the form x^m*(d+e*x)/(a+b*x+c*x^2) where m is an integer 
[x^4*(d + e*x)/(a + b*x + c*x^2), x, 5, ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*x)/c^4 - ((b*c*d - b^2*e + a*c*e)*x^2)/(2*c^3) + ((c*d - b*e)*x^3)/(3*c^2) + (e*x^4)/(4*c) + ((b^5*e - c*(b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d + 5*a*b^3*e - 5*a^2*b*c*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^5*sqrt(b^2 - 4*a*c)) - ((b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e)*log(a + b*x + c*x^2))/(2*c^5)],
[x^3*(d + e*x)/(a + b*x + c*x^2), x, 5, -(((b*c*d - b^2*e + a*c*e)*x)/c^3) + ((c*d - b*e)*x^2)/(2*c^2) + (e*x^3)/(3*c) - ((b^4*e - c*(b^3*d - 3*a*b*c*d + 4*a*b^2*e - 2*a^2*c*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^4*sqrt(b^2 - 4*a*c)) + ((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*log(a + b*x + c*x^2))/(2*c^4)],
[x^2*(d + e*x)/(a + b*x + c*x^2), x, 5, ((c*d - b*e)*x)/c^2 + (e*x^2)/(2*c) + ((b^3*e - c*(b^2*d - 2*a*c*d + 3*a*b*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^3*sqrt(b^2 - 4*a*c)) - ((b*c*d - b^2*e + a*c*e)*log(a + b*x + c*x^2))/(2*c^3)],
[x*(d + e*x)/(a + b*x + c*x^2), x, 4, (e*x)/c - ((b^2*e - c*(b*d + 2*a*e))*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(c^2*sqrt(b^2 - 4*a*c)) + ((c*d - b*e)*log(a + b*x + c*x^2))/(2*c^2)],
[(d + e*x)/(a + b*x + c*x^2), x, 2, -(((2*d - (b*e)/c)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/sqrt(b^2 - 4*a*c)) + (e*log(a + b*x + c*x^2))/(2*c)],
[(d + e*x)/(x*(a + b*x + c*x^2)), x, 5, ((b*d - 2*a*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a*sqrt(b^2 - 4*a*c)) + (d*log(x))/a - (d*log(a + b*x + c*x^2))/(2*a)],
[(d + e*x)/(x^2*(a + b*x + c*x^2)), x, 5, -(d/(a*x)) - ((b^2*d - 2*a*c*d - a*b*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^2*sqrt(b^2 - 4*a*c)) - ((b*d - a*e)*log(x))/a^2 + ((b*d - a*e)*log(a + b*x + c*x^2))/(2*a^2)],
[(d + e*x)/(x^3*(a + b*x + c*x^2)), x, 5, -(d/(2*a*x^2)) + (b*d - a*e)/(a^2*x) + ((b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^3*sqrt(b^2 - 4*a*c)) + ((b^2*d - a*c*d - a*b*e)*log(x))/a^3 - ((b^2*d - a*c*d - a*b*e)*log(a + b*x + c*x^2))/(2*a^3)],
[(d + e*x)/(x^4*(a + b*x + c*x^2)), x, 5, -(d/(3*a*x^3)) + (b*d - a*e)/(2*a^2*x^2) - (b^2*d - a*c*d - a*b*e)/(a^3*x) - ((b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*arctanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(a^4*sqrt(b^2 - 4*a*c)) - ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*log(x))/a^4 + ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*log(a + b*x + c*x^2))/(2*a^4)],


# Integrands of the form (a+b*x)^m/(a*c+(b*c+a*d)*x+b*d*x^2) where m is an integer 
[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2), x, 2, log(c + d*x)/d],
[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2), x, 4, (b*x)/d - ((b*c - a*d)*log(c + d*x))/d^2],
[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2), x, 5, -((b*(b*c - a*d)*x)/d^2) + (a + b*x)^2/(2*d) + ((b*c - a*d)^2*log(c + d*x))/d^3],

[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)), x, 6, -(1/((b*c - a*d)*(a + b*x))) - (d*log(a + b*x))/(b*c - a*d)^2 + (d*log(c + d*x))/(b*c - a*d)^2],
[1/((a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)), x, 7, -(1/(2*(b*c - a*d)*(a + b*x)^2)) + d/((b*c - a*d)^2*(a + b*x)) + (d^2*log(a + b*x))/(b*c - a*d)^3 - (d^2*log(c + d*x))/(b*c - a*d)^3],
[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)), x, 8, -(1/(3*(b*c - a*d)*(a + b*x)^3)) + d/(2*(b*c - a*d)^2*(a + b*x)^2) - d^2/((b*c - a*d)^3*(a + b*x)) - (d^3*log(a + b*x))/(b*c - a*d)^4 + (d^3*log(c + d*x))/(b*c - a*d)^4],


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

[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^2), x, 8, -(b/(2*(b*c - a*d)^2*(a + b*x)^2)) + (2*b*d)/((b*c - a*d)^3*(a + b*x)) + d^2/((b*c - a*d)^3*(c + d*x)) + (3*b*d^2*log(a + b*x))/(b*c - a*d)^4 - (3*b*d^2*log(c + d*x))/(b*c - a*d)^4],
[1/((a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)^2), x, 9, -(b/(3*(b*c - a*d)^2*(a + b*x)^3)) + (b*d)/((b*c - a*d)^3*(a + b*x)^2) - (3*b*d^2)/((b*c - a*d)^4*(a + b*x)) - d^3/((b*c - a*d)^4*(c + d*x)) - (4*b*d^3*log(a + b*x))/(b*c - a*d)^5 + (4*b*d^3*log(c + d*x))/(b*c - a*d)^5],
[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^2), x, 10, -(b/(4*(b*c - a*d)^2*(a + b*x)^4)) + (2*b*d)/(3*(b*c - a*d)^3*(a + b*x)^3) - (3*b*d^2)/(2*(b*c - a*d)^4*(a + b*x)^2) + (4*b*d^3)/((b*c - a*d)^5*(a + b*x)) + d^4/((b*c - a*d)^5*(c + d*x)) + (5*b*d^4*log(a + b*x))/(b*c - a*d)^6 - (5*b*d^4*log(c + d*x))/(b*c - a*d)^6],


# Integrands of the form (a+b*x)^m/(a*c+(b*c+a*d)*x+b*d*x^2)^3 where m is an integer 
[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^3, x, 8, -(b^2/((b*c - a*d)^3*(a + b*x))) - d/(2*(b*c - a*d)^2*(c + d*x)^2) - (2*b*d)/((b*c - a*d)^3*(c + d*x)) - (3*b^2*d*log(a + b*x))/(b*c - a*d)^4 + (3*b^2*d*log(c + d*x))/(b*c - a*d)^4],
[(a + b*x)^2/(a*c + (b*c + a*d)*x + b*d*x^2)^3, x, 7, 1/(2*(b*c - a*d)*(c + d*x)^2) + b/((b*c - a*d)^2*(c + d*x)) + (b^2*log(a + b*x))/(b*c - a*d)^3 - (b^2*log(c + d*x))/(b*c - a*d)^3],
[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2)^3, x, 2, -(1/(2*d*(c + d*x)^2))],

[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)^3), x, 10, -(b^2/(3*(b*c - a*d)^3*(a + b*x)^3)) + (3*b^2*d)/(2*(b*c - a*d)^4*(a + b*x)^2) - (6*b^2*d^2)/((b*c - a*d)^5*(a + b*x)) - d^3/(2*(b*c - a*d)^4*(c + d*x)^2) - (4*b*d^3)/((b*c - a*d)^5*(c + d*x)) - (10*b^2*d^3*log(a + b*x))/(b*c - a*d)^6 + (10*b^2*d^3*log(c + d*x))/(b*c - a*d)^6],
[1/((a + b*x)^2*(a*c + (b*c + a*d)*x + b*d*x^2)^3), x, 11, -(b^2/(4*(b*c - a*d)^3*(a + b*x)^4)) + (b^2*d)/((b*c - a*d)^4*(a + b*x)^3) - (3*b^2*d^2)/((b*c - a*d)^5*(a + b*x)^2) + (10*b^2*d^3)/((b*c - a*d)^6*(a + b*x)) + d^4/(2*(b*c - a*d)^5*(c + d*x)^2) + (5*b*d^4)/((b*c - a*d)^6*(c + d*x)) + (15*b^2*d^4*log(a + b*x))/(b*c - a*d)^7 - (15*b^2*d^4*log(c + d*x))/(b*c - a*d)^7],
[1/((a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^3), x, 12, -(b^2/(5*(b*c - a*d)^3*(a + b*x)^5)) + (3*b^2*d)/(4*(b*c - a*d)^4*(a + b*x)^4) - (2*b^2*d^2)/((b*c - a*d)^5*(a + b*x)^3) + (5*b^2*d^3)/((b*c - a*d)^6*(a + b*x)^2) - (15*b^2*d^4)/((b*c - a*d)^7*(a + b*x)) - d^5/(2*(b*c - a*d)^6*(c + d*x)^2) - (6*b*d^5)/((b*c - a*d)^7*(c + d*x)) - (21*b^2*d^5*log(a + b*x))/(b*c - a*d)^8 + (21*b^2*d^5*log(c + d*x))/(b*c - a*d)^8],


[(5 + x)/(9 + 12*x + 4*x^2), x, 4, -(7/(4*(3 + 2*x))) + (1/4)*log(3 + 2*x)],
[(a + b*x)/(4 - 4*x + x^2), x, 4, (a + 2*b)/(2 - x) + b*log(2 - x)],


# Integrands of the form (d+e*x)/(a+b*x+c*x^2) where m is an integer 
[(5 + 2*x)/(4 + 5*x + x^2), x, 1, log(4 + 5*x + x^2)],
[(7 + 3*x)/(8 + 6*x + x^2), x, 4, (1/2)*log(2 + x) + (5/2)*log(4 + x)],
[(-1 + 2*x)/(1 + 8*x + 4*x^2), x, 2, (1/2)*sqrt(3)*arctanh((2*(1 + x))/sqrt(3)) + (1/4)*log(1 + 8*x + 4*x^2)],
[(5 + 2*x)/(5 + 4*x + x^2), x, 2, arctan(2 + x) + log(5 + 4*x + x^2)],
[(-2 + 7*x)/(42 - 16*x + 2*x^2), x, 3, -((13*arctan((4 - x)/sqrt(5)))/sqrt(5)) + (7/4)*log(21 - 8*x + x^2)],


# Integrands of the form (a+b*x^n)^m*(c+d*x+e*x^2)^p where m, n and p are integers 
[(3 + 2*x)/(13 + 12*x + 4*x^2)^2, x, 1, -(1/(4*(13 + 12*x + 4*x^2)))],
[(4 + x)/(5 + 4*x + x^2)^2, x, 3, 3/(2*(5 + 4*x + x^2)) + x/(5 + 4*x + x^2) + arctan(2 + x)],
[(-1 + 3*x)/(1 + x + x^2)^2, x, 3, -(7/(3*(1 + x + x^2))) - (5*x)/(3*(1 + x + x^2)) - (10*arctan((1 + 2*x)/sqrt(3)))/(3*sqrt(3))],

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

[(1 + x^2)/(1 + x + x^2), x, 4, x + arctan((1 + 2*x)/sqrt(3))/sqrt(3) - log(1 + x + x^2)/2],
[(-1 + x^2)/(25 - 6*x + x^2), x, 4, x + 2*arctan(3/4 - x/4) + 3*log(25 - 6*x + x^2)],
[(-10 + 3*x^2)/(4 - 4*x + x^2), x, 4, 2/(2 - x) + 3*x + 12*log(2 - x)],
[(1 + x^2)/(2 - 3*x + x^2), x, 4, x - 2*log(1 - x) + 5*log(2 - x)],

[(1 - x^2)/(1 + x + x^2)^2, x, 1, x/(1 + x + x^2)],
[(8 + x^2)/(6 - 5*x + x^2), x, 4, x - 12*log(2 - x) + 17*log(3 - x)],

[(-1 + x^3)/(1 + x + x^2), x, 2, -x + x^2/2],
[(-3 + x^3)/(-7 - 6*x + x^2), x, 5, 6*x + x^2/2 + (85/2)*log(7 - x) + (1/2)*log(1 + x)],

[(1 + x^3)/(13 + 4*x + x^2)^2, x, 7, 67/(18*(13 + 4*x + x^2)) + (47*x)/(18*(13 + 4*x + x^2)) - (61/54)*arctan(2/3 + x/3) + (1/2)*log(13 + 4*x + x^2)],

[(2 + 2*x + x^2)/(2 + x), x, 4, x^2/2 + 2*log(2 + x)],
[(5 + 4*x + x^2)/(-2 + x), x, 4, 6*x + x^2/2 + 17*log(2 - x)],

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

[(2 + 12*x + 3*x^2)/(4 + x^2)^2, x, 4, -(6/(4 + x^2)) - (5*x)/(4*(4 + x^2)) + (7/8)*arctan(x/2)],
[(1 + 2*x + x^2)/(1 + x^2)^2, x, 4, -(1/(1 + x^2)) + arctan(x)],

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

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

[(b + 2*c*x)*(a + b*x + c*x^2)^n, x, 1, (a + b*x + c*x^2)^(1 + n)/(1 + n)],

[(b + 2*c*x)^5/(a + b*x + c*x^2)^3, x, 3, -((b + 2*c*x)^4/(2*(a + b*x + c*x^2)^2)) - (4*c*(b + 2*c*x)^2)/(a + b*x + c*x^2) + 16*c^2*log(a + b*x + c*x^2)],


[(2*((a/b)^(1/n) - x*cos(((-1 + 2*k)*Pi)/n)))/((a/b)^(2/n) + x^2 - 2*(a/b)^(1/n)*x*cos(((-1 + 2*k)*Pi)/n)), x, 3, (-cos(((1 - 2*k)*Pi)/n))*log((a/b)^(2/n) + x^2 - 2*(a/b)^(1/n)*x*cos(((1 - 2*k)*Pi)/n)) + 2*arctan(((x - (a/b)^(1/n)*cos(((1 - 2*k)*Pi)/n))*csc(((1 - 2*k)*Pi)/n))/(a/b)^(n^(-1)))*sin(((1 - 2*k)*Pi)/n)],


# ::Subsubsection::Closed:: 
#Miscellaneous integrands involving rational functions of quadratic trinomials


# Integrands of the form (a+b*x+c*x^2)^m*(d+e*x+f*x^2)^n where m and n are integers 
[(-4 + 3*x + x^2)/(-8 - 2*x + x^2), x, 4, x + 4*log(4 - x) + log(2 + x)],
[(7 + 5*x + 4*x^2)/(5 + 4*x + 4*x^2), x, 4, x + (3/8)*arctan(1/2 + x) + (1/8)*log(5 + 4*x + 4*x^2)],
[(7 - 3*x)/(-5 + 2*x + x^2), x, 2, -5*sqrt(2/3)*arctanh((1 + x)/sqrt(6)) - (3/2)*log(5 - 2*x - x^2)],
[(2 - x + x^2)/(-5 + 2*x + x^2), x, 4, x - 5*sqrt(2/3)*arctanh((1 + x)/sqrt(6)) - (3/2)*log(5 - 2*x - x^2)],

