lst:=[
# ::Package:: 

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


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


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


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


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


# Integrands of the form x^m*(a+b*x^2)^3 where m is an integer 
[x^7*(a + b*x^2)^3, x, 2, (a^3*x^8)/8 + (3/10)*a^2*b*x^10 + (1/4)*a*b^2*x^12 + (b^3*x^14)/14],
[x^6*(a + b*x^2)^3, x, 2, (a^3*x^7)/7 + (1/3)*a^2*b*x^9 + (3/11)*a*b^2*x^11 + (b^3*x^13)/13],
[x^5*(a + b*x^2)^3, x, 3, (a^3*x^6)/6 + (3/8)*a^2*b*x^8 + (3/10)*a*b^2*x^10 + (b^3*x^12)/12],
[x^4*(a + b*x^2)^3, x, 2, (a^3*x^5)/5 + (3/7)*a^2*b*x^7 + (1/3)*a*b^2*x^9 + (b^3*x^11)/11],
[x^3*(a + b*x^2)^3, x, 3, -((a*(a + b*x^2)^4)/(40*b^2)) + (x^2*(a + b*x^2)^4)/(10*b)],
[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^2)^3, x, 2, (a + b*x^2)^4/(8*b)],
[(a + b*x^2)^3, x, 2, a^3*x + a^2*b*x^3 + (3*a*b^2*x^5)/5 + (b^3*x^7)/7],
[(a + b*x^2)^3/x, x, 2, (3/2)*a^2*b*x^2 + (3/4)*a*b^2*x^4 + (b^3*x^6)/6 + a^3*log(x)],
[(a + b*x^2)^3/x^2, x, 2, -(a^3/x) + 3*a^2*b*x + a*b^2*x^3 + (b^3*x^5)/5],
[(a + b*x^2)^3/x^3, x, 2, -(a^3/(2*x^2)) + (3/2)*a*b^2*x^2 + (b^3*x^4)/4 + 3*a^2*b*log(x)],
[(a + b*x^2)^3/x^4, x, 2, -(a^3/(3*x^3)) - (3*a^2*b)/x + 3*a*b^2*x + (b^3*x^3)/3],
[(a + b*x^2)^3/x^5, x, 2, -(a^3/(4*x^4)) - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*log(x)],
[(a + b*x^2)^3/x^6, x, 2, -(a^3/(5*x^5)) - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x],
[(a + b*x^2)^3/x^7, x, 2, -(a^3/(6*x^6)) - (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) + b^3*log(x)],
[(a + b*x^2)^3/x^8, x, 2, -(a^3/(7*x^7)) - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x],
[(a + b*x^2)^3/x^9, x, 1, -((a + b*x^2)^4/(8*a*x^8))],
[(a + b*x^2)^3/x^10, x, 2, -(a^3/(9*x^9)) - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)],
[(a + b*x^2)^3/x^11, x, 2, -(a^3/(10*x^10)) - (3*a^2*b)/(8*x^8) - (a*b^2)/(2*x^6) - b^3/(4*x^4)],
[(a + b*x^2)^3/x^12, x, 2, -(a^3/(11*x^11)) - (a^2*b)/(3*x^9) - (3*a*b^2)/(7*x^7) - b^3/(5*x^5)],
[(a + b*x^2)^3/x^13, x, 2, -(a^3/(12*x^12)) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)],


# Integrands of the form x^m*(a+b*x^2)^4 where m is an integer 
[x^7*(a + b*x^2)^4, x, 2, (a^4*x^8)/8 + (2/5)*a^3*b*x^10 + (1/2)*a^2*b^2*x^12 + (2/7)*a*b^3*x^14 + (b^4*x^16)/16],
[x^6*(a + b*x^2)^4, x, 2, (a^4*x^7)/7 + (4/9)*a^3*b*x^9 + (6/11)*a^2*b^2*x^11 + (4/13)*a*b^3*x^13 + (b^4*x^15)/15],
[x^5*(a + b*x^2)^4, x, 3, (a^4*x^6)/6 + (1/2)*a^3*b*x^8 + (3/5)*a^2*b^2*x^10 + (1/3)*a*b^3*x^12 + (b^4*x^14)/14],
[x^4*(a + b*x^2)^4, x, 2, (a^4*x^5)/5 + (4/7)*a^3*b*x^7 + (2/3)*a^2*b^2*x^9 + (4/11)*a*b^3*x^11 + (b^4*x^13)/13],
[x^3*(a + b*x^2)^4, x, 3, -((a*(a + b*x^2)^5)/(60*b^2)) + (x^2*(a + b*x^2)^5)/(12*b)],
[x^2*(a + b*x^2)^4, x, 2, (a^4*x^3)/3 + (4*a^3*b*x^5)/5 + (6*a^2*b^2*x^7)/7 + (4*a*b^3*x^9)/9 + (b^4*x^11)/11],
[x*(a + b*x^2)^4, x, 2, (a + b*x^2)^5/(10*b)],
[(a + b*x^2)^4, x, 2, a^4*x + (4*a^3*b*x^3)/3 + (6*a^2*b^2*x^5)/5 + (4*a*b^3*x^7)/7 + (b^4*x^9)/9],
[(a + b*x^2)^4/x, x, 2, 2*a^3*b*x^2 + (3*a^2*b^2*x^4)/2 + (2*a*b^3*x^6)/3 + (b^4*x^8)/8 + a^4*log(x)],
[(a + b*x^2)^4/x^2, x, 2, -(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7],
[(a + b*x^2)^4/x^3, x, 2, -a^4/(2*x^2) + 3*a^2*b^2*x^2 + a*b^3*x^4 + (b^4*x^6)/6 + 4*a^3*b*log(x)],


# Integrands of the form x^m*(a+b*x^2)^5 where m is an integer 
[x^9*(a + b*x^2)^5, x, 2, (a^5*x^10)/10 + (5*a^4*b*x^12)/12 + (5*a^3*b^2*x^14)/7 + (5*a^2*b^3*x^16)/8 + (5*a*b^4*x^18)/18 + (b^5*x^20)/20],
[x^8*(a + b*x^2)^5, x, 2, (a^5*x^9)/9 + (5*a^4*b*x^11)/11 + (10*a^3*b^2*x^13)/13 + (2*a^2*b^3*x^15)/3 + (5*a*b^4*x^17)/17 + (b^5*x^19)/19],
[x^7*(a + b*x^2)^5, x, 2, (a^5*x^8)/8 + (1/2)*a^4*b*x^10 + (5/6)*a^3*b^2*x^12 + (5/7)*a^2*b^3*x^14 + (5/16)*a*b^4*x^16 + (b^5*x^18)/18],
[x^6*(a + b*x^2)^5, x, 2, (a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3*b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a*b^4*x^15)/3 + (b^5*x^17)/17],
[x^5*(a + b*x^2)^5, x, 4, (a^2*(a + b*x^2)^6)/(336*b^3) - (a*x^2*(a + b*x^2)^6)/(56*b^2) + (x^4*(a + b*x^2)^6)/(16*b)],
[x^4*(a + b*x^2)^5, x, 2, (a^5*x^5)/5 + (5*a^4*b*x^7)/7 + (10*a^3*b^2*x^9)/9 + (10*a^2*b^3*x^11)/11 + (5*a*b^4*x^13)/13 + (b^5*x^15)/15],
[x^3*(a + b*x^2)^5, x, 3, -((a*(a + b*x^2)^6)/(84*b^2)) + (x^2*(a + b*x^2)^6)/(14*b)],
[x^2*(a + b*x^2)^5, x, 2, (a^5*x^3)/3 + a^4*b*x^5 + (10*a^3*b^2*x^7)/7 + (10*a^2*b^3*x^9)/9 + (5*a*b^4*x^11)/11 + (b^5*x^13)/13],
[x*(a + b*x^2)^5, x, 2, (a + b*x^2)^6/(12*b)],
[(a + b*x^2)^5, x, 2, a^5*x + (5*a^4*b*x^3)/3 + 2*a^3*b^2*x^5 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^9)/9 + (b^5*x^11)/11],
[(a + b*x^2)^5/x, x, 2, (5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*log(x)],
[(a + b*x^2)^5/x^2, x, 2, -(a^5/x) + 5*a^4*b*x + (10*a^3*b^2*x^3)/3 + 2*a^2*b^3*x^5 + (5*a*b^4*x^7)/7 + (b^5*x^9)/9],
[(a + b*x^2)^5/x^3, x, 2, -a^5/(2*x^2) + 5*a^3*b^2*x^2 + (5*a^2*b^3*x^4)/2 + (5*a*b^4*x^6)/6 + (b^5*x^8)/8 + 5*a^4*b*log(x)],
[(a + b*x^2)^5/x^4, x, 2, -a^5/(3*x^3) - (5*a^4*b)/x + 10*a^3*b^2*x + (10*a^2*b^3*x^3)/3 + a*b^4*x^5 + (b^5*x^7)/7],
[(a + b*x^2)^5/x^5, x, 2, -a^5/(4*x^4) - (5*a^4*b)/(2*x^2) + 5*a^2*b^3*x^2 + (5*a*b^4*x^4)/4 + (b^5*x^6)/6 + 10*a^3*b^2*log(x)],
[(a + b*x^2)^5/x^6, x, 2, -a^5/(5*x^5) - (5*a^4*b)/(3*x^3) - (10*a^3*b^2)/x + 10*a^2*b^3*x + (5*a*b^4*x^3)/3 + (b^5*x^5)/5],
[(a + b*x^2)^5/x^7, x, 2, -(a^5/(6*x^6)) - (5*a^4*b)/(4*x^4) - (5*a^3*b^2)/x^2 + (5/2)*a*b^4*x^2 + (b^5*x^4)/4 + 10*a^2*b^3*log(x)],
[(a + b*x^2)^5/x^8, x, 2, -a^5/(7*x^7) - (a^4*b)/x^5 - (10*a^3*b^2)/(3*x^3) - (10*a^2*b^3)/x + 5*a*b^4*x + (b^5*x^3)/3],
[(a + b*x^2)^5/x^9, x, 2, -(a^5/(8*x^8)) - (5*a^4*b)/(6*x^6) - (5*a^3*b^2)/(2*x^4) - (5*a^2*b^3)/x^2 + (b^5*x^2)/2 + 5*a*b^4*log(x)],
[(a + b*x^2)^5/x^10, x, 2, -a^5/(9*x^9) - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x],
[(a + b*x^2)^5/x^11, x, 2, -(a^5/(10*x^10)) - (5*a^4*b)/(8*x^8) - (5*a^3*b^2)/(3*x^6) - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(2*x^2) + b^5*log(x)],
[(a + b*x^2)^5/x^12, x, 2, -(a^5/(11*x^11)) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x],
[(a + b*x^2)^5/x^13, x, 1, -(a + b*x^2)^6/(12*a*x^12)],
[(a + b*x^2)^5/x^14, x, 2, -a^5/(13*x^13) - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)],
[(a + b*x^2)^5/x^15, x, 2, -((a + b*x^2)^6/(14*a*x^14)) + (b*(a + b*x^2)^6)/(84*a^2*x^12)],
[(a + b*x^2)^5/x^16, x, 2, -a^5/(15*x^15) - (5*a^4*b)/(13*x^13) - (10*a^3*b^2)/(11*x^11) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(7*x^7) - b^5/(5*x^5)],
[(a + b*x^2)^5/x^17, x, 2, -(a^5/(16*x^16)) - (5*a^4*b)/(14*x^14) - (5*a^3*b^2)/(6*x^12) - (a^2*b^3)/x^10 - (5*a*b^4)/(8*x^8) - b^5/(6*x^6)],
[(a + b*x^2)^5/x^18, x, 2, -a^5/(17*x^17) - (a^4*b)/(3*x^15) - (10*a^3*b^2)/(13*x^13) - (10*a^2*b^3)/(11*x^11) - (5*a*b^4)/(9*x^9) - b^5/(7*x^7)],
[(a + b*x^2)^5/x^19, x, 2, -(a^5/(18*x^18)) - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(8*x^8)],
[(a + b*x^2)^5/x^20, x, 2, -a^5/(19*x^19) - (5*a^4*b)/(17*x^17) - (2*a^3*b^2)/(3*x^15) - (10*a^2*b^3)/(13*x^13) - (5*a*b^4)/(11*x^11) - b^5/(9*x^9)],
[(a + b*x^2)^5/x^21, x, 2, -(a^5/(20*x^20)) - (5*a^4*b)/(18*x^18) - (5*a^3*b^2)/(8*x^16) - (5*a^2*b^3)/(7*x^14) - (5*a*b^4)/(12*x^12) - b^5/(10*x^10)],
[(a + b*x^2)^5/x^22, x, 2, -a^5/(21*x^21) - (5*a^4*b)/(19*x^19) - (10*a^3*b^2)/(17*x^17) - (2*a^2*b^3)/(3*x^15) - (5*a*b^4)/(13*x^13) - b^5/(11*x^11)],