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

[1/((4 - 4*x + x^2)*(5 - 4*x + x^2)), x, 4, 1/(2 - x) + arctan(2 - x)],


[(-3 + x + x^2)/((-3 + x)*x^2), x, 4, -(1/x) + log(3 - x)],
[(1 + x + 4*x^2)/(x + 4*x^3), x, 4, arctan(2*x)/2 + log(x)],
[(1 - x + 3*x^2)/(-x^2 + x^3), x, 4, 1/x + 3*log(1 - x)],
[(4 + 3*x + x^2)/(x + x^2), x, 4, x + 4*log(x) - 2*log(1 + x)],
[(4 + x + 3*x^2)/(x + x^3), x, 6, arctan(x) + 4*log(x) - log(1 + x^2)/2],
[(x^2 + x^3)/(-2 + x + x^2), x, 5, x^2/2 + (2/3)*log(1 - x) + (4/3)*log(2 + x)],

[(7 - 4*x + 8*x^2)/((1 + 4*x)*(1 + x^2)), x, 4, -arctan(x) + 2*log(1 + 4*x)],
[x^2/((-1 + x)*(1 + 2*x + x^2)), x, 5, 1/(2*(1 + x)) + (1/4)*log(1 - x) + (3/4)*log(1 + x)],
[(-4 + 3*x + x^2)/((-1 + 2*x)^2*(3 + 2*x)), x, 5, -(9/(32*(1 - 2*x))) + (41/128)*log(1 - 2*x) - (25/128)*log(3 + 2*x)],
[(5 - 4*x + 3*x^2)/((-1 + x)*(1 + x^2)), x, 6, -3*arctan(x) + 2*log(1 - x) + (1/2)*log(1 + x^2)],
[(-1 - 2*x + x^2)/((-1 + x)^2*(1 + x^2)), x, 7, -(1/(1 - x)) + arctan(x) + log(1 - x) - (1/2)*log(1 + x^2)],
[(5 + x^3)/((10 - 6*x + x^2)*(1/2 - x + x^2)), x, 6, (-(261/221))*arctan(1 - 2*x) - (1026/221)*arctan(3 - x) + (56/221)*log(10 - 6*x + x^2) + (109/442)*log(1 - 2*x + 2*x^2)],

[(4 + 3*x + x^2)/((-3 + x)*(-2 + x)*(-1 + x)), x, 5, 4*log(1 - x) - 14*log(2 - x) + 11*log(3 - x)],
[(1 + 16*x)/((5 + x)^2*(-3 + 2*x)*(1 + x + x^2)), x, 7, -(79/(273*(5 + x))) + (451*arctan((1 + 2*x)/sqrt(3)))/(2793*sqrt(3)) + (200*log(3 - 2*x))/3211 + (2731*log(5 + x))/24843 - (481*log(1 + x + x^2))/5586],
[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2), x, 10, -(1/(288*(1 - x)^2)) - 7/(864*(1 - x)) + 325/(13248*(3 + 5*x + 4*x^2)) + (551*x)/(9936*(3 + 5*x + 4*x^2)) + (6023*arctan((5 + 8*x)/sqrt(23)))/(52992*sqrt(23)) + (11*log(1 - x))/2304 - (11*log(3 + 5*x + 4*x^2))/4608],


# Integrands of the form x^m/(a*x^n+b*x^p+c*x^q) where m, n, p and q are integers 
# In some of the following examples gcd cancellation should occur without also partial fraction        expansion, since that will result in unnecessary expansion. 
[x^4/(2 + 13*x + 15*x^2), x, 5, (139*x)/3375 - (13*x^2)/450 + x^3/45 - (16/567)*log(2 + 3*x) + log(1 + 5*x)/4375],
[x^3/(2 + 13*x + 15*x^2), x, 5, -((13*x)/225) + x^2/30 + (8/189)*log(2 + 3*x) - (1/875)*log(1 + 5*x)],
[x^2/(2 + 13*x + 15*x^2), x, 4, x/15 - (4/63)*log(2 + 3*x) + (1/175)*log(1 + 5*x)],
[x/(2 + 13*x + 15*x^2), x, 4, (2/21)*log(2 + 3*x) - (1/35)*log(1 + 5*x)],
[1/(2 + 13*x + 15*x^2), x, 1, (-(2/7))*arctanh(13/7 + (30*x)/7)],
[1/(x*(2 + 13*x + 15*x^2)), x, 5, log(x)/2 + (3/14)*log(2 + 3*x) - (5/7)*log(1 + 5*x)],
[1/(x^2*(2 + 13*x + 15*x^2)), x, 5, -(1/(2*x)) - (13*log(x))/4 - (9/28)*log(2 + 3*x) + (25/7)*log(1 + 5*x)],
[1/(x^3*(2 + 13*x + 15*x^2)), x, 5, -(1/(4*x^2)) + 13/(4*x) + (139*log(x))/8 + (27/56)*log(2 + 3*x) - (125/7)*log(1 + 5*x)],
[1/(x^4*(2 + 13*x + 15*x^2)), x, 5, -(1/(6*x^3)) + 13/(8*x^2) - 139/(8*x) - (1417*log(x))/16 - (81/112)*log(2 + 3*x) + (625/7)*log(1 + 5*x)],

[x^3/(2/x + 13 + 15*x), x, 5, (139*x)/3375 - (13*x^2)/450 + x^3/45 - (16/567)*log(2 + 3*x) + log(1 + 5*x)/4375],
[x^2/(2/x + 13 + 15*x), x, 5, -((13*x)/225) + x^2/30 + (8/189)*log(2 + 3*x) - (1/875)*log(1 + 5*x)],
[x/(2/x + 13 + 15*x), x, 4, x/15 - (4/63)*log(2 + 3*x) + (1/175)*log(1 + 5*x)],
[1/(2/x + 13 + 15*x), x, 4, (2/21)*log(2 + 3*x) - (1/35)*log(1 + 5*x)],
[1/(x*(2/x + 13 + 15*x)), x, 2, (-(2/7))*arctanh(13/7 + (30*x)/7)],
[1/(x^2*(2/x + 13 + 15*x)), x, 5, log(x)/2 + (3/14)*log(2 + 3*x) - (5/7)*log(1 + 5*x)],
[1/(x^3*(2/x + 13 + 15*x)), x, 5, -(1/(2*x)) - (13*log(x))/4 - (9/28)*log(2 + 3*x) + (25/7)*log(1 + 5*x)],
[1/(x^4*(2/x + 13 + 15*x)), x, 5, -(1/(4*x^2)) + 13/(4*x) + (139*log(x))/8 + (27/56)*log(2 + 3*x) - (125/7)*log(1 + 5*x)],
[1/(x^5*(2/x + 13 + 15*x)), x, 5, -(1/(6*x^3)) + 13/(8*x^2) - 139/(8*x) - (1417*log(x))/16 - (81/112)*log(2 + 3*x) + (625/7)*log(1 + 5*x)],

[x^2/(2/x^2 + 13/x + 15), x, 5, (139*x)/3375 - (13*x^2)/450 + x^3/45 - (16/567)*log(2 + 3*x) + log(1 + 5*x)/4375],
[x/(2/x^2 + 13/x + 15), x, 5, -((13*x)/225) + x^2/30 + (8/189)*log(2 + 3*x) - (1/875)*log(1 + 5*x)],
[1/(2/x^2 + 13/x + 15), x, 4, x/15 - (4/63)*log(2 + 3*x) + (1/175)*log(1 + 5*x)],
[1/(x*(2/x^2 + 13/x + 15)), x, 4, (2/21)*log(2 + 3*x) - (1/35)*log(1 + 5*x)],
[1/(x^2*(2/x^2 + 13/x + 15)), x, 2, (2/7)*arctanh(13/7 + 4/(7*x))],
[1/(x^3*(2/x^2 + 13/x + 15)), x, 5, log(x)/2 + (3/14)*log(2 + 3*x) - (5/7)*log(1 + 5*x)],
[1/(x^4*(2/x^2 + 13/x + 15)), x, 5, -(1/(2*x)) - (13*log(x))/4 - (9/28)*log(2 + 3*x) + (25/7)*log(1 + 5*x)],
[1/(x^5*(2/x^2 + 13/x + 15)), x, 5, -(1/(4*x^2)) + 13/(4*x) + (139*log(x))/8 + (27/56)*log(2 + 3*x) - (125/7)*log(1 + 5*x)],
[1/(x^6*(2/x^2 + 13/x + 15)), x, 5, -(1/(6*x^3)) + 13/(8*x^2) - 139/(8*x) - (1417*log(x))/16 - (81/112)*log(2 + 3*x) + (625/7)*log(1 + 5*x)],

[x^5/(2*x + 13*x^2 + 15*x^3), x, 6, (139*x)/3375 - (13*x^2)/450 + x^3/45 - (16/567)*log(2 + 3*x) + log(1 + 5*x)/4375],
[x^4/(2*x + 13*x^2 + 15*x^3), x, 6, -((13*x)/225) + x^2/30 + (8/189)*log(2 + 3*x) - (1/875)*log(1 + 5*x)],
[x^3/(2*x + 13*x^2 + 15*x^3), x, 5, x/15 - (4/63)*log(2 + 3*x) + (1/175)*log(1 + 5*x)],
[x^2/(2*x + 13*x^2 + 15*x^3), x, 5, (2/21)*log(2 + 3*x) - (1/35)*log(1 + 5*x)],
[x/(2*x + 13*x^2 + 15*x^3), x, 2, (-(2/7))*arctanh(13/7 + (30*x)/7)],
[1/(2*x + 13*x^2 + 15*x^3), x, 5, log(x)/2 + (3/14)*log(2 + 3*x) - (5/7)*log(1 + 5*x)],
[1/(x*(2*x + 13*x^2 + 15*x^3)), x, 5, -(1/(2*x)) - (13*log(x))/4 - (9/28)*log(2 + 3*x) + (25/7)*log(1 + 5*x)],
[1/(x^2*(2*x + 13*x^2 + 15*x^3)), x, 5, -(1/(4*x^2)) + 13/(4*x) + (139*log(x))/8 + (27/56)*log(2 + 3*x) - (125/7)*log(1 + 5*x)],
[1/(x^3*(2*x + 13*x^2 + 15*x^3)), x, 5, -(1/(6*x^3)) + 13/(8*x^2) - 139/(8*x) - (1417*log(x))/16 - (81/112)*log(2 + 3*x) + (625/7)*log(1 + 5*x)],


# ::Subsection::Closed:: 
#Integrands involving powers of quartic trinomials


# ::Subsubsection::Closed:: 
#Integrands of the form x^m (a+b x^2+c x^4)^n


# Integrands of the form x^m*(a+b*x^2+c*x^4) where m is an integer 
[x^4*(a + b*x^2 + c*x^4), x, 2, (a*x^5)/5 + (b*x^7)/7 + (c*x^9)/9],
[x^3*(a + b*x^2 + c*x^4), x, 2, (a*x^4)/4 + (b*x^6)/6 + (c*x^8)/8],
[x^2*(a + b*x^2 + c*x^4), x, 2, (a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7],
[x*(a + b*x^2 + c*x^4), x, 2, (a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6],
[(a + b*x^2 + c*x^4), x, 1, a*x + (b*x^3)/3 + (c*x^5)/5],
[(a + b*x^2 + c*x^4)/x, x, 2, (b*x^2)/2 + (c*x^4)/4 + a*log(x)],
[(a + b*x^2 + c*x^4)/x^2, x, 2, -(a/x) + b*x + (c*x^3)/3],
[(a + b*x^2 + c*x^4)/x^3, x, 2, -(a/(2*x^2)) + (c*x^2)/2 + b*log(x)],
[(a + b*x^2 + c*x^4)/x^4, x, 2, -(a/(3*x^3)) - b/x + c*x],


# Integrands of the form x^m*(a+b*x^2+c*x^4)^2 where m is an integer 
[x^4*(a + b*x^2 + c*x^4)^2, x, 2, (a^2*x^5)/5 + (2/7)*a*b*x^7 + (1/9)*(b^2 + 2*a*c)*x^9 + (2/11)*b*c*x^11 + (c^2*x^13)/13],
[x^3*(a + b*x^2 + c*x^4)^2, x, 3, (a^2*x^4)/4 + (1/3)*a*b*x^6 + (1/8)*(b^2 + 2*a*c)*x^8 + (1/5)*b*c*x^10 + (c^2*x^12)/12],
[x^2*(a + b*x^2 + c*x^4)^2, x, 2, (a^2*x^3)/3 + (2/5)*a*b*x^5 + (1/7)*(b^2 + 2*a*c)*x^7 + (2/9)*b*c*x^9 + (c^2*x^11)/11],
[x*(a + b*x^2 + c*x^4)^2, x, 3, (a^2*x^2)/2 + (1/2)*a*b*x^4 + (1/6)*(b^2 + 2*a*c)*x^6 + (1/4)*b*c*x^8 + (c^2*x^10)/10],
[(a + b*x^2 + c*x^4)^2, x, 2, a^2*x + (2/3)*a*b*x^3 + (1/5)*(b^2 + 2*a*c)*x^5 + (2/7)*b*c*x^7 + (c^2*x^9)/9],
[(a + b*x^2 + c*x^4)^2/x, x, 2, a*b*x^2 + (1/4)*(b^2 + 2*a*c)*x^4 + (1/3)*b*c*x^6 + (c^2*x^8)/8 + a^2*log(x)],
[(a + b*x^2 + c*x^4)^2/x^2, x, 2, -(a^2/x) + 2*a*b*x + (1/3)*(b^2 + 2*a*c)*x^3 + (2/5)*b*c*x^5 + (c^2*x^7)/7],
[(a + b*x^2 + c*x^4)^2/x^3, x, 2, -(a^2/(2*x^2)) + (1/2)*(b^2 + 2*a*c)*x^2 + (1/2)*b*c*x^4 + (c^2*x^6)/6 + 2*a*b*log(x)],
[(a + b*x^2 + c*x^4)^2/x^4, x, 2, -(a^2/(3*x^3)) - (2*a*b)/x + (b^2 + 2*a*c)*x + (2/3)*b*c*x^3 + (c^2*x^5)/5],
[(a + b*x^2 + c*x^4)^2/x^7, x, 2, -(a^2/(6*x^6)) - (a*b)/(2*x^4) - (b^2 + 2*a*c)/(2*x^2) + (c^2*x^2)/2 + 2*b*c*log(x)],
[(a + b*x^2 + c*x^4)^2/x^13, x, 2, -(a^2/(12*x^12)) - (a*b)/(5*x^10) - (b^2 + 2*a*c)/(8*x^8) - (b*c)/(3*x^6) - c^2/(4*x^4)],


# Integrands of the form x^m*(a+b*x^2+c*x^4)^3 where m is an integer 
[x^4*(a + b*x^2 + c*x^4)^3, x, 2, (a^3*x^5)/5 + (3/7)*a^2*b*x^7 + (1/3)*a*(b^2 + a*c)*x^9 + (1/11)*b*(b^2 + 6*a*c)*x^11 + (3/13)*c*(b^2 + a*c)*x^13 + (1/5)*b*c^2*x^15 + (c^3*x^17)/17],
[x^3*(a + b*x^2 + c*x^4)^3, x, 3, (a^3*x^4)/4 + (1/2)*a^2*b*x^6 + (3/8)*a*(b^2 + a*c)*x^8 + (1/10)*b*(b^2 + 6*a*c)*x^10 + (1/4)*c*(b^2 + a*c)*x^12 + (3/14)*b*c^2*x^14 + (c^3*x^16)/16],
[x^2*(a + b*x^2 + c*x^4)^3, x, 2, (a^3*x^3)/3 + (3/5)*a^2*b*x^5 + (3/7)*a*(b^2 + a*c)*x^7 + (1/9)*b*(b^2 + 6*a*c)*x^9 + (3/11)*c*(b^2 + a*c)*x^11 + (3/13)*b*c^2*x^13 + (c^3*x^15)/15],
[x*(a + b*x^2 + c*x^4)^3, x, 3, (a^3*x^2)/2 + (3/4)*a^2*b*x^4 + (1/2)*a*(b^2 + a*c)*x^6 + (1/8)*b*(b^2 + 6*a*c)*x^8 + (3/10)*c*(b^2 + a*c)*x^10 + (1/4)*b*c^2*x^12 + (c^3*x^14)/14],
[(a + b*x^2 + c*x^4)^3, x, 2, a^3*x + a^2*b*x^3 + (3/5)*a*(b^2 + a*c)*x^5 + (1/7)*b*(b^2 + 6*a*c)*x^7 + (1/3)*c*(b^2 + a*c)*x^9 + (3/11)*b*c^2*x^11 + (c^3*x^13)/13],
[(a + b*x^2 + c*x^4)^3/x, x, 2, (3/2)*a^2*b*x^2 + (3/4)*a*(b^2 + a*c)*x^4 + (1/6)*b*(b^2 + 6*a*c)*x^6 + (3/8)*c*(b^2 + a*c)*x^8 + (3/10)*b*c^2*x^10 + (c^3*x^12)/12 + a^3*log(x)],
[(a + b*x^2 + c*x^4)^3/x^2, x, 2, -(a^3/x) + 3*a^2*b*x + a*(b^2 + a*c)*x^3 + (1/5)*b*(b^2 + 6*a*c)*x^5 + (3/7)*c*(b^2 + a*c)*x^7 + (1/3)*b*c^2*x^9 + (c^3*x^11)/11],
[(a + b*x^2 + c*x^4)^3/x^3, x, 2, -(a^3/(2*x^2)) + (3/2)*a*(b^2 + a*c)*x^2 + (1/4)*b*(b^2 + 6*a*c)*x^4 + (1/2)*c*(b^2 + a*c)*x^6 + (3/8)*b*c^2*x^8 + (c^3*x^10)/10 + 3*a^2*b*log(x)],
[(a + b*x^2 + c*x^4)^3/x^4, x, 2, -(a^3/(3*x^3)) - (3*a^2*b)/x + 3*a*(b^2 + a*c)*x + (1/3)*b*(b^2 + 6*a*c)*x^3 + (3/5)*c*(b^2 + a*c)*x^5 + (3/7)*b*c^2*x^7 + (c^3*x^9)/9],


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

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


# Integrands of the form x^m/(a+b*x^2+c*x^4)^2 where m is an integer 
[x^4/(a + b*x^2 + c*x^4)^2, x, 5, -(x/(3*c*(a + b*x^2 + c*x^4))) - (x*(2*a - (2*b^2)/c - 3*b*x^2))/(6*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((b^2 + 4*a*c - b*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((b^2 + 4*a*c + b*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*sqrt(c)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[x^3/(a + b*x^2 + c*x^4)^2, x, 4, (2*a + b*x^2)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (b*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[x^2/(a + b*x^2 + c*x^4)^2, x, 4, -((x*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (sqrt(c)*(2*b - sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(2*b + sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[x/(a + b*x^2 + c*x^4)^2, x, 3, -((b + 2*c*x^2)/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (2*c*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(3/2)],
[1/(a + b*x^2 + c*x^4)^2, x, 4, (x*(b^2 - 2*a*c + b*c*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (sqrt(c)*(b^2 - 12*a*c + b*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(b^2 - 12*a*c - b*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[1/(x*(a + b*x^2 + c*x^4)^2), x, 11, 1/(4*a*(a + b*x^2 + c*x^4)) + (b*(b + 2*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (b*c*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a*(b^2 - 4*a*c)^(3/2)) + (b*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^2*sqrt(b^2 - 4*a*c)) + log(x)/a^2 - log(a + b*x^2 + c*x^4)/(4*a^2)],
[1/(x^2*(a + b*x^2 + c*x^4)^2), x, 10, -(1/(a^2*x)) - (x*(b^3 - 3*a*b*c + c*(b^2 - 2*a*c)*x^2))/(2*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (sqrt(c)*(3*b^3 - 16*a*b*c + 3*b^2*sqrt(b^2 - 4*a*c) - 10*a*c*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(3*b^3 - 16*a*b*c - 3*b^2*sqrt(b^2 - 4*a*c) + 10*a*c*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))), -(1/(a^2*x)) + (x*(-b^3 + 3*a*b*c + c*(-b^2 + 2*a*c)*x^2))/(2*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((c + (b*c)/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^2*sqrt(c)*sqrt(b - sqrt(b^2 - 4*a*c))) - (((-c)*(-b^2 + 2*a*c) + (b*c*(-b^2 + 2*a*c) + 2*c*(-b^3 + 3*a*b*c + 2*b*(b^2 - 4*a*c)))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*sqrt(c)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) - ((c - (b*c)/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^2*sqrt(c)*sqrt(b + sqrt(b^2 - 4*a*c))) - (((-c)*(-b^2 + 2*a*c) - (b*c*(-b^2 + 2*a*c) + 2*c*(-b^3 + 3*a*b*c + 2*b*(b^2 - 4*a*c)))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^2*sqrt(c)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[1/(x^3*(a + b*x^2 + c*x^4)^2), x, 11, -(1/(2*a^2*x^2)) - b/(4*a^2*(a + b*x^2 + c*x^4)) - ((b^2 - 2*a*c)*(b + 2*c*x^2))/(4*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (c*(b^2 - 2*a*c)*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(3/2)) - ((b^2 - a*c)*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a^3*sqrt(b^2 - 4*a*c)) - (2*b*log(x))/a^3 + (b*log(a + b*x^2 + c*x^4))/(2*a^3)],
[1/(x^4*(a + b*x^2 + c*x^4)^2), x, 10, -(1/(3*a^2*x^3)) + (2*b)/(a^3*x) + (x*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b*c*(b^2 - 3*a*c)*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (sqrt(c)*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*sqrt(b^2 - 4*a*c) - 19*a*b*c*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - 5*b^3*sqrt(b^2 - 4*a*c) + 19*a*b*c*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))), -(1/(3*a^2*x^3)) + (2*b)/(a^3*x) - (x*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + c*(2*a*b*c - b*(b^2 - a*c))*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((2*b*c + (-2*b^2*c + 2*c*(2*b^2 - a*c))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^3*sqrt(c)*sqrt(b - sqrt(b^2 - 4*a*c))) + (((-c)*(2*a*b*c - b*(b^2 - a*c)) + (b*c*(2*a*b*c - b*(b^2 - a*c)) + 2*c*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 2*(b^2 - 4*a*c)*(b^2 - a*c)))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*sqrt(c)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((2*b*c - (-2*b^2*c + 2*c*(2*b^2 - a*c))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^3*sqrt(c)*sqrt(b + sqrt(b^2 - 4*a*c))) + (((-c)*(2*a*b*c - b*(b^2 - a*c)) - (b*c*(2*a*b*c - b*(b^2 - a*c)) + 2*c*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 2*(b^2 - 4*a*c)*(b^2 - a*c)))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*sqrt(c)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c)))],


# Integrands of the form x^m/(a+b*x^2+c*x^4)^3 where m is an integer 
[x^4/(a + b*x^2 + c*x^4)^3, x, 6, -(x/(7*c*(a + b*x^2 + c*x^4)^2)) - (x*(2*a - (4*b^2)/c - 7*b*x^2))/(28*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x*(7*b^2 - 4*a*c + 12*b*c*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*sqrt(c)*(3*b^2 + 4*a*c - 2*b*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (3*sqrt(c)*(3*b^2 + 4*a*c + 2*b*sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(4*sqrt(2)*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[x^3/(a + b*x^2 + c*x^4)^3, x, 5, (2*a + b*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*b*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*b*c*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[x^2/(a + b*x^2 + c*x^4)^3, x, 5, -((x*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (x*(b^3 + 8*a*b*c + c*(b^2 + 20*a*c)*x^2))/(8*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (sqrt(c)*(b*(b^2 - 52*a*c) + sqrt(b^2 - 4*a*c)*(b^2 + 20*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(b*(b^2 - 52*a*c) - sqrt(b^2 - 4*a*c)*(b^2 + 20*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[x/(a + b*x^2 + c*x^4)^3, x, 4, -((b + 2*c*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)) + (3*c*(b + 2*c*x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (6*c^2*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(b^2 - 4*a*c)^(5/2)],
[1/(a + b*x^2 + c*x^4)^3, x, 5, (x*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*(3*b^4 - 25*a*b^2*c + 28*a^2*c^2 + 3*b*c*(b^2 - 8*a*c)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (sqrt(c)*(9*b^4 - 3*b*(b^2 - 8*a*c)*sqrt(b^2 - 4*a*c) - 2*(37*a*b^2*c - 28*a^2*c^2 + 2*(3*b^2 - 14*a*c)*(b^2 - 4*a*c)))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) + (sqrt(c)*(9*b^4 + 3*b*(b^2 - 8*a*c)*sqrt(b^2 - 4*a*c) - 2*(37*a*b^2*c - 28*a^2*c^2 + 2*(3*b^2 - 14*a*c)*(b^2 - 4*a*c)))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^2*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[1/(x*(a + b*x^2 + c*x^4)^3), x, 17, 1/(8*a*(a + b*x^2 + c*x^4)^2) + (b*(b + 2*c*x^2))/(8*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + 1/(4*a^2*(a + b*x^2 + c*x^4)) - (3*b*c*(b + 2*c*x^2))/(4*a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b + 2*c*x^2))/(4*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*b*c^2*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a*(b^2 - 4*a*c)^(5/2)) - (b*c*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(3/2)) + (b*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^3*sqrt(b^2 - 4*a*c)) + log(x)/a^3 - log(a + b*x^2 + c*x^4)/(4*a^3)],
[1/(x^2*(a + b*x^2 + c*x^4)^3), x, 15, -(1/(a^3*x)) - (x*(b^3 - 3*a*b*c + c*(b^2 - 2*a*c)*x^2))/(4*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x*(b^3 - 3*a*b*c + c*(b^2 - 2*a*c)*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (x*(3*b^5 - 24*a*b^3*c + 36*a^2*b*c^2 + c*(3*b^4 - 10*a*c*((23*b^2)/10 - 2*a*c))*x^2))/(8*a^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (sqrt(c)*(1 + b/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^3*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(b*(b^2 - 8*a*c) + sqrt(b^2 - 4*a*c)*(b^2 - 2*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*(b^2 - 4*a*c)^(3/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(sqrt(b^2 - 4*a*c)*(3*b^4 - a*c*(23*b^2 - 20*a*c)) - b*(9*b^4 - 71*a*b^2*c + 92*a^2*c^2 - 4*(3*b^2 - 13*a*c)*(b^2 - 4*a*c)))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^(5/2)*sqrt(b - sqrt(b^2 - 4*a*c))) - (sqrt(c)*(1 - b/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^3*sqrt(b + sqrt(b^2 - 4*a*c))) + (sqrt(c)*(b*(b^2 - 8*a*c) - sqrt(b^2 - 4*a*c)*(b^2 - 2*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^3*(b^2 - 4*a*c)^(3/2)*sqrt(b + sqrt(b^2 - 4*a*c))) - (sqrt(c)*(sqrt(b^2 - 4*a*c)*(3*b^4 - a*c*(23*b^2 - 20*a*c)) + b*(9*b^4 - 71*a*b^2*c + 92*a^2*c^2 - 4*(3*b^2 - 13*a*c)*(b^2 - 4*a*c)))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(8*sqrt(2)*a^3*(b^2 - 4*a*c)^(5/2)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[1/(x^3*(a + b*x^2 + c*x^4)^3), x, 17, -(1/(2*a^3*x^2)) - b/(8*a^2*(a + b*x^2 + c*x^4)^2) - ((b^2 - 2*a*c)*(b + 2*c*x^2))/(8*a^2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - b/(2*a^3*(a + b*x^2 + c*x^4)) + (3*c*(b^2 - 2*a*c)*(b + 2*c*x^2))/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - ((b^2 - a*c)*(b + 2*c*x^2))/(2*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (3*c^2*(b^2 - 2*a*c)*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a^2*(b^2 - 4*a*c)^(5/2)) - ((3*b^2 - 2*a*c)*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(2*a^4*sqrt(b^2 - 4*a*c)) + (2*c*(b^2 - a*c)*arctanh((b + 2*c*x^2)/sqrt(b^2 - 4*a*c)))/(a^3*(b^2 - 4*a*c)^(3/2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x^2 + c*x^4))/(4*a^4)],
[1/(x^4*(a + b*x^2 + c*x^4)^3), x, 15, -(1/(3*a^3*x^3)) + (3*b)/(a^4*x) - (x*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + c*(2*a*b*c - b*(b^2 - a*c))*x^2))/(4*a^3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (x*(2*a*b^2*c - b^2*(2*b^2 - a*c) + 2*a*c*(2*b^2 - a*c) + c*(4*a*b*c - b*(2*b^2 - a*c))*x^2))/(2*a^4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (1/(8*a^4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)))*(x*(-5*a*b*c*(2*a*b*c - b*(b^2 - a*c)) - b^2*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)) + 2*a*c*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)) + c*(-10*a*c*(2*a*b*c - b*(b^2 - a*c)) - b*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)))*x^2)) + ((3*b*c + (-3*b^2*c + 2*c*(3*b^2 - a*c))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^4*sqrt(c)*sqrt(b - sqrt(b^2 - 4*a*c))) + (((-c)*(4*a*b*c - b*(2*b^2 - a*c)) + (b*c*(4*a*b*c - b*(2*b^2 - a*c)) + 2*c*(2*a*b^2*c - b^2*(2*b^2 - a*c) + 2*a*c*(2*b^2 - a*c) + 2*(b^2 - 4*a*c)*(2*b^2 - a*c)))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^4*sqrt(c)*(b^2 - 4*a*c)*sqrt(b - sqrt(b^2 - 4*a*c))) + (1/(8*sqrt(2)*a^4*sqrt(c)*(b^2 - 4*a*c)^2*sqrt(b - sqrt(b^2 - 4*a*c))))*(((-c)*(-10*a*c*(2*a*b*c - b*(b^2 - a*c)) - b*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c))) + (1/sqrt(b^2 - 4*a*c))*(b*c*(-10*a*c*(2*a*b*c - b*(b^2 - a*c)) - b*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c))) + 2*c*(-5*a*b*c*(2*a*b*c - b*(b^2 - a*c)) - b^2*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)) + 2*a*c*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)) + 2*(b^2 - 4*a*c)*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)))))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c)))) + ((3*b*c - (-3*b^2*c + 2*c*(3*b^2 - a*c))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*a^4*sqrt(c)*sqrt(b + sqrt(b^2 - 4*a*c))) + (((-c)*(4*a*b*c - b*(2*b^2 - a*c)) - (b*c*(4*a*b*c - b*(2*b^2 - a*c)) + 2*c*(2*a*b^2*c - b^2*(2*b^2 - a*c) + 2*a*c*(2*b^2 - a*c) + 2*(b^2 - 4*a*c)*(2*b^2 - a*c)))/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(2*sqrt(2)*a^4*sqrt(c)*(b^2 - 4*a*c)*sqrt(b + sqrt(b^2 - 4*a*c))) + (1/(8*sqrt(2)*a^4*sqrt(c)*(b^2 - 4*a*c)^2*sqrt(b + sqrt(b^2 - 4*a*c))))*(((-c)*(-10*a*c*(2*a*b*c - b*(b^2 - a*c)) - b*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c))) - (1/sqrt(b^2 - 4*a*c))*(b*c*(-10*a*c*(2*a*b*c - b*(b^2 - a*c)) - b*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c))) + 2*c*(-5*a*b*c*(2*a*b*c - b*(b^2 - a*c)) - b^2*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)) + 2*a*c*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)) + 2*(b^2 - 4*a*c)*(a*b^2*c - b^2*(b^2 - a*c) + 2*a*c*(b^2 - a*c) + 4*(b^2 - 4*a*c)*(b^2 - a*c)))))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))],