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


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

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

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

# Formerly failed because both PosQ[(1-a)/a] and PosQ[-(1-a)/a] returned False. 
[1/(-1 + a + a*x^2), x, 1, -(arctanh((sqrt(a)*x)/sqrt(1 - a))/(sqrt(1 - a)*sqrt(a)))],


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

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


# Integrands of the form x^m/(a+b*x^2)^3 where m is an integer 
[x^16/(a + b*x^2)^3, x, 9, -((21*a^5*x)/b^8) + (5*a^4*x^3)/b^7 - (2*a^3*x^5)/b^6 + (6*a^2*x^7)/(7*b^5) - (a*x^9)/(3*b^4) + x^11/(11*b^3) + (a^7*x)/(4*b^8*(a + b*x^2)^2) - (29*a^6*x)/(8*b^8*(a + b*x^2)) + (195*a^(11/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*b^(17/2))],
[x^15/(a + b*x^2)^3, x, 7, (15*a^4*x^2)/(2*b^7) - (5*a^3*x^4)/(2*b^6) + (a^2*x^6)/b^5 - (3*a*x^8)/(8*b^4) + x^10/(10*b^3) + a^7/(4*b^8*(a + b*x^2)^2) - (7*a^6)/(2*b^8*(a + b*x^2)) - (21*a^5*log(a + b*x^2))/(2*b^8)],
[x^14/(a + b*x^2)^3, x, 9, (15*a^4*x)/b^7 - (10*a^3*x^3)/(3*b^6) + (6*a^2*x^5)/(5*b^5) - (3*a*x^7)/(7*b^4) + x^9/(9*b^3) - (a^6*x)/(4*b^7*(a + b*x^2)^2) + (25*a^5*x)/(8*b^7*(a + b*x^2)) - (143*a^(9/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*b^(15/2))],
[x^13/(a + b*x^2)^3, x, 7, -((5*a^3*x^2)/b^6) + (3*a^2*x^4)/(2*b^5) - (a*x^6)/(2*b^4) + x^8/(8*b^3) - a^6/(4*b^7*(a + b*x^2)^2) + (3*a^5)/(b^7*(a + b*x^2)) + (15*a^4*log(a + b*x^2))/(2*b^7)],
[x^12/(a + b*x^2)^3, x, 9, -((10*a^3*x)/b^6) + (2*a^2*x^3)/b^5 - (3*a*x^5)/(5*b^4) + x^7/(7*b^3) + (a^5*x)/(4*b^6*(a + b*x^2)^2) - (21*a^4*x)/(8*b^6*(a + b*x^2)) + (99*a^(7/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*b^(13/2))],
[x^11/(a + b*x^2)^3, x, 7, (3*a^2*x^2)/b^5 - (3*a*x^4)/(4*b^4) + x^6/(6*b^3) + a^5/(4*b^6*(a + b*x^2)^2) - (5*a^4)/(2*b^6*(a + b*x^2)) - (5*a^3*log(a + b*x^2))/b^6],
[x^10/(a + b*x^2)^3, x, 9, (6*a^2*x)/b^5 - (a*x^3)/b^4 + x^5/(5*b^3) - (a^4*x)/(4*b^5*(a + b*x^2)^2) + (17*a^3*x)/(8*b^5*(a + b*x^2)) - (63*a^(5/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*b^(11/2))],
[x^9/(a + b*x^2)^3, x, 7, -((3*a*x^2)/(2*b^4)) + x^4/(4*b^3) - a^4/(4*b^5*(a + b*x^2)^2) + (2*a^3)/(b^5*(a + b*x^2)) + (3*a^2*log(a + b*x^2))/b^5],
[x^8/(a + b*x^2)^3, x, 9, -((3*a*x)/b^4) + x^3/(3*b^3) + (a^3*x)/(4*b^4*(a + b*x^2)^2) - (13*a^2*x)/(8*b^4*(a + b*x^2)) + (35*a^(3/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*b^(9/2))],
[x^7/(a + b*x^2)^3, x, 6, x^2/(2*b^3) + a^3/(4*b^4*(a + b*x^2)^2) - (3*a^2)/(2*b^4*(a + b*x^2)) - (3*a*log(a + b*x^2))/(2*b^4)],
[x^6/(a + b*x^2)^3, x, 8, x/b^3 - (a^2*x)/(4*b^3*(a + b*x^2)^2) + (9*a*x)/(8*b^3*(a + b*x^2)) - (15*sqrt(a)*arctan((sqrt(b)*x)/sqrt(a)))/(8*b^(7/2))],
[x^5/(a + b*x^2)^3, x, 6, -(a^2/(4*b^3*(a + b*x^2)^2)) + a/(b^3*(a + b*x^2)) + log(a + b*x^2)/(2*b^3)],
[x^4/(a + b*x^2)^3, x, 8, (a*x)/(4*b^2*(a + b*x^2)^2) - (5*x)/(8*b^2*(a + b*x^2)) + (3*arctan((sqrt(b)*x)/sqrt(a)))/(8*sqrt(a)*b^(5/2))],
[x^3/(a + b*x^2)^3, x, 1, x^4/(4*a*(a + b*x^2)^2)],
[x^2/(a + b*x^2)^3, x, 3, -(x/(4*b*(a + b*x^2)^2)) + x/(8*a*b*(a + b*x^2)) + arctan((sqrt(b)*x)/sqrt(a))/(8*a^(3/2)*b^(3/2))],
[x/(a + b*x^2)^3, x, 2, -(1/(4*b*(a + b*x^2)^2))],
[1/(a + b*x^2)^3, x, 3, x/(4*a*(a + b*x^2)^2) + (3*x)/(8*a^2*(a + b*x^2)) + (3*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*sqrt(b))],
[1/(x*(a + b*x^2)^3), x, 9, 1/(4*a*(a + b*x^2)^2) + 1/(2*a^2*(a + b*x^2)) + log(x)/a^3 - log(a + b*x^2)/(2*a^3)],
[1/(x^2*(a + b*x^2)^3), x, 9, -(1/(a^3*x)) - (b*x)/(4*a^2*(a + b*x^2)^2) - (7*b*x)/(8*a^3*(a + b*x^2)) - (15*sqrt(b)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(7/2))],
[1/(x^3*(a + b*x^2)^3), x, 9, -(1/(2*a^3*x^2)) - b/(4*a^2*(a + b*x^2)^2) - b/(a^3*(a + b*x^2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x^2))/(2*a^4)],
[1/(x^4*(a + b*x^2)^3), x, 9, -(1/(3*a^3*x^3)) + (3*b)/(a^4*x) + (b^2*x)/(4*a^3*(a + b*x^2)^2) + (11*b^2*x)/(8*a^4*(a + b*x^2)) + (35*b^(3/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(9/2))],
[1/(x^5*(a + b*x^2)^3), x, 9, -(1/(4*a^3*x^4)) + (3*b)/(2*a^4*x^2) + b^2/(4*a^3*(a + b*x^2)^2) + (3*b^2)/(2*a^4*(a + b*x^2)) + (6*b^2*log(x))/a^5 - (3*b^2*log(a + b*x^2))/a^5],
[1/(x^6*(a + b*x^2)^3), x, 9, -(1/(5*a^3*x^5)) + b/(a^4*x^3) - (6*b^2)/(a^5*x) - (b^3*x)/(4*a^4*(a + b*x^2)^2) - (15*b^3*x)/(8*a^5*(a + b*x^2)) - (63*b^(5/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(11/2))],
[1/(x^7*(a + b*x^2)^3), x, 9, -(1/(6*a^3*x^6)) + (3*b)/(4*a^4*x^4) - (3*b^2)/(a^5*x^2) - b^3/(4*a^4*(a + b*x^2)^2) - (2*b^3)/(a^5*(a + b*x^2)) - (10*b^3*log(x))/a^6 + (5*b^3*log(a + b*x^2))/a^6],
[1/(x^8*(a + b*x^2)^3), x, 9, -(1/(7*a^3*x^7)) + (3*b)/(5*a^4*x^5) - (2*b^2)/(a^5*x^3) + (10*b^3)/(a^6*x) + (b^4*x)/(4*a^5*(a + b*x^2)^2) + (19*b^4*x)/(8*a^6*(a + b*x^2)) + (99*b^(7/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(13/2))],
[1/(x^9*(a + b*x^2)^3), x, 9, -(1/(8*a^3*x^8)) + b/(2*a^4*x^6) - (3*b^2)/(2*a^5*x^4) + (5*b^3)/(a^6*x^2) + b^4/(4*a^5*(a + b*x^2)^2) + (5*b^4)/(2*a^6*(a + b*x^2)) + (15*b^4*log(x))/a^7 - (15*b^4*log(a + b*x^2))/(2*a^7)],
[1/(x^10*(a + b*x^2)^3), x, 9, -(1/(9*a^3*x^9)) + (3*b)/(7*a^4*x^7) - (6*b^2)/(5*a^5*x^5) + (10*b^3)/(3*a^6*x^3) - (15*b^4)/(a^7*x) - (b^5*x)/(4*a^6*(a + b*x^2)^2) - (23*b^5*x)/(8*a^7*(a + b*x^2)) - (143*b^(9/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(15/2))],
[1/(x^11*(a + b*x^2)^3), x, 9, -(1/(10*a^3*x^10)) + (3*b)/(8*a^4*x^8) - b^2/(a^5*x^6) + (5*b^3)/(2*a^6*x^4) - (15*b^4)/(2*a^7*x^2) - b^5/(4*a^6*(a + b*x^2)^2) - (3*b^5)/(a^7*(a + b*x^2)) - (21*b^5*log(x))/a^8 + (21*b^5*log(a + b*x^2))/(2*a^8)],
[1/(x^12*(a + b*x^2)^3), x, 9, -(1/(11*a^3*x^11)) + b/(3*a^4*x^9) - (6*b^2)/(7*a^5*x^7) + (2*b^3)/(a^6*x^5) - (5*b^4)/(a^7*x^3) + (21*b^5)/(a^8*x) + (b^6*x)/(4*a^7*(a + b*x^2)^2) + (27*b^6*x)/(8*a^8*(a + b*x^2)) + (195*b^(11/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(17/2))],
[1/(x^13*(a + b*x^2)^3), x, 9, -(1/(12*a^3*x^12)) + (3*b)/(10*a^4*x^10) - (3*b^2)/(4*a^5*x^8) + (5*b^3)/(3*a^6*x^6) - (15*b^4)/(4*a^7*x^4) + (21*b^5)/(2*a^8*x^2) + b^6/(4*a^7*(a + b*x^2)^2) + (7*b^6)/(2*a^8*(a + b*x^2)) + (28*b^6*log(x))/a^9 - (14*b^6*log(a + b*x^2))/a^9],
[1/(x^14*(a + b*x^2)^3), x, 9, -(1/(13*a^3*x^13)) + (3*b)/(11*a^4*x^11) - (2*b^2)/(3*a^5*x^9) + (10*b^3)/(7*a^6*x^7) - (3*b^4)/(a^7*x^5) + (7*b^5)/(a^8*x^3) - (28*b^6)/(a^9*x) - (b^7*x)/(4*a^8*(a + b*x^2)^2) - (31*b^7*x)/(8*a^9*(a + b*x^2)) - (255*b^(13/2)*arctan((sqrt(b)*x)/sqrt(a)))/(8*a^(19/2))],
[1/(x^15*(a + b*x^2)^3), x, 9, -(1/(14*a^3*x^14)) + b/(4*a^4*x^12) - (3*b^2)/(5*a^5*x^10) + (5*b^3)/(4*a^6*x^8) - (5*b^4)/(2*a^7*x^6) + (21*b^5)/(4*a^8*x^4) - (14*b^6)/(a^9*x^2) - b^7/(4*a^8*(a + b*x^2)^2) - (4*b^7)/(a^9*(a + b*x^2)) - (36*b^7*log(x))/a^10 + (18*b^7*log(a + b*x^2))/a^10],


# Integrands of the form x^m/(a+b*x^2)^4 where m is an integer 
[x^3/(a + b*x^2)^4, x, 5, a/(6*b^2*(a + b*x^2)^3) - 1/(4*b^2*(a + b*x^2)^2)],
[x^2/(a + b*x^2)^4, x, 4, -(x/(6*b*(a + b*x^2)^3)) + x/(24*a*b*(a + b*x^2)^2) + x/(16*a^2*b*(a + b*x^2)) + arctan((sqrt(b)*x)/sqrt(a))/(16*a^(5/2)*b^(3/2))],
[x/(a + b*x^2)^4, x, 2, -1/(6*b*(a + b*x^2)^3)],
[(a + b*x^2)^(-4), x, 4, x/(6*a*(a + b*x^2)^3) + (5*x)/(24*a^2*(a + b*x^2)^2) + (5*x)/(16*a^3*(a + b*x^2)) + (5*arctan((sqrt(b)*x)/sqrt(a)))/(16*a^(7/2)*sqrt(b))],
[1/(x*(a + b*x^2)^4), x, 11, 1/(6*a*(a + b*x^2)^3) + 1/(4*a^2*(a + b*x^2)^2) + 1/(2*a^3*(a + b*x^2)) + log(x)/a^4 - log(a + b*x^2)/(2*a^4)],
[1/(x^2*(a + b*x^2)^4), x, 13, -(1/(a^4*x)) - (b*x)/(6*a^2*(a + b*x^2)^3) - (11*b*x)/(24*a^3*(a + b*x^2)^2) - (19*b*x)/(16*a^4*(a + b*x^2)) - (35*sqrt(b)*arctan((sqrt(b)*x)/sqrt(a)))/(16*a^(9/2))],
[1/(x^3*(a + b*x^2)^4), x, 11, -(1/(2*a^4*x^2)) - b/(6*a^2*(a + b*x^2)^3) - b/(2*a^3*(a + b*x^2)^2) - (3*b)/(2*a^4*(a + b*x^2)) - (4*b*log(x))/a^5 + (2*b*log(a + b*x^2))/a^5],


# Integrands of the form x^m/(a+b*x^2)^5 where m is an integer 
[x^3/(a + b*x^2)^5, x, 5, a/(8*b^2*(a + b*x^2)^4) - 1/(6*b^2*(a + b*x^2)^3)],
[x^2/(a + b*x^2)^5, x, 5, -(x/(8*b*(a + b*x^2)^4)) + x/(48*a*b*(a + b*x^2)^3) + (5*x)/(192*a^2*b*(a + b*x^2)^2) + (5*x)/(128*a^3*b*(a + b*x^2)) + (5*arctan((sqrt(b)*x)/sqrt(a)))/(128*a^(7/2)*b^(3/2))],
[x/(a + b*x^2)^5, x, 2, -1/(8*b*(a + b*x^2)^4)],
[(a + b*x^2)^(-5), x, 5, x/(8*a*(a + b*x^2)^4) + (7*x)/(48*a^2*(a + b*x^2)^3) + (35*x)/(192*a^3*(a + b*x^2)^2) + (35*x)/(128*a^4*(a + b*x^2)) + (35*arctan((sqrt(b)*x)/sqrt(a)))/(128*a^(9/2)*sqrt(b))],
[1/(x*(a + b*x^2)^5), x, 13, 1/(8*a*(a + b*x^2)^4) + 1/(6*a^2*(a + b*x^2)^3) + 1/(4*a^3*(a + b*x^2)^2) + 1/(2*a^4*(a + b*x^2)) + log(x)/a^5 - log(a + b*x^2)/(2*a^5)],
[1/(x^2*(a + b*x^2)^5), x, 18, -(1/(a^5*x)) - (b*x)/(8*a^2*(a + b*x^2)^4) - (5*b*x)/(16*a^3*(a + b*x^2)^3) - (41*b*x)/(64*a^4*(a + b*x^2)^2) - (187*b*x)/(128*a^5*(a + b*x^2)) - (315*sqrt(b)*arctan((sqrt(b)*x)/sqrt(a)))/(128*a^(11/2))],
[1/(x^3*(a + b*x^2)^5), x, 13, -(1/(2*a^5*x^2)) - b/(8*a^2*(a + b*x^2)^4) - b/(3*a^3*(a + b*x^2)^3) - (3*b)/(4*a^4*(a + b*x^2)^2) - (2*b)/(a^5*(a + b*x^2)) - (5*b*log(x))/a^6 + (5*b*log(a + b*x^2))/(2*a^6)],


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


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


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


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


[x^3/(a - b*x^2)^3, x, 1, x^4/(4*a*(a - b*x^2)^2)],
[x^2/(a - b*x^2)^3, x, 3, x/(4*b*(a - b*x^2)^2) - x/(8*a*b*(a - b*x^2)) - arctanh((sqrt(b)*x)/sqrt(a))/(8*a^(3/2)*b^(3/2))],
[x/(a - b*x^2)^3, x, 2, 1/(4*b*(a - b*x^2)^2)],
[1/(a - b*x^2)^3, x, 3, x/(4*a*(a - b*x^2)^2) + (3*x)/(8*a^2*(a - b*x^2)) + (3*arctanh((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*sqrt(b))],
[1/(x*(a - b*x^2)^3), x, 9, 1/(4*a*(a - b*x^2)^2) + 1/(2*a^2*(a - b*x^2)) + log(x)/a^3 - log(a - b*x^2)/(2*a^3)],
[1/(x^2*(a - b*x^2)^3), x, 9, -(1/(a^3*x)) + (b*x)/(4*a^2*(a - b*x^2)^2) + (7*b*x)/(8*a^3*(a - b*x^2)) + (15*sqrt(b)*arctanh((sqrt(b)*x)/sqrt(a)))/(8*a^(7/2))],
[1/(x^3*(a - b*x^2)^3), x, 9, -(1/(2*a^3*x^2)) + b/(4*a^2*(a - b*x^2)^2) + b/(a^3*(a - b*x^2)) + (3*b*log(x))/a^4 - (3*b*log(a - b*x^2))/(2*a^4)],


[x^3/(a - b*x^2)^4, x, 5, a/(6*b^2*(a - b*x^2)^3) - 1/(4*b^2*(a - b*x^2)^2)],
[x^2/(a - b*x^2)^4, x, 4, x/(6*b*(a - b*x^2)^3) - x/(24*a*b*(a - b*x^2)^2) - x/(16*a^2*b*(a - b*x^2)) - arctanh((sqrt(b)*x)/sqrt(a))/(16*a^(5/2)*b^(3/2))],
[x/(a - b*x^2)^4, x, 2, 1/(6*b*(a - b*x^2)^3)],
[1/(a - b*x^2)^4, x, 4, x/(6*a*(a - b*x^2)^3) + (5*x)/(24*a^2*(a - b*x^2)^2) + (5*x)/(16*a^3*(a - b*x^2)) + (5*arctanh((sqrt(b)*x)/sqrt(a)))/(16*a^(7/2)*sqrt(b))],
[1/(x*(a - b*x^2)^4), x, 11, 1/(6*a*(a - b*x^2)^3) + 1/(4*a^2*(a - b*x^2)^2) + 1/(2*a^3*(a - b*x^2)) + log(x)/a^4 - log(a - b*x^2)/(2*a^4)],
[1/(x^2*(a - b*x^2)^4), x, 13, -(1/(a^4*x)) + (b*x)/(6*a^2*(a - b*x^2)^3) + (11*b*x)/(24*a^3*(a - b*x^2)^2) + (19*b*x)/(16*a^4*(a - b*x^2)) + (35*sqrt(b)*arctanh((sqrt(b)*x)/sqrt(a)))/(16*a^(9/2))],
[1/(x^3*(a - b*x^2)^4), x, 11, -(1/(2*a^4*x^2)) + b/(6*a^2*(a - b*x^2)^3) + b/(2*a^3*(a - b*x^2)^2) + (3*b)/(2*a^4*(a - b*x^2)) + (4*b*log(x))/a^5 - (2*b*log(a - b*x^2))/a^5],


[x^3/(a - b*x^2)^5, x, 5, a/(8*b^2*(a - b*x^2)^4) - 1/(6*b^2*(a - b*x^2)^3)],
[x^2/(a - b*x^2)^5, x, 5, x/(8*b*(a - b*x^2)^4) - x/(48*a*b*(a - b*x^2)^3) - (5*x)/(192*a^2*b*(a - b*x^2)^2) - (5*x)/(128*a^3*b*(a - b*x^2)) - (5*arctanh((sqrt(b)*x)/sqrt(a)))/(128*a^(7/2)*b^(3/2))],
[x/(a - b*x^2)^5, x, 2, 1/(8*b*(a - b*x^2)^4)],
[1/(a - b*x^2)^5, x, 5, x/(8*a*(a - b*x^2)^4) + (7*x)/(48*a^2*(a - b*x^2)^3) + (35*x)/(192*a^3*(a - b*x^2)^2) + (35*x)/(128*a^4*(a - b*x^2)) + (35*arctanh((sqrt(b)*x)/sqrt(a)))/(128*a^(9/2)*sqrt(b))],
[1/(x*(a - b*x^2)^5), x, 13, 1/(8*a*(a - b*x^2)^4) + 1/(6*a^2*(a - b*x^2)^3) + 1/(4*a^3*(a - b*x^2)^2) + 1/(2*a^4*(a - b*x^2)) + log(x)/a^5 - log(a - b*x^2)/(2*a^5)],
[1/(x^2*(a - b*x^2)^5), x, 18, -(1/(a^5*x)) + (b*x)/(8*a^2*(a - b*x^2)^4) + (5*b*x)/(16*a^3*(a - b*x^2)^3) + (41*b*x)/(64*a^4*(a - b*x^2)^2) + (187*b*x)/(128*a^5*(a - b*x^2)) + (315*sqrt(b)*arctanh((sqrt(b)*x)/sqrt(a)))/(128*a^(11/2))],
[1/(x^3*(a - b*x^2)^5), x, 13, -(1/(2*a^5*x^2)) + b/(8*a^2*(a - b*x^2)^4) + b/(3*a^3*(a - b*x^2)^3) + (3*b)/(4*a^4*(a - b*x^2)^2) + (2*b)/(a^5*(a - b*x^2)) + (5*b*log(x))/a^6 - (5*b*log(a - b*x^2))/(2*a^6)],


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


# Integrands of the form (a+b*x)/(c+d*x^2) 
[(a + b*x)/(1 - x^2), x, 4, (-(1/2))*(a + b)*log(1 - x) + (1/2)*(a - b)*log(1 + x)],
[(a + b*x)/(c^2 - d^2*x^2), x, 4, -(((b*c + a*d)*log(c - d*x))/(2*c*d^2)) - ((b*c - a*d)*log(c + d*x))/(2*c*d^2)],

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


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


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

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


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

[1/((a + b*x)*(a^2 - b^2*x^2)^2), x, 7, 1/(8*a^3*b*(a - b*x)) - 1/(8*a^2*b*(a + b*x)^2) - 1/(4*a^3*b*(a + b*x)) + (3*arctanh((b*x)/a))/(8*a^4*b)],
[1/((a + b*x)^2*(a^2 - b^2*x^2)^2), x, 8, 1/(16*a^4*b*(a - b*x)) - 1/(12*a^2*b*(a + b*x)^3) - 1/(8*a^3*b*(a + b*x)^2) - 3/(16*a^4*b*(a + b*x)) + arctanh((b*x)/a)/(4*a^5*b)],
[1/((a + b*x)^3*(a^2 - b^2*x^2)^2), x, 9, 1/(32*a^5*b*(a - b*x)) - 1/(16*a^2*b*(a + b*x)^4) - 1/(12*a^3*b*(a + b*x)^3) - 3/(32*a^4*b*(a + b*x)^2) - 1/(8*a^5*b*(a + b*x)) + (5*arctanh((b*x)/a))/(32*a^6*b)],


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

[1/((a + b*x)*(a^2 - b^2*x^2)^3), x, 9, 1/(32*a^4*b*(a - b*x)^2) + 1/(8*a^5*b*(a - b*x)) - 1/(24*a^3*b*(a + b*x)^3) - 3/(32*a^4*b*(a + b*x)^2) - 3/(16*a^5*b*(a + b*x)) + (5*arctanh((b*x)/a))/(16*a^6*b)],
[1/((a + b*x)^2*(a^2 - b^2*x^2)^3), x, 10, 1/(64*a^5*b*(a - b*x)^2) + 5/(64*a^6*b*(a - b*x)) - 1/(32*a^3*b*(a + b*x)^4) - 1/(16*a^4*b*(a + b*x)^3) - 3/(32*a^5*b*(a + b*x)^2) - 5/(32*a^6*b*(a + b*x)) + (15*arctanh((b*x)/a))/(64*a^7*b)],
[1/((a + b*x)^3*(a^2 - b^2*x^2)^3), x, 11, 1/(128*a^6*b*(a - b*x)^2) + 3/(64*a^7*b*(a - b*x)) - 1/(40*a^3*b*(a + b*x)^5) - 3/(64*a^4*b*(a + b*x)^4) - 1/(16*a^5*b*(a + b*x)^3) - 5/(64*a^6*b*(a + b*x)^2) - 15/(128*a^7*b*(a + b*x)) + (21*arctanh((b*x)/a))/(128*a^8*b)],


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


# Integrands of the form (a+b*x^2)/(c+d*x^2) 
[(a + b*x^2)/(1 + x^2), x, 3, b*x + (a - b)*arctan(x)],
[(a + b*x^2)/(c + d*x^2), x, 3, (b*x)/d - ((b*c - a*d)*arctan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*d^(3/2))],
# Note: Algebraic expansion should not factor denominator into linears 
[(a + b*x^2)/(1 - x^2), x, 3, (-b)*x + (a + b)*arctanh(x)],
[(a + b*x^2)/(c - d*x^2), x, 3, -((b*x)/d) + ((b*c + a*d)*arctanh((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*d^(3/2))],