# Integrands of the form x^m/(a+b*x^n+c*x^(2*n)) where m and n>0 are integers 
[1/(a^2 + b + 2*a*x^2 + x^4), x, 3, arctan(x/sqrt(a - sqrt(-b)))/(2*sqrt(a - sqrt(-b))*sqrt(-b)) - arctan(x/sqrt(a + sqrt(-b)))/(2*sqrt(a + sqrt(-b))*sqrt(-b))],
[1/(-1 + a^2 + 2*a*x^2 + x^4), x, 3, -(arctan(x/sqrt(1 + a))/(2*sqrt(1 + a))) - arctanh(x/sqrt(1 - a))/(2*sqrt(1 - a))],
# Tests that NegativeQ[b^2-4*a*c] returns True to avoid I in antiderivative! 
[1/(1 + a^2 + 2*a*x^2 + x^4), x, 3, arctan((sqrt(2)*sqrt(1 + a^2 + a*sqrt(1 + a^2))*x)/((1 + a^2)^(1/4)*(sqrt(1 + a^2) - x^2)))/(2*sqrt(2)*(1 + a^2)^(1/4)*sqrt(1 + a^2 + a*sqrt(1 + a^2))) + arctanh((sqrt(2)*sqrt(1 + a^2 - a*sqrt(1 + a^2))*x)/((1 + a^2)^(1/4)*(sqrt(1 + a^2) + x^2)))/(2*sqrt(2)*(1 + a^2)^(1/4)*sqrt(1 + a^2 - a*sqrt(1 + a^2)))],

[1/(1 + 2*x^2 + x^4), x, 3, x/(2*(1 + x^2)) + arctan(x)/2],
[1/(4 - 5*x^2 + x^4), x, 3, (-(1/6))*arctanh(x/2) + arctanh(x)/3],
[1/(3 + 4*x^2 + x^4), x, 3, arctan(x)/2 - arctan(x/sqrt(3))/(2*sqrt(3))],
[1/(9 + 5*x^2 + x^4), x, 3, arctan((sqrt(11)*x)/(3 - x^2))/(6*sqrt(11)) + (1/6)*arctanh(x/(3 + x^2))],
[1/(1 - x^2 + x^4), x, 3, (1/2)*arctan(x/(1 - x^2)) + arctanh((sqrt(3)*x)/(1 + x^2))/(2*sqrt(3))],
[1/(2 + 2*x^2 + x^4), x, 3, (1/4)*sqrt(-1 + sqrt(2))*arctan((sqrt(2 + 2*sqrt(2))*x)/(sqrt(2) - x^2)) + (1/4)*sqrt(1 + sqrt(2))*arctanh((sqrt(-2 + 2*sqrt(2))*x)/(sqrt(2) + x^2))],

[x/(1 + 2*x^2 + x^4), x, 3, -(1/(2*(1 + x^2)))],
[x/(1 + x^2 + x^4), x, 2, arctan((1 + 2*x^2)/sqrt(3))/sqrt(3)],
[x/(81 - 18*x^2 + x^4), x, 3, 1/(2*(9 - x^2))],
[x/(10 + 2*x^2 + x^4), x, 2, (1/6)*arctan(1/3 + x^2/3)],

[x^2/(1 + 2*x^2 + x^4), x, 3, -(x/(2*(1 + x^2))) + arctan(x)/2],
[x^2/(20 + 9*x^2 + x^4), x, 3, -2*arctan(x/2) + sqrt(5)*arctan(x/sqrt(5))],
[x^2/(1 - x^2 + x^4), x, 3, (1/2)*arctan(x/(1 - x^2)) - arctanh((sqrt(3)*x)/(1 + x^2))/(2*sqrt(3))],
[x^2/(2 - 2*x^2 + x^4), x, 3, (1/4)*sqrt(2 + 2*sqrt(2))*arctan((sqrt(-2 + 2*sqrt(2))*x)/(sqrt(2) - x^2)) - (1/4)*sqrt(-2 + 2*sqrt(2))*arctanh((sqrt(2 + 2*sqrt(2))*x)/(sqrt(2) + x^2))],
[x^2/(1 + (-1 + x^2)^2), x, 4, (1/4)*sqrt(2 + 2*sqrt(2))*arctan((sqrt(-2 + 2*sqrt(2))*x)/(sqrt(2) - x^2)) - (1/4)*sqrt(-2 + 2*sqrt(2))*arctanh((sqrt(2 + 2*sqrt(2))*x)/(sqrt(2) + x^2))],

[x^3/(1 + 2*x^2 + x^4), x, 6, 1/(2*(1 + x^2)) + (1/2)*log(1 + x^2)],
[x^3/(16 - 8*x^2 + x^4), x, 6, 2/(4 - x^2) + (1/2)*log(4 - x^2)],

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


# ::Subsubsection::Closed:: 
#Integrands of the form (d+e x^2) (a+b x^2+c x^4)^n


# Integrands of the form (d+e*x^2)/(a+b*x^2+c*x^4) 
[(1 + x^2)/(1 + x^2)^2, x, 1, arctan(x)],
[(1 + x^2)/(1 + 2*x^2 + x^4), x, 2, arctan(x)],
[(3 + 2*x^2)/(1 - 2*x^2 + x^4), x, 4, (5*x)/(2*(1 - x^2)) + arctanh(x)/2],

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

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

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

[(d + e*x^2)/(a + b*x^2 + c*x^4), x, 3, ((e + (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(b - sqrt(b^2 - 4*a*c))) + ((e - (2*c*d - b*e)/sqrt(b^2 - 4*a*c))*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - 4*a*c))))/(sqrt(2)*sqrt(c)*sqrt(b + sqrt(b^2 - 4*a*c)))],
[(d + e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4), x, 1, (e*arctan((sqrt(b + (2*c*d)/e)*e*x)/(sqrt(c)*(d - e*x^2))))/(sqrt(c)*sqrt(b + (2*c*d)/e))],
[(d + e*x^2)/(b*x^2 + c*(d^2/e^2 + x^4)), x, 8, (sqrt(2)*sqrt(c)*d*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - (4*c^2*d^2)/e^2))))/(sqrt(b - sqrt(b^2 - (4*c^2*d^2)/e^2))*sqrt(b^2 - (4*c^2*d^2)/e^2)) - (sqrt(b - sqrt(b^2 - (4*c^2*d^2)/e^2))*e*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b - sqrt(b^2 - (4*c^2*d^2)/e^2))))/(sqrt(2)*sqrt(c)*sqrt(b^2 - (4*c^2*d^2)/e^2)) - (sqrt(2)*sqrt(c)*d*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - (4*c^2*d^2)/e^2))))/(sqrt(b + sqrt(b^2 - (4*c^2*d^2)/e^2))*sqrt(b^2 - (4*c^2*d^2)/e^2)) + (sqrt(b + sqrt(b^2 - (4*c^2*d^2)/e^2))*e*arctan((sqrt(2)*sqrt(c)*x)/sqrt(b + sqrt(b^2 - (4*c^2*d^2)/e^2))))/(sqrt(2)*sqrt(c)*sqrt(b^2 - (4*c^2*d^2)/e^2))],


# Integrands of the form (d+e*x^2)/(a+b*x^2+c*x^4)^2 
[(a + b*x^2)/(1 + x^2 + x^4)^2, x, 4, (x*(a + b - (a - 2*b)*x^2))/(6*(1 + x^2 + x^4)) + ((4*a + b)*arctan((sqrt(3)*x)/(1 - x^2)))/(12*sqrt(3)) + (1/4)*(2*a - b)*arctanh(x/(1 + x^2))],
[(a + b*x^2)/(2 + x^2 + x^4)^2, x, 4, (x*(3*a + 2*b - (a - 4*b)*x^2))/(28*(2 + x^2 + x^4)) - ((2*(a - 4*b) - sqrt(2)*(11*a - 2*b))*arctan((sqrt(1 + 2*sqrt(2))*x)/(sqrt(2) - x^2)))/(112*sqrt(1 + 2*sqrt(2))) + ((2*(a - 4*b) + sqrt(2)*(11*a - 2*b))*arctanh((sqrt(-1 + 2*sqrt(2))*x)/(sqrt(2) + x^2)))/(112*sqrt(-1 + 2*sqrt(2)))],
[(a + b*x^2)/(c + d*x^2 + e*x^4)^2, x, 4, -((x*(b*c*d - a*d^2 + 2*a*c*e + (2*b*c - a*d)*e*x^2))/(2*c*(d^2 - 4*c*e)*(c + d*x^2 + e*x^4))) + (sqrt(e)*(d*(4*b*c + a*d) - 12*a*c*e - (2*b*c - a*d)*sqrt(d^2 - 4*c*e))*arctan((sqrt(2)*sqrt(e)*x)/sqrt(d - sqrt(d^2 - 4*c*e))))/(2*sqrt(2)*c*(d^2 - 4*c*e)^(3/2)*sqrt(d - sqrt(d^2 - 4*c*e))) - (sqrt(e)*(d*(4*b*c + a*d) - 12*a*c*e + (2*b*c - a*d)*sqrt(d^2 - 4*c*e))*arctan((sqrt(2)*sqrt(e)*x)/sqrt(d + sqrt(d^2 - 4*c*e))))/(2*sqrt(2)*c*(d^2 - 4*c*e)^(3/2)*sqrt(d + sqrt(d^2 - 4*c*e)))],


# ::Subsubsection::Closed:: 
#Integrands involving powers of symmetric higher trinomials


[1/(2 + x^3 + x^6), x, 11, (I*arctan((2^(2/3)*(1 - I*sqrt(7))^(1/3) - 4*x)/(2^(2/3)*sqrt(3)*(1 - I*sqrt(7))^(1/3))))/(sqrt(21)*((1/2)*(1 - I*sqrt(7)))^(2/3)) - (I*arctan(((1 - I*sqrt(7))^(1/3)*(2^(2/3)*(1 + I*sqrt(7))^(1/3) - 4*x))/(2*2^(2/3)*sqrt(3))))/(sqrt(21)*((1/2)*(1 + I*sqrt(7)))^(2/3)) - (I*log(2^(2/3)*(1 - I*sqrt(7))^(1/3) + 2*x))/(3*sqrt(7)*((1/2)*(1 - I*sqrt(7)))^(2/3)) + (I*log(2^(2/3)*(1 + I*sqrt(7))^(1/3) + 2*x))/(3*sqrt(7)*((1/2)*(1 + I*sqrt(7)))^(2/3)) + (I*log(2^(1/3)*(1 - I*sqrt(7))^(2/3) - 2^(2/3)*(1 - I*sqrt(7))^(1/3)*x + 2*x^2))/(3*2^(1/3)*sqrt(7)*(1 - I*sqrt(7))^(2/3)) - (I*log((sqrt(2) + I*sqrt(14))^(2/3) - 2^(2/3)*(1 + I*sqrt(7))^(1/3)*x + 2*x^2))/(3*sqrt(7)*(sqrt(2) + I*sqrt(14))^(2/3))],
[x^2/(2 + x^3 + x^6), x, 2, (2*arctan((1 + 2*x^3)/sqrt(7)))/(3*sqrt(7))],
[x^3/(2 + x^3 + x^6), x, 11, -(((7 + I*sqrt(7))*arctan((2^(2/3)*(1 - I*sqrt(7))^(1/3) - 4*x)/(2^(2/3)*sqrt(3)*(1 - I*sqrt(7))^(1/3))))/(7*2^(1/3)*sqrt(3)*(1 - I*sqrt(7))^(2/3))) - ((7 - I*sqrt(7))*arctan(((1 - I*sqrt(7))^(1/3)*(2^(2/3)*(1 + I*sqrt(7))^(1/3) - 4*x))/(2*2^(2/3)*sqrt(3))))/(7*sqrt(3)*(sqrt(2) + I*sqrt(14))^(2/3)) + (I*((1/2)*(1 - I*sqrt(7)))^(1/3)*log(2^(2/3)*(1 - I*sqrt(7))^(1/3) + 2*x))/(3*sqrt(7)) + ((7 - I*sqrt(7))*log(2^(2/3)*(1 + I*sqrt(7))^(1/3) + 2*x))/(21*(sqrt(2) + I*sqrt(14))^(2/3)) - ((7 + I*sqrt(7))*log(2^(1/3)*(1 - I*sqrt(7))^(2/3) - 2^(2/3)*(1 - I*sqrt(7))^(1/3)*x + 2*x^2))/(42*2^(1/3)*(1 - I*sqrt(7))^(2/3)) - ((7 - I*sqrt(7))*log((sqrt(2) + I*sqrt(14))^(2/3) - 2^(2/3)*(1 + I*sqrt(7))^(1/3)*x + 2*x^2))/(42*(sqrt(2) + I*sqrt(14))^(2/3))],