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


# Integrands of the form (a+b*x^2)/(c+d*x^2)^3 
[(1 + x^2)/(16 + x^2)^3, x, 5, -((15*x)/(64*(16 + x^2)^2)) + (19*x)/(2048*(16 + x^2)) + (19*arctan(x/4))/8192],
[(a + b*x^2)/(c + d*x^2)^3, x, 5, (a*x)/(4*c*(c + d*x^2)^2) - (b*x)/(4*d*(c + d*x^2)^2) + ((b*c + 3*a*d)*x)/(8*c^2*d*(c + d*x^2)) + ((b*c + 3*a*d)*arctan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*d^(3/2))],


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


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


[(-1 + x^2)^4/(1 + x^2)^5, x, 10, -(x/(5*(1 + x^2)^4)) - (11*x^3)/(5*(1 + x^2)^4) - x^5/(1 + x^2)^4 - x^7/(1 + x^2)^4 + x/(5*(1 + x^2)^3) + x/(4*(1 + x^2)^2) + (3*x)/(8*(1 + x^2)) + (3*arctan(x))/8],


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


# Integrands of the form x^m*(a+b*x^2)/(c+d*x^2)^n where m and n>0 are integers 
[x^5*(a + b*x^2)/(c + d*x^2), x, 5, (c*(b*c - a*d)*x^2)/(2*d^3) - ((b*c - a*d)*x^4)/(4*d^2) + (b*x^6)/(6*d) - (c^2*(b*c - a*d)*log(c + d*x^2))/(2*d^4)],
[x^4*(a + b*x^2)/(c + d*x^2), x, 4, (c*(b*c - a*d)*x)/d^3 - ((b*c - a*d)*x^3)/(3*d^2) + (b*x^5)/(5*d) - (c^(3/2)*(b*c - a*d)*arctan((sqrt(d)*x)/sqrt(c)))/d^(7/2)],
[x^3*(a + b*x^2)/(c + d*x^2), x, 5, -(((b*c - a*d)*x^2)/(2*d^2)) + (b*x^4)/(4*d) + (c*(b*c - a*d)*log(c + d*x^2))/(2*d^3)],
[x^2*(a + b*x^2)/(c + d*x^2), x, 4, -(((b*c - a*d)*x)/d^2) + (b*x^3)/(3*d) + (sqrt(c)*(b*c - a*d)*arctan((sqrt(d)*x)/sqrt(c)))/d^(5/2)],
[x*(a + b*x^2)/(c + d*x^2), x, 4, (b*x^2)/(2*d) - ((b*c - a*d)*log(c + d*x^2))/(2*d^2)],
[(a + b*x^2)/(c + d*x^2), x, 3, (b*x)/d - ((b*c - a*d)*arctan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*d^(3/2))],
[(a + b*x^2)/(x*(c + d*x^2)), x, 5, (a*log(x))/c + ((b*c - a*d)*log(c + d*x^2))/(2*c*d)],
[(a + b*x^2)/(x^2*(c + d*x^2)), x, 4, -(a/(c*x)) + ((b*c - a*d)*arctan((sqrt(d)*x)/sqrt(c)))/(c^(3/2)*sqrt(d))],
[(a + b*x^2)/(x^3*(c + d*x^2)), x, 5, -(a/(2*c*x^2)) + ((b*c - a*d)*log(x))/c^2 - ((b*c - a*d)*log(c + d*x^2))/(2*c^2)],
[(a + b*x^2)/(x^4*(c + d*x^2)), x, 4, -(a/(3*c*x^3)) - (b*c - a*d)/(c^2*x) - (sqrt(d)*(b*c - a*d)*arctan((sqrt(d)*x)/sqrt(c)))/c^(5/2)],
[(a + b*x^2)/(x^5*(c + d*x^2)), x, 5, -(a/(4*c*x^4)) - (b*c - a*d)/(2*c^2*x^2) - (d*(b*c - a*d)*log(x))/c^3 + (d*(b*c - a*d)*log(c + d*x^2))/(2*c^3)],

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


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


# ::Subsection::Closed:: 
#Integrands involving powers of cubic binomials


# ::Subsubsection::Closed:: 
#Integrands of the form x^m (a + b x^3)^n


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


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


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

[(a + b*x^3)^3/x^13, x, 1, -((a + b*x^3)^4/(12*a*x^12))],
[(a + b*x^3)^3/x^14, x, 2, -(a^3/(13*x^13)) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(7*x^7) - b^3/(4*x^4)],
[(a + b*x^3)^3/x^15, x, 2, -(a^3/(14*x^14)) - (3*a^2*b)/(11*x^11) - (3*a*b^2)/(8*x^8) - b^3/(5*x^5)],
[(a + b*x^3)^3/x^16, x, 2, -(a^3/(15*x^15)) - (a^2*b)/(4*x^12) - (a*b^2)/(3*x^9) - b^3/(6*x^6)],


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

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

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


# Integrands of the form x^m/(a+b*x^3)^2 where m is an integer 
[x^4/(a + b*x^3)^2, x, 5, -(x^2/(3*b*(a + b*x^3))) - (2*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(1/3)*b^(5/3)) - (2*log(a^(1/3) + b^(1/3)*x))/(9*a^(1/3)*b^(5/3)) + log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(9*a^(1/3)*b^(5/3))],
[x^3/(a + b*x^3)^2, x, 5, -(x/(3*b*(a + b*x^3))) - arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3)))/(3*sqrt(3)*a^(2/3)*b^(4/3)) + log(a^(1/3) + b^(1/3)*x)/(9*a^(2/3)*b^(4/3)) - log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(18*a^(2/3)*b^(4/3))],
[x^2/(a + b*x^3)^2, x, 2, -(1/(3*b*(a + b*x^3)))],
[x/(a + b*x^3)^2, x, 5, x^2/(3*a*(a + b*x^3)) - arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3)))/(3*sqrt(3)*a^(4/3)*b^(2/3)) - log(a^(1/3) + b^(1/3)*x)/(9*a^(4/3)*b^(2/3)) + log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(18*a^(4/3)*b^(2/3))],
[1/(a + b*x^3)^2, x, 5, x/(3*a*(a + b*x^3)) - (2*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(1/3)) + (2*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*b^(1/3)) - log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(9*a^(5/3)*b^(1/3))],
[1/(x*(a + b*x^3)^2), x, 7, 1/(3*a*(a + b*x^3)) + log(x)/a^2 - log(a + b*x^3)/(3*a^2)],
[1/(x^2*(a + b*x^3)^2), x, 12, -(1/(a^2*x)) - (b*x^2)/(3*a^2*(a + b*x^3)) + (4*b^(1/3)*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(7/3)) + (4*b^(1/3)*log(a^(1/3) + b^(1/3)*x))/(9*a^(7/3)) - (2*b^(1/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(9*a^(7/3))],
[1/(x^3*(a + b*x^3)^2), x, 12, -(1/(2*a^2*x^2)) - (b*x)/(3*a^2*(a + b*x^3)) + (5*b^(2/3)*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(8/3)) - (5*b^(2/3)*log(a^(1/3) + b^(1/3)*x))/(9*a^(8/3)) + (5*b^(2/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(18*a^(8/3))],
[1/(x^4*(a + b*x^3)^2), x, 7, -(1/(3*a^2*x^3)) - b/(3*a^2*(a + b*x^3)) - (2*b*log(x))/a^3 + (2*b*log(a + b*x^3))/(3*a^3)],


# Integrands of the form x^m/(a+b*x^3)^3 where m is an integer 
[x^11/(a*x^2 + b*x^5)^3, x, 2, x^6/(6*a*(a + b*x^3)^2)],
[x^5/(a + b*x^3)^3, x, 1, x^6/(6*a*(a + b*x^3)^2)],
[x^4/(a + b*x^3)^3, x, 6, -(x^2/(6*b*(a + b*x^3)^2)) + x^2/(9*a*b*(a + b*x^3)) - arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3)))/(9*sqrt(3)*a^(4/3)*b^(5/3)) - log(a^(1/3) + b^(1/3)*x)/(27*a^(4/3)*b^(5/3)) + log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(54*a^(4/3)*b^(5/3))],
[x^3/(a + b*x^3)^3, x, 6, -(x/(6*b*(a + b*x^3)^2)) + x/(18*a*b*(a + b*x^3)) - arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3)))/(9*sqrt(3)*a^(5/3)*b^(4/3)) + log(a^(1/3) + b^(1/3)*x)/(27*a^(5/3)*b^(4/3)) - log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(54*a^(5/3)*b^(4/3))],
[x^2/(a + b*x^3)^3, x, 2, -(1/(6*b*(a + b*x^3)^2))],
[x/(a + b*x^3)^3, x, 6, x^2/(6*a*(a + b*x^3)^2) + (2*x^2)/(9*a^2*(a + b*x^3)) - (2*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(7/3)*b^(2/3)) - (2*log(a^(1/3) + b^(1/3)*x))/(27*a^(7/3)*b^(2/3)) + log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(27*a^(7/3)*b^(2/3))],
[1/(a + b*x^3)^3, x, 6, x/(6*a*(a + b*x^3)^2) + (5*x)/(18*a^2*(a + b*x^3)) - (5*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(8/3)*b^(1/3)) + (5*log(a^(1/3) + b^(1/3)*x))/(27*a^(8/3)*b^(1/3)) - (5*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(54*a^(8/3)*b^(1/3))],
[1/(x*(a + b*x^3)^3), x, 9, 1/(6*a*(a + b*x^3)^2) + 1/(3*a^2*(a + b*x^3)) + log(x)/a^3 - log(a + b*x^3)/(3*a^3)],
[1/(x^2*(a + b*x^3)^3), x, 18, -(1/(a^3*x)) - (b*x^2)/(6*a^2*(a + b*x^3)^2) - (5*b*x^2)/(9*a^3*(a + b*x^3)) + (14*b^(1/3)*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(10/3)) + (14*b^(1/3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(10/3)) - (7*b^(1/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(27*a^(10/3))],
[1/(x^3*(a + b*x^3)^3), x, 18, -(1/(2*a^3*x^2)) - (b*x)/(6*a^2*(a + b*x^3)^2) - (11*b*x)/(18*a^3*(a + b*x^3)) + (20*b^(2/3)*arctan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(11/3)) - (20*b^(2/3)*log(a^(1/3) + b^(1/3)*x))/(27*a^(11/3)) + (10*b^(2/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(27*a^(11/3))],
[1/(x^4*(a + b*x^3)^3), x, 9, -(1/(3*a^3*x^3)) - b/(6*a^2*(a + b*x^3)^2) - (2*b)/(3*a^3*(a + b*x^3)) - (3*b*log(x))/a^4 + (b*log(a + b*x^3))/a^4],


# ::Subsubsection::Closed:: 
#Integrands of the form (a+b x)^m / (c + d x^3)


[(a + b*x)^2/(c + d*x^3), x, 12, -((2*a*b*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(1/3)*d^(2/3))) - (a^2*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(1/3)) - (2*a*b*log(c^(1/3) + d^(1/3)*x))/(3*c^(1/3)*d^(2/3)) + (a^2*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(1/3)) + (a*b*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(3*c^(1/3)*d^(2/3)) - (a^2*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(1/3)) + (b^2*log(c + d*x^3))/(3*d)],
[(a + b*x)^3/(c + d*x^3), x, 12, (b^3*x)/d - (sqrt(3)*a^2*b*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(c^(1/3)*d^(2/3)) + ((b^3*c - a^3*d)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(4/3)) - (a^2*b*log(c^(1/3) + d^(1/3)*x))/(c^(1/3)*d^(2/3)) - ((b^3*c - a^3*d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(4/3)) + (a^2*b*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(2*c^(1/3)*d^(2/3)) + ((b^3*c - a^3*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(4/3)) + (a*b^2*log(c + d*x^3))/d],
[(a + b*x)^4/(c + d*x^3), x, 13, (4*a*b^3*x)/d + (b^4*x^2)/(2*d) + (b*(b^3*c - 4*a^3*d)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(1/3)*d^(5/3)) + (a*(4*b^3*c - a^3*d)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(4/3)) + (b*(b^3*c - 4*a^3*d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(1/3)*d^(5/3)) - (a*(4*b^3*c - a^3*d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(4/3)) - (b*(b^3*c - 4*a^3*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(1/3)*d^(5/3)) + (a*(4*b^3*c - a^3*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(4/3)) + (2*a^2*b^2*log(c + d*x^3))/d],