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


[1/(2 + x^4 + x^8), x, 7, -(((sqrt(2) - sqrt(4 - sqrt(2)))*arctan((sqrt(2*sqrt(2) - sqrt(4 - sqrt(2)))*x)/(2^(1/8)*(2^(1/4) - x^2))))/(4*2^(7/8)*sqrt((-1 + 2*sqrt(2))*(2*sqrt(2) - sqrt(4 - sqrt(2)))))) + ((sqrt(2) + sqrt(4 - sqrt(2)))*arctan((sqrt(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))*x)/(2^(1/4) - x^2)))/(8*sqrt((-1 + 2*sqrt(2))*(2*2^(1/4) + sqrt(-1 + 2*sqrt(2))))) - ((sqrt(2) - sqrt(4 - sqrt(2)))*arctanh((sqrt(2*sqrt(2) - sqrt(4 - sqrt(2)))*x)/(2^(1/8)*(2^(1/4) + x^2))))/(4*2^(7/8)*sqrt((-1 + 2*sqrt(2))*(2*sqrt(2) - sqrt(4 - sqrt(2))))) + ((sqrt(2) + sqrt(4 - sqrt(2)))*arctanh((sqrt(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))*x)/(2^(1/4) + x^2)))/(8*sqrt((-1 + 2*sqrt(2))*(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))))],
[x/(2 + x^4 + x^8), x, 4, arctan((sqrt(1 + 2*sqrt(2))*x^2)/(sqrt(2) - x^4))/(4*sqrt(2 + 4*sqrt(2))) + arctanh((sqrt(-1 + 2*sqrt(2))*x^2)/(sqrt(2) + x^4))/(4*sqrt(-2 + 4*sqrt(2)))],
[x^2/(2 + x^4 + x^8), x, 7, arctan((sqrt(2*sqrt(2) - sqrt(4 - sqrt(2)))*x)/(2^(1/8)*(2^(1/4) - x^2)))/(4*2^(1/8)*sqrt((-1 + 2*sqrt(2))*(2*sqrt(2) - sqrt(4 - sqrt(2))))) - arctan((sqrt(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))*x)/(2^(1/4) - x^2))/(4*2^(1/4)*sqrt((-1 + 2*sqrt(2))*(2*2^(1/4) + sqrt(-1 + 2*sqrt(2))))) - arctanh((sqrt(2*sqrt(2) - sqrt(4 - sqrt(2)))*x)/(2^(1/8)*(2^(1/4) + x^2)))/(4*2^(1/8)*sqrt((-1 + 2*sqrt(2))*(2*sqrt(2) - sqrt(4 - sqrt(2))))) + arctanh((sqrt(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))*x)/(2^(1/4) + x^2))/(4*2^(1/4)*sqrt((-1 + 2*sqrt(2))*(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))))],
[x^3/(2 + x^4 + x^8), x, 2, arctan((1 + 2*x^4)/sqrt(7))/(2*sqrt(7))],
[x^4/(2 + x^4 + x^8), x, 7, arctan((sqrt(2*sqrt(2) - sqrt(4 - sqrt(2)))*x)/(2^(1/8)*(2^(1/4) - x^2)))/(2*2^(7/8)*sqrt((-1 + 2*sqrt(2))*(2*sqrt(2) - sqrt(4 - sqrt(2))))) - arctan((sqrt(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))*x)/(2^(1/4) - x^2))/(4*sqrt((-1 + 2*sqrt(2))*(2*2^(1/4) + sqrt(-1 + 2*sqrt(2))))) + arctanh((sqrt(2*sqrt(2) - sqrt(4 - sqrt(2)))*x)/(2^(1/8)*(2^(1/4) + x^2)))/(2*2^(7/8)*sqrt((-1 + 2*sqrt(2))*(2*sqrt(2) - sqrt(4 - sqrt(2))))) - arctanh((sqrt(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))*x)/(2^(1/4) + x^2))/(4*sqrt((-1 + 2*sqrt(2))*(2*2^(1/4) + sqrt(-1 + 2*sqrt(2)))))],

[x^3/(2 + 3*x^4 + x^8), x, 2, -arctanh(3 + 2*x^4)/2],


# {1/(2 + x^5 + x^10), x, 19, (I*2^(3/10)*Sqrt[(1/7)*(5 + Sqrt[5])]*ArcTan[((1 - Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 + Sqrt[5]]*(1 - I*Sqrt[7])^(1/5))])/(5*(1 - I*Sqrt[7])^(4/5)) + (I*2^(3/10)*Sqrt[(1/7)*(5 - Sqrt[5])]*ArcTan[((1 + Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 - Sqrt[5]]*(1 - I*Sqrt[7])^(1/5))])/(5*(1 - I*Sqrt[7])^(4/5)) - (I*2^(3/10)*Sqrt[(1/7)*(5 + Sqrt[5])]*ArcTan[((1 - Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 + Sqrt[5]]*(1 + I*Sqrt[7])^(1/5))])/(5*(1 + I*Sqrt[7])^(4/5)) - (I*2^(3/10)*Sqrt[(1/7)*(5 - Sqrt[5])]*ArcTan[((1 + Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 - Sqrt[5]]*(1 + I*Sqrt[7])^(1/5))])/(5*(1 + I*Sqrt[7])^(4/5)) - (I*Log[((1/2)*(1 - I*Sqrt[7]))^(1/5) + x])/(5*Sqrt[7]*((1/2)*(1 - I*Sqrt[7]))^(4/5)) + (I*Log[((1/2)*(1 + I*Sqrt[7]))^(1/5) + x])/(5*Sqrt[7]*((1/2)*(1 + I*Sqrt[7]))^(4/5)) + (I*(1 - Sqrt[5])*Log[2^(3/5)*(1 - I*Sqrt[7])^(2/5) - (1 - Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5)*x + 2*x^2])/(10*2^(1/5)*Sqrt[7]*(1 - I*Sqrt[7])^(4/5)) + (I*(1 + Sqrt[5])*Log[2^(3/5)*(1 - I*Sqrt[7])^(2/5) - (1 + Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5)*x + 2*x^2])/(10*2^(1/5)*Sqrt[7]*(1 - I*Sqrt[7])^(4/5)) - (I*(1 - Sqrt[5])*Log[2^(3/5)*(1 + I*Sqrt[7])^(2/5) - (1 - Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5)*x + 2*x^2])/(10*2^(1/5)*Sqrt[7]*(1 + I*Sqrt[7])^(4/5)) - (I*(1 + Sqrt[5])*Log[2^(3/5)*(1 + I*Sqrt[7])^(2/5) - (1 + Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5)*x + 2*x^2])/(10*2^(1/5)*Sqrt[7]*(1 + I*Sqrt[7])^(4/5))} 
[x^4/(2 + x^5 + x^10), x, 2, (2*arctan((1 + 2*x^5)/sqrt(7)))/(5*sqrt(7))],
# {x^5/(2 + x^5 + x^10), x, 19, -((I*Sqrt[(1/7)*(5 + Sqrt[5])]*(1 - I*Sqrt[7])^(1/5)*ArcTan[((1 - Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 + Sqrt[5]]*(1 - I*Sqrt[7])^(1/5))])/(5*2^(7/10))) - (I*Sqrt[(1/7)*(5 - Sqrt[5])]*(1 - I*Sqrt[7])^(1/5)*ArcTan[((1 + Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 - Sqrt[5]]*(1 - I*Sqrt[7])^(1/5))])/(5*2^(7/10)) + (I*Sqrt[(1/7)*(5 + Sqrt[5])]*(1 + I*Sqrt[7])^(1/5)*ArcTan[((1 - Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 + Sqrt[5]]*(1 + I*Sqrt[7])^(1/5))])/(5*2^(7/10)) + (I*Sqrt[(1/7)*(5 - Sqrt[5])]*(1 + I*Sqrt[7])^(1/5)*ArcTan[((1 + Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5) - 4*x)/(2^(3/10)*Sqrt[5 - Sqrt[5]]*(1 + I*Sqrt[7])^(1/5))])/(5*2^(7/10)) + (I*((1/2)*(1 - I*Sqrt[7]))^(1/5)*Log[((1/2)*(1 - I*Sqrt[7]))^(1/5) + x])/(5*Sqrt[7]) - (I*((1/2)*(1 + I*Sqrt[7]))^(1/5)*Log[((1/2)*(1 + I*Sqrt[7]))^(1/5) + x])/(5*Sqrt[7]) - (I*(1 - Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5)*Log[2^(3/5)*(1 - I*Sqrt[7])^(2/5) - (1 - Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5)*x + 2*x^2])/(20*Sqrt[7]) - (I*(1 + Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5)*Log[2^(3/5)*(1 - I*Sqrt[7])^(2/5) - (1 + Sqrt[5])*((1/2)*(1 - I*Sqrt[7]))^(1/5)*x + 2*x^2])/(20*Sqrt[7]) + (I*(1 - Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5)*Log[2^(3/5)*(1 + I*Sqrt[7])^(2/5) - (1 - Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5)*x + 2*x^2])/(20*Sqrt[7]) + (I*(1 + Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5)*Log[2^(3/5)*(1 + I*Sqrt[7])^(2/5) - (1 + Sqrt[5])*((1/2)*(1 + I*Sqrt[7]))^(1/5)*x + 2*x^2])/(20*Sqrt[7])} 

[1/(x*(1 + x^5 + x^10)), x, 3, -arctan((1 + 2*x^5)/sqrt(3))/(5*sqrt(3)) + log(x) - log(1 + x^5 + x^10)/10],
[1/(x + x^6 + x^11), x, 4, -arctan((1 + 2*x^5)/sqrt(3))/(5*sqrt(3)) + log(x) - log(1 + x^5 + x^10)/10],


# ::Subsection::Closed:: 
#Integrands involving quotients of polynomials


[(e - 2*f*x^2)/(e^2 + 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4), x, 1, arctan((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f))],
[(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4), x, 1, arctanh((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f))],

[(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6), x, 1, arctan((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))],
[(e - 4*f*x^3)/(e^2 - 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6), x, 1, arctanh((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))],

[(e - 2*f*(-1 + n)*x^n)/(e^2 + 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctan((2*sqrt(d)*sqrt(f)*(-1 + n)*x)/(e*(-1 + n) + 2*f*(-1 + n)*x^n))/(2*sqrt(d)*sqrt(f))],
[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctanh((2*sqrt(d)*sqrt(f)*(-1 + n)*x)/(e*(-1 + n) + 2*f*(-1 + n)*x^n))/(2*sqrt(d)*sqrt(f))],


[x/(e^2 + 4*e*f*x^2 + 4*d*f*x^4 + 4*f^2*x^4), x, 2, arctan((sqrt(f)*(e + 2*(d + f)*x^2))/(sqrt(d)*e))/(4*sqrt(d)*e*sqrt(f))],
[x/(e^2 + 4*e*f*x^2 - 4*d*f*x^4 + 4*f^2*x^4), x, 2, -(arctanh((sqrt(f)*(e - 2*(d - f)*x^2))/(sqrt(d)*e))/(4*sqrt(d)*e*sqrt(f)))],

[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^6), x, 1, arctan((2*sqrt(d)*sqrt(f)*x^3)/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f))],
[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6), x, 1, arctanh((2*sqrt(d)*sqrt(f)*x^3)/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f))],

[(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^(2 + 2*m)), x, 1, arctan((2*sqrt(d)*sqrt(f)*(1 - m^2)*x^(1 + m))/(e*(1 - m)*(1 + m) + 2*f*(1 - m^2)*x^2))/(2*sqrt(d)*sqrt(f))],
[(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^(2 + 2*m)), x, 1, arctanh((2*sqrt(d)*sqrt(f)*(1 - m^2)*x^(1 + m))/(e*(1 - m)*(1 + m) + 2*f*(1 - m^2)*x^2))/(2*sqrt(d)*sqrt(f))],


[(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 + 4*d*f*x^4 + 4*f^2*x^6), x, 1, arctan((2*sqrt(d)*sqrt(f)*x^2)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))],
[(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 - 4*d*f*x^4 + 4*f^2*x^6), x, 1, arctanh((2*sqrt(d)*sqrt(f)*x^2)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))],

[x^2/(e^2 + 4*e*f*x^3 + 4*d*f*x^6 + 4*f^2*x^6), x, 2, arctan((sqrt(f)*(e + 2*(d + f)*x^3))/(sqrt(d)*e))/(6*sqrt(d)*e*sqrt(f))],
[x^2/(e^2 + 4*e*f*x^3 - 4*d*f*x^6 + 4*f^2*x^6), x, 2, -(arctanh((sqrt(f)*(e - 2*(d - f)*x^3))/(sqrt(d)*e))/(6*sqrt(d)*e*sqrt(f)))],

[(x^m*(e*(1 + m) + 2*f*(-2 + m)*x^3))/(e^2 + 4*e*f*x^3 + 4*f^2*x^6 + 4*d*f*x^(2 + 2*m)), x, 1, arctan((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))],
[(x^m*(e*(1 + m) + 2*f*(-2 + m)*x^3))/(e^2 + 4*e*f*x^3 + 4*f^2*x^6 - 4*d*f*x^(2 + 2*m)), x, 1, arctanh((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))],


[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctan((2*sqrt(d)*sqrt(f)*(1 + m)*(1 + m - n)*x^(1 + m))/(e*(1 + m)*(1 + m - n) + 2*f*(1 + m)*(1 + m - n)*x^n))/(2*sqrt(d)*sqrt(f))],
[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctanh((2*sqrt(d)*sqrt(f)*(1 + m)*(1 + m - n)*x^(1 + m))/(e*(1 + m)*(1 + m - n) + 2*f*(1 + m)*(1 + m - n)*x^n))/(2*sqrt(d)*sqrt(f))],