[(1 + x)^2/(c + d*x^3), x, 12, -((2*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(1/3)*d^(2/3))) - arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3)))/(sqrt(3)*c^(2/3)*d^(1/3)) - (2*log(c^(1/3) + d^(1/3)*x))/(3*c^(1/3)*d^(2/3)) + log(c^(1/3) + d^(1/3)*x)/(3*c^(2/3)*d^(1/3)) + log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/(3*c^(1/3)*d^(2/3)) - log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/(6*c^(2/3)*d^(1/3)) + log(c + d*x^3)/(3*d)],
[(1 + x)^3/(c + d*x^3), x, 12, x/d + ((c - d)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(4/3)) - (sqrt(3)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(c^(1/3)*d^(2/3)) - ((c - d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(4/3)) - log(c^(1/3) + d^(1/3)*x)/(c^(1/3)*d^(2/3)) + ((c - d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(4/3)) + log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/(2*c^(1/3)*d^(2/3)) + log(c + d*x^3)/d],
[(1 + x)^4/(c + d*x^3), x, 13, (4*x)/d + x^2/(2*d) + ((c - 4*d)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(1/3)*d^(5/3)) + ((4*c - d)*arctan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(4/3)) + ((c - 4*d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(1/3)*d^(5/3)) - ((4*c - d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(4/3)) - ((c - 4*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(1/3)*d^(5/3)) + ((4*c - d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(4/3)) + (2*log(c + d*x^3))/d],


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b (c+d x)^3)


# Integrands of the form x^m/(a+b*(c+d*x)^3) 
[x^3/(a + b*(c + d*x)^3), x, 13, (c + d*x)/(b*d^4) - (sqrt(3)*c^2*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(a^(1/3)*b^(2/3)*d^4) + ((a + b*c^3)*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(4/3)*d^4) - (c^2*log(a^(1/3) + b^(1/3)*(c + d*x)))/(a^(1/3)*b^(2/3)*d^4) - ((a + b*c^3)*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(2/3)*b^(4/3)*d^4) + (c^2*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(2*a^(1/3)*b^(2/3)*d^4) + ((a + b*c^3)*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(2/3)*b^(4/3)*d^4) - (c*log(a + b*(c + d*x)^3))/(b*d^4)],
[x^2/(a + b*(c + d*x)^3), x, 13, (2*c*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(1/3)*b^(2/3)*d^3) - (c^2*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(1/3)*d^3) + (2*c*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(1/3)*b^(2/3)*d^3) + (c^2*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(2/3)*b^(1/3)*d^3) - (c*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(3*a^(1/3)*b^(2/3)*d^3) - (c^2*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(2/3)*b^(1/3)*d^3) + log(a + b*(c + d*x)^3)/(3*b*d^3)],
[x/(a + b*(c + d*x)^3), x, 11, -(arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3)))/(sqrt(3)*a^(1/3)*b^(2/3)*d^2)) + (c*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(1/3)*d^2) - log(a^(1/3) + b^(1/3)*(c + d*x))/(3*a^(1/3)*b^(2/3)*d^2) - (c*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(2/3)*b^(1/3)*d^2) + log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(6*a^(1/3)*b^(2/3)*d^2) + (c*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(2/3)*b^(1/3)*d^2)],
[1/(a + b*(c + d*x)^3), x, 5, -(arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3)))/(sqrt(3)*a^(2/3)*b^(1/3)*d)) + log(a^(1/3) + b^(1/3)*(c + d*x))/(3*a^(2/3)*b^(1/3)*d) - log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(6*a^(2/3)*b^(1/3)*d)],
[1/(x*(a + b*(c + d*x)^3)), x, 14, (b^(1/3)*c*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(1/3)*(a + b*c^3)) + (b^(2/3)*c^2*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*(a + b*c^3)) + log((-d)*x)/(a + b*c^3) + (b^(1/3)*c*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(1/3)*(a + b*c^3)) - (b^(2/3)*c^2*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(2/3)*(a + b*c^3)) - (b^(1/3)*c*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(1/3)*(a + b*c^3)) + (b^(2/3)*c^2*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(2/3)*(a + b*c^3)) - log(a + b*(c + d*x)^3)/(3*(a + b*c^3))],
[1/(x^2*(a + b*(c + d*x)^3)), x, 15, -(1/((a + b*c^3)*x)) + (b^(1/3)*(a - 2*b*c^3)*d*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(1/3)*(a + b*c^3)^2) + (b^(2/3)*c*(2*a - b*c^3)*d*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*(a + b*c^3)^2) - (3*b*c^2*d*log((-d)*x))/(a + b*c^3)^2 + (b^(1/3)*(a - 2*b*c^3)*d*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(1/3)*(a + b*c^3)^2) - (b^(2/3)*c*(2*a - b*c^3)*d*log(a^(1/3) + b^(1/3)*(c + d*x)))/(3*a^(2/3)*(a + b*c^3)^2) - (b^(1/3)*(a - 2*b*c^3)*d*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(1/3)*(a + b*c^3)^2) + (b^(2/3)*c*(2*a - b*c^3)*d*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(6*a^(2/3)*(a + b*c^3)^2) + (b*c^2*d*log(a + b*(c + d*x)^3))/(a + b*c^3)^2],
# {1/(x^3*(a + b*(c + d*x)^3)), x, 10, -(1/(2*(a + b*c^3)*x^2)) + (3*b*c^2*d)/((a + b*c^3)^2*x) - (3*b*c*(a - 2*b*c^3)*d^2*Log[(-d)*x])/(a + b*c^3)^3 - (1/(2*a^(1/3)*(a + b*c^3)^3))*(b^(4/3)*c^2*(2*a - b*c^3)*d^2*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x))/a^(1/3))/Sqrt[3]] + 2*Log[a^(1/3)/b^(1/3) + c + d*x] - Log[a^(2/3)/b^(2/3) - (a^(1/3)*(c + d*x))/b^(1/3) + (c + d*x)^2])) + (1/(6*a^(2/3)*(a + b*c^3)^3))*(b^(2/3)*(a^2 - 7*a*b*c^3 + b^2*c^6)*d^2*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x))/a^(1/3))/Sqrt[3]] - 2*Log[a^(1/3)/b^(1/3) + c + d*x] + Log[a^(2/3)/b^(2/3) - (a^(1/3)*(c + d*x))/b^(1/3) + (c + d*x)^2])) + (b*c*(a - 2*b*c^3)*d^2*Log[a + b*(c + d*x)^3])/(a + b*c^3)^3} 


# ::Subsubsection::Closed:: 
#Integrands of the form (c+d x)^m (a + b (c+d x)^3)^n


# Integrands of the form (c+d*x)^3*(a+b*(c+d*x)^3)^n where n is an integer 
[(c + d*x)^3*(a + b*(c + d*x)^3), x, 3, (a*(c + d*x)^4)/(4*d) + (b*(c + d*x)^7)/(7*d)],
[(c + d*x)^3*(a + b*(c + d*x)^3)^2, x, 3, (a^2*(c + d*x)^4)/(4*d) + (2*a*b*(c + d*x)^7)/(7*d) + (b^2*(c + d*x)^10)/(10*d)],
[(c + d*x)^3*(a + b*(c + d*x)^3)^3, x, 3, (1/4)*a^3*((c + d*x)^4/d) + (3/(7*d))*a^2*b*(c + d*x)^7 + (3/(10*d))*a*b^2*(c + d*x)^10 + (1/(13*d))*b^3*(c + d*x)^13],


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


# Integrands of the form (c + d*x)^m/(a+b*(c + d*x)^3)^2 where m is an integer 
[(c + d*x)^4/(a + b*(c + d*x)^3)^2, x, 6, -((c + d*x)^2/(3*b*d*(a + b*(c + d*x)^3))) - (2*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(1/3)*b^(5/3)*d) - (2*log(a^(1/3) + b^(1/3)*(c + d*x)))/(9*a^(1/3)*b^(5/3)*d) + log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(9*a^(1/3)*b^(5/3)*d)],
[(c + d*x)^3/(a + b*(c + d*x)^3)^2, x, 6, -((c + d*x)/(3*b*d*(a + b*(c + d*x)^3))) - arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3)))/(3*sqrt(3)*a^(2/3)*b^(4/3)*d) + log(a^(1/3) + b^(1/3)*(c + d*x))/(9*a^(2/3)*b^(4/3)*d) - log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(18*a^(2/3)*b^(4/3)*d)],
[(c + d*x)^2/(a + b*(c + d*x)^3)^2, x, 2, -(1/(3*b*d*(a + b*(c + d*x)^3)))],
[(c + d*x)/(a + b*(c + d*x)^3)^2, x, 6, (c + d*x)^2/(3*a*d*(a + b*(c + d*x)^3)) - arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3)))/(3*sqrt(3)*a^(4/3)*b^(2/3)*d) - log(a^(1/3) + b^(1/3)*(c + d*x))/(9*a^(4/3)*b^(2/3)*d) + log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(18*a^(4/3)*b^(2/3)*d)],
[1/(a + b*(c + d*x)^3)^2, x, 6, (c + d*x)/(3*a*d*(a + b*(c + d*x)^3)) - (2*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(1/3)*d) + (2*log(a^(1/3) + b^(1/3)*(c + d*x)))/(9*a^(5/3)*b^(1/3)*d) - log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(9*a^(5/3)*b^(1/3)*d)],
[1/((c + d*x)*(a + b*(c + d*x)^3)^2), x, 8, 1/(3*a*d*(a + b*(c + d*x)^3)) + log(c + d*x)/(a^2*d) - log(a + b*(c + d*x)^3)/(3*a^2*d)],
[1/((c + d*x)^2*(a + b*(c + d*x)^3)^2), x, 13, -(1/(a^2*d*(c + d*x))) - (b*(c + d*x)^2)/(3*a^2*d*(a + b*(c + d*x)^3)) + (4*b^(1/3)*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(7/3)*d) + (4*b^(1/3)*log(a^(1/3) + b^(1/3)*(c + d*x)))/(9*a^(7/3)*d) - (2*b^(1/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(9*a^(7/3)*d)],
[1/((c + d*x)^3*(a + b*(c + d*x)^3)^2), x, 13, -(1/(2*a^2*d*(c + d*x)^2)) - (b*(c + d*x))/(3*a^2*d*(a + b*(c + d*x)^3)) + (5*b^(2/3)*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(8/3)*d) - (5*b^(2/3)*log(a^(1/3) + b^(1/3)*(c + d*x)))/(9*a^(8/3)*d) + (5*b^(2/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(18*a^(8/3)*d)],
[1/((c + d*x)^4*(a + b*(c + d*x)^3)^2), x, 8, -(1/(3*a^2*d*(c + d*x)^3)) - b/(3*a^2*d*(a + b*(c + d*x)^3)) - (2*b*log(c + d*x))/(a^3*d) + (2*b*log(a + b*(c + d*x)^3))/(3*a^3*d)],


# Integrands of the form (c + d*x)^m/(a+b*(c + d*x)^3)^3 where m is an integer 
[(c + d*x)^4/(a + b*(c + d*x)^3)^3, x, 7, -((c + d*x)^2/(6*b*d*(a + b*(c + d*x)^3)^2)) + (c + d*x)^2/(9*a*b*d*(a + b*(c + d*x)^3)) - arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3)))/(9*sqrt(3)*a^(4/3)*b^(5/3)*d) - log(a^(1/3) + b^(1/3)*(c + d*x))/(27*a^(4/3)*b^(5/3)*d) + log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(54*a^(4/3)*b^(5/3)*d)],
[(c + d*x)^3/(a + b*(c + d*x)^3)^3, x, 7, -((c + d*x)/(6*b*d*(a + b*(c + d*x)^3)^2)) + (c + d*x)/(18*a*b*d*(a + b*(c + d*x)^3)) - arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3)))/(9*sqrt(3)*a^(5/3)*b^(4/3)*d) + log(a^(1/3) + b^(1/3)*(c + d*x))/(27*a^(5/3)*b^(4/3)*d) - log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(54*a^(5/3)*b^(4/3)*d)],
[(c + d*x)^2/(a + b*(c + d*x)^3)^3, x, 2, -(1/(6*b*d*(a + b*(c + d*x)^3)^2))],
[(c + d*x)/(a + b*(c + d*x)^3)^3, x, 7, (c + d*x)^2/(6*a*d*(a + b*(c + d*x)^3)^2) + (2*(c + d*x)^2)/(9*a^2*d*(a + b*(c + d*x)^3)) - (2*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(7/3)*b^(2/3)*d) - (2*log(a^(1/3) + b^(1/3)*(c + d*x)))/(27*a^(7/3)*b^(2/3)*d) + log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2)/(27*a^(7/3)*b^(2/3)*d)],
[1/(a + b*(c + d*x)^3)^3, x, 7, (c + d*x)/(6*a*d*(a + b*(c + d*x)^3)^2) + (5*(c + d*x))/(18*a^2*d*(a + b*(c + d*x)^3)) - (5*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(8/3)*b^(1/3)*d) + (5*log(a^(1/3) + b^(1/3)*(c + d*x)))/(27*a^(8/3)*b^(1/3)*d) - (5*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(54*a^(8/3)*b^(1/3)*d)],
[1/((c + d*x)*(a + b*(c + d*x)^3)^3), x, 10, 1/(6*a*d*(a + b*(c + d*x)^3)^2) + 1/(3*a^2*d*(a + b*(c + d*x)^3)) + log(c + d*x)/(a^3*d) - log(a + b*(c + d*x)^3)/(3*a^3*d)],
[1/((c + d*x)^2*(a + b*(c + d*x)^3)^3), x, 19, -(1/(a^3*d*(c + d*x))) - (b*(c + d*x)^2)/(6*a^2*d*(a + b*(c + d*x)^3)^2) - (5*b*(c + d*x)^2)/(9*a^3*d*(a + b*(c + d*x)^3)) + (14*b^(1/3)*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(10/3)*d) + (14*b^(1/3)*log(a^(1/3) + b^(1/3)*(c + d*x)))/(27*a^(10/3)*d) - (7*b^(1/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(27*a^(10/3)*d)],
[1/((c + d*x)^3*(a + b*(c + d*x)^3)^3), x, 19, -(1/(2*a^3*d*(c + d*x)^2)) - (b*(c + d*x))/(6*a^2*d*(a + b*(c + d*x)^3)^2) - (11*b*(c + d*x))/(18*a^3*d*(a + b*(c + d*x)^3)) + (20*b^(2/3)*arctan((a^(1/3) - 2*b^(1/3)*(c + d*x))/(sqrt(3)*a^(1/3))))/(9*sqrt(3)*a^(11/3)*d) - (20*b^(2/3)*log(a^(1/3) + b^(1/3)*(c + d*x)))/(27*a^(11/3)*d) + (10*b^(2/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2))/(27*a^(11/3)*d)],
[1/((c + d*x)^4*(a + b*(c + d*x)^3)^3), x, 10, -(1/(3*a^3*d*(c + d*x)^3)) - b/(6*a^2*d*(a + b*(c + d*x)^3)^2) - (2*b)/(3*a^3*d*(a + b*(c + d*x)^3)) - (3*b*log(c + d*x))/(a^4*d) + (b*log(a + b*(c + d*x)^3))/(a^4*d)],


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


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


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

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

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


[x^3/(2 + 3*x^4)^2, x, 2, -(1/(12*(2 + 3*x^4)))],
[x^2/(2 + 3*x^4)^2, x, 8, x^3/(8*(2 + 3*x^4)) - arctan(1 - 6^(1/4)*x)/(16*6^(3/4)) + arctan(1 + 6^(1/4)*x)/(16*6^(3/4)) + log(sqrt(6) - 6^(3/4)*x + 3*x^2)/(32*6^(3/4)) - log(sqrt(6) + 6^(3/4)*x + 3*x^2)/(32*6^(3/4))],
[x/(2 + 3*x^4)^2, x, 3, x^2/(8*(2 + 3*x^4)) + arctan(sqrt(3/2)*x^2)/(8*sqrt(6))],
[1/(2 + 3*x^4)^2, x, 8, x/(8*(2 + 3*x^4)) - (3^(3/4)*arctan(1 - 6^(1/4)*x))/(32*2^(1/4)) + (3^(3/4)*arctan(1 + 6^(1/4)*x))/(32*2^(1/4)) - (3^(3/4)*log(sqrt(6) - 6^(3/4)*x + 3*x^2))/(64*2^(1/4)) + (3^(3/4)*log(sqrt(6) + 6^(3/4)*x + 3*x^2))/(64*2^(1/4))],
[1/(x*(2 + 3*x^4)^2), x, 7, 1/(8*(2 + 3*x^4)) + log(x)/4 - (1/16)*log(2 + 3*x^4)],
[1/(x^2*(2 + 3*x^4)^2), x, 18, -(1/(4*x)) - (3*x^3)/(16*(2 + 3*x^4)) + (5*3^(1/4)*arctan(1 - 6^(1/4)*x))/(32*2^(3/4)) - (5*3^(1/4)*arctan(1 + 6^(1/4)*x))/(32*2^(3/4)) - (5*3^(1/4)*log(sqrt(6) - 6^(3/4)*x + 3*x^2))/(64*2^(3/4)) + (5*3^(1/4)*log(sqrt(6) + 6^(3/4)*x + 3*x^2))/(64*2^(3/4))],
[1/(x^3*(2 + 3*x^4)^2), x, 8, -(1/(8*x^2)) - (3*x^2)/(16*(2 + 3*x^4)) - (3/16)*sqrt(3/2)*arctan(sqrt(3/2)*x^2)],

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


[x^3/(a + b*x^4)^3, x, 2, -(1/(8*b*(a + b*x^4)^2))],


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


[x^3*(a + b*x^4)^2, x, 2, (a + b*x^4)^3/(12*b)],
[x^3*(a + b*x^4)^3, x, 2, (a + b*x^4)^4/(16*b)],


# ::Subsubsection::Closed:: 
#Integrands of the form Pn[x] / (a + b x^4)


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

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


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

[(1 + x + x^2 + x^3)/(1 - x^4), x, 2, -log(1 - x)],
[(1 + x + x^2 + x^3)/(a - b*x^4), x, 9, -(((sqrt(a) - sqrt(b))*arctan((b^(1/4)*x)/a^(1/4)))/(2*a^(3/4)*b^(3/4))) + ((sqrt(a) + sqrt(b))*arctanh((b^(1/4)*x)/a^(1/4)))/(2*a^(3/4)*b^(3/4)) + arctanh((sqrt(b)*x^2)/sqrt(a))/(2*sqrt(a)*sqrt(b)) - log(a - b*x^4)/(4*b)],
[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4), x, 9, ((sqrt(b)*c - sqrt(a)*e)*arctan((b^(1/4)*x)/a^(1/4)))/(2*a^(3/4)*b^(3/4)) + ((sqrt(b)*c + sqrt(a)*e)*arctanh((b^(1/4)*x)/a^(1/4)))/(2*a^(3/4)*b^(3/4)) + (d*arctanh((sqrt(b)*x^2)/sqrt(a)))/(2*sqrt(a)*sqrt(b)) - (f*log(a - b*x^4))/(4*b)],


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


# Integrands of the form x^m/(a+b*(c+d*x)^4) 
[x^3/(a + b*(c + d*x)^4), x, 12, (3*c^2*arctan((sqrt(b)*(c + d*x)^2)/sqrt(a)))/(2*sqrt(a)*sqrt(b)*d^4) + (c*(3*sqrt(a) + sqrt(b)*c^2)*arctan((a^(1/4) - sqrt(2)*b^(1/4)*(c + d*x))/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(3/4)*d^4) - (c*(3*sqrt(a) + sqrt(b)*c^2)*arctan((a^(1/4) + sqrt(2)*b^(1/4)*(c + d*x))/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(3/4)*d^4) - (c*(3*sqrt(a) - sqrt(b)*c^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*(c + d*x) + sqrt(b)*(c + d*x)^2))/(4*sqrt(2)*a^(3/4)*b^(3/4)*d^4) + (c*(3*sqrt(a) - sqrt(b)*c^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*(c + d*x) + sqrt(b)*(c + d*x)^2))/(4*sqrt(2)*a^(3/4)*b^(3/4)*d^4) + log(a + b*(c + d*x)^4)/(4*b*d^4)],
[x^2/(a + b*(c + d*x)^4), x, 10, -((c*arctan((sqrt(b)*(c + d*x)^2)/sqrt(a)))/(sqrt(a)*sqrt(b)*d^3)) - ((sqrt(a) + sqrt(b)*c^2)*arctan((a^(1/4) - sqrt(2)*b^(1/4)*(c + d*x))/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(3/4)*d^3) + ((sqrt(a) + sqrt(b)*c^2)*arctan((a^(1/4) + sqrt(2)*b^(1/4)*(c + d*x))/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(3/4)*d^3) + ((sqrt(a) - sqrt(b)*c^2)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*(c + d*x) + sqrt(b)*(c + d*x)^2))/(4*sqrt(2)*a^(3/4)*b^(3/4)*d^3) - ((sqrt(a) - sqrt(b)*c^2)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*(c + d*x) + sqrt(b)*(c + d*x)^2))/(4*sqrt(2)*a^(3/4)*b^(3/4)*d^3)],
[x/(a + b*(c + d*x)^4), x, 8, (c*arctan((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(3/4)*b^(1/4)*d^2) + arctan((sqrt(b)*(c + d*x)^2)/sqrt(a))/(2*sqrt(a)*sqrt(b)*d^2) + (c*arctanh((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(3/4)*b^(1/4)*d^2)],
[1/(a + b*(c + d*x)^4), x, 4, -(arctan((b^(1/4)*(c + d*x))/(-a)^(1/4))/(2*(-a)^(3/4)*b^(1/4)*d)) - arctanh((b^(1/4)*(c + d*x))/(-a)^(1/4))/(2*(-a)^(3/4)*b^(1/4)*d)],
[1/(x*(a + b*(c + d*x)^4)), x, 14, -((b^(1/4)*c*arctan((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(1/4)*(a + b*c^4))) + (b^(3/4)*c^3*arctan((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(3/4)*(a + b*c^4)) - (sqrt(b)*c^2*arctan((sqrt(b)*(c + d*x)^2)/sqrt(a)))/(2*sqrt(a)*(a + b*c^4)) + (b^(1/4)*c*arctanh((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(1/4)*(a + b*c^4)) + (b^(3/4)*c^3*arctanh((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(3/4)*(a + b*c^4)) + log((-d)*x)/(a + b*c^4) - log(a + b*(c + d*x)^4)/(4*(a + b*c^4))],
[1/(x^2*(a + b*(c + d*x)^4)), x, 15, -(1/((a + b*c^4)*x)) - (b^(1/4)*(a - 3*b*c^4)*d*arctan((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(1/4)*(a + b*c^4)^2) + (b^(3/4)*c^2*(3*a - b*c^4)*d*arctan((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(3/4)*(a + b*c^4)^2) - (sqrt(b)*c*(a - b*c^4)*d*arctan((sqrt(b)*(c + d*x)^2)/sqrt(a)))/(sqrt(a)*(a + b*c^4)^2) + (b^(1/4)*(a - 3*b*c^4)*d*arctanh((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(1/4)*(a + b*c^4)^2) + (b^(3/4)*c^2*(3*a - b*c^4)*d*arctanh((b^(1/4)*(c + d*x))/(-a)^(1/4)))/(2*(-a)^(3/4)*(a + b*c^4)^2) - (4*b*c^3*d*log((-d)*x))/(a + b*c^4)^2 + (b*c^3*d*log(a + b*(c + d*x)^4))/(a + b*c^4)^2],


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


# Integrands of the form (c+d*x)^3*(a+b*(c+d*x)^4)^n where n is an integer 
[(c + d*x)^3*(a + b*(c + d*x)^4)^3, x, 2, (a + b*(c + d*x)^4)^4/(16*b*d)],
[(c + d*x)^3*(a + b*(c + d*x)^4)^2, x, 2, (a + b*(c + d*x)^4)^3/(12*b*d)],
[(c + d*x)^3*(a + b*(c + d*x)^4), x, 2, (a*(c + d*x)^4)/(4*d) + (b*(c + d*x)^8)/(8*d)],
[(c + d*x)^3/(a + b*(c + d*x)^4), x, 2, log(a + b*(c + d*x)^4)/(4*b*d)],
[(c + d*x)^3/(a + b*(c + d*x)^4)^2, x, 2, -(1/(4*b*d*(a + b*(c + d*x)^4)))],
[(c + d*x)^3/(a + b*(c + d*x)^4)^3, x, 2, -(1/(8*b*d*(a + b*(c + d*x)^4)^2))],


# ::Subsubsection::Closed:: 
#Miscellaneous integrands involving quartic binomials


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


# ::Subsection::Closed:: 
#Integrands involving powers of quintic binomials


# ::Subsubsection::Closed:: 
#Integrands of the form x^m (a + b x^5)^n


# Integrands of the form x^m/(1+x^5) where m is an integer 
[x^4/(1 + x^5), x, 2, (1/5)*log(1 + x^5)],
[x^3/(1 + x^5), x, 9, (-(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)],
[x^2/(1 + x^5), x, 9, (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)],
[x/(1 + x^5), x, 9, (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/(1 + x^5), x, 9, (-(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/(x*(1 + x^5)), x, 1, (-(2/5))*arctanh(1 + 2*x^5)],
[1/(x^2*(1 + x^5)), x, 12, -(1/x) + (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/(x^3*(1 + x^5)), x, 12, -(1/(2*x^2)) - (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/(x^4*(1 + x^5)), x, 12, -(1/(3*x^3)) - (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)],