# ::Subsection::Closed:: 
#Problems from Calculus textbooks


# ::Subsubsection::Closed:: 
#Anton Calculus, 4th Edition


[(1 - 5*x^2)/(x^3*(1 + x^2)), x, 5, -(1/(2*x^2)) - 6*log(x) + 3*log(1 + x^2)],
[(2*x)/((-1 + x)*(5 + x^2)), x, 7, (1/3)*sqrt(5)*arctan(x/sqrt(5)) + (1/3)*log(1 - x) - (1/6)*log(5 + x^2)],
[(-x + 4*x^3)/(5 + x^2)^2, x, 6, 21/(2*(5 + x^2)) + 2*log(5 + x^2)],
[(1 - 3*x^4)/((-2 + x)*(1 + x^2)^2), x, 9, -(1/(5*(1 + x^2))) + (2*x)/(5*(1 + x^2)) - (46*arctan(x))/25 - (47/25)*log(2 - x) - (14/25)*log(1 + x^2)],
[(-9 - 9*x + 2*x^2)/(-9*x + x^3), x, 5, -2*arctanh(1 - (2*x)/3) + 2*log(3 + x)],
[(2 + x^2)/(2 + x), x, 4, -2*x + x^2/2 + 6*log(2 + x)],
[(1 + 2*x^2 + x^5)/(-x + x^3), x, 5, x + x^3/3 + 2*arctanh(1 + 2*x) + 2*log(1 - x)],
[(3 + 2*x^2)/((-1 + x)^2*x), x, 5, 5/(1 - x) - log(1 - x) + 3*log(x)],
[(-1 + 2*x^2)/((-1 + 4*x)*(1 + x^2)), x, 6, (3*arctan(x))/17 - (7/34)*log(1 - 4*x) + (6/17)*log(1 + x^2)],
[(9 + x + 3*x^2 + x^3)/((1 + x^2)*(3 + x^2)), x, 5, 3*arctan(x) + (1/2)*log(3 + x^2)],
[(2 + x + x^2 + x^3)/((1 + x^2)*(2 + x^2)), x, 5, arctan(x) + log(2 + x^2)/2],
[(-3 + 2*x - 3*x^2 + x^3)/(1 + x^2), x, 5, -3*x + x^2/2 + (1/2)*log(1 + x^2)],
[(x + 10*x^2 + 6*x^3 + x^4)/(10 + 6*x + x^2), x, 5, x^3/3 - 3*arctan(3 + x) + log(10 + 6*x + x^2)/2],
[(4 + 4*x + 4*x^2 + 4*x^3 + x^4 + x^5)/(2 + x^2)^3, x, 6, arctan(x/sqrt(2))/sqrt(2) + (1/2)*log(2 + x^2)],
[1/(-18 + 27*x - 7*x^2 - 3*x^3 + x^4), x, 6, (1/8)*log(1 - x) - (1/5)*log(2 - x) + (1/12)*log(3 - x) - (1/120)*log(3 + x)],
[1/(-1 + 4*x - 4*x^2 + 16*x^3), x, 6, (-(1/10))*arctan(2*x) + (1/5)*log(1 - 4*x) - (1/10)*log(1 + 4*x^2)],
[(1 + x^3)/(-2 + x), x, 4, 4*x + x^2 + x^3/3 + 9*log(2 - x)],
[(-3 + x)/(-1 + x^3), x, 5, (4*arctan((1 + 2*x)/sqrt(3)))/sqrt(3) - (2/3)*log(1 - x) + (1/3)*log(1 + x + x^2)],
[1/((-3 + x)*(4 + x^2)), x, 6, (-(3/26))*arctan(x/2) + (1/13)*log(3 - x) - (1/26)*log(4 + x^2)],
[(-2 + 3*x^6)/(x*(5 + 2*x^6)), x, 5, (-2*log(x))/5 + (19*log(5 + 2*x^6))/60],


# ::Subsubsection::Closed:: 
#Apostol Calculus, Volume 1, 2nd Edition, Section 6.25


[(3 + 2*x)/((-2 + x)*(5 + x)), x, 4, log(2 - x) + log(5 + x)],
[x/((1 + x)*(2 + x)*(3 + x)), x, 5, (-(1/2))*log(1 + x) + 2*log(2 + x) - (3/2)*log(3 + x)],
[x/(2 - 3*x + x^3), x, 5, 1/(3*(1 - x)) + (2/9)*log(1 - x) - (2/9)*log(2 + x)],
[(-6 + 2*x + x^4)/(-2*x + x^2 + x^3), x, 5, -x + x^2/2 - log(1 - x) + 3*log(x) + log(2 + x)],
[(7 + 8*x^3)/((1 + x)*(1 + 2*x)^3), x, 5, -(3/(1 + 2*x)^2) + 3/(1 + 2*x) + log(1 + x)],
[(1 + x + 4*x^2)/(-1 + x^3), x, 4, 2*log(1 - x) + log(1 + x + x^2)],
[x^4/(4 + 5*x^2 + x^4), x, 4, x - (8/3)*arctan(x/2) + arctan(x)/3],
[1/((1 + x)*(2 + x)^2*(3 + x)^3), x, 8, 1/(2 + x) + 1/(4*(3 + x)^2) + 5/(4*(3 + x)) + (1/8)*log(1 + x) + 2*log(2 + x) - (17/8)*log(3 + x)],
[x/(-1 + x^2), x, 2, (1/2)*log(1 - x^2)],
[(-3 + x)/(2*x + 3*x^2 + x^3), x, 5, -((3*log(x))/2) + 4*log(1 + x) - (5/2)*log(2 + x)],
[(-1 + x^2)^(-2), x, 2, x/(2*(1 - x^2)) + arctanh(x)/2],
[(1 + x)/(-1 + x^3), x, 4, (2/3)*log(1 - x) - (1/3)*log(1 + x + x^2)],
[(1 + x^4)/(x*(1 + x^2)^2), x, 5, (1 + x^2)^(-1) + log(x)],
[(1 - x^3)/(x*(1 + x^2)), x, 6, -x + arctan(x) + log(x) - log(1 + x^2)/2],
[(-1 + 4*x^5)/(1 + x + x^5)^2, x, 1, -(x/(1 + x + x^5))],

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


# ::Subsubsection::Closed:: 
#Ayres Calculus, 1964 edition


[x^2*(4 - x^2)^2, x, 2, (16*x^3)/3 - (8*x^5)/5 + x^7/7],
[x*(1 - x^3)^2, x, 2, x^2/2 - (2*x^5)/5 + x^8/8],


[(-4 + 5*x^2 + x^3)/x^2, x, 2, 4/x + 5*x + x^2/2],
[(3*x - 4*x^2 + 3*x^3)/(1 + x^2), x, 4, -4*x + (3*x^2)/2 + 4*arctan(x)],
[(-x^2 + 2*x^4)/(1 + 2*x^2), x, 4, -x + x^3/3 + arctan(sqrt(2)*x)/sqrt(2)],
[(5 + 3*x)/(1 - x - x^2 + x^3), x, 4, 4/(1 - x) + arctanh(x)],
[(-1 - x - x^3 + x^4)/(-x^2 + x^3), x, 5, -(1/x) + x^2/2 - 4*arctanh(1 - 2*x)],
[(2 + x + x^2 + x^3)/(2 + 3*x^2 + x^4), x, 5, arctan(x) + log(2 + x^2)/2],
[(-4 + 8*x - 4*x^2 + 4*x^3 - x^4 + x^5)/(2 + x^2)^3, x, 7, -(1/(2 + x^2)^2) - arctan(x/sqrt(2))/sqrt(2) + (1/2)*log(2 + x^2)],
[(-1 + x)/(3 - 4*x + 3*x^2), x, 2, arctan((2 - 3*x)/sqrt(5))/(3*sqrt(5)) + (1/6)*log(3 - 4*x + 3*x^2)],
[(-1 - 3*x + x^2)/(-2*x + x^2 + x^3), x, 5, -log(1 - x) + log(x)/2 + (3/2)*log(2 + x)],
[(3 + x + x^2 + x^3)/((1 + x^2)*(3 + x^2)), x, 5, arctan(x) + log(3 + x^2)/2],
[(3 - x + 3*x^2 - 2*x^3 + x^4)/(3*x - 2*x^2 + x^3), x, 4, x^2/2 + log(x) - (1/2)*log(3 - 2*x + x^2)],
[(-1 + x + x^3)/(1 + x^2)^2, x, 6, -(x/(2*(1 + x^2))) - arctan(x)/2 + (1/2)*log(1 + x^2)],
[(1 + 2*x - x^2 + 8*x^3 + x^4)/((x + x^2)*(1 + x^3)), x, 7, -(3/(1 + x)) - (2*arctan((1 - 2*x)/sqrt(3)))/sqrt(3) + log(x) - 2*log(1 + x) + log(1 - x + x^2)],
[(15 - 5*x + x^2 + x^3)/((5 + x^2)*(3 + 2*x + x^2)), x, 5, (-sqrt(5))*arctan(x/sqrt(5)) + (5*arctan((1 + x)/sqrt(2)))/sqrt(2) + (1/2)*log(3 + 2*x + x^2)],
[(-3 + 25*x + 23*x^2 + 32*x^3 + 15*x^4 + 7*x^5 + x^6)/((1 + x^2)^2*(2 + x + x^2)^2), x, 8, -3/(1 + x^2) + (2 + x + x^2)^(-1) + log(1 + x^2) - log(2 + x + x^2)],


# ::Subsubsection::Closed:: 
#Edwards and Penney Calculus


[(2 + x^3)^2, x, 2, 4*x + x^4 + x^7/7],


[(-4 + x^2)/(2 + x), x, 2, -2*x + x^2/2],
[(x + 2*x^3)/(x^2 + x^4)^3, x, 2, -(1/(4*x^4*(1 + x^2)^2))],
[(2 - x^2)/(1 - 6*x + x^3)^5, x, 2, 1/(12*(1 - 6*x + x^3)^4)],
[1/((1 + x)*(1 + x^2)), x, 6, arctan(x)/2 + (1/2)*log(1 + x) - (1/4)*log(1 + x^2)],
[1/((1 + x^2)*(4 + x^2)), x, 4, (-(1/6))*arctan(x/2) + arctan(x)/3],
[(a + b*x^3)/(1 + x^2), x, 6, (b*x^2)/2 + a*arctan(x) - (1/2)*b*log(1 + x^2)],
[(2*x + x^2)/(1 + x)^2, x, 3, x + (1 + x)^(-1)],
[(x + x^2)/((4 + x)*(-4 + x^2)), x, 4, (-(1/2))*arctanh(x/2) + log(4 + x)],
[x/((1 + x)*(1 + x^2)), x, 6, arctan(x)/2 - (1/2)*log(1 + x) + (1/4)*log(1 + x^2)],
[(-10 + x^2)/(4 + 9*x^2 + 2*x^4), x, 3, arctan(x/2) - (3*arctan(sqrt(2)*x))/sqrt(2)],
[(4 + x^2)/((1 + x^2)*(2 + x^2)), x, 4, 3*arctan(x) - sqrt(2)*arctan(x/sqrt(2))],
[(5 - 4*x + 3*x^2 + x^4)/((-1 + x)^2*(1 + x^2)), x, 7, 5/(2*(1 - x)) + x + 2*arctan(x) + (1/2)*log(1 - x) + (3/4)*log(1 + x^2)],
[(2*x^2 + x^4)/(-1 + x^3), x, 6, x^2/2 + arctan((1 + 2*x)/sqrt(3))/sqrt(3) + log(1 - x) + (1/2)*log(1 + x + x^2)],
[(31 + 5*x)/(11 - 4*x + 3*x^2), x, 2, -((103*arctan((2 - 3*x)/sqrt(29)))/(3*sqrt(29))) + (5/6)*log(11 - 4*x + 3*x^2)],
[(1 + x^4)/(2 + x^2), x, 4, -2*x + x^3/3 + (5*arctan(x/sqrt(2)))/sqrt(2)],
[(1 - x + 4*x^3)/(1 + x^3), x, 5, 4*x + (4*arctan((1 - 2*x)/sqrt(3)))/sqrt(3) - (2/3)*log(1 + x) + (1/3)*log(1 - x + x^2)],
[(2 + 2*x + x^4)/(x^4 + x^5), x, 4, -2/(3*x^3) + log(1 + x)],
[(-1 - 5*x + 2*x^2)/(2 - x - 2*x^2 + x^3), x, 5, 2*log(1 - x) - log(2 - x) + log(1 + x)],
[(4 + 3*x^2 + 2*x^3)/(1 + x)^4, x, 5, -5/(3*(1 + x)^3) + 3/(1 + x) + 2*log(1 + x)],
[(2 + x + x^3)/(1 + 2*x^2 + x^4), x, 6, x/(1 + x^2) + arctan(x) + log(1 + x^2)/2],
[(1 + 2*x + x^2 + x^3)/(1 + 2*x^2 + x^4), x, 7, -(1/(2*(1 + x^2))) + arctan(x) + (1/2)*log(1 + x^2)],


[(3 + 4*x)/((1 + x^2)*(2 + x^2)), x, 8, 3*arctan(x) - (3*arctan(x/sqrt(2)))/sqrt(2) + 2*log(1 + x^2) - 2*log(2 + x^2)],
[(2 + x)/((1 + x^2)*(4 + x^2)), x, 8, (-(1/3))*arctan(x/2) + (2*arctan(x))/3 + (1/6)*log(1 + x^2) - (1/6)*log(4 + x^2)],


# ::Subsubsection::Closed:: 
#Grossman Calculus


[(-x + x^3)/(6 + 2*x), x, 4, 4*x - (3*x^2)/4 + x^3/6 - 12*log(3 + x)],
[(7 - 2*x + 3*x^2 - x^3 + x^4)/(2 + x), x, 4, -20*x + (9*x^2)/2 - x^3 + x^4/4 + 47*log(2 + x)],
[(3 + x^2)/(-3 + x^2), x, 3, x - 2*sqrt(3)*arctanh(x/sqrt(3))],
[(-1 + x^2)/(1 + x^2), x, 3, x - 2*arctan(x)],
[(2 - x + x^3)/(-7 - 6*x + x^2), x, 5, 6*x + x^2/2 + (169/4)*log(7 - x) - (1/4)*log(1 + x)],
[(-1 + x^5)/(-1 + x^2), x, 5, x^2/2 + x^4/4 + log(1 + x)],
[(5 + 2*x - x^2 + x^3)/(1 + x + x^2), x, 5, -2*x + x^2/2 + (11*arctan((1 + 2*x)/sqrt(3)))/sqrt(3) + (3/2)*log(1 + x + x^2)],
[(-3 + x - 2*x^3 + x^4)/(10 - 8*x + 2*x^2), x, 5, (3*x)/2 + x^2/2 + x^3/6 + 6*arctan(2 - x) + (3/4)*log(5 - 4*x + x^2)],
[(1 + 2*x + 3*x^2 + x^3)/((-3 + x)*(-2 + x)*(-1 + x)), x, 5, x + (7/2)*log(1 - x) - 25*log(2 - x) + (61/2)*log(3 - x)],
[(-2 + x^2 + x^3)/x^4, x, 2, 2/(3*x^3) - x^(-1) + log(x)],
[(2 - 7*x + x^2 - x^3 + x^4)/(-24 - 14*x + x^2 + x^3), x, 6, -2*x + x^2/2 + (13/3)*log(4 - x) - (22/3)*log(2 + x) + 20*log(3 + x)],
[(2 + x^2)/((-1 + x)^2*x*(1 + x)), x, 6, 3/(2*(1 - x)) - (5/4)*log(1 - x) + 2*log(x) - (3/4)*log(1 + x)],
[1/((2 + x)*(1 + x^2)), x, 6, (2*arctan(x))/5 + (1/5)*log(2 + x) - (1/10)*log(1 + x^2)],
[(3 + x^2 + x^3)/(2 + x^2)^2, x, 8, 1/(2 + x^2) + x/(4*(2 + x^2)) + (5*arctan(x/sqrt(2)))/(4*sqrt(2)) + (1/2)*log(2 + x^2)],
[x/((1 + x)^2*(1 + x^2)), x, 4, 1/(2*(1 + x)) + arctan(x)/2],
[(-35 + 70*x - 4*x^2 + 2*x^3)/((26 - 10*x + x^2)*(17 - 2*x + x^2)), x, 6, -((15033*arctan(5 - x))/1025) + (4607*arctan(1/4 - x/4))/4100 + (1003*log(26 - 10*x + x^2))/1025 + (22*log(17 - 2*x + x^2))/1025],
[x^2/((-1 + x)^2*(1 + x)^2), x, 3, x/(2*(1 - x^2)) - arctanh(x)/2],
[(2 + x^2)/((-5 + x)*(-3 + x)*(4 + x)), x, 5, (-(11/14))*log(3 - x) + (3/2)*log(5 - x) + (2/7)*log(4 + x)],
[(-3 + x^2)/(-1 + x^3), x, 5, sqrt(3)*arctan((1 + 2*x)/sqrt(3)) - (2*log(1 - x))/3 + (5*log(1 + x + x^2))/6],
[x^4/((-1 + x)*(2 + x^2)), x, 7, x + x^2/2 - (2/3)*sqrt(2)*arctan(x/sqrt(2)) + (1/3)*log(1 - x) - (2/3)*log(2 + x^2)],