# Integrands of the form x^m/(1-x^5) where m is an integer 
[x^4/(1 - x^5), x, 2, (-(1/5))*log(1 - x^5)],
[x^3/(1 - x^5), x, 9, (-(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)],
[x^2/(1 - x^5), x, 9, (-(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)],
[x/(1 - x^5), x, 9, (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/(1 - x^5), x, 9, (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/(x*(1 - x^5)), x, 1, (-(2/5))*arctanh(1 - 2*x^5)],
[1/(x^2*(1 - x^5)), x, 12, -(1/x) - (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/(x^3*(1 - x^5)), x, 12, -(1/(2*x^2)) - (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/(x^4*(1 - x^5)), x, 12, -(1/(3*x^3)) + (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)],


# Integrands of the form x^m/(a+b*x^5) where m is an integer 
[x^4/(a + b*x^5), x, 2, log(a + b*x^5)/(5*b)],
[1/(a + b*x^5), x, 9, -((sqrt(10 + 2*sqrt(5))*arctan(((1 - sqrt(5))*a^(1/5) - 4*b^(1/5)*x)/(sqrt(10 + 2*sqrt(5))*a^(1/5))))/(10*a^(4/5)*b^(1/5))) - (sqrt(10 - 2*sqrt(5))*arctan(((1 + sqrt(5))*a^(1/5) - 4*b^(1/5)*x)/(sqrt(10 - 2*sqrt(5))*a^(1/5))))/(10*a^(4/5)*b^(1/5)) + log(a^(1/5) + b^(1/5)*x)/(5*a^(4/5)*b^(1/5)) - ((1 - sqrt(5))*log(2*a^(2/5) - (1 - sqrt(5))*a^(1/5)*b^(1/5)*x + 2*b^(2/5)*x^2))/(20*a^(4/5)*b^(1/5)) - ((1 + sqrt(5))*log(2*a^(2/5) - (1 + sqrt(5))*a^(1/5)*b^(1/5)*x + 2*b^(2/5)*x^2))/(20*a^(4/5)*b^(1/5))],
[1/(x*(a + b*x^5)), x, 1, log(x)/a - log(a + b*x^5)/(5*a)],


# ::Subsection::Closed:: 
#Integrands involving powers of sextic binomials


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b x^6)


# Integrands of the form x^m/(1-x^6) where m is an integer 
[x^8/(1 - x^6), x, 4, -(x^3/3) + arctanh(x^3)/3],
[x^7/(1 - x^6), x, 7, -(x^2/2) + arctan((1 + 2*x^2)/sqrt(3))/(2*sqrt(3)) - (1/6)*log(1 - x^2) + (1/12)*log(1 + x^2 + x^4)],
[x^6/(1 - x^6), x, 7, -x + arctan((sqrt(3)*x)/(1 - x^2))/(2*sqrt(3)) + arctanh(x)/3 + (1/6)*arctanh(x/(1 + x^2))],
[x^5/(1 - x^6), x, 2, (-(1/6))*log(1 - x^6)],
[x^4/(1 - x^6), x, 5, -(arctan((sqrt(3)*x)/(1 - x^2))/(2*sqrt(3))) + arctanh(x)/3 + (1/6)*arctanh(x/(1 + x^2))],
[x^3/(1 - x^6), x, 5, -(arctan((1 + 2*x^2)/sqrt(3))/(2*sqrt(3))) - (1/6)*log(1 - x^2) + (1/12)*log(1 + x^2 + x^4)],
[x^2/(1 - x^6), x, 2, arctanh(x^3)/3],
[x/(1 - x^6), x, 5, arctan((1 + 2*x^2)/sqrt(3))/(2*sqrt(3)) - (1/6)*log(1 - x^2) + (1/12)*log(1 + x^2 + x^4)],
[1/(1 - x^6), x, 5, arctan((sqrt(3)*x)/(1 - x^2))/(2*sqrt(3)) + arctanh(x)/3 + (1/6)*arctanh(x/(1 + x^2))],
[1/(x*(1 - x^6)), x, 1, (-(1/3))*arctanh(1 - 2*x^6)],
[1/(x^2*(1 - x^6)), x, 8, -(1/x) - arctan((sqrt(3)*x)/(1 - x^2))/(2*sqrt(3)) + arctanh(x)/3 + (1/6)*arctanh(x/(1 + x^2))],
[1/(x^3*(1 - x^6)), x, 8, -(1/(2*x^2)) - arctan((1 + 2*x^2)/sqrt(3))/(2*sqrt(3)) - (1/6)*log(1 - x^2) + (1/12)*log(1 + x^2 + x^4)],
[1/(x^4*(1 - x^6)), x, 5, -(1/(3*x^3)) + arctanh(x^3)/3],
[1/(x^5*(1 - x^6)), x, 8, -(1/(4*x^4)) + arctan((1 + 2*x^2)/sqrt(3))/(2*sqrt(3)) - (1/6)*log(1 - x^2) + (1/12)*log(1 + x^2 + x^4)],


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


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

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


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b x^6)^2


[x^8/(a + b*x^6)^2, x, 3, -x^3/(6*b*(a + b*x^6)) + arctan((sqrt(b)*x^3)/sqrt(a))/(6*sqrt(a)*b^(3/2))],
[x^7/(a + b*x^6)^2, x, 6, -(x^2/(6*b*(a + b*x^6))) - arctan((a^(1/3) - 2*b^(1/3)*x^2)/(sqrt(3)*a^(1/3)))/(6*sqrt(3)*a^(2/3)*b^(4/3)) + log(a^(1/3) + b^(1/3)*x^2)/(18*a^(2/3)*b^(4/3)) - log(a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4)/(36*a^(2/3)*b^(4/3))],
[x^5/(a + b*x^6)^2, x, 2, -(1/(6*b*(a + b*x^6)))],
[x^3/(a + b*x^6)^2, x, 6, x^4/(6*a*(a + b*x^6)) - arctan((a^(1/3) - 2*b^(1/3)*x^2)/(sqrt(3)*a^(1/3)))/(6*sqrt(3)*a^(4/3)*b^(2/3)) - log(a^(1/3) + b^(1/3)*x^2)/(18*a^(4/3)*b^(2/3)) + log(a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4)/(36*a^(4/3)*b^(2/3))],
[x^2/(a + b*x^6)^2, x, 3, x^3/(6*a*(a + b*x^6)) + arctan((sqrt(b)*x^3)/sqrt(a))/(6*a^(3/2)*sqrt(b))],
[x/(a + b*x^6)^2, x, 6, x^2/(6*a*(a + b*x^6)) - arctan((a^(1/3) - 2*b^(1/3)*x^2)/(sqrt(3)*a^(1/3)))/(3*sqrt(3)*a^(5/3)*b^(1/3)) + log(a^(1/3) + b^(1/3)*x^2)/(9*a^(5/3)*b^(1/3)) - log(a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4)/(18*a^(5/3)*b^(1/3))],
[1/(x*(a + b*x^6)^2), x, 7, 1/(6*a*(a + b*x^6)) + log(x)/a^2 - log(a + b*x^6)/(6*a^2)],
[1/(x^3*(a + b*x^6)^2), x, 14, -(1/(2*a^2*x^2)) - (b*x^4)/(6*a^2*(a + b*x^6)) + (2*b^(1/3)*arctan((a^(1/3) - 2*b^(1/3)*x^2)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(7/3)) + (2*b^(1/3)*log(a^(1/3) + b^(1/3)*x^2))/(9*a^(7/3)) - (b^(1/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4))/(9*a^(7/3))],
[1/(x^4*(a + b*x^6)^2), x, 8, -(1/(3*a^2*x^3)) - (b*x^3)/(6*a^2*(a + b*x^6)) - (sqrt(b)*arctan((sqrt(b)*x^3)/sqrt(a)))/(2*a^(5/2))],


# ::Subsubsection::Closed:: 
#Miscellaneous integrands involving sextic binomials


[(1 + x^6)/(-1 + x^6), x, 7, x - arctan((sqrt(3)*x)/(1 - x^2))/sqrt(3) - (2*arctanh(x))/3 - (1/3)*arctanh(x/(1 + x^2))],
# Requires simplification of integrand before overly aggressive expansion 
[(1/x^3 + x^3)/(-(1/x^3) + x^3), x, 7, x - arctan((sqrt(3)*x)/(1 - x^2))/sqrt(3) - (2*arctanh(x))/3 - (1/3)*arctanh(x/(1 + x^2)), x + arctan((1 - 2*x)/sqrt(3))/sqrt(3) - arctan((1 + 2*x)/sqrt(3))/sqrt(3) - (2*arctanh(x))/3 + (1/6)*log(1 - x + x^2) - (1/6)*log(1 + x + x^2)],


# ::Subsection::Closed:: 
#Integrands involving powers of higher degree binomials


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b x^7)


[1/(1 - x^7), x, 9, (2/7)*arctan(sec(Pi/14)*(x + sin(Pi/14)))*cos(Pi/14) + (2/7)*arctan(sec((3*Pi)/14)*(x - sin((3*Pi)/14)))*cos((3*Pi)/14) - (1/7)*log(1 - x) + (1/7)*cos(Pi/7)*log(1 + x^2 + 2*x*cos(Pi/7)) + (1/7)*log(1 + x^2 + 2*x*sin(Pi/14))*sin(Pi/14) + (2/7)*arctan((x + cos(Pi/7))*csc(Pi/7))*sin(Pi/7) - (1/7)*log(1 + x^2 - 2*x*sin((3*Pi)/14))*sin((3*Pi)/14)],
[1/(1 + x^7), x, 9, (2/7)*arctan(sec(Pi/14)*(x - sin(Pi/14)))*cos(Pi/14) + (2/7)*arctan(sec((3*Pi)/14)*(x + sin((3*Pi)/14)))*cos((3*Pi)/14) + (1/7)*log(1 + x) - (1/7)*cos(Pi/7)*log(1 + x^2 - 2*x*cos(Pi/7)) - (1/7)*log(1 + x^2 - 2*x*sin(Pi/14))*sin(Pi/14) + (2/7)*arctan((x - cos(Pi/7))*csc(Pi/7))*sin(Pi/7) + (1/7)*log(1 + x^2 + 2*x*sin((3*Pi)/14))*sin((3*Pi)/14)],
# {1/(a + b*x^7), x, 0, (2*ArcTan[(b^(1/7)*x*Sec[Pi/14])/a^(1/7) - Tan[Pi/14]]*Cos[Pi/14] + 2*ArcTan[(b^(1/7)*x*Sec[(3*Pi)/14])/a^(1/7) + Tan[(3*Pi)/14]]*Cos[(3*Pi)/14] + Log[a^(1/7)/b^(1/7) + x] - Cos[Pi/7]*Log[a^(2/7)/b^(2/7) + x^2 - (2*a^(1/7)*x*Cos[Pi/7])/b^(1/7)] - Log[a^(2/7)/b^(2/7) + x^2 - (2*a^(1/7)*x*Sin[Pi/14])/b^(1/7)]*Sin[Pi/14] - 2*ArcTan[Cot[Pi/7] - (b^(1/7)*x*Csc[Pi/7])/a^(1/7)]*Sin[Pi/7] + Log[a^(2/7)/b^(2/7) + x^2 + (2*a^(1/7)*x*Sin[(3*Pi)/14])/b^(1/7)]*Sin[(3*Pi)/14])/(7*a^(6/7)*b^(1/7))} 


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b x^8)


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

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


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b x^10)


[1/(1 - x^10), x, 9, (1/20)*sqrt(10 - 2*sqrt(5))*arctan((sqrt(10 - 2*sqrt(5))*x)/(2*(1 - x^2))) + (1/20)*sqrt(10 + 2*sqrt(5))*arctan((sqrt(10 + 2*sqrt(5))*x)/(2*(1 - x^2))) + arctanh(x)/5 + (1/20)*(1 - sqrt(5))*arctanh(((1 - sqrt(5))*x)/(2*(1 + x^2))) + (1/20)*(1 + sqrt(5))*arctanh(((1 + sqrt(5))*x)/(2*(1 + x^2)))],


# ::Subsubsection::Closed:: 
#Integrands of the form x^m / (a + b x^12)


[x^5/(9 + x^12), x, 2, arctan(x^6/3)/18],
[x^5/(9 - x^12), x, 2, arctanh(x^6/3)/18],


# ::Subsection::Closed:: 
#Integrands involving powers of improper binomials


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


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


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


# Integrands of the form x^m/(a+b/x)^3 where m is an integer 
[x^4/(a + b/x)^3, x, 6, (15*b^4*x)/a^7 - (5*b^3*x^2)/a^6 + (2*b^2*x^3)/a^5 - (3*b*x^4)/(4*a^4) + x^5/(5*a^3) + b^7/(2*a^8*(b + a*x)^2) - (7*b^6)/(a^8*(b + a*x)) - (21*b^5*log(b + a*x))/a^8],
[x^3/(a + b/x)^3, x, 6, -((10*b^3*x)/a^6) + (3*b^2*x^2)/a^5 - (b*x^3)/a^4 + x^4/(4*a^3) - b^6/(2*a^7*(b + a*x)^2) + (6*b^5)/(a^7*(b + a*x)) + (15*b^4*log(b + a*x))/a^7],
[x^2/(a + b/x)^3, x, 6, (6*b^2*x)/a^5 - (3*b*x^2)/(2*a^4) + x^3/(3*a^3) + b^5/(2*a^6*(b + a*x)^2) - (5*b^4)/(a^6*(b + a*x)) - (10*b^3*log(b + a*x))/a^6],
[x/(a + b/x)^3, x, 6, -((3*b*x)/a^4) + x^2/(2*a^3) - b^4/(2*a^5*(b + a*x)^2) + (4*b^3)/(a^5*(b + a*x)) + (6*b^2*log(b + a*x))/a^5],
[1/(a + b/x)^3, x, 5, x/a^3 + b^3/(2*a^4*(b + a*x)^2) - (3*b^2)/(a^4*(b + a*x)) - (3*b*log(b + a*x))/a^4],
[1/(x*(a + b/x)^3), x, 5, -(b^2/(2*a^3*(b + a*x)^2)) + (2*b)/(a^3*(b + a*x)) + log(b + a*x)/a^3],
[1/(x^2*(a + b/x)^3), x, 2, 1/(2*b*(a + b/x)^2)],
[1/(x^3*(a + b/x)^3), x, 1, -(1/(2*a*(b + a*x)^2))],
[1/(x^4*(a + b/x)^3), x, 6, 1/(2*b*(b + a*x)^2) + 1/(b^2*(b + a*x)) + log(x)/b^3 - log(b + a*x)/b^3],


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


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


# Integrands of the form x^m/(a+b/x^2)^3 where m is an integer 
[x^4/(a + b/x^2)^3, x, 9, (6*b^2*x)/a^5 - (b*x^3)/a^4 + x^5/(5*a^3) - (b^4*x)/(4*a^5*(b + a*x^2)^2) + (17*b^3*x)/(8*a^5*(b + a*x^2)) - (63*b^(5/2)*arctan((sqrt(a)*x)/sqrt(b)))/(8*a^(11/2))],
[x^3/(a + b/x^2)^3, x, 7, -((3*b*x^2)/(2*a^4)) + x^4/(4*a^3) - b^4/(4*a^5*(b + a*x^2)^2) + (2*b^3)/(a^5*(b + a*x^2)) + (3*b^2*log(b + a*x^2))/a^5],
[x^2/(a + b/x^2)^3, x, 9, -((3*b*x)/a^4) + x^3/(3*a^3) + (b^3*x)/(4*a^4*(b + a*x^2)^2) - (13*b^2*x)/(8*a^4*(b + a*x^2)) + (35*b^(3/2)*arctan((sqrt(a)*x)/sqrt(b)))/(8*a^(9/2))],
[x/(a + b/x^2)^3, x, 6, x^2/(2*a^3) + b^3/(4*a^4*(b + a*x^2)^2) - (3*b^2)/(2*a^4*(b + a*x^2)) - (3*b*log(b + a*x^2))/(2*a^4)],
[1/(a + b/x^2)^3, x, 8, x/a^3 - (b^2*x)/(4*a^3*(b + a*x^2)^2) + (9*b*x)/(8*a^3*(b + a*x^2)) - (15*sqrt(b)*arctan((sqrt(a)*x)/sqrt(b)))/(8*a^(7/2))],
[1/(x*(a + b/x^2)^3), x, 8, -(b^2/(4*a^3*(b + a*x^2)^2)) + b/(a^3*(b + a*x^2)) + log(b + a*x^2)/(2*a^3)],
[1/(x^2*(a + b/x^2)^3), x, 4, -(x^3/(4*a*(b + a*x^2)^2)) - (3*x)/(8*a^2*(b + a*x^2)) - (3*arctan(sqrt(b)/(sqrt(a)*x)))/(8*a^(5/2)*sqrt(b))],
[1/(x^3*(a + b/x^2)^3), x, 2, 1/(4*b*(a + b/x^2)^2)],
[1/(x^4*(a + b/x^2)^3), x, 4, -(x/(4*a*(b + a*x^2)^2)) + x/(8*a*b*(b + a*x^2)) + arctan((sqrt(a)*x)/sqrt(b))/(8*a^(3/2)*b^(3/2))],


# ::Subsubsection::Closed:: 
#Integrands of the form x^m (a + b/x^p)^n where p>2


# Integrands of the form 1/(a+b/x^n) where n>0 is an integer 
[1/(a + b/x^3), x, 6, x/a + (b^(1/3)*arctan((b^(1/3) - 2*a^(1/3)*x)/(sqrt(3)*b^(1/3))))/(sqrt(3)*a^(4/3)) - (b^(1/3)*log(b^(1/3) + a^(1/3)*x))/(3*a^(4/3)) + (b^(1/3)*log(b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2))/(6*a^(4/3))],
[1/(a + b/x^4), x, 5, x/a - ((-b)^(1/4)*arctan((a^(1/4)*x)/(-b)^(1/4)))/(2*a^(5/4)) - ((-b)^(1/4)*arctanh((a^(1/4)*x)/(-b)^(1/4)))/(2*a^(5/4))],
[1/(a + b/x^5), x, 11, x/a + (sqrt(10 + 2*sqrt(5))*b^(1/5)*arctan(((1 - sqrt(5))*b^(1/5) - 4*a^(1/5)*x)/(sqrt(10 + 2*sqrt(5))*b^(1/5))))/(10*a^(6/5)) + (sqrt(10 - 2*sqrt(5))*b^(1/5)*arctan(((1 + sqrt(5))*b^(1/5) - 4*a^(1/5)*x)/(sqrt(10 - 2*sqrt(5))*b^(1/5))))/(10*a^(6/5)) - (b^(1/5)*log(b^(1/5) + a^(1/5)*x))/(5*a^(6/5)) + ((1 - sqrt(5))*b^(1/5)*log(2*b^(2/5) - (1 - sqrt(5))*a^(1/5)*b^(1/5)*x + 2*a^(2/5)*x^2))/(20*a^(6/5)) + ((1 + sqrt(5))*b^(1/5)*log(2*b^(2/5) - (1 + sqrt(5))*a^(1/5)*b^(1/5)*x + 2*a^(2/5)*x^2))/(20*a^(6/5))],
[1/(a + b/x^6), x, 7, x/a - (sqrt(sqrt(3)*sqrt(-b^(2/3)) - b^(1/3))*arctan((sqrt(2)*a^(1/6)*x)/sqrt(sqrt(3)*sqrt(-b^(2/3)) - b^(1/3))))/(3*sqrt(2)*a^(7/6)) - (b^(1/6)*arctan((a^(1/6)*x)/b^(1/6)))/(3*a^(7/6)) - (sqrt(sqrt(3)*sqrt(-b^(2/3)) + b^(1/3))*arctanh((sqrt(2)*a^(1/6)*x)/sqrt(sqrt(3)*sqrt(-b^(2/3)) + b^(1/3))))/(3*sqrt(2)*a^(7/6))],
# {1/(a + b/x^7), x, 2, x/a - (b^(1/7)*(2*ArcTan[(a^(1/7)*x*Sec[Pi/14])/b^(1/7) - Tan[Pi/14]]*Cos[Pi/14] + 2*ArcTan[(a^(1/7)*x*Sec[(3*Pi)/14])/b^(1/7) + Tan[(3*Pi)/14]]*Cos[(3*Pi)/14] + Log[b^(1/7)/a^(1/7) + x] - Cos[Pi/7]*Log[b^(2/7)/a^(2/7) + x^2 - (2*b^(1/7)*x*Cos[Pi/7])/a^(1/7)] - Log[b^(2/7)/a^(2/7) + x^2 - (2*b^(1/7)*x*Sin[Pi/14])/a^(1/7)]*Sin[Pi/14] - 2*ArcTan[Cot[Pi/7] - (a^(1/7)*x*Csc[Pi/7])/b^(1/7)]*Sin[Pi/7] + Log[b^(2/7)/a^(2/7) + x^2 + (2*b^(1/7)*x*Sin[(3*Pi)/14])/a^(1/7)]*Sin[(3*Pi)/14]))/(7*a^(8/7))} 
# {1/(a + b/x^8), x, 2, x/a - (b^(1/8)*(2*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) - Tan[Pi/8]]*Cos[Pi/8] + 2*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - Cos[Pi/8]*Log[b^(1/4)/a^(1/4) + x^2 - (2*b^(1/8)*x*Cos[Pi/8])/a^(1/8)] + Cos[Pi/8]*Log[b^(1/4)/a^(1/4) + x^2 + (2*b^(1/8)*x*Cos[Pi/8])/a^(1/8)] - 2*ArcTan[Cot[Pi/8] - (a^(1/8)*x*Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] + 2*ArcTan[Cot[Pi/8] + (a^(1/8)*x*Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] - Log[b^(1/4)/a^(1/4) + x^2 - (2*b^(1/8)*x*Sin[Pi/8])/a^(1/8)]*Sin[Pi/8] + Log[b^(1/4)/a^(1/4) + x^2 + (2*b^(1/8)*x*Sin[Pi/8])/a^(1/8)]*Sin[Pi/8]))/(8*a^(9/8))} 


# ::Subsection::Closed:: 
#Miscellaneous integrands involving powers of binomials


# Integrands of the form x^m*(a+b*x^n)^p where n*p+n+m+1=0 
[(a + b*x^n)^p/x^(n*p + n + 1), x, 1, -((x^(-n - n*p)*(a + b*x^n)^(1 + p))/(a*n*(1 + p)))],
[(a + b*x^n)^8/x^(n*8 + n + 1), x, 1, -((a + b*x^n)^9/(x^(9*n)*(9*a*n)))],
[(a + b*x^3)^p/x^(3*p + 3 + 1), x, 1, -((x^(-3 - 3*p)*(a + b*x^3)^(1 + p))/(3*a*(1 + p)))],
[(a + b*x^3)^8/x^(3*8 + 3 + 1), x, 1, -((a + b*x^3)^9/(27*a*x^27))],
[(a + b*x^n)^(-1)/x^(n*(-1) + n + 1), x, 1, log(x)/a - log(a + b*x^n)/(a*n)],
[(a + b*x^3)^(-1)/x^(3*(-1) + 3 + 1), x, 1, log(x)/a - log(a + b*x^3)/(3*a)],


[(-1 + x)/(1 - x + x^2), x, 2, arctan((1 - 2*x)/sqrt(3))/sqrt(3) + (1/2)*log(1 - x + x^2)],
# Need to cancel gcd to get simpler answer. 
[(-1 + x^2)/(1 + x^3), x, 3, arctan((1 - 2*x)/sqrt(3))/sqrt(3) + (1/2)*log(1 - x + x^2)],


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


# Integrands of the form 1/(a+b*(c+d*x)^2)^n where n is an integer 
[1/(a + b*(c + d*x)^2), x, 2, arctan((sqrt(b)*(c + d*x))/sqrt(a))/(sqrt(a)*sqrt(b)*d)],
[1/(a + b*(c + d*x)^2)^2, x, 3, (c + d*x)/(2*a*d*(a + b*c^2 + 2*b*c*d*x + b*d^2*x^2)) + arctan((sqrt(b)*(c + d*x))/sqrt(a))/(2*a^(3/2)*sqrt(b)*d)],
[1/(a + b*(c + d*x)^2)^3, x, 4, (c + d*x)/(4*a*d*(a + b*c^2 + 2*b*c*d*x + b*d^2*x^2)^2) + (3*(c + d*x))/(8*a^2*d*(a + b*c^2 + 2*b*c*d*x + b*d^2*x^2)) + (3*arctan((sqrt(b)*(c + d*x))/sqrt(a)))/(8*a^(5/2)*sqrt(b)*d)],

[1/(1 + (c + d*x)^2), x, 2, arctan(c + d*x)/d],
[1/(1 + (c + d*x)^2)^2, x, 3, (c + d*x)/(2*d*(1 + c^2 + 2*c*d*x + d^2*x^2)) + arctan(c + d*x)/(2*d)],
[1/(1 + (c + d*x)^2)^3, x, 4, (c + d*x)/(4*d*(1 + c^2 + 2*c*d*x + d^2*x^2)^2) + (3*(c + d*x))/(8*d*(1 + c^2 + 2*c*d*x + d^2*x^2)) + (3*arctan(c + d*x))/(8*d)],

[1/(1 - (c + d*x)^2), x, 2, arctanh(c + d*x)/d],
[1/(1 - (c + d*x)^2)^2, x, 3, (c + d*x)/(2*d*(1 - c^2 - 2*c*d*x - d^2*x^2)) + arctanh(c + d*x)/(2*d)],
[1/(1 - (c + d*x)^2)^3, x, 4, (c + d*x)/(4*d*(1 - c^2 - 2*c*d*x - d^2*x^2)^2) + (3*(c + d*x))/(8*d*(1 - c^2 - 2*c*d*x - d^2*x^2)) + (3*arctanh(c + d*x))/(8*d)],

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