# ::Subsubsection::Closed:: 
#Hughes, Hallet, Gleason, et al Calculus, 2nd Edition


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


# ::Subsubsection::Closed:: 
#Spivak Calculus


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


# ::Subsubsection::Closed:: 
#Stewart Calculus


[(x + x^3)/(-1 + x), x, 4, 2*x + x^2/2 + x^3/3 + 2*log(1 - x)],
[(-1 + 2*x + x^2)/(-2*x + 3*x^2 + 2*x^3), x, 5, (1/10)*log(1 - 2*x) + log(x)/2 - (1/10)*log(2 + x)],
[(1 + 4*x - 2*x^2 + x^4)/(1 - x - x^2 + x^3), x, 5, 2/(1 - x) + x + x^2/2 - 2*arctanh(x)],
[(4 - x + 2*x^2)/(4*x + x^3), x, 6, (-(1/2))*arctan(x/2) + log(x) + (1/2)*log(4 + x^2)],
[(1 + x^2 + x^3)/((-1 + x)*x*(1 + x^2)^3*(1 + x + x^2)), x, 16, 1/(8*(1 + x^2)^2) + x/(8*(1 + x^2)^2) - 3/(8*(1 + x^2)) + (9*x)/(16*(1 + x^2)) + (7*arctan(x))/16 - arctan((1 + 2*x)/sqrt(3))/sqrt(3) + (1/8)*log(1 - x) - log(x) + (15/16)*log(1 + x^2) - (1/2)*log(1 + x + x^2)],
[(1 - 3*x + 2*x^2 - x^3)/(1 + x^2)^2, x, 8, 1/(1 + x^2) - x/(2*(1 + x^2)) + (3*arctan(x))/2 - (1/2)*log(1 + x^2)],
[(1 - 3*x + 2*x^2 - x^3)/(x*(1 + x^2)^2), x, 9, -(1/(2*(1 + x^2))) - x/(1 + x^2) - 2*arctan(x) + log(x) - (1/2)*log(1 + x^2)],
[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3), x, 6, -(12/(1375*(3 + 5*x)^2)) + 201/(15125*(3 + 5*x)) + (20*log(6 - x))/3993 + (1493*log(3 + 5*x))/499125],
[(1 - x - x^2 + x^3 + x^4)/(-x + x^3), x, 5, x + x^2/2 + (1/2)*log(1 - x) - log(x) + (1/2)*log(1 + x)],
[(-2 + x^2)/(x*(2 + x^2)), x, 5, -log(x) + log(2 + x^2)],
[(2 - 4*x^2 + x^3)/((1 + x^2)*(2 + x^2)), x, 8, 6*arctan(x) - 5*sqrt(2)*arctan(x/sqrt(2)) - log(1 + x^2)/2 + log(2 + x^2)],
[(1 + x^2 + x^4)/((1 + x^2)*(4 + x^2)^2), x, 6, -((13*x)/(24*(4 + x^2))) + (25/144)*arctan(x/2) + arctan(x)/9],
[(1 + x^2 + x^3)/(2*x^2 + x^3 + x^4), x, 5, -(1/(2*x)) + arctan((1 + 2*x)/sqrt(7))/(4*sqrt(7)) - log(x)/4 + (5/8)*log(2 + x + x^2)],
[(1 - 12*x + x^2 + x^3)/(-12 + x + x^2), x, 5, x^2/2 - (2/7)*arctanh((1/7)*(1 + 2*x)), x^2/2 + (1/7)*log(3 - x) - (1/7)*log(4 + x)],
[(-3 + 5*x + 6*x^2)/(-3*x + 2*x^2 + x^3), x, 5, 2*log(1 - x) + log(x) + 3*log(3 + x)],
[(-2 + 3*x + 5*x^2)/(2*x^2 + x^3), x, 4, x^(-1) + 2*log(x) + 3*log(2 + x)],
[(18 - 2*x - 4*x^2)/(-6 + x + 4*x^2 + x^3), x, 5, log(1 - x) - 2*log(2 + x) - 3*log(3 + x)],
[(2*x + x^2)/(4 + 3*x^2 + x^3), x, 2, log(4 + 3*x^2 + x^3)/3],
[(-x + 2*x^3)/(1 - x^2 + x^4), x, 2, log(1 - x^2 + x^4)/2],
[(-4 + 6*x - x^2 + 3*x^3)/((1 + x^2)*(2 + x^2)), x, 6, -3*arctan(x) + sqrt(2)*arctan(x/sqrt(2)) + (3*log(1 + x^2))/2],
[(1 + x - 2*x^2 + x^3)/(4 + 5*x^2 + x^4), x, 6, (-(3/2))*arctan(x/2) + arctan(x) + (1/2)*log(4 + x^2)],
[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 13*x^4 + 30*x^5), x, 7, (3988*arctan((1 + 2*x)/sqrt(19)))/(13685*sqrt(19)) - (3146*log(7 - 3*x))/80155 - (334/323)*log(1 + 2*x) + (4822*log(2 + 5*x))/4879 + (11049*log(5 + x + x^2))/260015],
[(8 - 13*x^2 - 7*x^3 + 12*x^5)/(4 - 20*x + 41*x^2 - 80*x^3 + 116*x^4 - 80*x^5 + 100*x^6), x, 10, 5828/(9075*(2 - 5*x)) - 313/(1452*(1 + 2*x^2)) - (251*x)/(726*(1 + 2*x^2)) + (503*arctan(sqrt(2)*x))/(7986*sqrt(2)) - (59096*log(2 - 5*x))/99825 + (2843*log(1 + 2*x^2))/7986],
[(1 + x + x^3)/(4*x + 2*x^2 + x^4), x, 2, (1/4)*log(4*x + 2*x^2 + x^4)],
[1/(1 + x + x^2 + x^3), x, 6, arctan(x)/2 + (1/2)*log(1 + x) - (1/4)*log(1 + x^2)],
[x/((1 + x^2)*(4 + x^2)), x, 2, (-(1/3))*arctanh(5/3 + (2*x^2)/3)],


# ::Subsubsection::Closed:: 
#Thomas Calculus, 8th Edition


[(-3 + x)*(-7 + 4*x^2), x, 2, 21*x - (7*x^2)/2 - 4*x^3 + x^4],
[(-2 + 7*x)^3, x, 1, (1/28)*(2 - 7*x)^4],


[(-1 + x^3)/(-1 + x), x, 2, x + x^2/2 + x^3/3],
[(-7 + 4*x^2)/(3 + 2*x), x, 4, -3*x + x^2 + log(3 + 2*x)],
[(1 + x)/((-1 + x)*x^2), x, 4, 1/x + 4*arctanh(1 - 2*x)],
[(9 + x^4)/(x^2*(9 + x^2)), x, 4, -x^(-1) + x - (10*arctan(x/3))/3],
[(4*x^2 + x^3)/(x + x^3), x, 6, x - arctan(x) + 2*log(1 + x^2)],
[(2*x + x^4)/(1 + x^2), x, 6, -x + x^3/3 + arctan(x) + log(1 + x^2)],
[(4 + 4*x)/(x^2*(1 + x^2)), x, 6, -4/x - 4*arctan(x) + 4*log(x) - 2*log(1 + x^2)],
[(-x + x^3)/((-1 + x)^2*(1 + x^2)), x, 5, arctan(x) + log(1 - x)],
[(2 + 5*x + 3*x^2 + 2*x^3)/(1 + x + x^2), x, 4, x + x^2 + log(1 + x + x^2)],
[(2 + 2*x)/((-1 + x)^3*(1 + x^2)), x, 5, -(1/(1 - x)^2) - 1/(1 - x) + arctan(x)],
[(3 - 4*x - 5*x^2 + 3*x^3)/(x^3*(-1 + x + x^2)), x, 5, 3/(2*x^2) - 1/x - arctanh((1 + 2*x)/sqrt(5))/sqrt(5) + 3*log(x) - (3/2)*log(1 - x - x^2)],
[(-4 + x^2)/(2 - 5*x + x^3), x, 3, -(arctanh((1 + x)/sqrt(2))/sqrt(2)) + (1/2)*log(1 - 2*x - x^2)],
[(2 + x)/(-1 + 2*x + x^2), x, 2, -(arctanh((1 + x)/sqrt(2))/sqrt(2)) + (1/2)*log(1 - 2*x - x^2)],
[(4 + 8*x + 5*x^2 + 2*x^3)/(2 + 2*x + x^2)^2, x, 5, -(1/(2 + 2*x + x^2)) - arctan(1 + x) + log(2 + 2*x + x^2)],
[(1 - 3*x + 2*x^2 - 4*x^3 + x^4)/(1 + 3*x^2 + 3*x^4 + x^6), x, 7, -(1/(4*(1 + x^2)^2)) + 2/(1 + x^2) + arctan(x)],
[((-1 + x)^4*x^4)/(1 + x^2), x, 4, 4*x - (4*x^3)/3 + x^5 - (2*x^6)/3 + x^7/7 - 4*arctan(x)],
[(-20*x + 4*x^2)/(9 - 10*x^2 + x^4), x, 6, log(1 - x) - (1/2)*log(3 - x) + (3/2)*log(1 + x) - 2*log(3 + x)],
[(24 + 8*x)/(x*(-4 + x^2)), x, 5, 5*log(2 - x) - 6*log(x) + log(2 + x)],
[(-1 + x + 4*x^3)/((-1 + x)*x^2*(1 + x^2)), x, 7, -x^(-1) + arctan(x) + 2*log(1 - x) - log(1 + x^2)],


# ::Subsection::Closed:: 
#Problems from integration competitions


# ::Subsubsection::Closed:: 
#MIT Integration Competition


[(-1 + 3*x - 3*x^2 + x^3)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4), x, 7, 8/(3*(1 + x)^3) - 6/(1 + x)^2 + 6/(1 + x) + log(1 + x)],
[1/(4*x^2 + 4*x^3 + x^4), x, 6, -(1/(4*x)) - 1/(4*(2 + x)) + (1/2)*arctanh(1 + x)],
[(1 + x^2)/(1 + x), x, 4, -x + x^2/2 + 2*log(1 + x)],


# ::Subsubsection::Closed:: 
#North Texas University Integration Competition


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


# ::Subsubsection::Closed:: 
#University of Wisconsin Integration Competition


[((3 - sqrt(37))/2 + x)*((3 + sqrt(37))/2 + x), x, 2, -7*x + (3*x^2)/2 + x^3/3],
[(3*x + 3*x^2 + x^3)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4), x, 5, 1/(3*(1 + x)^3) + log(1 + x)],


# ::Subsection::Closed:: 
#Miscellaneous problems


[(2 + x)/(x + x^2), x, 4, 2*log(x) - log(1 + x)],
[(x^3 + x^4)/(1 + x^2), x, 6, -x + x^2/2 + x^3/3 + arctan(x) - (1/2)*log(1 + x^2)],
[(4 + x^2)/(2 + x), x, 4, -2*x + x^2/2 + 8*log(2 + x)],
[(-32 + 36*x - 42*x^2 + 21*x^3 - 10*x^4 + 3*x^5)/(x*(1 + x^2)*(4 + x^2)^2), x, 9, (4 + x^2)^(-1) + arctan(x/2)/2 + 2*arctan(x) - 2*log(x) + log(4 + x^2)],


[(-1 + x^4 + 7*x^5 + x^9)/(-7 + 6*x^4 + x^8), x, 12, x^2/2 - arctan((7^(1/4) - sqrt(2)*x)/7^(1/4))/(2*sqrt(2)*7^(3/4)) + arctan((7^(1/4) + sqrt(2)*x)/7^(1/4))/(2*sqrt(2)*7^(3/4)) + (1/4)*log(1 - x) + (1/4)*log(1 + x) - (1/4)*log(1 + x^2) - log(sqrt(7) - sqrt(2)*7^(1/4)*x + x^2)/(4*sqrt(2)*7^(3/4)) + log(sqrt(7) + sqrt(2)*7^(1/4)*x + x^2)/(4*sqrt(2)*7^(3/4))],
[(1 + x^3 + x^6)/(x + x^5), x, 12, x^2/2 - arctan(x^2)/2 - arctan(1 - sqrt(2)*x)/(2*sqrt(2)) + arctan(1 + sqrt(2)*x)/(2*sqrt(2)) + log(x) + log(1 - sqrt(2)*x + x^2)/(4*sqrt(2)) - log(1 + sqrt(2)*x + x^2)/(4*sqrt(2)) - (1/4)*log(1 + x^4)],
[1/(-7 + sqrt(5) - 4*x^2), x, 1, -(arctan((2*x)/sqrt(7 - sqrt(5)))/(2*sqrt(7 - sqrt(5))))],
[(11*x + 2*x^3)/(3 + 2*x^2 + x^4)^2, x, 5, 5/(8*(3 + 2*x^2 + x^4)) + (9*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*arctan((1 + x^2)/sqrt(2)))/(8*sqrt(2))],
[(-x + 2*x^3 + 4*x^5)/(3 + 2*x^2 + x^4)^2, x, 6, (5 - 7*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*arctan((1 + x^2)/sqrt(2)))/(8*sqrt(2)), 5/(8*(3 + 2*x^2 + x^4)) - (7*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*arctan((1 + x^2)/sqrt(2)))/(8*sqrt(2))],
[(x + x^5)/(1 + 2*x^2 + 2*x^4)^3, x, 7, (3 + 4*x^2)/(16*(1 + 2*x^2 + 2*x^4)^2) + (1 + 2*x^2)/(2 + 4*x^2 + 4*x^4) + arctan(1 + 2*x^2), 3/(16*(1 + 2*x^2 + 2*x^4)^2) + x^2/(4*(1 + 2*x^2 + 2*x^4)^2) + (1 + 2*x^2)/(2*(1 + 2*x^2 + 2*x^4)) + arctan(1 + 2*x^2)],
[1/(1 + sqrt(5) - x^2 + sqrt(5)*x^2), x, 2, (-(1/2))*arctan((1/2)*(1 - sqrt(5))*x)],
[x^11/(2 + 3*x^4 + x^8), x, 6, x^4/4 + (1/4)*log(1 + x^4) - log(2 + x^4)],
[(-11 + 6*x)/((-1 + 2*x)*(-1 + x^2)), x, 5, (16/3)*log(1 - 2*x) - (5/2)*log(1 - x) - (17/6)*log(1 + x)],
[(a + b*x)^3/((c + d*x)*(e + f*x)), x, 5, -((b^2*(b*d*e + b*c*f - 3*a*d*f)*x)/(d^2*f^2)) + (b^3*x^2)/(2*d*f) - ((b*c - a*d)^3*log(c + d*x))/(d^3*(d*e - c*f)) + ((b*e - a*f)^3*log(e + f*x))/(f^3*(d*e - c*f))],


# Note: This test problem formerly caused stack overflow because the degree of the polynomial        was not properly reduced by the Ostrogradskiy-Hermite method code. 
# {(a + 2*b*x - a*x^2)^4/(-1 + x^2)^5, x, 14, -((4*a*b*(3*a^2 - 2*b^2))/(3*(1 - x^2)^4)) + (11*a^4*x)/(5*(1 - x^2)^4) - (48*a^2*b^2*x)/(5*(1 - x^2)^4) + (6*b^4*x)/(5*(1 - x^2)^4) + (4*a*b*(9*a^2 - 8*b^2)*x^2)/(3*(1 - x^2)^4) - ((73*a^4 - 264*a^2*b^2 + 48*b^4)*x^3)/(15*(1 - x^2)^4) - (4*a*b*(3*a^2 - 2*b^2)*x^4)/(1 - x^2)^4 + (a^2*(11*a^2 - 24*b^2)*x^5)/(3*(1 - x^2)^4) + (4*a^3*b*x^6)/(1 - x^2)^4 - (a^4*x^7)/(1 - x^2)^4 - ((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/(15*(1 - x^2)^3) - ((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/(12*(1 - x^2)^2) - ((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/(8*(1 - x^2)) - (1/8)*(8*a^4 - 24*a^2*b^2 + 3*b^4)*ArcTanh[x]} 


# Integrands of the form x^m/(a*x+b*x^2)^n where m and are integers 
[x^3/(a*x + b*x^2), x, 5, -((a*x)/b^2) + x^2/(2*b) + (a^2*log(a + b*x))/b^3],
[x^2/(a*x + b*x^2), x, 4, x/b - (a*log(a + b*x))/b^2],
[x/(a*x + b*x^2), x, 2, log(a + b*x)/b],
[1/(a*x + b*x^2), x, 2, log(x)/a - log(a + b*x)/a],
[1/(x*(a*x + b*x^2)), x, 5, -(1/(a*x)) - (b*log(x))/a^2 + (b*log(a + b*x))/a^2],
[1/(x^2*(a*x + b*x^2)), x, 5, -(1/(2*a*x^2)) + b/(a^2*x) + (b^2*log(x))/a^3 - (b^2*log(a + b*x))/a^3],
[1/(x^3*(a*x + b*x^2)), x, 5, -(1/(3*a*x^3)) + b/(2*a^2*x^2) - b^2/(a^3*x) - (b^3*log(x))/a^4 + (b^3*log(a + b*x))/a^4],

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


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

[1/(x  + x^2)^2, x, 6, -(1/x) - 1/(1 + x) + 4*arctanh(1 + 2*x)],
[1/(x  + x^2)^3, x, 7, -(1/(2*x^2)) + 3/x + 1/(2*(1 + x)^2) + 3/(1 + x) - 12*arctanh(1 + 2*x)],


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

# In some of the following examples gcd cancellation should occur without also partial fraction        expansion, since that will result in unnecessary expansion. 
[x^5/(x - x^3), x, 5, -x - x^3/3 + arctanh(x)],
[x^4/(x - x^3), x, 5, -(x^2/2) - (1/2)*log(1 - x^2)],
[x^3/(x - x^3), x, 4, -x + arctanh(x)],
[x^2/(x - x^3), x, 3, (-(1/2))*log(1 - x^2)],
[x/(x - x^3), x, 2, arctanh(x)],
[1/(x - x^3), x, 2, -arctanh(1 - 2*x^2)],
[1/(x*(x - x^3)), x, 5, -(1/x) + arctanh(x)],
[1/(x^2*(x - x^3)), x, 6, -(1/(2*x^2)) + log(x) - (1/2)*log(1 - x^2)],
[1/(x^3*(x - x^3)), x, 5, -(1/(3*x^3)) - 1/x + arctanh(x)],
[1/(x^4*(x - x^3)), x, 6, -(1/(4*x^4)) - 1/(2*x^2) + log(x) - (1/2)*log(1 - x^2)],


# Integrands of the form x^m/(a*x^2+b*x^4) where m is an integer 
[x^4/(a*x^2 + b*x^4), x, 4, x/b - (sqrt(a)*arctan((sqrt(b)*x)/sqrt(a)))/b^(3/2)],
[x^3/(a*x^2 + b*x^4), x, 3, log(a + b*x^2)/(2*b)],
[x^2/(a*x^2 + b*x^4), x, 2, arctan((sqrt(b)*x)/sqrt(a))/(sqrt(a)*sqrt(b))],
[x/(a*x^2 + b*x^4), x, 2, log(x)/a - log(a + b*x^2)/(2*a)],
[1/(a*x^2 + b*x^4), x, 5, -(1/(a*x)) - (sqrt(b)*arctan((sqrt(b)*x)/sqrt(a)))/a^(3/2)],
[1/(x*(a*x^2 + b*x^4)), x, 6, -(1/(2*a*x^2)) - (b*log(x))/a^2 + (b*log(a + b*x^2))/(2*a^2)],
[1/(x^2*(a*x^2 + b*x^4)), x, 5, -(1/(3*a*x^3)) + b/(a^2*x) + (b^(3/2)*arctan((sqrt(b)*x)/sqrt(a)))/a^(5/2)],
[1/(x^3*(a*x^2 + b*x^4)), x, 6, -(1/(4*a*x^4)) + b/(2*a^2*x^2) + (b^2*log(x))/a^3 - (b^2*log(a + b*x^2))/(2*a^3)],
[1/(x^4*(a*x^2 + b*x^4)), x, 5, -(1/(5*a*x^5)) + b/(3*a^2*x^3) - b^2/(a^3*x) - (b^(5/2)*arctan((sqrt(b)*x)/sqrt(a)))/a^(7/2)],


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


# Integrands of the form 1/(a*x^m + b*x^n) where m and n are integers 
[1/(a*x + b*x), x, 3, log(x)/(a + b)],
[1/(a*x^2 + b*x^2), x, 3, -(1/((a + b)*x))],

[1/(-x^3 + x^6), x, 8, 1/(2*x^2) - arctan((1 + 2*x)/sqrt(3))/sqrt(3) + (1/3)*log(1 - x) - (1/6)*log(1 + x + x^2)],


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


# Integrands of the form (a+b*x^m)/(c*x^n+d*x^p) where m, n and p are integers 
[(1 + x)/(2*x + x^2), x, 2, (1/2)*log(2*x + x^2)],
[(1 + x^2)/(3*x + x^3), x, 2, (1/3)*log(3*x + x^3)],
[(a + 2*b*x)/(a*x + b*x^2), x, 2, log(a*x+b*x^2)],
[(a + 3*b*x^2)/(a*x + b*x^3), x, 2, log(a*x+b*x^3)],

[(6 + 3*x)/(3*x + x^2), x, 4, 2*log(x) + log(3 + x)],
[(1 + x^2)/(-x + x^2), x, 4, x + 2*log(1 - x) - log(x)],

[(-2 + 4*x)/(-x + x^3), x, 5, log(1 - x) + 2*log(x) - 3*log(1 + x)],
[(4 + x)/(4*x + x^3), x, 6, (1/2)*arctan(x/2) + log(x) - (1/2)*log(4 + x^2)],
[(-1 + x^2)/(-2*x + x^3), x, 5, log(x)/2 + (1/4)*log(2 - x^2)],
[(1 + x^3)/(-x + x^3), x, 4, x + 2*arctanh(1 - 2*x)],
[(1 + x^3)/(-x^2 + x^3), x, 4, x^(-1) + x + 2*log(1 - x) - log(x)],
[(-1 + x^5)/(-x + x^3), x, 4, x + x^3/3 - 2*arctanh(1 + 2*x)],
[(-3 + 2*x)/(-x^2 + x^3), x, 4, -(3/x) - 2*arctanh(1 - 2*x)],
[(1 - 2*x^2)/(x^2 - x^4), x, 4, -1/x - arctanh(x)],

[(4 + 3*x^4)/(5*x + 2*x^5), x, 5, (4*log(x))/5 + (7*log(5 + 2*x^4))/40],
[(1 + x^4)/(x^3 + x^5), x, 5, -1/(2*x^2) - log(x) + log(1 + x^2)],
[(1 + x^6)/(x - x^7), x, 5, log(x) - log(1 - x^6)/3],
[(8 + 5*x^10)/(2*x - x^11), x, 5, 4*log(x) - (9/10)*log(2 - x^10)],
[(8 + 5*x^10)/(x - x^11), x, 5, 8*log(x) - (13*log(1 - x^10))/10],


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


# Integrands of the form (a+b*x^m)/(c*x^n+d*x^p+e*x^q) where m, n, p and q are integers 
[(x^2*(-2 + x^3))/(1 - x^3 + x^6), x, 3, arctan((1 - 2*x^3)/sqrt(3))/sqrt(3) + (1/6)*log(1 - x^3 + x^6)],
[(1 + x^3)/(x*(1 - x^3 + x^6)), x, 3, -(arctan((1 - 2*x^3)/sqrt(3))/sqrt(3)) + log(x) - (1/6)*log(1 - x^3 + x^6)],
[(1 + x^3)/(x - x^4 + x^7), x, 6, -(arctan((1 - 2*x^3)/sqrt(3))/sqrt(3)) + log(x) - (1/6)*log(1 - x^3 + x^6)],
[(1 + x^2)/(x + 2*x^2 + x^3), x, 4, 2/(1 + x) + log(x)],
[(1 + x)/(-6*x + x^2 + x^3), x, 5, (3/10)*log(2 - x) - log(x)/6 - (2/15)*log(3 + x)],
[(2 + 4*x)/(x^2 + 2*x^3 + x^4), x, 4, -(2/(x*(1 + x)))],
[(1 + x^5)/(-10*x - 3*x^2 + x^3), x, 5, 19*x + (3*x^2)/2 + x^3/3 + (3126*log(5 - x))/35 - log(x)/10 - (31*log(2 + x))/14],


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


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

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

[2/(-1 + x) + (1 + 2*x)/(1 + x + x^2), x, 3, 2*log(1 - x) + log(1 + x + x^2)],
[(1 + 4*x + 3*x^2)/(4 + 7*x + 2*x^2)^2, x, 2, -(1/(4 + 7*x + 2*x^2)) - (3*x)/(2*(4 + 7*x + 2*x^2))],

[(b*c - a*d - 2*a*e*x - b*e*x^2 - 3*a*f*x^2 - 2*b*f*x^3)/(c + d*x + e*x^2 + f*x^3)^2, x, 3, a/(c + d*x + e*x^2 + f*x^3) + (b*x)/(c + d*x + e*x^2 + f*x^3)],
[(9 - 40*x - 18*x^2 + 174*x^4 + 24*x^5 + 26*x^6 - 39*x^8)/(3 + 2*x^2 + x^4)^3, x, 5, 2/(3 + 2*x^2 + x^4)^2 + (3*x)/(3 + 2*x^2 + x^4)^2 - (4*x^2)/(3 + 2*x^2 + x^4)^2 + (13*x^5)/(3 + 2*x^2 + x^4)^2],
[(-3 + 10*x + 4*x^3 - 30*x^5)/(3 + x + x^4)^3 - (3*(1 + 4*x^3)*(2 - 3*x + 5*x^2 + x^4 - 5*x^6))/(3 + x + x^4)^4, x, -13, (2 - 3*x + 5*x^2 + x^4 - 5*x^6)/(3 + x + x^4)^3],
[(15 - 5*x + x^2 + x^3)/((5 + x^2)*(3 + 2*x + x^2)), x, 5, (-sqrt(5))*arctan(x/sqrt(5)) + (5*arctan((1 + x)/sqrt(2)))/sqrt(2) + (1/2)*log(3 + 2*x + x^2)],
[1/(1/x^2 + x^3), x, 10, (1/10)*sqrt(10 - 2*sqrt(5))*arctan((1 - sqrt(5) - 4*x)/sqrt(10 + 2*sqrt(5))) - (1/10)*sqrt(10 + 2*sqrt(5))*arctan((1 + sqrt(5) - 4*x)/sqrt(10 - 2*sqrt(5))) + (1/5)*log(1 + x) - (1/20)*(1 + sqrt(5))*log(2 - (1 - sqrt(5))*x + 2*x^2) - (1/20)*(1 - sqrt(5))*log(2 - (1 + sqrt(5))*x + 2*x^2)],

[1/((c + d*x)*(a^2 + 2*a*b*x + b^2*x^2)), x, 6, -(1/((b*c - a*d)*(a + b*x))) - (d*log(a + b*x))/(b*c - a*d)^2 + (d*log(c + d*x))/(b*c - a*d)^2],
[(2 - x^2)/(1 - 6*x + x^3)^5, x, 2, 1/(12*(1 - 6*x + x^3)^4)],

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


[(e - 2*f*(-1 + n)*x^n)/(e^2 + 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctan((2*sqrt(d)*sqrt(f)*(-1 + n)*x)/(e*(-1 + n) + 2*f*(-1 + n)*x^n))/(2*sqrt(d)*sqrt(f))],
[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctanh((2*sqrt(d)*sqrt(f)*(-1 + n)*x)/(e*(-1 + n) + 2*f*(-1 + n)*x^n))/(2*sqrt(d)*sqrt(f))],


[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctan((2*sqrt(d)*sqrt(f)*(1 + m)*(1 + m - n)*x^(1 + m))/(e*(1 + m)*(1 + m - n) + 2*f*(1 + m)*(1 + m - n)*x^n))/(2*sqrt(d)*sqrt(f))],
[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 1, arctanh((2*sqrt(d)*sqrt(f)*(1 + m)*(1 + m - n)*x^(1 + m))/(e*(1 + m)*(1 + m - n) + 2*f*(1 + m)*(1 + m - n)*x^n))/(2*sqrt(d)*sqrt(f))]
]:
