# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.3 General\1.1.3.2 (c x)^m (a+b x^n)^p.txt"

lst:=[

# Integrands of the form (c x)^m (b x^n)^p

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

# Integrands of the form (c x)^m (b x^2)^p

# Integrands of the form x^m (b x^2)^(p/2)

# p>0
[x^3*sqrt(b*x^2),x,2,1/5*x^4*sqrt(b*x^2)],
[x^2*sqrt(b*x^2),x,2,1/4*x^3*sqrt(b*x^2)],
[x*sqrt(b*x^2),x,2,1/3*x^2*sqrt(b*x^2)],
[sqrt(b*x^2),x,2,1/2*x*sqrt(b*x^2)],
[sqrt(b*x^2)/x,x,2,sqrt(b*x^2)],
[sqrt(b*x^2)/x^2,x,2,log(x)*sqrt(b*x^2)/x],
[sqrt(b*x^2)/x^3,x,2,-sqrt(b*x^2)/x^2],
[sqrt(b*x^2)/x^4,x,2,-1/2*sqrt(b*x^2)/x^3],
[sqrt(b*x^2)/x^5,x,2,-1/3*sqrt(b*x^2)/x^4],
[x^2*sqrt(x^2),x,2,1/4*x^3*sqrt(x^2)],
[x^2*(b*x^2)^(3/2),x,2,1/6*b*x^5*sqrt(b*x^2)],
[x*(b*x^2)^(3/2),x,2,1/5*b*x^4*sqrt(b*x^2)],
[(b*x^2)^(3/2),x,2,1/4*b*x^3*sqrt(b*x^2)],
[(b*x^2)^(3/2)/x,x,2,1/3*b*x^2*sqrt(b*x^2)],
[(b*x^2)^(3/2)/x^2,x,2,1/2*b*x*sqrt(b*x^2)],
[(b*x^2)^(3/2)/x^3,x,2,b*sqrt(b*x^2)],
[(b*x^2)^(3/2)/x^4,x,2,b*log(x)*sqrt(b*x^2)/x],
[(b*x^2)^(3/2)/x^5,x,2,-b*sqrt(b*x^2)/x^2],
[(b*x^2)^(3/2)/x^6,x,2,-1/2*b*sqrt(b*x^2)/x^3],
[(b*x^2)^(3/2)/x^7,x,2,-1/3*b*sqrt(b*x^2)/x^4],
[(x^2)^(5/2),x,2,1/6*x^5*sqrt(x^2)],
[x*(b*x^2)^(5/2),x,2,1/7*b^2*x^6*sqrt(b*x^2)],
[(b*x^2)^(5/2),x,2,1/6*b^2*x^5*sqrt(b*x^2)],
[(b*x^2)^(5/2)/x,x,2,1/5*b^2*x^4*sqrt(b*x^2)],
[(b*x^2)^(5/2)/x^2,x,2,1/4*b^2*x^3*sqrt(b*x^2)],
[(b*x^2)^(5/2)/x^3,x,2,1/3*b^2*x^2*sqrt(b*x^2)],
[(b*x^2)^(5/2)/x^4,x,2,1/2*b^2*x*sqrt(b*x^2)],
[(b*x^2)^(5/2)/x^5,x,2,b^2*sqrt(b*x^2)],
[(b*x^2)^(5/2)/x^6,x,2,b^2*log(x)*sqrt(b*x^2)/x],
[(b*x^2)^(5/2)/x^7,x,2,-b^2*sqrt(b*x^2)/x^2],
[(b*x^2)^(5/2)/x^8,x,2,-1/2*b^2*sqrt(b*x^2)/x^3],
[(b*x^2)^(5/2)/x^9,x,2,-1/3*b^2*sqrt(b*x^2)/x^4],
[(x^2)^(7/2),x,2,1/8*x^7*sqrt(x^2)],

# p<0
[x^3/sqrt(b*x^2),x,2,1/3*x^4/sqrt(b*x^2)],
[x/sqrt(b*x^2),x,2,x^2/sqrt(b*x^2)],
[1/(x*sqrt(b*x^2)),x,2,(-1)/sqrt(b*x^2)],
[1/(x^3*sqrt(b*x^2)),x,2,(-1/3)/(x^2*sqrt(b*x^2))],
[x^2/sqrt(b*x^2),x,2,1/2*x^3/sqrt(b*x^2)],
[1/sqrt(b*x^2),x,2,x*log(x)/sqrt(b*x^2)],
[1/(x^2*sqrt(b*x^2)),x,2,(-1/2)/(x*sqrt(b*x^2))],
[x^5/(b*x^2)^(3/2),x,2,1/3*x^4/(b*sqrt(b*x^2))],
[x^3/(b*x^2)^(3/2),x,2,x^2/(b*sqrt(b*x^2))],
[x/(b*x^2)^(3/2),x,2,(-1)/(b*sqrt(b*x^2))],
[1/(x*(b*x^2)^(3/2)),x,2,(-1/3)/(b*x^2*sqrt(b*x^2))],
[1/(x^3*(b*x^2)^(3/2)),x,2,(-1/5)/(b*x^4*sqrt(b*x^2))],
[x^6/(b*x^2)^(3/2),x,2,1/4*x^5/(b*sqrt(b*x^2))],
[x^4/(b*x^2)^(3/2),x,2,1/2*x^3/(b*sqrt(b*x^2))],
[x^2/(b*x^2)^(3/2),x,2,x*log(x)/(b*sqrt(b*x^2))],
[1/(b*x^2)^(3/2),x,2,(-1/2)/(b*x*sqrt(b*x^2))],
[1/(x^2*(b*x^2)^(3/2)),x,2,(-1/4)/(b*x^3*sqrt(b*x^2))],
[x^7/(b*x^2)^(5/2),x,2,1/3*x^4/(b^2*sqrt(b*x^2))],
[x^5/(b*x^2)^(5/2),x,2,x^2/(b^2*sqrt(b*x^2))],
[x^3/(b*x^2)^(5/2),x,2,(-1)/(b^2*sqrt(b*x^2))],
[x/(b*x^2)^(5/2),x,2,(-1/3)/(b^2*x^2*sqrt(b*x^2))],
[1/(x*(b*x^2)^(5/2)),x,2,(-1/5)/(b^2*x^4*sqrt(b*x^2))],
[x^6/(b*x^2)^(5/2),x,2,1/2*x^3/(b^2*sqrt(b*x^2))],
[x^4/(b*x^2)^(5/2),x,2,x*log(x)/(b^2*sqrt(b*x^2))],
[x^2/(b*x^2)^(5/2),x,2,(-1/2)/(b^2*x*sqrt(b*x^2))],
[1/(b*x^2)^(5/2),x,2,(-1/4)/(b^2*x^3*sqrt(b*x^2))],
[1/(x^2*(b*x^2)^(5/2)),x,2,(-1/6)/(b^2*x^5*sqrt(b*x^2))],

# Integrands of the form (c x)^(m/2) (b x^2)^(p/2)

# Integrands of the form (c x)^m (b x^2)^(p/2) with m symbolic
[(c*x)^m*(b*x^2)^(3/2),x,3,b*(c*x)^(4+m)*sqrt(b*x^2)/(c^4*(4+m)*x)],
[(c*x)^m*(b*x^2)^(1/2),x,3,(c*x)^(2+m)*sqrt(b*x^2)/(c^2*(2+m)*x)],
[(c*x)^m/(b*x^2)^(1/2),x,3,x*(c*x)^m/(m*sqrt(b*x^2))],
[(c*x)^m/(b*x^2)^(3/2),x,3,-c^2*x*(c*x)^(-2+m)/(b*(2-m)*sqrt(b*x^2))],

# Integrands of the form (c x)^m (b x^2)^p with p symbolic
[x^m*(b*x^2)^p,x,2,x^(1+m)*(b*x^2)^p/(1+m+2*p)],
[(c*x)^m*(b*x^2)^p,x,3,x*(c*x)^m*(b*x^2)^p/(1+m+2*p)],
[x^(-1-2*p)*(x^2)^p,x,2,(x^2)^p*log(x)/x^(2*p)],
[x^3*(b*x^2)^p,x,2,1/2*x^4*(b*x^2)^p/(2+p)],
[x^2*(b*x^2)^p,x,2,x^3*(b*x^2)^p/(3+2*p)],
[x*(b*x^2)^p,x,2,1/2*x^2*(b*x^2)^p/(1+p)],
[(b*x^2)^p,x,2,x*(b*x^2)^p/(1+2*p)],
[(b*x^2)^p/x,x,2,1/2*(b*x^2)^p/p],
[(b*x^2)^p/x^2,x,2,-(b*x^2)^p/((1-2*p)*x)],
[(b*x^2)^p/x^3,x,2,-1/2*(b*x^2)^p/((1-p)*x^2)],
[(b*x^2)^p/x^4,x,2,-(b*x^2)^p/((3-2*p)*x^3)],

# Integrands of the form (c x)^m (b x^n)^p

# Integrands of the form (b x^n)^(p/2)

# p>0
[sqrt(b*x^3),x,2,2/5*x*sqrt(b*x^3)],
[sqrt(b*x^2),x,2,1/2*x*sqrt(b*x^2)],
[sqrt(b*x),x,1,2/3*(b*x)^(3/2)/b],
[sqrt(b/x),x,2,2*x*sqrt(b/x)],
[sqrt(b/x^2),x,2,x*log(x)*sqrt(b/x^2)],
[sqrt(b/x^3),x,2,-2*x*sqrt(b/x^3)],
[(b*x^3)^(3/2),x,2,2/11*b*x^4*sqrt(b*x^3)],
[(b*x^2)^(3/2),x,2,1/4*b*x^3*sqrt(b*x^2)],
[(b*x)^(3/2),x,1,2/5*(b*x)^(5/2)/b],
[(b/x)^(3/2),x,2,-2*b*sqrt(b/x)],
[(b/x^2)^(3/2),x,2,-1/2*b*sqrt(b/x^2)/x],
[(b/x^3)^(3/2),x,2,-2/7*b*sqrt(b/x^3)/x^2],

# p<0
[1/sqrt(b*x^3),x,2,-2*x/sqrt(b*x^3)],
[1/sqrt(b*x^2),x,2,x*log(x)/sqrt(b*x^2)],
[1/sqrt(b*x),x,1,2*sqrt(b*x)/b],
[1/sqrt(b/x),x,2,2/3*x/sqrt(b/x)],
[1/sqrt(b/x^2),x,2,1/2*x/sqrt(b/x^2)],
[1/sqrt(b/x^3),x,2,2/5*x/sqrt(b/x^3)],
[1/(b*x^3)^(3/2),x,2,(-2/7)/(b*x^2*sqrt(b*x^3))],
[1/(b*x^2)^(3/2),x,2,(-1/2)/(b*x*sqrt(b*x^2))],
[1/(b*x)^(3/2),x,1,(-2)/(b*sqrt(b*x))],
[1/(b/x)^(3/2),x,2,2/5*x^2/(b*sqrt(b/x))],
[1/(b/x^2)^(3/2),x,2,1/4*x^3/(b*sqrt(b/x^2))],
[1/(b/x^3)^(3/2),x,2,2/11*x^4/(b*sqrt(b/x^3))],

# Integrands of the form (b x^n)^(p/3)

# p>0
[(b*x^n)^(1/3),x,2,3*x*(b*x^n)^(1/3)/(3+n)],
[(b*x^3)^(1/3),x,2,1/2*x*(b*x^3)^(1/3)],
[(b*x^2)^(1/3),x,2,3/5*x*(b*x^2)^(1/3)],
[(b*x)^(1/3),x,1,3/4*(b*x)^(4/3)/b],
[(b/x)^(1/3),x,2,3/2*(b/x)^(1/3)*x],
[(b/x^2)^(1/3),x,2,3*(b/x^2)^(1/3)*x],
[(b/x^3)^(1/3),x,2,(b/x^3)^(1/3)*x*log(x)],
[(b/x^4)^(1/3),x,2,-3*(b/x^4)^(1/3)*x],
[(b*x^n)^(2/3),x,2,3*x*(b*x^n)^(2/3)/(3+2*n)],
[(b*x^2)^(2/3),x,2,3/7*x*(b*x^2)^(2/3)],
[(b*x)^(2/3),x,1,3/5*(b*x)^(5/3)/b],
[(b/x)^(2/3),x,2,3*(b/x)^(2/3)*x],
[(b/x^2)^(2/3),x,2,-3*(b/x^2)^(2/3)*x],
[(b/x^3)^(2/3),x,2,-(b/x^3)^(2/3)*x],
[(b/x^4)^(2/3),x,2,-3/5*(b/x^4)^(2/3)*x],

# p<0
[1/(b*x^n)^(1/3),x,2,3*x/((3-n)*(b*x^n)^(1/3))],
[1/(b*x^4)^(1/3),x,2,-3*x/(b*x^4)^(1/3)],
[1/(b*x^3)^(1/3),x,2,x*log(x)/(b*x^3)^(1/3)],
[1/(b*x^2)^(1/3),x,2,3*x/(b*x^2)^(1/3)],
[1/(b*x)^(1/3),x,1,3/2*(b*x)^(2/3)/b],
[1/(b/x)^(1/3),x,2,3/4*x/(b/x)^(1/3)],
[1/(b/x^2)^(1/3),x,2,3/5*x/(b/x^2)^(1/3)],
[1/(b/x^3)^(1/3),x,2,1/2*x/(b/x^3)^(1/3)],
[1/(b*x^n)^(2/3),x,2,3*x/((3-2*n)*(b*x^n)^(2/3))],
[1/(b*x^3)^(2/3),x,2,-x/(b*x^3)^(2/3)],
[1/(b*x^2)^(2/3),x,2,-3*x/(b*x^2)^(2/3)],
[1/(b*x)^(2/3),x,1,3*(b*x)^(1/3)/b],
[1/(b/x)^(2/3),x,2,3/5*x/(b/x)^(2/3)],
[1/(b/x^2)^(2/3),x,2,3/7*x/(b/x^2)^(2/3)],
[1/(b/x^3)^(2/3),x,2,1/3*x/(b/x^3)^(2/3)],

# Integrands of the form x^m (b x^n)^(p/2)

# p>0
[x^2*sqrt(b*x^n),x,2,2*x^3*sqrt(b*x^n)/(6+n)],
[x*sqrt(b*x^n),x,2,2*x^2*sqrt(b*x^n)/(4+n)],
[sqrt(b*x^n),x,2,2*x*sqrt(b*x^n)/(2+n)],
[sqrt(b*x^n)/x,x,2,2*sqrt(b*x^n)/n],
[sqrt(b*x^n)/x^2,x,2,-2*sqrt(b*x^n)/((2-n)*x)],
[sqrt(b*x^n)/x^3,x,2,-2*sqrt(b*x^n)/((4-n)*x^2)],
[x*(b*x^n)^(3/2),x,2,2*b*x^(2+n)*sqrt(b*x^n)/(4+3*n)],
[(b*x^n)^(3/2),x,2,2*b*x^(1+n)*sqrt(b*x^n)/(2+3*n)],
[(b*x^n)^(3/2)/x,x,2,2/3*b*x^n*sqrt(b*x^n)/n],
[(b*x^n)^(3/2)/x^2,x,2,-2*b*x^(-1+n)*sqrt(b*x^n)/(2-3*n)],
[(b*x^n)^(3/2)/x^3,x,2,-2*b*x^(-2+n)*sqrt(b*x^n)/(4-3*n)],
[(b*x^n)^(3/2)/x^4,x,2,-2/3*b*x^(-3+n)*sqrt(b*x^n)/(2-n)],

# p<0
[x^2/sqrt(b*x^n),x,2,2*x^3/((6-n)*sqrt(b*x^n))],
[x/sqrt(b*x^n),x,2,2*x^2/((4-n)*sqrt(b*x^n))],
[1/sqrt(b*x^n),x,2,2*x/((2-n)*sqrt(b*x^n))],
[1/(x*sqrt(b*x^n)),x,2,(-2)/(n*sqrt(b*x^n))],
[1/(x^2*sqrt(b*x^n)),x,2,(-2)/((2+n)*x*sqrt(b*x^n))],
[1/(x^3*sqrt(b*x^n)),x,2,(-2)/((4+n)*x^2*sqrt(b*x^n))],
[x^2/(b*x^n)^(3/2),x,2,2/3*x^(3-n)/(b*(2-n)*sqrt(b*x^n))],
[x/(b*x^n)^(3/2),x,2,2*x^(2-n)/(b*(4-3*n)*sqrt(b*x^n))],
[1/(b*x^n)^(3/2),x,2,2*x^(1-n)/(b*(2-3*n)*sqrt(b*x^n))],
[1/(x*(b*x^n)^(3/2)),x,2,(-2/3)/(b*n*x^n*sqrt(b*x^n))],
[1/(x^2*(b*x^n)^(3/2)),x,2,-2*x^(-1-n)/(b*(2+3*n)*sqrt(b*x^n))],
[1/(x^3*(b*x^n)^(3/2)),x,2,-2*x^(-2-n)/(b*(4+3*n)*sqrt(b*x^n))],
[1/(x^4*(b*x^n)^(3/2)),x,2,-2/3*x^(-3-n)/(b*(2+n)*sqrt(b*x^n))],

# Integrands of the form (c x)^m (b x^n)^(p/2) with m symbolic
[x^m/(a*x^n)^(3/2),x,2,2*x^(1+m-n)/(a*(2+2*m-3*n)*sqrt(a*x^n))],
[(c*x)^m/(a*x^n)^(3/2),x,3,2*x^(1-n)*(c*x)^m/(a*(2+2*m-3*n)*sqrt(a*x^n))],
[x^m*(b*x^n)^(3/2),x,2,2*b*x^(1+m+n)*sqrt(b*x^n)/(2+2*m+3*n)],
[x^m*(b*x^n)^(1/2),x,2,2*x^(1+m)*sqrt(b*x^n)/(2+2*m+n)],
[x^m/(b*x^n)^(1/2),x,2,2*x^(1+m)/((2+2*m-n)*sqrt(b*x^n))],
[x^m/(b*x^n)^(3/2),x,2,2*x^(1+m-n)/(b*(2+2*m-3*n)*sqrt(b*x^n))],
[(c*x)^m*(b*x^n)^(5/2),x,3,2*(c*x)^(1+m)*(b*x^n)^(5/2)/(c*(2+2*m+5*n)),2*b^2*x^(1+2*n)*(c*x)^m*sqrt(b*x^n)/(2+2*m+5*n)],
[(c*x)^m*(b*x^n)^(3/2),x,3,2*b*x^(1+n)*(c*x)^m*sqrt(b*x^n)/(2+2*m+3*n)],
[(c*x)^m*(b*x^n)^(1/2),x,3,2*(c*x)^(1+m)*sqrt(b*x^n)/(c*(2+2*m+n)),2*x*(c*x)^m*sqrt(b*x^n)/(2+2*m+n)],
[(c*x)^m/(b*x^n)^(1/2),x,3,2*x*(c*x)^m/((2+2*m-n)*sqrt(b*x^n))],
[(c*x)^m/(b*x^n)^(3/2),x,3,2*x^(1-n)*(c*x)^m/(b*(2+2*m-3*n)*sqrt(b*x^n))],
[(c*x)^m/(b*x^n)^(5/2),x,3,2*x^(1-2*n)*(c*x)^m/(b^2*(2+2*m-5*n)*sqrt(b*x^n))],
[x^(-1-3/2*n)*(b*x^n)^(3/2),x,2,b*log(x)*sqrt(b*x^n)/x^(1/2*n)],
[x^(-1-1/2*n)*(b*x^n)^(1/2),x,2,log(x)*sqrt(b*x^n)/x^(1/2*n)],
[x^(-1+1/2*n)/(b*x^n)^(1/2),x,2,x^(1/2*n)*log(x)/sqrt(b*x^n)],
[x^(-1+3/2*n)/(b*x^n)^(3/2),x,2,x^(1/2*n)*log(x)/(b*sqrt(b*x^n))],

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

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

# Integrands of the form (c x)^m (a+b x^n)^p when n>2 is an integer

# Integrands of the form (c x)^m (a+b x^3)^p

# Integrands of the form x^m (a+b x^3)^p

# p>0
[x^3*(a+b*x^3),x,2,1/4*a*x^4+1/7*b*x^7],
[x^2*(a+b*x^3),x,2,1/3*a*x^3+1/6*b*x^6],
[x*(a+b*x^3),x,2,1/2*a*x^2+1/5*b*x^5],
[a+b*x^3,x,1,a*x+1/4*b*x^4],
[(a+b*x^3)/x,x,2,1/3*b*x^3+a*log(x)],
[(a+b*x^3)/x^2,x,2,-a/x+1/2*b*x^2],
[(a+b*x^3)/x^3,x,2,-1/2*a/x^2+b*x],
[(a+b*x^3)/x^4,x,2,-1/3*a/x^3+b*log(x)],
[(a+b*x^3)/x^5,x,2,-1/4*a/x^4-b/x],
[(a+b*x^3)/x^6,x,2,-1/5*a/x^5-1/2*b/x^2],
[(a+b*x^3)/x^7,x,2,-1/6*a/x^6-1/3*b/x^3],
[(a+b*x^3)/x^8,x,2,-1/7*a/x^7-1/4*b/x^4],
[x^4*(a+b*x^3)^2,x,2,1/5*a^2*x^5+1/4*a*b*x^8+1/11*b^2*x^11],
[x^3*(a+b*x^3)^2,x,2,1/4*a^2*x^4+2/7*a*b*x^7+1/10*b^2*x^10],
[x^2*(a+b*x^3)^2,x,1,1/9*(a+b*x^3)^3/b],
[x*(a+b*x^3)^2,x,2,1/2*a^2*x^2+2/5*a*b*x^5+1/8*b^2*x^8],
[(a+b*x^3)^2,x,2,a^2*x+1/2*a*b*x^4+1/7*b^2*x^7],
[(a+b*x^3)^2/x,x,3,2/3*a*b*x^3+1/6*b^2*x^6+a^2*log(x)],
[(a+b*x^3)^2/x^2,x,2,-a^2/x+a*b*x^2+1/5*b^2*x^5],
[(a+b*x^3)^2/x^3,x,2,-1/2*a^2/x^2+2*a*b*x+1/4*b^2*x^4],
[(a+b*x^3)^2/x^4,x,3,-1/3*a^2/x^3+1/3*b^2*x^3+2*a*b*log(x)],
[(a+b*x^3)^2/x^5,x,2,-1/4*a^2/x^4-2*a*b/x+1/2*b^2*x^2],
[(a+b*x^3)^2/x^6,x,2,-1/5*a^2/x^5-a*b/x^2+b^2*x],
[(a+b*x^3)^2/x^7,x,3,-1/6*a^2/x^6-2/3*a*b/x^3+b^2*log(x)],
[(a+b*x^3)^2/x^8,x,2,-1/7*a^2/x^7-1/2*a*b/x^4-b^2/x],
[(a+b*x^3)^2/x^9,x,2,-1/8*a^2/x^8-2/5*a*b/x^5-1/2*b^2/x^2],
[(a+b*x^3)^2/x^10,x,1,-1/9*(a+b*x^3)^3/(a*x^9)],
[(a+b*x^3)^2/x^11,x,2,-1/10*a^2/x^10-2/7*a*b/x^7-1/4*b^2/x^4],
[(a+b*x^3)^2/x^12,x,2,-1/11*a^2/x^11-1/4*a*b/x^8-1/5*b^2/x^5],
[(a+b*x^3)^2/x^13,x,3,-1/12*a^2/x^12-2/9*a*b/x^9-1/6*b^2/x^6],
[x^14*(a+b*x^3)^3,x,3,1/15*a^3*x^15+1/6*a^2*b*x^18+1/7*a*b^2*x^21+1/24*b^3*x^24],
[x^11*(a+b*x^3)^3,x,3,1/12*a^3*x^12+1/5*a^2*b*x^15+1/6*a*b^2*x^18+1/21*b^3*x^21],
[x^8*(a+b*x^3)^3,x,3,1/9*a^3*x^9+1/4*a^2*b*x^12+1/5*a*b^2*x^15+1/18*b^3*x^18],
[x^5*(a+b*x^3)^3,x,3,-1/12*a*(a+b*x^3)^4/b^2+1/15*(a+b*x^3)^5/b^2],
[x^2*(a+b*x^3)^3,x,1,1/12*(a+b*x^3)^4/b],
[(a+b*x^3)^3/x,x,3,a^2*b*x^3+1/2*a*b^2*x^6+1/9*b^3*x^9+a^3*log(x)],
[(a+b*x^3)^3/x^4,x,3,-1/3*a^3/x^3+a*b^2*x^3+1/6*b^3*x^6+3*a^2*b*log(x)],
[(a+b*x^3)^3/x^7,x,3,-1/6*a^3/x^6-a^2*b/x^3+1/3*b^3*x^3+3*a*b^2*log(x)],
[(a+b*x^3)^3/x^10,x,3,-1/9*a^3/x^9-1/2*a^2*b/x^6-a*b^2/x^3+b^3*log(x)],
[(a+b*x^3)^3/x^13,x,1,-1/12*(a+b*x^3)^4/(a*x^12)],
[(a+b*x^3)^3/x^16,x,3,-1/15*(a+b*x^3)^4/(a*x^15)+1/60*b*(a+b*x^3)^4/(a^2*x^12)],
[(a+b*x^3)^3/x^19,x,3,-1/18*a^3/x^18-1/5*a^2*b/x^15-1/4*a*b^2/x^12-1/9*b^3/x^9],
[(a+b*x^3)^3/x^22,x,3,-1/21*a^3/x^21-1/6*a^2*b/x^18-1/5*a*b^2/x^15-1/12*b^3/x^12],
[x^4*(a+b*x^3)^3,x,2,1/5*a^3*x^5+3/8*a^2*b*x^8+3/11*a*b^2*x^11+1/14*b^3*x^14],
[x^3*(a+b*x^3)^3,x,2,1/4*a^3*x^4+3/7*a^2*b*x^7+3/10*a*b^2*x^10+1/13*b^3*x^13],
[x*(a+b*x^3)^3,x,2,1/2*a^3*x^2+3/5*a^2*b*x^5+3/8*a*b^2*x^8+1/11*b^3*x^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+1/10*b^3*x^10],
[(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+1/8*b^3*x^8],
[(a+b*x^3)^3/x^3,x,2,-1/2*a^3/x^2+3*a^2*b*x+3/4*a*b^2*x^4+1/7*b^3*x^7],
[(a+b*x^3)^3/x^5,x,2,-1/4*a^3/x^4-3*a^2*b/x+3/2*a*b^2*x^2+1/5*b^3*x^5],
[(a+b*x^3)^3/x^6,x,2,-1/5*a^3/x^5-3/2*a^2*b/x^2+3*a*b^2*x+1/4*b^3*x^4],
[(a+b*x^3)^3/x^8,x,2,-1/7*a^3/x^7-3/4*a^2*b/x^4-3*a*b^2/x+1/2*b^3*x^2],
[x^17*(a+b*x^3)^5,x,3,1/18*a^5*x^18+5/21*a^4*b*x^21+5/12*a^3*b^2*x^24+10/27*a^2*b^3*x^27+1/6*a*b^4*x^30+1/33*b^5*x^33],
[x^14*(a+b*x^3)^5,x,3,1/15*a^5*x^15+5/18*a^4*b*x^18+10/21*a^3*b^2*x^21+5/12*a^2*b^3*x^24+5/27*a*b^4*x^27+1/30*b^5*x^30],
[x^11*(a+b*x^3)^5,x,3,-1/18*a^3*(a+b*x^3)^6/b^4+1/7*a^2*(a+b*x^3)^7/b^4-1/8*a*(a+b*x^3)^8/b^4+1/27*(a+b*x^3)^9/b^4],
[x^8*(a+b*x^3)^5,x,3,1/18*a^2*(a+b*x^3)^6/b^3-2/21*a*(a+b*x^3)^7/b^3+1/24*(a+b*x^3)^8/b^3],
[x^5*(a+b*x^3)^5,x,3,-1/18*a*(a+b*x^3)^6/b^2+1/21*(a+b*x^3)^7/b^2],
[x^2*(a+b*x^3)^5,x,1,1/18*(a+b*x^3)^6/b],
[(a+b*x^3)^5/x,x,3,5/3*a^4*b*x^3+5/3*a^3*b^2*x^6+10/9*a^2*b^3*x^9+5/12*a*b^4*x^12+1/15*b^5*x^15+a^5*log(x)],
[(a+b*x^3)^5/x^4,x,3,-1/3*a^5/x^3+10/3*a^3*b^2*x^3+5/3*a^2*b^3*x^6+5/9*a*b^4*x^9+1/12*b^5*x^12+5*a^4*b*log(x)],
[(a+b*x^3)^5/x^7,x,3,-1/6*a^5/x^6-5/3*a^4*b/x^3+10/3*a^2*b^3*x^3+5/6*a*b^4*x^6+1/9*b^5*x^9+10*a^3*b^2*log(x)],
[(a+b*x^3)^5/x^10,x,3,-1/9*a^5/x^9-5/6*a^4*b/x^6-10/3*a^3*b^2/x^3+5/3*a*b^4*x^3+1/6*b^5*x^6+10*a^2*b^3*log(x)],
[(a+b*x^3)^5/x^13,x,3,-1/12*a^5/x^12-5/9*a^4*b/x^9-5/3*a^3*b^2/x^6-10/3*a^2*b^3/x^3+1/3*b^5*x^3+5*a*b^4*log(x)],
[(a+b*x^3)^5/x^16,x,3,-1/15*a^5/x^15-5/12*a^4*b/x^12-10/9*a^3*b^2/x^9-5/3*a^2*b^3/x^6-5/3*a*b^4/x^3+b^5*log(x)],
[(a+b*x^3)^5/x^19,x,1,-1/18*(a+b*x^3)^6/(a*x^18)],
[(a+b*x^3)^5/x^22,x,3,-1/21*(a+b*x^3)^6/(a*x^21)+1/126*b*(a+b*x^3)^6/(a^2*x^18)],
[(a+b*x^3)^5/x^25,x,4,-1/24*(a+b*x^3)^6/(a*x^24)+1/84*b*(a+b*x^3)^6/(a^2*x^21)-1/504*b^2*(a+b*x^3)^6/(a^3*x^18)],
[(a+b*x^3)^5/x^28,x,3,-1/27*a^5/x^27-5/24*a^4*b/x^24-10/21*a^3*b^2/x^21-5/9*a^2*b^3/x^18-1/3*a*b^4/x^15-1/12*b^5/x^12],
[(a+b*x^3)^5/x^31,x,3,-1/30*a^5/x^30-5/27*a^4*b/x^27-5/12*a^3*b^2/x^24-10/21*a^2*b^3/x^21-5/18*a*b^4/x^18-1/15*b^5/x^15],
[x^4*(a+b*x^3)^5,x,2,1/5*a^5*x^5+5/8*a^4*b*x^8+10/11*a^3*b^2*x^11+5/7*a^2*b^3*x^14+5/17*a*b^4*x^17+1/20*b^5*x^20],
[x^3*(a+b*x^3)^5,x,2,1/4*a^5*x^4+5/7*a^4*b*x^7+a^3*b^2*x^10+10/13*a^2*b^3*x^13+5/16*a*b^4*x^16+1/19*b^5*x^19],
[x*(a+b*x^3)^5,x,2,1/2*a^5*x^2+a^4*b*x^5+5/4*a^3*b^2*x^8+10/11*a^2*b^3*x^11+5/14*a*b^4*x^14+1/17*b^5*x^17],
[(a+b*x^3)^5,x,2,a^5*x+5/4*a^4*b*x^4+10/7*a^3*b^2*x^7+a^2*b^3*x^10+5/13*a*b^4*x^13+1/16*b^5*x^16],
[(a+b*x^3)^5/x^2,x,2,-a^5/x+5/2*a^4*b*x^2+2*a^3*b^2*x^5+5/4*a^2*b^3*x^8+5/11*a*b^4*x^11+1/14*b^5*x^14],
[(a+b*x^3)^5/x^3,x,2,-1/2*a^5/x^2+5*a^4*b*x+5/2*a^3*b^2*x^4+10/7*a^2*b^3*x^7+1/2*a*b^4*x^10+1/13*b^5*x^13],
[(a+b*x^3)^5/x^5,x,2,-1/4*a^5/x^4-5*a^4*b/x+5*a^3*b^2*x^2+2*a^2*b^3*x^5+5/8*a*b^4*x^8+1/11*b^5*x^11],
[(a+b*x^3)^5/x^6,x,2,-1/5*a^5/x^5-5/2*a^4*b/x^2+10*a^3*b^2*x+5/2*a^2*b^3*x^4+5/7*a*b^4*x^7+1/10*b^5*x^10],
[(a+b*x^3)^5/x^8,x,2,-1/7*a^5/x^7-5/4*a^4*b/x^4-10*a^3*b^2/x+5*a^2*b^3*x^2+a*b^4*x^5+1/8*b^5*x^8],
[(a+b*x^3)^5/x^9,x,2,-1/8*a^5/x^8-a^4*b/x^5-5*a^3*b^2/x^2+10*a^2*b^3*x+5/4*a*b^4*x^4+1/7*b^5*x^7],
[x^20*(a+b*x^3)^8,x,3,1/27*a^6*(a+b*x^3)^9/b^7-1/5*a^5*(a+b*x^3)^10/b^7+5/11*a^4*(a+b*x^3)^11/b^7-5/9*a^3*(a+b*x^3)^12/b^7+5/13*a^2*(a+b*x^3)^13/b^7-1/7*a*(a+b*x^3)^14/b^7+1/45*(a+b*x^3)^15/b^7],
[x^17*(a+b*x^3)^8,x,3,-1/27*a^5*(a+b*x^3)^9/b^6+1/6*a^4*(a+b*x^3)^10/b^6-10/33*a^3*(a+b*x^3)^11/b^6+5/18*a^2*(a+b*x^3)^12/b^6-5/39*a*(a+b*x^3)^13/b^6+1/42*(a+b*x^3)^14/b^6],
[x^14*(a+b*x^3)^8,x,3,1/27*a^4*(a+b*x^3)^9/b^5-2/15*a^3*(a+b*x^3)^10/b^5+2/11*a^2*(a+b*x^3)^11/b^5-1/9*a*(a+b*x^3)^12/b^5+1/39*(a+b*x^3)^13/b^5],
[x^11*(a+b*x^3)^8,x,3,-1/27*a^3*(a+b*x^3)^9/b^4+1/10*a^2*(a+b*x^3)^10/b^4-1/11*a*(a+b*x^3)^11/b^4+1/36*(a+b*x^3)^12/b^4],
[x^8*(a+b*x^3)^8,x,3,1/27*a^2*(a+b*x^3)^9/b^3-1/15*a*(a+b*x^3)^10/b^3+1/33*(a+b*x^3)^11/b^3],
[x^5*(a+b*x^3)^8,x,3,-1/27*a*(a+b*x^3)^9/b^2+1/30*(a+b*x^3)^10/b^2],
[x^2*(a+b*x^3)^8,x,1,1/27*(a+b*x^3)^9/b],
[(a+b*x^3)^8/x,x,3,8/3*a^7*b*x^3+14/3*a^6*b^2*x^6+56/9*a^5*b^3*x^9+35/6*a^4*b^4*x^12+56/15*a^3*b^5*x^15+14/9*a^2*b^6*x^18+8/21*a*b^7*x^21+1/24*b^8*x^24+a^8*log(x)],
[(a+b*x^3)^8/x^4,x,3,-1/3*a^8/x^3+28/3*a^6*b^2*x^3+28/3*a^5*b^3*x^6+70/9*a^4*b^4*x^9+14/3*a^3*b^5*x^12+28/15*a^2*b^6*x^15+4/9*a*b^7*x^18+1/21*b^8*x^21+8*a^7*b*log(x)],
[(a+b*x^3)^8/x^7,x,3,-1/6*a^8/x^6-8/3*a^7*b/x^3+56/3*a^5*b^3*x^3+35/3*a^4*b^4*x^6+56/9*a^3*b^5*x^9+7/3*a^2*b^6*x^12+8/15*a*b^7*x^15+1/18*b^8*x^18+28*a^6*b^2*log(x)],
[(a+b*x^3)^8/x^10,x,3,-1/9*a^8/x^9-4/3*a^7*b/x^6-28/3*a^6*b^2/x^3+70/3*a^4*b^4*x^3+28/3*a^3*b^5*x^6+28/9*a^2*b^6*x^9+2/3*a*b^7*x^12+1/15*b^8*x^15+56*a^5*b^3*log(x)],
[(a+b*x^3)^8/x^13,x,3,-1/12*a^8/x^12-8/9*a^7*b/x^9-14/3*a^6*b^2/x^6-56/3*a^5*b^3/x^3+56/3*a^3*b^5*x^3+14/3*a^2*b^6*x^6+8/9*a*b^7*x^9+1/12*b^8*x^12+70*a^4*b^4*log(x)],
[(a+b*x^3)^8/x^16,x,3,-1/15*a^8/x^15-2/3*a^7*b/x^12-28/9*a^6*b^2/x^9-28/3*a^5*b^3/x^6-70/3*a^4*b^4/x^3+28/3*a^2*b^6*x^3+4/3*a*b^7*x^6+1/9*b^8*x^9+56*a^3*b^5*log(x)],
[(a+b*x^3)^8/x^19,x,3,-1/18*a^8/x^18-8/15*a^7*b/x^15-7/3*a^6*b^2/x^12-56/9*a^5*b^3/x^9-35/3*a^4*b^4/x^6-56/3*a^3*b^5/x^3+8/3*a*b^7*x^3+1/6*b^8*x^6+28*a^2*b^6*log(x)],
[(a+b*x^3)^8/x^22,x,3,-1/21*a^8/x^21-4/9*a^7*b/x^18-28/15*a^6*b^2/x^15-14/3*a^5*b^3/x^12-70/9*a^4*b^4/x^9-28/3*a^3*b^5/x^6-28/3*a^2*b^6/x^3+1/3*b^8*x^3+8*a*b^7*log(x)],
[(a+b*x^3)^8/x^25,x,3,-1/24*a^8/x^24-8/21*a^7*b/x^21-14/9*a^6*b^2/x^18-56/15*a^5*b^3/x^15-35/6*a^4*b^4/x^12-56/9*a^3*b^5/x^9-14/3*a^2*b^6/x^6-8/3*a*b^7/x^3+b^8*log(x)],
[(a+b*x^3)^8/x^28,x,1,-1/27*(a+b*x^3)^9/(a*x^27)],
[(a+b*x^3)^8/x^31,x,3,-1/30*(a+b*x^3)^9/(a*x^30)+1/270*b*(a+b*x^3)^9/(a^2*x^27)],
[(a+b*x^3)^8/x^34,x,4,-1/33*(a+b*x^3)^9/(a*x^33)+1/165*b*(a+b*x^3)^9/(a^2*x^30)-1/1485*b^2*(a+b*x^3)^9/(a^3*x^27)],
[(a+b*x^3)^8/x^37,x,5,-1/36*(a+b*x^3)^9/(a*x^36)+1/132*b*(a+b*x^3)^9/(a^2*x^33)-1/660*b^2*(a+b*x^3)^9/(a^3*x^30)+1/5940*b^3*(a+b*x^3)^9/(a^4*x^27)],
[(a+b*x^3)^8/x^40,x,6,-1/39*(a+b*x^3)^9/(a*x^39)+1/117*b*(a+b*x^3)^9/(a^2*x^36)-1/429*b^2*(a+b*x^3)^9/(a^3*x^33)+1/2145*b^3*(a+b*x^3)^9/(a^4*x^30)-1/19305*b^4*(a+b*x^3)^9/(a^5*x^27)],
[(a+b*x^3)^8/x^43,x,3,-1/42*a^8/x^42-8/39*a^7*b/x^39-7/9*a^6*b^2/x^36-56/33*a^5*b^3/x^33-7/3*a^4*b^4/x^30-56/27*a^3*b^5/x^27-7/6*a^2*b^6/x^24-8/21*a*b^7/x^21-1/18*b^8/x^18],
[(a+b*x^3)^8/x^46,x,3,-1/45*a^8/x^45-4/21*a^7*b/x^42-28/39*a^6*b^2/x^39-14/9*a^5*b^3/x^36-70/33*a^4*b^4/x^33-28/15*a^3*b^5/x^30-28/27*a^2*b^6/x^27-1/3*a*b^7/x^24-1/21*b^8/x^21],
[x^4*(a+b*x^3)^8,x,2,1/5*a^8*x^5+a^7*b*x^8+28/11*a^6*b^2*x^11+4*a^5*b^3*x^14+70/17*a^4*b^4*x^17+14/5*a^3*b^5*x^20+28/23*a^2*b^6*x^23+4/13*a*b^7*x^26+1/29*b^8*x^29],
[x^3*(a+b*x^3)^8,x,2,1/4*a^8*x^4+8/7*a^7*b*x^7+14/5*a^6*b^2*x^10+56/13*a^5*b^3*x^13+35/8*a^4*b^4*x^16+56/19*a^3*b^5*x^19+14/11*a^2*b^6*x^22+8/25*a*b^7*x^25+1/28*b^8*x^28],
[x*(a+b*x^3)^8,x,2,1/2*a^8*x^2+8/5*a^7*b*x^5+7/2*a^6*b^2*x^8+56/11*a^5*b^3*x^11+5*a^4*b^4*x^14+56/17*a^3*b^5*x^17+7/5*a^2*b^6*x^20+8/23*a*b^7*x^23+1/26*b^8*x^26],
[(a+b*x^3)^8,x,2,a^8*x+2*a^7*b*x^4+4*a^6*b^2*x^7+28/5*a^5*b^3*x^10+70/13*a^4*b^4*x^13+7/2*a^3*b^5*x^16+28/19*a^2*b^6*x^19+4/11*a*b^7*x^22+1/25*b^8*x^25],
[(a+b*x^3)^8/x^2,x,2,-a^8/x+4*a^7*b*x^2+28/5*a^6*b^2*x^5+7*a^5*b^3*x^8+70/11*a^4*b^4*x^11+4*a^3*b^5*x^14+28/17*a^2*b^6*x^17+2/5*a*b^7*x^20+1/23*b^8*x^23],
[(a+b*x^3)^8/x^3,x,2,-1/2*a^8/x^2+8*a^7*b*x+7*a^6*b^2*x^4+8*a^5*b^3*x^7+7*a^4*b^4*x^10+56/13*a^3*b^5*x^13+7/4*a^2*b^6*x^16+8/19*a*b^7*x^19+1/22*b^8*x^22],
[(a+b*x^3)^8/x^5,x,2,-1/4*a^8/x^4-8*a^7*b/x+14*a^6*b^2*x^2+56/5*a^5*b^3*x^5+35/4*a^4*b^4*x^8+56/11*a^3*b^5*x^11+2*a^2*b^6*x^14+8/17*a*b^7*x^17+1/20*b^8*x^20],
[(a+b*x^3)^8/x^6,x,2,-1/5*a^8/x^5-4*a^7*b/x^2+28*a^6*b^2*x+14*a^5*b^3*x^4+10*a^4*b^4*x^7+28/5*a^3*b^5*x^10+28/13*a^2*b^6*x^13+1/2*a*b^7*x^16+1/19*b^8*x^19],
[(a+b*x^3)^8/x^8,x,2,-1/7*a^8/x^7-2*a^7*b/x^4-28*a^6*b^2/x+28*a^5*b^3*x^2+14*a^4*b^4*x^5+7*a^3*b^5*x^8+28/11*a^2*b^6*x^11+4/7*a*b^7*x^14+1/17*b^8*x^17],
[(a+b*x^3)^8/x^9,x,2,-1/8*a^8/x^8-8/5*a^7*b/x^5-14*a^6*b^2/x^2+56*a^5*b^3*x+35/2*a^4*b^4*x^4+8*a^3*b^5*x^7+14/5*a^2*b^6*x^10+8/13*a*b^7*x^13+1/16*b^8*x^16],

# p<0
[x^8/(a+b*x^3),x,3,-1/3*a*x^3/b^2+1/6*x^6/b+1/3*a^2*log(a+b*x^3)/b^3],
[x^5/(a+b*x^3),x,3,1/3*x^3/b-1/3*a*log(a+b*x^3)/b^2],
[x^2/(a+b*x^3),x,1,1/3*log(a+b*x^3)/b],
[1/(x*(a+b*x^3)),x,4,log(x)/a-1/3*log(a+b*x^3)/a],
[1/(x^4*(a+b*x^3)),x,3,(-1/3)/(a*x^3)-b*log(x)/a^2+1/3*b*log(a+b*x^3)/a^2],
[x^4/(a+b*x^3),x,7,1/2*x^2/b+1/3*a^(2/3)*log(a^(1/3)+b^(1/3)*x)/b^(5/3)-1/6*a^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(5/3)+a^(2/3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3))],
[x^3/(a+b*x^3),x,7,x/b-1/3*a^(1/3)*log(a^(1/3)+b^(1/3)*x)/b^(4/3)+1/6*a^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(4/3)+a^(1/3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(4/3)*sqrt(3))],
[x/(a+b*x^3),x,6,-1/3*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(2/3))+1/6*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(2/3))-arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*sqrt(3))],
[1/(a+b*x^3),x,6,1/3*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(1/3))-1/6*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))-arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*sqrt(3))],
[1/(x^2*(a+b*x^3)),x,7,(-1)/(a*x)+1/3*b^(1/3)*log(a^(1/3)+b^(1/3)*x)/a^(4/3)-1/6*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)+b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*sqrt(3))],
[1/(x^3*(a+b*x^3)),x,7,(-1/2)/(a*x^2)-1/3*b^(2/3)*log(a^(1/3)+b^(1/3)*x)/a^(5/3)+1/6*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)+b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3))],
[x^8/(a+b*x^3)^2,x,3,1/3*x^3/b^2-1/3*a^2/(b^3*(a+b*x^3))-2/3*a*log(a+b*x^3)/b^3],
[x^5/(a+b*x^3)^2,x,3,1/3*a/(b^2*(a+b*x^3))+1/3*log(a+b*x^3)/b^2],
[x^2/(a+b*x^3)^2,x,1,(-1/3)/(b*(a+b*x^3))],
[1/(x*(a+b*x^3)^2),x,3,1/3/(a*(a+b*x^3))+log(x)/a^2-1/3*log(a+b*x^3)/a^2],
[1/(x^4*(a+b*x^3)^2),x,3,(-1/3)/(a^2*x^3)-1/3*b/(a^2*(a+b*x^3))-2*b*log(x)/a^3+2/3*b*log(a+b*x^3)/a^3],
[x^4/(a+b*x^3)^2,x,7,-1/3*x^2/(b*(a+b*x^3))-2/9*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(5/3))+1/9*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(5/3))-2/3*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(5/3)*sqrt(3))],
[x^3/(a+b*x^3)^2,x,7,-1/3*x/(b*(a+b*x^3))+1/9*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(4/3))-1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(4/3))-1/3*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(4/3)*sqrt(3))],
[x/(a+b*x^3)^2,x,7,1/3*x^2/(a*(a+b*x^3))-1/9*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(2/3))+1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(2/3))-1/3*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(2/3)*sqrt(3))],
[1/(a+b*x^3)^2,x,7,1/3*x/(a*(a+b*x^3))+2/9*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(1/3))-1/9*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(1/3))-2/3*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(1/3)*sqrt(3))],
[1/(x^2*(a+b*x^3)^2),x,8,(-4/3)/(a^2*x)+1/3/(a*x*(a+b*x^3))+4/9*b^(1/3)*log(a^(1/3)+b^(1/3)*x)/a^(7/3)-2/9*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(7/3)+4/3*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*sqrt(3))],
[1/(x^3*(a+b*x^3)^2),x,8,(-5/6)/(a^2*x^2)+1/3/(a*x^2*(a+b*x^3))-5/9*b^(2/3)*log(a^(1/3)+b^(1/3)*x)/a^(8/3)+5/18*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)+5/3*b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*sqrt(3))],
[x^11/(a+b*x^3)^3,x,3,1/3*x^3/b^3+1/6*a^3/(b^4*(a+b*x^3)^2)-a^2/(b^4*(a+b*x^3))-a*log(a+b*x^3)/b^4],
[x^8/(a+b*x^3)^3,x,3,-1/6*a^2/(b^3*(a+b*x^3)^2)+2/3*a/(b^3*(a+b*x^3))+1/3*log(a+b*x^3)/b^3],
[x^5/(a+b*x^3)^3,x,1,1/6*x^6/(a*(a+b*x^3)^2)],
[x^2/(a+b*x^3)^3,x,1,(-1/6)/(b*(a+b*x^3)^2)],
[1/(x*(a+b*x^3)^3),x,3,1/6/(a*(a+b*x^3)^2)+1/3/(a^2*(a+b*x^3))+log(x)/a^3-1/3*log(a+b*x^3)/a^3],
[1/(x^4*(a+b*x^3)^3),x,3,(-1/3)/(a^3*x^3)-1/6*b/(a^2*(a+b*x^3)^2)-2/3*b/(a^3*(a+b*x^3))-3*b*log(x)/a^4+b*log(a+b*x^3)/a^4],
[x^7/(a+b*x^3)^3,x,8,-1/6*x^5/(b*(a+b*x^3)^2)-5/18*x^2/(b^2*(a+b*x^3))-5/27*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(8/3))+5/54*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(8/3))-5/9*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(8/3)*sqrt(3))],
[x^6/(a+b*x^3)^3,x,8,-1/6*x^4/(b*(a+b*x^3)^2)-2/9*x/(b^2*(a+b*x^3))+2/27*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(7/3))-1/27*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(7/3))-2/9*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(7/3)*sqrt(3))],
[x^4/(a+b*x^3)^3,x,8,-1/6*x^2/(b*(a+b*x^3)^2)+1/9*x^2/(a*b*(a+b*x^3))-1/27*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(5/3))+1/54*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(5/3))-1/9*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(5/3)*sqrt(3))],
[x^3/(a+b*x^3)^3,x,8,-1/6*x/(b*(a+b*x^3)^2)+1/18*x/(a*b*(a+b*x^3))+1/27*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(4/3))-1/54*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(4/3))-1/9*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(4/3)*sqrt(3))],
[x/(a+b*x^3)^3,x,8,1/6*x^2/(a*(a+b*x^3)^2)+2/9*x^2/(a^2*(a+b*x^3))-2/27*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(2/3))+1/27*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(2/3))-2/9*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(2/3)*sqrt(3))],
[1/(a+b*x^3)^3,x,8,1/6*x/(a*(a+b*x^3)^2)+5/18*x/(a^2*(a+b*x^3))+5/27*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(1/3))-5/54*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(1/3))-5/9*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(1/3)*sqrt(3))],
[x^8/(a-b*x^3),x,3,-1/3*a*x^3/b^2-1/6*x^6/b-1/3*a^2*log(a-b*x^3)/b^3],
[x^5/(a-b*x^3),x,3,-1/3*x^3/b-1/3*a*log(a-b*x^3)/b^2],
[x^2/(a-b*x^3),x,1,-1/3*log(a-b*x^3)/b],
[1/(x*(a-b*x^3)),x,4,log(x)/a-1/3*log(a-b*x^3)/a],
[1/(x^4*(a-b*x^3)),x,3,(-1/3)/(a*x^3)+b*log(x)/a^2-1/3*b*log(a-b*x^3)/a^2],
[x^4/(a-b*x^3),x,7,-1/2*x^2/b-1/3*a^(2/3)*log(a^(1/3)-b^(1/3)*x)/b^(5/3)+1/6*a^(2/3)*log(a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(5/3)-a^(2/3)*arctan((a^(1/3)+2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3))],
[x^3/(a-b*x^3),x,7,-x/b-1/3*a^(1/3)*log(a^(1/3)-b^(1/3)*x)/b^(4/3)+1/6*a^(1/3)*log(a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(4/3)+a^(1/3)*arctan((a^(1/3)+2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(4/3)*sqrt(3))],
[x/(a-b*x^3),x,6,-1/3*log(a^(1/3)-b^(1/3)*x)/(a^(1/3)*b^(2/3))+1/6*log(a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(2/3))-arctan((a^(1/3)+2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*sqrt(3))],
[1/(a-b*x^3),x,6,-1/3*log(a^(1/3)-b^(1/3)*x)/(a^(2/3)*b^(1/3))+1/6*log(a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))+arctan((a^(1/3)+2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*sqrt(3))],
[1/(x^2*(a-b*x^3)),x,7,(-1)/(a*x)-1/3*b^(1/3)*log(a^(1/3)-b^(1/3)*x)/a^(4/3)+1/6*b^(1/3)*log(a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)-b^(1/3)*arctan((a^(1/3)+2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*sqrt(3))],
[1/(x^3*(a-b*x^3)),x,7,(-1/2)/(a*x^2)-1/3*b^(2/3)*log(a^(1/3)-b^(1/3)*x)/a^(5/3)+1/6*b^(2/3)*log(a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)+b^(2/3)*arctan((a^(1/3)+2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3))],
[1/(1+a+b*x^3),x,6,1/3*log((1+a)^(1/3)+b^(1/3)*x)/((1+a)^(2/3)*b^(1/3))-1/6*log((1+a)^(2/3)-(1+a)^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+a)^(2/3)*b^(1/3))-arctan((1-2*b^(1/3)*x/(1+a)^(1/3))/sqrt(3))/((1+a)^(2/3)*b^(1/3)*sqrt(3))],
[1/(1+a-b*x^3),x,6,-1/3*log((1+a)^(1/3)-b^(1/3)*x)/((1+a)^(2/3)*b^(1/3))+1/6*log((1+a)^(2/3)+(1+a)^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+a)^(2/3)*b^(1/3))+arctan((1+2*b^(1/3)*x/(1+a)^(1/3))/sqrt(3))/((1+a)^(2/3)*b^(1/3)*sqrt(3))],
[1/(-1+a+b*x^3),x,6,1/3*log((1-a)^(1/3)-b^(1/3)*x)/((1-a)^(2/3)*b^(1/3))-1/6*log((1-a)^(2/3)+(1-a)^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1-a)^(2/3)*b^(1/3))-arctan((1+2*b^(1/3)*x/(1-a)^(1/3))/sqrt(3))/((1-a)^(2/3)*b^(1/3)*sqrt(3))],
[1/(-1+a-b*x^3),x,6,-1/3*log((1-a)^(1/3)+b^(1/3)*x)/((1-a)^(2/3)*b^(1/3))+1/6*log((1-a)^(2/3)-(1-a)^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1-a)^(2/3)*b^(1/3))+arctan((1-2*b^(1/3)*x/(1-a)^(1/3))/sqrt(3))/((1-a)^(2/3)*b^(1/3)*sqrt(3))],

# Integrands of the form x^(m/2) (a+b x^3)^p

# p>0

# p<0
[sqrt(x)/(1+x^3),x,3,2/3*arctan(x^(3/2))],

# Integrands of the form x^m (a+b x^3)^(p/2)

# p>0
[x^11*sqrt(a+b*x^3),x,3,-2/9*a^3*(a+b*x^3)^(3/2)/b^4+2/5*a^2*(a+b*x^3)^(5/2)/b^4-2/7*a*(a+b*x^3)^(7/2)/b^4+2/27*(a+b*x^3)^(9/2)/b^4],
[x^8*sqrt(a+b*x^3),x,3,2/9*a^2*(a+b*x^3)^(3/2)/b^3-4/15*a*(a+b*x^3)^(5/2)/b^3+2/21*(a+b*x^3)^(7/2)/b^3],
[x^5*sqrt(a+b*x^3),x,3,-2/9*a*(a+b*x^3)^(3/2)/b^2+2/15*(a+b*x^3)^(5/2)/b^2],
[x^2*sqrt(a+b*x^3),x,1,2/9*(a+b*x^3)^(3/2)/b],
[sqrt(a+b*x^3)/x,x,4,-2/3*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+2/3*sqrt(a+b*x^3)],
[sqrt(a+b*x^3)/x^4,x,4,-1/3*b*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)-1/3*sqrt(a+b*x^3)/x^3],
[sqrt(a+b*x^3)/x^7,x,5,1/12*b^2*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/6*sqrt(a+b*x^3)/x^6-1/12*b*sqrt(a+b*x^3)/(a*x^3)],
[sqrt(a+b*x^3)/x^10,x,6,-1/24*b^3*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(5/2)-1/9*sqrt(a+b*x^3)/x^9-1/36*b*sqrt(a+b*x^3)/(a*x^6)+1/24*b^2*sqrt(a+b*x^3)/(a^2*x^3)],
[x^6*sqrt(a+b*x^3),x,4,-48/935*a^2*x*sqrt(a+b*x^3)/b^2+6/187*a*x^4*sqrt(a+b*x^3)/b+2/17*x^7*sqrt(a+b*x^3)+32/935*3^(3/4)*a^3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(7/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3*sqrt(a+b*x^3),x,3,6/55*a*x*sqrt(a+b*x^3)/b+2/11*x^4*sqrt(a+b*x^3)-4/55*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b*x^3),x,2,2/5*x*sqrt(a+b*x^3)+2/5*3^(3/4)*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b*x^3)/x^3,x,2,-1/2*sqrt(a+b*x^3)/x^2+1/2*3^(3/4)*b^(2/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b*x^3)/x^6,x,3,-1/5*sqrt(a+b*x^3)/x^5-3/20*b*sqrt(a+b*x^3)/(a*x^2)-1/20*3^(3/4)*b^(5/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b*x^3)/x^9,x,4,-1/8*sqrt(a+b*x^3)/x^8-3/80*b*sqrt(a+b*x^3)/(a*x^5)+21/320*b^2*sqrt(a+b*x^3)/(a^2*x^2)+7/320*3^(3/4)*b^(8/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^2*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^7*sqrt(a+b*x^3),x,6,-60/1729*a^2*x^2*sqrt(a+b*x^3)/b^2+6/247*a*x^5*sqrt(a+b*x^3)/b+2/19*x^8*sqrt(a+b*x^3)+240/1729*a^3*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+80/1729*3^(3/4)*a^(10/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-120/1729*3^(1/4)*a^(10/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4*sqrt(a+b*x^3),x,5,6/91*a*x^2*sqrt(a+b*x^3)/b+2/13*x^5*sqrt(a+b*x^3)-24/91*a^2*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-8/91*3^(3/4)*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+12/91*3^(1/4)*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x*sqrt(a+b*x^3),x,4,2/7*x^2*sqrt(a+b*x^3)+6/7*a*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+2/7*3^(3/4)*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-3/7*3^(1/4)*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b*x^3)/x^2,x,4,-sqrt(a+b*x^3)/x+3*b^(1/3)*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))+3^(3/4)*a^(1/3)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-3/2*3^(1/4)*a^(1/3)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b*x^3)/x^5,x,5,-1/4*sqrt(a+b*x^3)/x^4-3/8*b*sqrt(a+b*x^3)/(a*x)+3/8*b^(4/3)*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+1/4*3^(3/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(2)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-3/16*3^(1/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^11*(a+b*x^3)^(3/2),x,3,-2/15*a^3*(a+b*x^3)^(5/2)/b^4+2/7*a^2*(a+b*x^3)^(7/2)/b^4-2/9*a*(a+b*x^3)^(9/2)/b^4+2/33*(a+b*x^3)^(11/2)/b^4],
[x^8*(a+b*x^3)^(3/2),x,3,2/15*a^2*(a+b*x^3)^(5/2)/b^3-4/21*a*(a+b*x^3)^(7/2)/b^3+2/27*(a+b*x^3)^(9/2)/b^3],
[x^5*(a+b*x^3)^(3/2),x,3,-2/15*a*(a+b*x^3)^(5/2)/b^2+2/21*(a+b*x^3)^(7/2)/b^2],
[x^2*(a+b*x^3)^(3/2),x,1,2/15*(a+b*x^3)^(5/2)/b],
[(a+b*x^3)^(3/2)/x,x,5,2/9*(a+b*x^3)^(3/2)-2/3*a^(3/2)*arctanh(sqrt(a+b*x^3)/sqrt(a))+2/3*a*sqrt(a+b*x^3)],
[(a+b*x^3)^(3/2)/x^4,x,5,-1/3*(a+b*x^3)^(3/2)/x^3-b*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+b*sqrt(a+b*x^3)],
[(a+b*x^3)^(3/2)/x^7,x,5,-1/6*(a+b*x^3)^(3/2)/x^6-1/4*b^2*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)-1/4*b*sqrt(a+b*x^3)/x^3],
[x^6*(a+b*x^3)^(3/2),x,5,2/23*x^7*(a+b*x^3)^(3/2)-432/21505*a^3*x*sqrt(a+b*x^3)/b^2+54/4301*a^2*x^4*sqrt(a+b*x^3)/b+18/391*a*x^7*sqrt(a+b*x^3)+288/21505*3^(3/4)*a^4*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(7/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3*(a+b*x^3)^(3/2),x,4,2/17*x^4*(a+b*x^3)^(3/2)+54/935*a^2*x*sqrt(a+b*x^3)/b+18/187*a*x^4*sqrt(a+b*x^3)-36/935*3^(3/4)*a^3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2),x,3,2/11*x*(a+b*x^3)^(3/2)+18/55*a*x*sqrt(a+b*x^3)+18/55*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)/x^3,x,3,-1/2*(a+b*x^3)^(3/2)/x^2+9/10*b*x*sqrt(a+b*x^3)+9/10*3^(3/4)*a*b^(2/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)/x^6,x,3,-1/5*(a+b*x^3)^(3/2)/x^5-9/20*b*sqrt(a+b*x^3)/x^2+9/20*3^(3/4)*b^(5/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^7*(a+b*x^3)^(3/2),x,7,2/25*x^8*(a+b*x^3)^(3/2)-108/8645*a^3*x^2*sqrt(a+b*x^3)/b^2+54/6175*a^2*x^5*sqrt(a+b*x^3)/b+18/475*a*x^8*sqrt(a+b*x^3)+432/8645*a^4*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+144/8645*3^(3/4)*a^(13/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-216/8645*3^(1/4)*a^(13/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4*(a+b*x^3)^(3/2),x,6,2/19*x^5*(a+b*x^3)^(3/2)+54/1729*a^2*x^2*sqrt(a+b*x^3)/b+18/247*a*x^5*sqrt(a+b*x^3)-216/1729*a^3*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-72/1729*3^(3/4)*a^(10/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+108/1729*3^(1/4)*a^(10/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x*(a+b*x^3)^(3/2),x,5,2/13*x^2*(a+b*x^3)^(3/2)+18/91*a*x^2*sqrt(a+b*x^3)+54/91*a^2*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+18/91*3^(3/4)*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-27/91*3^(1/4)*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)/x^2,x,5,-(a+b*x^3)^(3/2)/x+9/7*b*x^2*sqrt(a+b*x^3)+27/7*a*b^(1/3)*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))+9/7*3^(3/4)*a^(4/3)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-27/14*3^(1/4)*a^(4/3)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)/x^5,x,5,-1/4*(a+b*x^3)^(3/2)/x^4-9/8*b*sqrt(a+b*x^3)/x+27/8*b^(4/3)*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))+9/4*3^(3/4)*a^(1/3)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(2)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-27/16*3^(1/4)*a^(1/3)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],

# p<0
[x^11/sqrt(a+b*x^3),x,3,2/3*a^2*(a+b*x^3)^(3/2)/b^4-2/5*a*(a+b*x^3)^(5/2)/b^4+2/21*(a+b*x^3)^(7/2)/b^4-2/3*a^3*sqrt(a+b*x^3)/b^4],
[x^8/sqrt(a+b*x^3),x,3,-4/9*a*(a+b*x^3)^(3/2)/b^3+2/15*(a+b*x^3)^(5/2)/b^3+2/3*a^2*sqrt(a+b*x^3)/b^3],
[x^5/sqrt(a+b*x^3),x,3,2/9*(a+b*x^3)^(3/2)/b^2-2/3*a*sqrt(a+b*x^3)/b^2],
[x^2/sqrt(a+b*x^3),x,1,2/3*sqrt(a+b*x^3)/b],
[1/(x*sqrt(a+b*x^3)),x,3,-2/3*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)],
[1/(x^4*sqrt(a+b*x^3)),x,4,1/3*b*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/3*sqrt(a+b*x^3)/(a*x^3)],
[1/(x^7*sqrt(a+b*x^3)),x,5,-1/4*b^2*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(5/2)-1/6*sqrt(a+b*x^3)/(a*x^6)+1/4*b*sqrt(a+b*x^3)/(a^2*x^3)],
[x^6/sqrt(a+b*x^3),x,3,-16/55*a*x*sqrt(a+b*x^3)/b^2+2/11*x^4*sqrt(a+b*x^3)/b+32/55*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(7/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3/sqrt(a+b*x^3),x,2,2/5*x*sqrt(a+b*x^3)/b-4/5*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/sqrt(a+b*x^3),x,1,2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^3*sqrt(a+b*x^3)),x,2,-1/2*sqrt(a+b*x^3)/(a*x^2)-1/2*b^(2/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^6*sqrt(a+b*x^3)),x,3,-1/5*sqrt(a+b*x^3)/(a*x^5)+7/20*b*sqrt(a+b*x^3)/(a^2*x^2)+7/20*b^(5/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^2*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^7/sqrt(a+b*x^3),x,5,-20/91*a*x^2*sqrt(a+b*x^3)/b^2+2/13*x^5*sqrt(a+b*x^3)/b+80/91*a^2*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+80/91*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-40/91*3^(1/4)*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4/sqrt(a+b*x^3),x,4,2/7*x^2*sqrt(a+b*x^3)/b-8/7*a*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-8/7*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+4/7*3^(1/4)*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/sqrt(a+b*x^3),x,3,2*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+2*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^2*sqrt(a+b*x^3)),x,4,-sqrt(a+b*x^3)/(a*x)+b^(1/3)*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-1/2*3^(1/4)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^5*sqrt(a+b*x^3)),x,5,-1/4*sqrt(a+b*x^3)/(a*x^4)+5/8*b*sqrt(a+b*x^3)/(a^2*x)-5/8*b^(4/3)*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-5/4*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(5/3)*sqrt(2)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+5/16*3^(1/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^11/(a+b*x^3)^(3/2),x,3,-2/3*a*(a+b*x^3)^(3/2)/b^4+2/15*(a+b*x^3)^(5/2)/b^4+2/3*a^3/(b^4*sqrt(a+b*x^3))+2*a^2*sqrt(a+b*x^3)/b^4],
[x^8/(a+b*x^3)^(3/2),x,3,2/9*(a+b*x^3)^(3/2)/b^3-2/3*a^2/(b^3*sqrt(a+b*x^3))-4/3*a*sqrt(a+b*x^3)/b^3],
[x^5/(a+b*x^3)^(3/2),x,3,2/3*a/(b^2*sqrt(a+b*x^3))+2/3*sqrt(a+b*x^3)/b^2],
[x^2/(a+b*x^3)^(3/2),x,1,(-2/3)/(b*sqrt(a+b*x^3))],
[1/(x*(a+b*x^3)^(3/2)),x,4,-2/3*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)+2/3/(a*sqrt(a+b*x^3))],
[1/(x^4*(a+b*x^3)^(3/2)),x,5,b*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(5/2)-b/(a^2*sqrt(a+b*x^3))+(-1/3)/(a*x^3*sqrt(a+b*x^3))],
[1/(x^7*(a+b*x^3)^(3/2)),x,6,-5/4*b^2*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(7/2)+5/4*b^2/(a^3*sqrt(a+b*x^3))+(-1/6)/(a*x^6*sqrt(a+b*x^3))+5/12*b/(a^2*x^3*sqrt(a+b*x^3))],
[x^6/(a+b*x^3)^(3/2),x,3,-2/3*x^4/(b*sqrt(a+b*x^3))+16/15*x*sqrt(a+b*x^3)/b^2-32/15*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(7/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3/(a+b*x^3)^(3/2),x,2,-2/3*x/(b*sqrt(a+b*x^3))+4/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(a+b*x^3)^(3/2),x,2,2/3*x/(a*sqrt(a+b*x^3))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^3*(a+b*x^3)^(3/2)),x,3,2/3/(a*x^2*sqrt(a+b*x^3))-7/6*sqrt(a+b*x^3)/(a^2*x^2)-7/6*b^(2/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^2*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^6*(a+b*x^3)^(3/2)),x,4,2/3/(a*x^5*sqrt(a+b*x^3))-13/15*sqrt(a+b*x^3)/(a^2*x^5)+91/60*b*sqrt(a+b*x^3)/(a^3*x^2)+91/60*b^(5/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^3*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^7/(a+b*x^3)^(3/2),x,5,-2/3*x^5/(b*sqrt(a+b*x^3))+20/21*x^2*sqrt(a+b*x^3)/b^2-80/21*a*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-80/21*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+40/7*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4/(a+b*x^3)^(3/2),x,4,-2/3*x^2/(b*sqrt(a+b*x^3))+8/3*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+8/3*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-4*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/(a+b*x^3)^(3/2),x,4,2/3*x^2/(a*sqrt(a+b*x^3))-2/3*sqrt(a+b*x^3)/(a*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(2/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^2*(a+b*x^3)^(3/2)),x,5,2/3/(a*x*sqrt(a+b*x^3))-5/3*sqrt(a+b*x^3)/(a^2*x)+5/3*b^(1/3)*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+5/3*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-5/2*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^5*(a+b*x^3)^(3/2)),x,6,2/3/(a*x^4*sqrt(a+b*x^3))-11/12*sqrt(a+b*x^3)/(a^2*x^4)+55/24*b*sqrt(a+b*x^3)/(a^3*x)-55/24*b^(4/3)*sqrt(a+b*x^3)/(a^3*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-55/12*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(8/3)*sqrt(2)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+55/16*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^11/sqrt(1+x^3),x,3,2/3*(1+x^3)^(3/2)-2/5*(1+x^3)^(5/2)+2/21*(1+x^3)^(7/2)-2/3*sqrt(1+x^3)],
[x^8/sqrt(1+x^3),x,3,-4/9*(1+x^3)^(3/2)+2/15*(1+x^3)^(5/2)+2/3*sqrt(1+x^3)],
[x^5/sqrt(1+x^3),x,3,2/9*(1+x^3)^(3/2)-2/3*sqrt(1+x^3)],
[x^2/sqrt(1+x^3),x,1,2/3*sqrt(1+x^3)],
[1/(x*sqrt(1+x^3)),x,3,-2/3*arctanh(sqrt(1+x^3))],
[1/(x^4*sqrt(1+x^3)),x,4,1/3*arctanh(sqrt(1+x^3))-1/3*sqrt(1+x^3)/x^3],
[1/(x^7*sqrt(1+x^3)),x,5,-1/4*arctanh(sqrt(1+x^3))-1/6*sqrt(1+x^3)/x^6+1/4*sqrt(1+x^3)/x^3],
[1/(x^10*sqrt(1+x^3)),x,6,5/24*arctanh(sqrt(1+x^3))-1/9*sqrt(1+x^3)/x^9+5/36*sqrt(1+x^3)/x^6-5/24*sqrt(1+x^3)/x^3],
[x^6/sqrt(1+x^3),x,3,-16/55*x*sqrt(1+x^3)+2/11*x^4*sqrt(1+x^3)+32/55*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^3/sqrt(1+x^3),x,2,2/5*x*sqrt(1+x^3)-4/5*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/sqrt(1+x^3),x,1,2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^3*sqrt(1+x^3)),x,2,-1/2*sqrt(1+x^3)/x^2-1/2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^6*sqrt(1+x^3)),x,3,-1/5*sqrt(1+x^3)/x^5+7/20*sqrt(1+x^3)/x^2+7/20*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^7/sqrt(1+x^3),x,5,-20/91*x^2*sqrt(1+x^3)+2/13*x^5*sqrt(1+x^3)+80/91*sqrt(1+x^3)/(1+x+sqrt(3))+80/91*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-40/91*3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^4/sqrt(1+x^3),x,4,2/7*x^2*sqrt(1+x^3)-8/7*sqrt(1+x^3)/(1+x+sqrt(3))-8/7*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+4/7*3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x/sqrt(1+x^3),x,3,2*sqrt(1+x^3)/(1+x+sqrt(3))+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^2*sqrt(1+x^3)),x,4,-sqrt(1+x^3)/x+sqrt(1+x^3)/(1+x+sqrt(3))+(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-1/2*3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^5*sqrt(1+x^3)),x,5,-1/4*sqrt(1+x^3)/x^4+5/8*sqrt(1+x^3)/x-5/8*sqrt(1+x^3)/(1+x+sqrt(3))-5/4*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(2)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+5/16*3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^11/sqrt(1-x^3),x,3,2/3*(1-x^3)^(3/2)-2/5*(1-x^3)^(5/2)+2/21*(1-x^3)^(7/2)-2/3*sqrt(1-x^3)],
[x^8/sqrt(1-x^3),x,3,4/9*(1-x^3)^(3/2)-2/15*(1-x^3)^(5/2)-2/3*sqrt(1-x^3)],
[x^5/sqrt(1-x^3),x,3,2/9*(1-x^3)^(3/2)-2/3*sqrt(1-x^3)],
[x^2/sqrt(1-x^3),x,1,-2/3*sqrt(1-x^3)],
[1/(x*sqrt(1-x^3)),x,3,-2/3*arctanh(sqrt(1-x^3))],
[1/(x^4*sqrt(1-x^3)),x,4,-1/3*arctanh(sqrt(1-x^3))-1/3*sqrt(1-x^3)/x^3],
[1/(x^7*sqrt(1-x^3)),x,5,-1/4*arctanh(sqrt(1-x^3))-1/6*sqrt(1-x^3)/x^6-1/4*sqrt(1-x^3)/x^3],
[1/(x^10*sqrt(1-x^3)),x,6,-5/24*arctanh(sqrt(1-x^3))-1/9*sqrt(1-x^3)/x^9-5/36*sqrt(1-x^3)/x^6-5/24*sqrt(1-x^3)/x^3],
[x^6/sqrt(1-x^3),x,3,-16/55*x*sqrt(1-x^3)-2/11*x^4*sqrt(1-x^3)-32/55*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x^3/sqrt(1-x^3),x,2,-2/5*x*sqrt(1-x^3)-4/5*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/sqrt(1-x^3),x,1,-2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/(x^3*sqrt(1-x^3)),x,2,-1/2*sqrt(1-x^3)/x^2-1/2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/(x^6*sqrt(1-x^3)),x,3,-1/5*sqrt(1-x^3)/x^5-7/20*sqrt(1-x^3)/x^2-7/20*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x^7/sqrt(1-x^3),x,5,-20/91*x^2*sqrt(1-x^3)-2/13*x^5*sqrt(1-x^3)+80/91*sqrt(1-x^3)/(1-x+sqrt(3))+80/91*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-40/91*3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x^4/sqrt(1-x^3),x,4,-2/7*x^2*sqrt(1-x^3)+8/7*sqrt(1-x^3)/(1-x+sqrt(3))+8/7*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-4/7*3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x/sqrt(1-x^3),x,3,2*sqrt(1-x^3)/(1-x+sqrt(3))+2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/(x^2*sqrt(1-x^3)),x,4,-sqrt(1-x^3)/x-sqrt(1-x^3)/(1-x+sqrt(3))-(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+1/2*3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/(x^5*sqrt(1-x^3)),x,5,-1/4*sqrt(1-x^3)/x^4-5/8*sqrt(1-x^3)/x-5/8*sqrt(1-x^3)/(1-x+sqrt(3))-5/4*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(2)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+5/16*3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x^11/sqrt(-1+x^3),x,3,2/3*(-1+x^3)^(3/2)+2/5*(-1+x^3)^(5/2)+2/21*(-1+x^3)^(7/2)+2/3*sqrt(-1+x^3)],
[x^8/sqrt(-1+x^3),x,3,4/9*(-1+x^3)^(3/2)+2/15*(-1+x^3)^(5/2)+2/3*sqrt(-1+x^3)],
[x^5/sqrt(-1+x^3),x,3,2/9*(-1+x^3)^(3/2)+2/3*sqrt(-1+x^3)],
[x^2/sqrt(-1+x^3),x,1,2/3*sqrt(-1+x^3)],
[1/(x*sqrt(-1+x^3)),x,3,2/3*arctan(sqrt(-1+x^3))],
[1/(x^4*sqrt(-1+x^3)),x,4,1/3*arctan(sqrt(-1+x^3))+1/3*sqrt(-1+x^3)/x^3],
[1/(x^7*sqrt(-1+x^3)),x,5,1/4*arctan(sqrt(-1+x^3))+1/6*sqrt(-1+x^3)/x^6+1/4*sqrt(-1+x^3)/x^3],
[1/(x^10*sqrt(-1+x^3)),x,6,5/24*arctan(sqrt(-1+x^3))+1/9*sqrt(-1+x^3)/x^9+5/36*sqrt(-1+x^3)/x^6+5/24*sqrt(-1+x^3)/x^3],
[x^6/sqrt(-1+x^3),x,3,16/55*x*sqrt(-1+x^3)+2/11*x^4*sqrt(-1+x^3)-32/55*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x^3/sqrt(-1+x^3),x,2,2/5*x*sqrt(-1+x^3)-4/5*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[1/sqrt(-1+x^3),x,1,-2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[1/(x^3*sqrt(-1+x^3)),x,2,1/2*sqrt(-1+x^3)/x^2-1/2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[1/(x^6*sqrt(-1+x^3)),x,3,1/5*sqrt(-1+x^3)/x^5+7/20*sqrt(-1+x^3)/x^2-7/20*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x^7/sqrt(-1+x^3),x,5,20/91*x^2*sqrt(-1+x^3)+2/13*x^5*sqrt(-1+x^3)-80/91*sqrt(-1+x^3)/(1-x-sqrt(3))-80/91*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))+40/91*3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x^4/sqrt(-1+x^3),x,4,2/7*x^2*sqrt(-1+x^3)-8/7*sqrt(-1+x^3)/(1-x-sqrt(3))-8/7*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))+4/7*3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x/sqrt(-1+x^3),x,3,-2*sqrt(-1+x^3)/(1-x-sqrt(3))-2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))+3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[1/(x^2*sqrt(-1+x^3)),x,4,sqrt(-1+x^3)/x+sqrt(-1+x^3)/(1-x-sqrt(3))+(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-1/2*3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[1/(x^5*sqrt(-1+x^3)),x,5,1/4*sqrt(-1+x^3)/x^4+5/8*sqrt(-1+x^3)/x+5/8*sqrt(-1+x^3)/(1-x-sqrt(3))+5/4*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(2)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-5/16*3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x^11/sqrt(-1-x^3),x,3,2/3*(-1-x^3)^(3/2)+2/5*(-1-x^3)^(5/2)+2/21*(-1-x^3)^(7/2)+2/3*sqrt(-1-x^3)],
[x^8/sqrt(-1-x^3),x,3,-4/9*(-1-x^3)^(3/2)-2/15*(-1-x^3)^(5/2)-2/3*sqrt(-1-x^3)],
[x^5/sqrt(-1-x^3),x,3,2/9*(-1-x^3)^(3/2)+2/3*sqrt(-1-x^3)],
[x^2/sqrt(-1-x^3),x,1,-2/3*sqrt(-1-x^3)],
[1/(x*sqrt(-1-x^3)),x,3,2/3*arctan(sqrt(-1-x^3))],
[1/(x^4*sqrt(-1-x^3)),x,4,-1/3*arctan(sqrt(-1-x^3))+1/3*sqrt(-1-x^3)/x^3],
[1/(x^7*sqrt(-1-x^3)),x,5,1/4*arctan(sqrt(-1-x^3))+1/6*sqrt(-1-x^3)/x^6-1/4*sqrt(-1-x^3)/x^3],
[1/(x^10*sqrt(-1-x^3)),x,6,-5/24*arctan(sqrt(-1-x^3))+1/9*sqrt(-1-x^3)/x^9-5/36*sqrt(-1-x^3)/x^6+5/24*sqrt(-1-x^3)/x^3],
[x^6/sqrt(-1-x^3),x,3,16/55*x*sqrt(-1-x^3)-2/11*x^4*sqrt(-1-x^3)+32/55*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x^3/sqrt(-1-x^3),x,2,-2/5*x*sqrt(-1-x^3)-4/5*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[1/sqrt(-1-x^3),x,1,2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[1/(x^3*sqrt(-1-x^3)),x,2,1/2*sqrt(-1-x^3)/x^2-1/2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[1/(x^6*sqrt(-1-x^3)),x,3,1/5*sqrt(-1-x^3)/x^5-7/20*sqrt(-1-x^3)/x^2+7/20*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x^7/sqrt(-1-x^3),x,5,20/91*x^2*sqrt(-1-x^3)-2/13*x^5*sqrt(-1-x^3)-80/91*sqrt(-1-x^3)/(1+x-sqrt(3))-80/91*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+40/91*3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x^4/sqrt(-1-x^3),x,4,-2/7*x^2*sqrt(-1-x^3)+8/7*sqrt(-1-x^3)/(1+x-sqrt(3))+8/7*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))-4/7*3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x/sqrt(-1-x^3),x,3,-2*sqrt(-1-x^3)/(1+x-sqrt(3))-2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[1/(x^2*sqrt(-1-x^3)),x,4,sqrt(-1-x^3)/x-sqrt(-1-x^3)/(1+x-sqrt(3))-(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+1/2*3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[1/(x^5*sqrt(-1-x^3)),x,5,1/4*sqrt(-1-x^3)/x^4-5/8*sqrt(-1-x^3)/x+5/8*sqrt(-1-x^3)/(1+x-sqrt(3))+5/4*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(2)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))-5/16*3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],

# Integrands of the form x^(m/2) (a+b x^3)^(p/2)

# Integrands of the form x^m (a+b x^3)^(p/3)

# p>0
[x^11*(a+b*x^3)^(1/3),x,3,-1/4*a^3*(a+b*x^3)^(4/3)/b^4+3/7*a^2*(a+b*x^3)^(7/3)/b^4-3/10*a*(a+b*x^3)^(10/3)/b^4+1/13*(a+b*x^3)^(13/3)/b^4],
[x^8*(a+b*x^3)^(1/3),x,3,1/4*a^2*(a+b*x^3)^(4/3)/b^3-2/7*a*(a+b*x^3)^(7/3)/b^3+1/10*(a+b*x^3)^(10/3)/b^3],
[x^5*(a+b*x^3)^(1/3),x,3,-1/4*a*(a+b*x^3)^(4/3)/b^2+1/7*(a+b*x^3)^(7/3)/b^2],
[x^2*(a+b*x^3)^(1/3),x,1,1/4*(a+b*x^3)^(4/3)/b],
[(a+b*x^3)^(1/3)/x,x,6,(a+b*x^3)^(1/3)-1/2*a^(1/3)*log(x)+1/2*a^(1/3)*log(a^(1/3)-(a+b*x^3)^(1/3))-a^(1/3)*arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/sqrt(3)],
[(a+b*x^3)^(1/3)/x^4,x,6,-1/3*(a+b*x^3)^(1/3)/x^3-1/6*b*log(x)/a^(2/3)+1/6*b*log(a^(1/3)-(a+b*x^3)^(1/3))/a^(2/3)-1/3*b*arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/(a^(2/3)*sqrt(3))],
[x^4*(a+b*x^3)^(1/3),x,3,1/18*a*x^2*(a+b*x^3)^(1/3)/b+1/6*x^5*(a+b*x^3)^(1/3)+1/18*a^2*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(5/3)+1/9*a^2*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(5/3)*sqrt(3))],
[x*(a+b*x^3)^(1/3),x,2,1/3*x^2*(a+b*x^3)^(1/3)-1/6*a*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-1/3*a*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[(a+b*x^3)^(1/3)/x^2,x,2,-(a+b*x^3)^(1/3)/x-1/2*b^(1/3)*log(b^(1/3)*x-(a+b*x^3)^(1/3))-b^(1/3)*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/sqrt(3)],
[(a+b*x^3)^(1/3)/x^5,x,1,-1/4*(a+b*x^3)^(4/3)/(a*x^4)],
[(a+b*x^3)^(1/3)/x^8,x,2,-1/7*(a+b*x^3)^(4/3)/(a*x^7)+3/28*b*(a+b*x^3)^(4/3)/(a^2*x^4)],
[(a+b*x^3)^(1/3)/x^11,x,3,-1/10*(a+b*x^3)^(4/3)/(a*x^10)+3/35*b*(a+b*x^3)^(4/3)/(a^2*x^7)-9/140*b^2*(a+b*x^3)^(4/3)/(a^3*x^4)],
[(a+b*x^3)^(1/3)/x^14,x,4,-1/13*(a+b*x^3)^(4/3)/(a*x^13)+9/130*b*(a+b*x^3)^(4/3)/(a^2*x^10)-27/455*b^2*(a+b*x^3)^(4/3)/(a^3*x^7)+81/1820*b^3*(a+b*x^3)^(4/3)/(a^4*x^4)],
[x^3*(a+b*x^3)^(1/3),x,2,1/4*x^4*(a+b*x^3)^(4/3)*hypergeom([1,8/3],[7/3],-b*x^3/a)/a,1/4*x^4*(a+b*x^3)^(1/3)*hypergeom([-1/3,4/3],[7/3],-b*x^3/a)/(1+b*x^3/a)^(1/3)],
[(a+b*x^3)^(1/3),x,2,x*(a+b*x^3)^(4/3)*hypergeom([1,5/3],[4/3],-b*x^3/a)/a,x*(a+b*x^3)^(1/3)*hypergeom([-1/3,1/3],[4/3],-b*x^3/a)/(1+b*x^3/a)^(1/3)],
[(a+b*x^3)^(1/3)/x^3,x,2,-1/2*(a+b*x^3)^(4/3)*hypergeom([2/3,1],[1/3],-b*x^3/a)/(a*x^2),-1/2*(a+b*x^3)^(1/3)*hypergeom([-2/3,-1/3],[1/3],-b*x^3/a)/(x^2*(1+b*x^3/a)^(1/3))],
[(a+b*x^3)^(1/3)/x^6,x,2,-1/5*(a+b*x^3)^(4/3)*hypergeom([-1/3,1],[-2/3],-b*x^3/a)/(a*x^5),-1/5*(a+b*x^3)^(1/3)*hypergeom([-5/3,-1/3],[-2/3],-b*x^3/a)/(x^5*(1+b*x^3/a)^(1/3))],
[x^11*(a+b*x^3)^(2/3),x,3,-1/5*a^3*(a+b*x^3)^(5/3)/b^4+3/8*a^2*(a+b*x^3)^(8/3)/b^4-3/11*a*(a+b*x^3)^(11/3)/b^4+1/14*(a+b*x^3)^(14/3)/b^4],
[x^8*(a+b*x^3)^(2/3),x,3,1/5*a^2*(a+b*x^3)^(5/3)/b^3-1/4*a*(a+b*x^3)^(8/3)/b^3+1/11*(a+b*x^3)^(11/3)/b^3],
[x^5*(a+b*x^3)^(2/3),x,3,-1/5*a*(a+b*x^3)^(5/3)/b^2+1/8*(a+b*x^3)^(8/3)/b^2],
[x^2*(a+b*x^3)^(2/3),x,1,1/5*(a+b*x^3)^(5/3)/b],
[(a+b*x^3)^(2/3)/x,x,6,1/2*(a+b*x^3)^(2/3)-1/2*a^(2/3)*log(x)+1/2*a^(2/3)*log(a^(1/3)-(a+b*x^3)^(1/3))+a^(2/3)*arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/sqrt(3)],
[(a+b*x^3)^(2/3)/x^4,x,6,-1/3*(a+b*x^3)^(2/3)/x^3-1/3*b*log(x)/a^(1/3)+1/3*b*log(a^(1/3)-(a+b*x^3)^(1/3))/a^(1/3)+2/3*b*arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/(a^(1/3)*sqrt(3))],
[x^4*(a+b*x^3)^(2/3),x,2,1/5*x^5*(a+b*x^3)^(5/3)*hypergeom([1,10/3],[8/3],-b*x^3/a)/a,1/5*x^5*(a+b*x^3)^(2/3)*hypergeom([-2/3,5/3],[8/3],-b*x^3/a)/(1+b*x^3/a)^(2/3)],
[x*(a+b*x^3)^(2/3),x,2,1/2*x^2*(a+b*x^3)^(5/3)*hypergeom([1,7/3],[5/3],-b*x^3/a)/a,1/2*x^2*(a+b*x^3)^(2/3)*hypergeom([-2/3,2/3],[5/3],-b*x^3/a)/(1+b*x^3/a)^(2/3)],
[(a+b*x^3)^(2/3)/x^2,x,2,-(a+b*x^3)^(5/3)*hypergeom([1,4/3],[2/3],-b*x^3/a)/(a*x),-(a+b*x^3)^(2/3)*hypergeom([-2/3,-1/3],[2/3],-b*x^3/a)/(x*(1+b*x^3/a)^(2/3))],
[(a+b*x^3)^(2/3)/x^5,x,2,-1/4*(a+b*x^3)^(5/3)*hypergeom([1/3,1],[-1/3],-b*x^3/a)/(a*x^4),-1/4*(a+b*x^3)^(2/3)*hypergeom([-4/3,-2/3],[-1/3],-b*x^3/a)/(x^4*(1+b*x^3/a)^(2/3))],
[x^3*(a+b*x^3)^(2/3),x,3,1/9*a*x*(a+b*x^3)^(2/3)/b+1/6*x^4*(a+b*x^3)^(2/3)+1/18*a^2*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(4/3)-1/9*a^2*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(4/3)*sqrt(3))],
[(a+b*x^3)^(2/3),x,2,1/3*x*(a+b*x^3)^(2/3)-1/3*a*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(1/3)+2/3*a*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))],
[(a+b*x^3)^(2/3)/x^3,x,2,-1/2*(a+b*x^3)^(2/3)/x^2-1/2*b^(2/3)*log(-b^(1/3)*x+(a+b*x^3)^(1/3))+b^(2/3)*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/sqrt(3)],
[(a+b*x^3)^(2/3)/x^6,x,1,-1/5*(a+b*x^3)^(5/3)/(a*x^5)],
[(a+b*x^3)^(2/3)/x^9,x,2,-1/8*(a+b*x^3)^(5/3)/(a*x^8)+3/40*b*(a+b*x^3)^(5/3)/(a^2*x^5)],
[(a+b*x^3)^(2/3)/x^12,x,3,-1/11*(a+b*x^3)^(5/3)/(a*x^11)+3/44*b*(a+b*x^3)^(5/3)/(a^2*x^8)-9/220*b^2*(a+b*x^3)^(5/3)/(a^3*x^5)],
[x^8*(1-x^3)^(6/5),x,3,-5/33*(1-x^3)^(11/5)+5/24*(1-x^3)^(16/5)-5/63*(1-x^3)^(21/5)],

# p<0
[x^11/(a+b*x^3)^(1/3),x,3,-1/2*a^3*(a+b*x^3)^(2/3)/b^4+3/5*a^2*(a+b*x^3)^(5/3)/b^4-3/8*a*(a+b*x^3)^(8/3)/b^4+1/11*(a+b*x^3)^(11/3)/b^4],
[x^8/(a+b*x^3)^(1/3),x,3,1/2*a^2*(a+b*x^3)^(2/3)/b^3-2/5*a*(a+b*x^3)^(5/3)/b^3+1/8*(a+b*x^3)^(8/3)/b^3],
[x^5/(a+b*x^3)^(1/3),x,3,-1/2*a*(a+b*x^3)^(2/3)/b^2+1/5*(a+b*x^3)^(5/3)/b^2],
[x^2/(a+b*x^3)^(1/3),x,1,1/2*(a+b*x^3)^(2/3)/b],
[1/(x*(a+b*x^3)^(1/3)),x,5,-1/2*log(x)/a^(1/3)+1/2*log(a^(1/3)-(a+b*x^3)^(1/3))/a^(1/3)+arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/(a^(1/3)*sqrt(3))],
[1/(x^4*(a+b*x^3)^(1/3)),x,6,-1/3*(a+b*x^3)^(2/3)/(a*x^3)+1/6*b*log(x)/a^(4/3)-1/6*b*log(a^(1/3)-(a+b*x^3)^(1/3))/a^(4/3)-1/3*b*arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/(a^(4/3)*sqrt(3))],
[x^7/(a+b*x^3)^(1/3),x,2,1/8*x^8*(a+b*x^3)^(2/3)*hypergeom([1,10/3],[11/3],-b*x^3/a)/a,1/8*x^8*(1+b*x^3/a)^(1/3)*hypergeom([1/3,8/3],[11/3],-b*x^3/a)/(a+b*x^3)^(1/3)],
[x^4/(a+b*x^3)^(1/3),x,2,1/5*x^5*(a+b*x^3)^(2/3)*hypergeom([1,7/3],[8/3],-b*x^3/a)/a,1/5*x^5*(1+b*x^3/a)^(1/3)*hypergeom([1/3,5/3],[8/3],-b*x^3/a)/(a+b*x^3)^(1/3)],
[x/(a+b*x^3)^(1/3),x,2,1/2*x^2*(a+b*x^3)^(2/3)*hypergeom([1,4/3],[5/3],-b*x^3/a)/a,1/2*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3,2/3],[5/3],-b*x^3/a)/(a+b*x^3)^(1/3)],
[1/(x^2*(a+b*x^3)^(1/3)),x,2,-(a+b*x^3)^(2/3)*hypergeom([1/3,1],[2/3],-b*x^3/a)/(a*x),-(1+b*x^3/a)^(1/3)*hypergeom([-1/3,1/3],[2/3],-b*x^3/a)/(x*(a+b*x^3)^(1/3))],
[1/(x^5*(a+b*x^3)^(1/3)),x,2,-1/4*(a+b*x^3)^(2/3)*hypergeom([-2/3,1],[-1/3],-b*x^3/a)/(a*x^4),-1/4*(1+b*x^3/a)^(1/3)*hypergeom([-4/3,1/3],[-1/3],-b*x^3/a)/(x^4*(a+b*x^3)^(1/3))],
[x^3/(a+b*x^3)^(1/3),x,2,1/3*x*(a+b*x^3)^(2/3)/b+1/6*a*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(4/3)-1/3*a*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(4/3)*sqrt(3))],
[1/(a+b*x^3)^(1/3),x,1,-1/2*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(1/3)+arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))],
[1/(x^3*(a+b*x^3)^(1/3)),x,1,-1/2*(a+b*x^3)^(2/3)/(a*x^2)],
[1/(x^6*(a+b*x^3)^(1/3)),x,2,-1/5*(a+b*x^3)^(2/3)/(a*x^5)+3/10*b*(a+b*x^3)^(2/3)/(a^2*x^2)],
[1/(x^9*(a+b*x^3)^(1/3)),x,3,-1/8*(a+b*x^3)^(2/3)/(a*x^8)+3/20*b*(a+b*x^3)^(2/3)/(a^2*x^5)-9/40*b^2*(a+b*x^3)^(2/3)/(a^3*x^2)],
[1/(x^12*(a+b*x^3)^(1/3)),x,4,-1/11*(a+b*x^3)^(2/3)/(a*x^11)+9/88*b*(a+b*x^3)^(2/3)/(a^2*x^8)-27/220*b^2*(a+b*x^3)^(2/3)/(a^3*x^5)+81/440*b^3*(a+b*x^3)^(2/3)/(a^4*x^2)],
[x^11/(a+b*x^3)^(2/3),x,3,-a^3*(a+b*x^3)^(1/3)/b^4+3/4*a^2*(a+b*x^3)^(4/3)/b^4-3/7*a*(a+b*x^3)^(7/3)/b^4+1/10*(a+b*x^3)^(10/3)/b^4],
[x^8/(a+b*x^3)^(2/3),x,3,a^2*(a+b*x^3)^(1/3)/b^3-1/2*a*(a+b*x^3)^(4/3)/b^3+1/7*(a+b*x^3)^(7/3)/b^3],
[x^5/(a+b*x^3)^(2/3),x,3,-a*(a+b*x^3)^(1/3)/b^2+1/4*(a+b*x^3)^(4/3)/b^2],
[x^2/(a+b*x^3)^(2/3),x,1,(a+b*x^3)^(1/3)/b],
[1/(x*(a+b*x^3)^(2/3)),x,5,-1/2*log(x)/a^(2/3)+1/2*log(a^(1/3)-(a+b*x^3)^(1/3))/a^(2/3)-arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/(a^(2/3)*sqrt(3))],
[1/(x^4*(a+b*x^3)^(2/3)),x,6,-1/3*(a+b*x^3)^(1/3)/(a*x^3)+1/3*b*log(x)/a^(5/3)-1/3*b*log(a^(1/3)-(a+b*x^3)^(1/3))/a^(5/3)+2/3*b*arctan((a^(1/3)+2*(a+b*x^3)^(1/3))/(a^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3))],
[x^7/(a+b*x^3)^(2/3),x,3,-5/18*a*x^2*(a+b*x^3)^(1/3)/b^2+1/6*x^5*(a+b*x^3)^(1/3)/b-5/18*a^2*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(8/3)-5/9*a^2*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(8/3)*sqrt(3))],
[x^4/(a+b*x^3)^(2/3),x,2,1/3*x^2*(a+b*x^3)^(1/3)/b+1/3*a*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(5/3)+2/3*a*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(5/3)*sqrt(3))],
[x/(a+b*x^3)^(2/3),x,1,-1/2*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[1/(x^2*(a+b*x^3)^(2/3)),x,1,-(a+b*x^3)^(1/3)/(a*x)],
[1/(x^5*(a+b*x^3)^(2/3)),x,2,-1/4*(a+b*x^3)^(1/3)/(a*x^4)+3/4*b*(a+b*x^3)^(1/3)/(a^2*x)],
[1/(x^8*(a+b*x^3)^(2/3)),x,3,-1/7*(a+b*x^3)^(1/3)/(a*x^7)+3/14*b*(a+b*x^3)^(1/3)/(a^2*x^4)-9/14*b^2*(a+b*x^3)^(1/3)/(a^3*x)],
[1/(x^11*(a+b*x^3)^(2/3)),x,4,-1/10*(a+b*x^3)^(1/3)/(a*x^10)+9/70*b*(a+b*x^3)^(1/3)/(a^2*x^7)-27/140*b^2*(a+b*x^3)^(1/3)/(a^3*x^4)+81/140*b^3*(a+b*x^3)^(1/3)/(a^4*x)],
[x^6/(a+b*x^3)^(2/3),x,2,1/7*x^7*(a+b*x^3)^(1/3)*hypergeom([1,8/3],[10/3],-b*x^3/a)/a,1/7*x^7*(1+b*x^3/a)^(2/3)*hypergeom([2/3,7/3],[10/3],-b*x^3/a)/(a+b*x^3)^(2/3)],
[x^3/(a+b*x^3)^(2/3),x,2,1/4*x^4*(a+b*x^3)^(1/3)*hypergeom([1,5/3],[7/3],-b*x^3/a)/a,1/4*x^4*(1+b*x^3/a)^(2/3)*hypergeom([2/3,4/3],[7/3],-b*x^3/a)/(a+b*x^3)^(2/3)],
[1/(a+b*x^3)^(2/3),x,2,x*(a+b*x^3)^(1/3)*hypergeom([2/3,1],[4/3],-b*x^3/a)/a,x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)],
[1/(x^3*(a+b*x^3)^(2/3)),x,2,-1/2*(a+b*x^3)^(1/3)*hypergeom([-1/3,1],[1/3],-b*x^3/a)/(a*x^2),-1/2*(1+b*x^3/a)^(2/3)*hypergeom([-2/3,2/3],[1/3],-b*x^3/a)/(x^2*(a+b*x^3)^(2/3))],
[1/(x^6*(a+b*x^3)^(2/3)),x,2,-1/5*(a+b*x^3)^(1/3)*hypergeom([-4/3,1],[-2/3],-b*x^3/a)/(a*x^5),-1/5*(1+b*x^3/a)^(2/3)*hypergeom([-5/3,2/3],[-2/3],-b*x^3/a)/(x^5*(a+b*x^3)^(2/3))],
[1/(a-b*x^3)^(1/3),x,1,1/2*log(b^(1/3)*x+(a-b*x^3)^(1/3))/b^(1/3)-arctan((1-2*b^(1/3)*x/(a-b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))],
[1/(2+x^3)^(1/3),x,1,-1/2*log(-x+(2+x^3)^(1/3))+arctan((1+2*x/(2+x^3)^(1/3))/sqrt(3))/sqrt(3)],
[x/(1-x^3)^(2/3),x,1,-1/2*log(-x-(1-x^3)^(1/3))-arctan((1-2*x/(1-x^3)^(1/3))/sqrt(3))/sqrt(3)],

# Integrands of the form x^m (a+b x^3)^(p/4)
[x^2/(2+x^3)^(1/4),x,1,4/9*(2+x^3)^(3/4)],

# Integrands of the form (c x)^m (a+b x^3)^p with m symbolic
[x^m*(a+b*x^3)^8,x,2,a^8*x^(1+m)/(1+m)+8*a^7*b*x^(4+m)/(4+m)+28*a^6*b^2*x^(7+m)/(7+m)+56*a^5*b^3*x^(10+m)/(10+m)+70*a^4*b^4*x^(13+m)/(13+m)+56*a^3*b^5*x^(16+m)/(16+m)+28*a^2*b^6*x^(19+m)/(19+m)+8*a*b^7*x^(22+m)/(22+m)+b^8*x^(25+m)/(25+m)],
[x^m*(a+b*x^3)^5,x,2,a^5*x^(1+m)/(1+m)+5*a^4*b*x^(4+m)/(4+m)+10*a^3*b^2*x^(7+m)/(7+m)+10*a^2*b^3*x^(10+m)/(10+m)+5*a*b^4*x^(13+m)/(13+m)+b^5*x^(16+m)/(16+m)],
[x^m*(a+b*x^3)^3,x,2,a^3*x^(1+m)/(1+m)+3*a^2*b*x^(4+m)/(4+m)+3*a*b^2*x^(7+m)/(7+m)+b^3*x^(10+m)/(10+m)],
[x^m*(a+b*x^3)^2,x,2,a^2*x^(1+m)/(1+m)+2*a*b*x^(4+m)/(4+m)+b^2*x^(7+m)/(7+m)],
[x^m*(a+b*x^3),x,2,a*x^(1+m)/(1+m)+b*x^(4+m)/(4+m)],
[x^m/(a+b*x^3),x,1,x^(1+m)*hypergeom([1,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(a*(1+m))],
[x^m/(a+b*x^3)^2,x,1,x^(1+m)*hypergeom([2,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(a^2*(1+m))],
[x^m/(a+b*x^3)^3,x,1,x^(1+m)*hypergeom([3,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(a^3*(1+m))],
[x^m*(a+b*x^3)^(3/2),x,2,a*x^(1+m)*hypergeom([-3/2,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)*sqrt(a+b*x^3)/((1+m)*sqrt(1+b*x^3/a))],
[x^m*sqrt(a+b*x^3),x,2,x^(1+m)*hypergeom([-1/2,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)*sqrt(a+b*x^3)/((1+m)*sqrt(1+b*x^3/a))],
[x^m/sqrt(a+b*x^3),x,2,x^(1+m)*hypergeom([1,1/6*(5+2*m)],[1/3*(4+m)],-b*x^3/a)*sqrt(a+b*x^3)/(a*(1+m)),x^(1+m)*hypergeom([1/2,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)*sqrt(1+b*x^3/a)/((1+m)*sqrt(a+b*x^3))],
[x^m/(a+b*x^3)^(3/2),x,2,x^(1+m)*hypergeom([3/2,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)*sqrt(1+b*x^3/a)/(a*(1+m)*sqrt(a+b*x^3))],
[(c*x)^m*(a+b*x^3)^(4/3),x,2,a*(c*x)^(1+m)*(a+b*x^3)^(1/3)*hypergeom([-4/3,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(c*(1+m)*(1+b*x^3/a)^(1/3))],
[(c*x)^m*(a+b*x^3)^(2/3),x,2,(c*x)^(1+m)*(a+b*c^3*x^3)^(5/3)*hypergeom([1,1/3*(6+m)],[1/3*(4+m)],-b*c^3*x^3/a)/(a*c*(1+m)),(c*x)^(1+m)*(a+b*x^3)^(2/3)*hypergeom([-2/3,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(c*(1+m)*(1+b*x^3/a)^(2/3))],
[(c*x)^m*(a+b*x^3)^(1/3),x,2,(c*x)^(1+m)*(a+b*c^3*x^3)^(4/3)*hypergeom([1,1/3*(5+m)],[1/3*(4+m)],-b*c^3*x^3/a)/(a*c*(1+m)),(c*x)^(1+m)*(a+b*x^3)^(1/3)*hypergeom([-1/3,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(c*(1+m)*(1+b*x^3/a)^(1/3))],

# Integrands of the form (c x)^m (a+b x^3)^p with p symbolic
[(c*x)^m*(a+b*x^3)^p,x,2,(c*x)^(1+m)*(a+b*x^3)^p*hypergeom([1/3*(1+m),-p],[1/3*(4+m)],-b*x^3/a)/(c*(1+m)*(1+b*x^3/a)^p)],
[x^2*(a+b*x^3)^p,x,1,1/3*(a+b*x^3)^(1+p)/(b*(1+p))],
[x^5*(a+b*x^3)^p,x,3,-1/3*a*(a+b*x^3)^(1+p)/(b^2*(1+p))+1/3*(a+b*x^3)^(2+p)/(b^2*(2+p))],
[x^8*(a+b*x^3)^p,x,3,1/3*a^2*(a+b*x^3)^(1+p)/(b^3*(1+p))-2/3*a*(a+b*x^3)^(2+p)/(b^3*(2+p))+1/3*(a+b*x^3)^(3+p)/(b^3*(3+p))],
[x^11*(a+b*x^3)^p,x,3,-1/3*a^3*(a+b*x^3)^(1+p)/(b^4*(1+p))+a^2*(a+b*x^3)^(2+p)/(b^4*(2+p))-a*(a+b*x^3)^(3+p)/(b^4*(3+p))+1/3*(a+b*x^3)^(4+p)/(b^4*(4+p))],

# Integrands of the form (c x)^m (a+b x^4)^p

# Integrands of the form x^m (a+b x^4)^p

# p>0
[x^m*(a+b*x^4),x,2,a*x^(1+m)/(1+m)+b*x^(5+m)/(5+m)],
[x^5*(a+b*x^4),x,2,1/6*a*x^6+1/10*b*x^10],
[x^4*(a+b*x^4),x,2,1/5*a*x^5+1/9*b*x^9],
[x^3*(a+b*x^4),x,2,1/4*a*x^4+1/8*b*x^8],
[x^2*(a+b*x^4),x,2,1/3*a*x^3+1/7*b*x^7],
[x*(a+b*x^4),x,2,1/2*a*x^2+1/6*b*x^6],
[a+b*x^4,x,1,a*x+1/5*b*x^5],
[(a+b*x^4)/x,x,2,1/4*b*x^4+a*log(x)],
[(a+b*x^4)/x^2,x,2,-a/x+1/3*b*x^3],
[(a+b*x^4)/x^3,x,2,-1/2*a/x^2+1/2*b*x^2],
[(a+b*x^4)/x^4,x,2,-1/3*a/x^3+b*x],
[(a+b*x^4)/x^5,x,2,-1/4*a/x^4+b*log(x)],
[(a+b*x^4)/x^6,x,2,-1/5*a/x^5-b/x],
[(a+b*x^4)/x^7,x,2,-1/6*a/x^6-1/2*b/x^2],
[(a+b*x^4)/x^8,x,2,-1/7*a/x^7-1/3*b/x^3],
[(a+b*x^4)/x^9,x,2,-1/8*a/x^8-1/4*b/x^4],
[(a+b*x^4)/x^10,x,2,-1/9*a/x^9-1/5*b/x^5],
[x^m*(a+b*x^4)^2,x,2,a^2*x^(1+m)/(1+m)+2*a*b*x^(5+m)/(5+m)+b^2*x^(9+m)/(9+m)],
[x^5*(a+b*x^4)^2,x,2,1/6*a^2*x^6+1/5*a*b*x^10+1/14*b^2*x^14],
[x^4*(a+b*x^4)^2,x,2,1/5*a^2*x^5+2/9*a*b*x^9+1/13*b^2*x^13],
[x^3*(a+b*x^4)^2,x,1,1/12*(a+b*x^4)^3/b],
[x^2*(a+b*x^4)^2,x,2,1/3*a^2*x^3+2/7*a*b*x^7+1/11*b^2*x^11],
[x*(a+b*x^4)^2,x,2,1/2*a^2*x^2+1/3*a*b*x^6+1/10*b^2*x^10],
[(a+b*x^4)^2,x,2,a^2*x+2/5*a*b*x^5+1/9*b^2*x^9],
[(a+b*x^4)^2/x,x,3,1/2*a*b*x^4+1/8*b^2*x^8+a^2*log(x)],
[(a+b*x^4)^2/x^2,x,2,-a^2/x+2/3*a*b*x^3+1/7*b^2*x^7],
[(a+b*x^4)^2/x^3,x,2,-1/2*a^2/x^2+a*b*x^2+1/6*b^2*x^6],
[(a+b*x^4)^2/x^4,x,2,-1/3*a^2/x^3+2*a*b*x+1/5*b^2*x^5],
[(a+b*x^4)^2/x^5,x,3,-1/4*a^2/x^4+1/4*b^2*x^4+2*a*b*log(x)],
[x^m*(a+b*x^4)^3,x,2,a^3*x^(1+m)/(1+m)+3*a^2*b*x^(5+m)/(5+m)+3*a*b^2*x^(9+m)/(9+m)+b^3*x^(13+m)/(13+m)],
[x^5*(a+b*x^4)^3,x,2,1/6*a^3*x^6+3/10*a^2*b*x^10+3/14*a*b^2*x^14+1/18*b^3*x^18],
[x^4*(a+b*x^4)^3,x,2,1/5*a^3*x^5+1/3*a^2*b*x^9+3/13*a*b^2*x^13+1/17*b^3*x^17],
[x^3*(a+b*x^4)^3,x,1,1/16*(a+b*x^4)^4/b],
[x^2*(a+b*x^4)^3,x,2,1/3*a^3*x^3+3/7*a^2*b*x^7+3/11*a*b^2*x^11+1/15*b^3*x^15],
[x*(a+b*x^4)^3,x,2,1/2*a^3*x^2+1/2*a^2*b*x^6+3/10*a*b^2*x^10+1/14*b^3*x^14],
[(a+b*x^4)^3,x,2,a^3*x+3/5*a^2*b*x^5+1/3*a*b^2*x^9+1/13*b^3*x^13],
[(a+b*x^4)^3/x,x,3,3/4*a^2*b*x^4+3/8*a*b^2*x^8+1/12*b^3*x^12+a^3*log(x)],
[(a+b*x^4)^3/x^2,x,2,-a^3/x+a^2*b*x^3+3/7*a*b^2*x^7+1/11*b^3*x^11],
[(a+b*x^4)^3/x^3,x,2,-1/2*a^3/x^2+3/2*a^2*b*x^2+1/2*a*b^2*x^6+1/10*b^3*x^10],
[(a+b*x^4)^3/x^4,x,2,-1/3*a^3/x^3+3*a^2*b*x+3/5*a*b^2*x^5+1/9*b^3*x^9],
[(a+b*x^4)^3/x^5,x,3,-1/4*a^3/x^4+3/4*a*b^2*x^4+1/8*b^3*x^8+3*a^2*b*log(x)],

# p<0
[x^9/(a+c*x^4),x,4,-1/2*a*x^2/c^2+1/6*x^6/c+1/2*a^(3/2)*arctan(x^2*sqrt(c)/sqrt(a))/c^(5/2)],
[x^7/(a+c*x^4),x,3,1/4*x^4/c-1/4*a*log(a+c*x^4)/c^2],
[x^5/(a+c*x^4),x,3,1/2*x^2/c-1/2*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(a)/c^(3/2)],
[x^3/(a+c*x^4),x,1,1/4*log(a+c*x^4)/c],
[x/(a+c*x^4),x,2,1/2*arctan(x^2*sqrt(c)/sqrt(a))/(sqrt(a)*sqrt(c))],
[1/(x*(a+c*x^4)),x,4,log(x)/a-1/4*log(a+c*x^4)/a],
[1/(x^3*(a+c*x^4)),x,3,(-1/2)/(a*x^2)-1/2*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/a^(3/2)],
[1/(x^5*(a+c*x^4)),x,3,(-1/4)/(a*x^4)-c*log(x)/a^2+1/4*c*log(a+c*x^4)/a^2],
[1/(x^7*(a+c*x^4)),x,4,(-1/6)/(a*x^6)+1/2*c/(a^2*x^2)+1/2*c^(3/2)*arctan(x^2*sqrt(c)/sqrt(a))/a^(5/2)],
[x^4/(a+c*x^4),x,10,x/c+1/2*a^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(c^(5/4)*sqrt(2))-1/2*a^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(c^(5/4)*sqrt(2))+1/4*a^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(c^(5/4)*sqrt(2))-1/4*a^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(c^(5/4)*sqrt(2))],
[x^2/(a+c*x^4),x,9,-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(1/4)*c^(3/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(1/4)*c^(3/4)*sqrt(2))+1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(1/4)*c^(3/4)*sqrt(2))-1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(1/4)*c^(3/4)*sqrt(2))],
[1/(a+c*x^4),x,9,-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))],
[1/(x^2*(a+c*x^4)),x,10,(-1)/(a*x)+1/2*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(5/4)*sqrt(2))-1/2*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(5/4)*sqrt(2))-1/4*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(5/4)*sqrt(2))+1/4*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(5/4)*sqrt(2))],
[1/(x^4*(a+c*x^4)),x,10,(-1/3)/(a*x^3)+1/2*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*sqrt(2))-1/2*c^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*sqrt(2))+1/4*c^(3/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*sqrt(2))-1/4*c^(3/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*sqrt(2))],
[x^11/(a+c*x^4)^2,x,3,1/4*x^4/c^2-1/4*a^2/(c^3*(a+c*x^4))-1/2*a*log(a+c*x^4)/c^3],
[x^9/(a+c*x^4)^2,x,4,3/4*x^2/c^2-1/4*x^6/(c*(a+c*x^4))-3/4*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(a)/c^(5/2)],
[x^7/(a+c*x^4)^2,x,3,1/4*a/(c^2*(a+c*x^4))+1/4*log(a+c*x^4)/c^2],
[x^5/(a+c*x^4)^2,x,3,-1/4*x^2/(c*(a+c*x^4))+1/4*arctan(x^2*sqrt(c)/sqrt(a))/(c^(3/2)*sqrt(a))],
[x^3/(a+c*x^4)^2,x,1,(-1/4)/(c*(a+c*x^4))],
[x/(a+c*x^4)^2,x,3,1/4*x^2/(a*(a+c*x^4))+1/4*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*sqrt(c))],
[1/(x*(a+c*x^4)^2),x,3,1/4/(a*(a+c*x^4))+log(x)/a^2-1/4*log(a+c*x^4)/a^2],
[1/(x^3*(a+c*x^4)^2),x,4,(-3/4)/(a^2*x^2)+1/4/(a*x^2*(a+c*x^4))-3/4*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/a^(5/2)],
[x^6/(a+c*x^4)^2,x,10,-1/4*x^3/(c*(a+c*x^4))-3/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(1/4)*c^(7/4)*sqrt(2))+3/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(1/4)*c^(7/4)*sqrt(2))+3/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(1/4)*c^(7/4)*sqrt(2))-3/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(1/4)*c^(7/4)*sqrt(2))],
[x^4/(a+c*x^4)^2,x,10,-1/4*x/(c*(a+c*x^4))-1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(5/4)*sqrt(2))+1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(5/4)*sqrt(2))-1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(5/4)*sqrt(2))+1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(5/4)*sqrt(2))],
[x^2/(a+c*x^4)^2,x,10,1/4*x^3/(a*(a+c*x^4))-1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(5/4)*c^(3/4)*sqrt(2))+1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(5/4)*c^(3/4)*sqrt(2))+1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(5/4)*c^(3/4)*sqrt(2))-1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(5/4)*c^(3/4)*sqrt(2))],
[1/(a+c*x^4)^2,x,10,1/4*x/(a*(a+c*x^4))-3/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))+3/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))-3/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))+3/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))],
[1/(x^2*(a+c*x^4)^2),x,11,(-5/4)/(a^2*x)+1/4/(a*x*(a+c*x^4))+5/8*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(9/4)*sqrt(2))-5/8*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(9/4)*sqrt(2))-5/16*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(9/4)*sqrt(2))+5/16*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(9/4)*sqrt(2))],
[1/(x^4*(a+c*x^4)^2),x,11,(-7/12)/(a^2*x^3)+1/4/(a*x^3*(a+c*x^4))+7/8*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*sqrt(2))-7/8*c^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*sqrt(2))+7/16*c^(3/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*sqrt(2))-7/16*c^(3/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*sqrt(2))],
[x^11/(a+c*x^4)^3,x,3,-1/8*a^2/(c^3*(a+c*x^4)^2)+1/2*a/(c^3*(a+c*x^4))+1/4*log(a+c*x^4)/c^3],
[x^9/(a+c*x^4)^3,x,4,-1/8*x^6/(c*(a+c*x^4)^2)-3/16*x^2/(c^2*(a+c*x^4))+3/16*arctan(x^2*sqrt(c)/sqrt(a))/(c^(5/2)*sqrt(a))],
[x^7/(a+c*x^4)^3,x,1,1/8*x^8/(a*(a+c*x^4)^2)],
[x^5/(a+c*x^4)^3,x,4,-1/8*x^2/(c*(a+c*x^4)^2)+1/16*x^2/(a*c*(a+c*x^4))+1/16*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*c^(3/2))],
[x^3/(a+c*x^4)^3,x,1,(-1/8)/(c*(a+c*x^4)^2)],
[x/(a+c*x^4)^3,x,4,1/8*x^2/(a*(a+c*x^4)^2)+3/16*x^2/(a^2*(a+c*x^4))+3/16*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*sqrt(c))],
[1/(x*(a+c*x^4)^3),x,3,1/8/(a*(a+c*x^4)^2)+1/4/(a^2*(a+c*x^4))+log(x)/a^3-1/4*log(a+c*x^4)/a^3],
[1/(x^3*(a+c*x^4)^3),x,5,(-15/16)/(a^3*x^2)+1/8/(a*x^2*(a+c*x^4)^2)+5/16/(a^2*x^2*(a+c*x^4))-15/16*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/a^(7/2)],
[x^10/(a+c*x^4)^3,x,11,-1/8*x^7/(c*(a+c*x^4)^2)-7/32*x^3/(c^2*(a+c*x^4))-21/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(1/4)*c^(11/4)*sqrt(2))+21/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(1/4)*c^(11/4)*sqrt(2))+21/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(1/4)*c^(11/4)*sqrt(2))-21/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(1/4)*c^(11/4)*sqrt(2))],
[x^8/(a+c*x^4)^3,x,11,-1/8*x^5/(c*(a+c*x^4)^2)-5/32*x/(c^2*(a+c*x^4))-5/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(9/4)*sqrt(2))+5/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(9/4)*sqrt(2))-5/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(9/4)*sqrt(2))+5/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(9/4)*sqrt(2))],
[x^6/(a+c*x^4)^3,x,11,-1/8*x^3/(c*(a+c*x^4)^2)+3/32*x^3/(a*c*(a+c*x^4))-3/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(5/4)*c^(7/4)*sqrt(2))+3/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(5/4)*c^(7/4)*sqrt(2))+3/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(5/4)*c^(7/4)*sqrt(2))-3/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(5/4)*c^(7/4)*sqrt(2))],
[x^4/(a+c*x^4)^3,x,11,-1/8*x/(c*(a+c*x^4)^2)+1/32*x/(a*c*(a+c*x^4))-3/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(5/4)*sqrt(2))+3/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(5/4)*sqrt(2))-3/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(5/4)*sqrt(2))+3/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(5/4)*sqrt(2))],
[x^2/(a+c*x^4)^3,x,11,1/8*x^3/(a*(a+c*x^4)^2)+5/32*x^3/(a^2*(a+c*x^4))-5/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(9/4)*c^(3/4)*sqrt(2))+5/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(9/4)*c^(3/4)*sqrt(2))+5/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(9/4)*c^(3/4)*sqrt(2))-5/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(9/4)*c^(3/4)*sqrt(2))],
[1/(a+c*x^4)^3,x,11,1/8*x/(a*(a+c*x^4)^2)+7/32*x/(a^2*(a+c*x^4))-21/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*c^(1/4)*sqrt(2))+21/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*c^(1/4)*sqrt(2))-21/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*c^(1/4)*sqrt(2))+21/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*c^(1/4)*sqrt(2))],
[1/(x^2*(a+c*x^4)^3),x,12,(-45/32)/(a^3*x)+1/8/(a*x*(a+c*x^4)^2)+9/32/(a^2*x*(a+c*x^4))+45/64*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(13/4)*sqrt(2))-45/64*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(13/4)*sqrt(2))-45/128*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(13/4)*sqrt(2))+45/128*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(13/4)*sqrt(2))],
[x^9/(2+3*x^4),x,4,-1/9*x^2+1/18*x^6+1/9*arctan(x^2*sqrt(3/2))*sqrt(2/3)],
[x^7/(2+3*x^4),x,3,1/12*x^4-1/18*log(2+3*x^4)],
[x^5/(2+3*x^4),x,3,1/6*x^2-1/3*arctan(x^2*sqrt(3/2))/sqrt(6)],
[x^3/(2+3*x^4),x,1,1/12*log(2+3*x^4)],
[x/(2+3*x^4),x,2,1/2*arctan(x^2*sqrt(3/2))/sqrt(6)],
[1/(x*(2+3*x^4)),x,4,1/2*log(x)-1/8*log(2+3*x^4)],
[1/(x^3*(2+3*x^4)),x,3,(-1/4)/x^2-1/4*arctan(x^2*sqrt(3/2))*sqrt(3/2)],
[x^6/(2+3*x^4),x,10,1/9*x^3+1/3*arctan(1-6^(1/4)*x)/6^(3/4)-1/3*arctan(1+6^(1/4)*x)/6^(3/4)-1/6*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)+1/6*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)],
[x^4/(2+3*x^4),x,10,1/3*x+1/6*arctan(1-6^(1/4)*x)/6^(1/4)-1/6*arctan(1+6^(1/4)*x)/6^(1/4)+1/12*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)-1/12*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)],
[x^2/(2+3*x^4),x,9,-1/2*arctan(1-6^(1/4)*x)/6^(3/4)+1/2*arctan(1+6^(1/4)*x)/6^(3/4)+1/4*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)-1/4*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)],
[1/(2+3*x^4),x,9,-1/4*arctan(1-6^(1/4)*x)/6^(1/4)+1/4*arctan(1+6^(1/4)*x)/6^(1/4)-1/8*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/8*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)],
[1/(x^2*(2+3*x^4)),x,10,(-1/2)/x+1/4*3^(1/4)*arctan(1-6^(1/4)*x)/2^(3/4)-1/4*3^(1/4)*arctan(1+6^(1/4)*x)/2^(3/4)-1/8*3^(1/4)*log(-6^(3/4)*x+3*x^2+sqrt(6))/2^(3/4)+1/8*3^(1/4)*log(6^(3/4)*x+3*x^2+sqrt(6))/2^(3/4)],
[x^3/(2+3*x^4)^2,x,1,(-1/12)/(2+3*x^4)],
[x/(2+3*x^4)^2,x,3,1/8*x^2/(2+3*x^4)+1/8*arctan(x^2*sqrt(3/2))/sqrt(6)],
[1/(x*(2+3*x^4)^2),x,3,1/8/(2+3*x^4)+1/4*log(x)-1/16*log(2+3*x^4)],
[1/(x^3*(2+3*x^4)^2),x,4,(-3/16)/x^2+1/8/(x^2*(2+3*x^4))-3/16*arctan(x^2*sqrt(3/2))*sqrt(3/2)],
[x^4/(2+3*x^4)^2,x,10,-1/12*x/(2+3*x^4)-1/48*arctan(1-6^(1/4)*x)/6^(1/4)+1/48*arctan(1+6^(1/4)*x)/6^(1/4)-1/96*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/96*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)],
[x^2/(2+3*x^4)^2,x,10,1/8*x^3/(2+3*x^4)-1/16*arctan(1-6^(1/4)*x)/6^(3/4)+1/16*arctan(1+6^(1/4)*x)/6^(3/4)+1/32*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)-1/32*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)],
[1/(2+3*x^4)^2,x,10,1/8*x/(2+3*x^4)-1/32*3^(3/4)*arctan(1-6^(1/4)*x)/2^(1/4)+1/32*3^(3/4)*arctan(1+6^(1/4)*x)/2^(1/4)-1/64*3^(3/4)*log(-6^(3/4)*x+3*x^2+sqrt(6))/2^(1/4)+1/64*3^(3/4)*log(6^(3/4)*x+3*x^2+sqrt(6))/2^(1/4)],
[1/(x^2*(2+3*x^4)^2),x,11,(-5/16)/x+1/8/(x*(2+3*x^4))+5/32*3^(1/4)*arctan(1-6^(1/4)*x)/2^(3/4)-5/32*3^(1/4)*arctan(1+6^(1/4)*x)/2^(3/4)-5/64*3^(1/4)*log(-6^(3/4)*x+3*x^2+sqrt(6))/2^(3/4)+5/64*3^(1/4)*log(6^(3/4)*x+3*x^2+sqrt(6))/2^(3/4)],
[x^2/(3+x^4),x,9,-1/2*arctan(1-x*sqrt(2)/3^(1/4))/(3^(1/4)*sqrt(2))+1/2*arctan(1+x*sqrt(2)/3^(1/4))/(3^(1/4)*sqrt(2))+1/4*log(x^2-3^(1/4)*x*sqrt(2)+sqrt(3))/(3^(1/4)*sqrt(2))-1/4*log(x^2+3^(1/4)*x*sqrt(2)+sqrt(3))/(3^(1/4)*sqrt(2))],
[1/(1+a+(-1+a)*x^4),x,3,1/2*arctan((1-a)^(1/4)*x/(1+a)^(1/4))/((1-a^2)^(1/4)*sqrt(1+a))+1/2*arctanh((1-a)^(1/4)*x/(1+a)^(1/4))/((1-a^2)^(1/4)*sqrt(1+a))],

#  Following pairs of integrands are equal. 
[1/(2*a+2*b+x^4),x,3,-1/2*arctan(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(3/4)*(-a-b)^(3/4))-1/2*arctanh(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(3/4)*(-a-b)^(3/4))],
[1/(2*(a+b)+x^4),x,3,-1/2*arctan(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(3/4)*(-a-b)^(3/4))-1/2*arctanh(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(3/4)*(-a-b)^(3/4))],
[x/(2*a+2*b+x^4),x,2,1/2*arctan(x^2/(sqrt(2)*sqrt(a+b)))/(sqrt(2)*sqrt(a+b))],
[x/(2*(a+b)+x^4),x,2,1/2*arctan(x^2/(sqrt(2)*sqrt(a+b)))/(sqrt(2)*sqrt(a+b))],
[x^2/(2*a+2*b+x^4),x,3,1/2*arctan(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(1/4)*(-a-b)^(1/4))-1/2*arctanh(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(1/4)*(-a-b)^(1/4))],
[x^2/(2*(a+b)+x^4),x,3,1/2*arctan(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(1/4)*(-a-b)^(1/4))-1/2*arctanh(x/(2^(1/4)*(-a-b)^(1/4)))/(2^(1/4)*(-a-b)^(1/4))],
[x^3/(2*a+2*b+x^4),x,1,1/4*log(2*(a+b)+x^4)],
[x^3/(2*(a+b)+x^4),x,1,1/4*log(2*(a+b)+x^4)],

# Integrands of the form x^(m/2) (a+b x^4)^p

# p>0
[x^(5/2)*(a+c*x^4),x,2,2/7*a*x^(7/2)+2/15*c*x^(15/2)],
[x^(3/2)*(a+c*x^4),x,2,2/5*a*x^(5/2)+2/13*c*x^(13/2)],
[(a+c*x^4)*sqrt(x),x,2,2/3*a*x^(3/2)+2/11*c*x^(11/2)],
[(a+c*x^4)/sqrt(x),x,2,2/9*c*x^(9/2)+2*a*sqrt(x)],
[(a+c*x^4)/x^(3/2),x,2,2/7*c*x^(7/2)-2*a/sqrt(x)],
[(a+c*x^4)/x^(5/2),x,2,-2/3*a/x^(3/2)+2/5*c*x^(5/2)],
[(a+c*x^4)/x^(7/2),x,2,-2/5*a/x^(5/2)+2/3*c*x^(3/2)],
[x^(5/2)*(a+c*x^4)^2,x,2,2/7*a^2*x^(7/2)+4/15*a*c*x^(15/2)+2/23*c^2*x^(23/2)],
[x^(3/2)*(a+c*x^4)^2,x,2,2/5*a^2*x^(5/2)+4/13*a*c*x^(13/2)+2/21*c^2*x^(21/2)],
[(a+c*x^4)^2*sqrt(x),x,2,2/3*a^2*x^(3/2)+4/11*a*c*x^(11/2)+2/19*c^2*x^(19/2)],
[(a+c*x^4)^2/sqrt(x),x,2,4/9*a*c*x^(9/2)+2/17*c^2*x^(17/2)+2*a^2*sqrt(x)],
[(a+c*x^4)^2/x^(3/2),x,2,4/7*a*c*x^(7/2)+2/15*c^2*x^(15/2)-2*a^2/sqrt(x)],
[(a+c*x^4)^2/x^(5/2),x,2,-2/3*a^2/x^(3/2)+4/5*a*c*x^(5/2)+2/13*c^2*x^(13/2)],
[(a+c*x^4)^2/x^(7/2),x,2,-2/5*a^2/x^(5/2)+4/3*a*c*x^(3/2)+2/11*c^2*x^(11/2)],
[x^(5/2)*(a+c*x^4)^3,x,2,2/7*a^3*x^(7/2)+2/5*a^2*c*x^(15/2)+6/23*a*c^2*x^(23/2)+2/31*c^3*x^(31/2)],
[x^(3/2)*(a+c*x^4)^3,x,2,2/5*a^3*x^(5/2)+6/13*a^2*c*x^(13/2)+2/7*a*c^2*x^(21/2)+2/29*c^3*x^(29/2)],
[(a+c*x^4)^3*sqrt(x),x,2,2/3*a^3*x^(3/2)+6/11*a^2*c*x^(11/2)+6/19*a*c^2*x^(19/2)+2/27*c^3*x^(27/2)],
[(a+c*x^4)^3/sqrt(x),x,2,2/3*a^2*c*x^(9/2)+6/17*a*c^2*x^(17/2)+2/25*c^3*x^(25/2)+2*a^3*sqrt(x)],
[(a+c*x^4)^3/x^(3/2),x,2,6/7*a^2*c*x^(7/2)+2/5*a*c^2*x^(15/2)+2/23*c^3*x^(23/2)-2*a^3/sqrt(x)],
[(a+c*x^4)^3/x^(5/2),x,2,-2/3*a^3/x^(3/2)+6/5*a^2*c*x^(5/2)+6/13*a*c^2*x^(13/2)+2/21*c^3*x^(21/2)],
[(a+c*x^4)^3/x^(7/2),x,2,-2/5*a^3/x^(5/2)+2*a^2*c*x^(3/2)+6/11*a*c^2*x^(11/2)+2/19*c^3*x^(19/2)],

# p<0
[x^(9/2)/(a+c*x^4),x,15,2/3*x^(3/2)/c+1/2*(-a)^(3/8)*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/c^(11/8)-1/2*(-a)^(3/8)*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/c^(11/8)+1/2*(-a)^(3/8)*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/(c^(11/8)*sqrt(2))-1/2*(-a)^(3/8)*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/(c^(11/8)*sqrt(2))-1/4*(-a)^(3/8)*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/(c^(11/8)*sqrt(2))+1/4*(-a)^(3/8)*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/(c^(11/8)*sqrt(2))],
[x^(7/2)/(a+c*x^4),x,15,-1/2*(-a)^(1/8)*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/c^(9/8)-1/2*(-a)^(1/8)*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/c^(9/8)+1/2*(-a)^(1/8)*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/(c^(9/8)*sqrt(2))-1/2*(-a)^(1/8)*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/(c^(9/8)*sqrt(2))+1/4*(-a)^(1/8)*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/(c^(9/8)*sqrt(2))-1/4*(-a)^(1/8)*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/(c^(9/8)*sqrt(2))+2*sqrt(x)/c],
[x^(5/2)/(a+c*x^4),x,14,1/2*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(7/8))-1/2*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(7/8))-1/2*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(7/8)*sqrt(2))+1/2*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(7/8)*sqrt(2))+1/4*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(1/8)*c^(7/8)*sqrt(2))-1/4*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(1/8)*c^(7/8)*sqrt(2))],
[x^(3/2)/(a+c*x^4),x,14,-1/2*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(5/8))-1/2*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(5/8))-1/2*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(5/8)*sqrt(2))+1/2*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(5/8)*sqrt(2))-1/4*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(3/8)*c^(5/8)*sqrt(2))+1/4*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(3/8)*c^(5/8)*sqrt(2))],
[sqrt(x)/(a+c*x^4),x,14,1/2*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(3/8))-1/2*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(3/8))+1/2*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(3/8)*sqrt(2))-1/2*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(3/8)*sqrt(2))-1/4*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(5/8)*c^(3/8)*sqrt(2))+1/4*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(5/8)*c^(3/8)*sqrt(2))],
[1/((a+c*x^4)*sqrt(x)),x,14,-1/2*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(1/8))-1/2*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(1/8))+1/2*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(1/8)*sqrt(2))-1/2*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(1/8)*sqrt(2))+1/4*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(7/8)*c^(1/8)*sqrt(2))-1/4*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(7/8)*c^(1/8)*sqrt(2))],
[1/(x^(3/2)*(a+c*x^4)),x,15,1/2*c^(1/8)*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/(-a)^(9/8)-1/2*c^(1/8)*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/(-a)^(9/8)-1/2*c^(1/8)*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*sqrt(2))+1/2*c^(1/8)*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*sqrt(2))+1/4*c^(1/8)*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(9/8)*sqrt(2))-1/4*c^(1/8)*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(9/8)*sqrt(2))+(-2)/(a*sqrt(x))],
[1/(x^(5/2)*(a+c*x^4)),x,15,(-2/3)/(a*x^(3/2))-1/2*c^(3/8)*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/(-a)^(11/8)-1/2*c^(3/8)*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/(-a)^(11/8)-1/2*c^(3/8)*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*sqrt(2))+1/2*c^(3/8)*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*sqrt(2))-1/4*c^(3/8)*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(11/8)*sqrt(2))+1/4*c^(3/8)*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(11/8)*sqrt(2))],
[x^(13/2)/(a+c*x^4)^2,x,15,-1/4*x^(7/2)/(c*(a+c*x^4))+7/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(15/8))-7/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(15/8))-7/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(15/8)*sqrt(2))+7/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(1/8)*c^(15/8)*sqrt(2))+7/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(1/8)*c^(15/8)*sqrt(2))-7/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(1/8)*c^(15/8)*sqrt(2))],
[x^(11/2)/(a+c*x^4)^2,x,15,-1/4*x^(5/2)/(c*(a+c*x^4))-5/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(13/8))-5/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(13/8))-5/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(13/8)*sqrt(2))+5/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(3/8)*c^(13/8)*sqrt(2))-5/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(3/8)*c^(13/8)*sqrt(2))+5/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(3/8)*c^(13/8)*sqrt(2))],
[x^(9/2)/(a+c*x^4)^2,x,15,-1/4*x^(3/2)/(c*(a+c*x^4))+3/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(11/8))-3/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(11/8))+3/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(11/8)*sqrt(2))-3/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(5/8)*c^(11/8)*sqrt(2))-3/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(5/8)*c^(11/8)*sqrt(2))+3/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(5/8)*c^(11/8)*sqrt(2))],
[x^(7/2)/(a+c*x^4)^2,x,15,-1/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(9/8))-1/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(9/8))+1/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(9/8)*sqrt(2))-1/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(9/8)*sqrt(2))+1/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(7/8)*c^(9/8)*sqrt(2))-1/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(7/8)*c^(9/8)*sqrt(2))-1/4*sqrt(x)/(c*(a+c*x^4))],
[x^(5/2)/(a+c*x^4)^2,x,15,1/4*x^(7/2)/(a*(a+c*x^4))-1/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(7/8))+1/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(7/8))+1/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(7/8)*sqrt(2))-1/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(7/8)*sqrt(2))-1/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(9/8)*c^(7/8)*sqrt(2))+1/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(9/8)*c^(7/8)*sqrt(2))],
[x^(3/2)/(a+c*x^4)^2,x,15,1/4*x^(5/2)/(a*(a+c*x^4))+3/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(5/8))+3/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(5/8))+3/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(5/8)*sqrt(2))-3/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(5/8)*sqrt(2))+3/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(11/8)*c^(5/8)*sqrt(2))-3/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(11/8)*c^(5/8)*sqrt(2))],
[sqrt(x)/(a+c*x^4)^2,x,15,1/4*x^(3/2)/(a*(a+c*x^4))-5/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(3/8))+5/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(3/8))-5/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(3/8)*sqrt(2))+5/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(3/8)*sqrt(2))+5/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(13/8)*c^(3/8)*sqrt(2))-5/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(13/8)*c^(3/8)*sqrt(2))],
[1/((a+c*x^4)^2*sqrt(x)),x,15,7/16*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(1/8))+7/16*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(1/8))-7/16*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(1/8)*sqrt(2))+7/16*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(1/8)*sqrt(2))-7/32*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(15/8)*c^(1/8)*sqrt(2))+7/32*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(15/8)*c^(1/8)*sqrt(2))+1/4*sqrt(x)/(a*(a+c*x^4))],
[1/(x^(3/2)*(a+c*x^4)^2),x,16,-9/16*c^(1/8)*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/(-a)^(17/8)+9/16*c^(1/8)*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/(-a)^(17/8)+9/16*c^(1/8)*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(17/8)*sqrt(2))-9/16*c^(1/8)*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(17/8)*sqrt(2))-9/32*c^(1/8)*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(17/8)*sqrt(2))+9/32*c^(1/8)*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(17/8)*sqrt(2))+(-9/4)/(a^2*sqrt(x))+1/4/(a*(a+c*x^4)*sqrt(x))],
[x^(15/2)/(a+c*x^4)^3,x,16,-1/8*x^(9/2)/(c*(a+c*x^4)^2)-9/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(17/8))-9/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(17/8))+9/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(17/8)*sqrt(2))-9/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(7/8)*c^(17/8)*sqrt(2))+9/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(7/8)*c^(17/8)*sqrt(2))-9/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(7/8)*c^(17/8)*sqrt(2))-9/64*sqrt(x)/(c^2*(a+c*x^4))],
[x^(13/2)/(a+c*x^4)^3,x,16,-1/8*x^(7/2)/(c*(a+c*x^4)^2)+7/64*x^(7/2)/(a*c*(a+c*x^4))-7/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(15/8))+7/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(15/8))+7/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(15/8)*sqrt(2))-7/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(9/8)*c^(15/8)*sqrt(2))-7/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(9/8)*c^(15/8)*sqrt(2))+7/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(9/8)*c^(15/8)*sqrt(2))],
[x^(11/2)/(a+c*x^4)^3,x,16,-1/8*x^(5/2)/(c*(a+c*x^4)^2)+5/64*x^(5/2)/(a*c*(a+c*x^4))+15/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(13/8))+15/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(13/8))+15/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(13/8)*sqrt(2))-15/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(11/8)*c^(13/8)*sqrt(2))+15/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(11/8)*c^(13/8)*sqrt(2))-15/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(11/8)*c^(13/8)*sqrt(2))],
[x^(9/2)/(a+c*x^4)^3,x,16,-1/8*x^(3/2)/(c*(a+c*x^4)^2)+3/64*x^(3/2)/(a*c*(a+c*x^4))-15/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(11/8))+15/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(11/8))-15/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(11/8)*sqrt(2))+15/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(13/8)*c^(11/8)*sqrt(2))+15/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(13/8)*c^(11/8)*sqrt(2))-15/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(13/8)*c^(11/8)*sqrt(2))],
[x^(7/2)/(a+c*x^4)^3,x,16,7/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(9/8))+7/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(9/8))-7/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(9/8)*sqrt(2))+7/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(15/8)*c^(9/8)*sqrt(2))-7/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(15/8)*c^(9/8)*sqrt(2))+7/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(15/8)*c^(9/8)*sqrt(2))-1/8*sqrt(x)/(c*(a+c*x^4)^2)+1/64*sqrt(x)/(a*c*(a+c*x^4))],
[x^(5/2)/(a+c*x^4)^3,x,16,1/8*x^(7/2)/(a*(a+c*x^4)^2)+9/64*x^(7/2)/(a^2*(a+c*x^4))+9/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(17/8)*c^(7/8))-9/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(17/8)*c^(7/8))-9/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(17/8)*c^(7/8)*sqrt(2))+9/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(17/8)*c^(7/8)*sqrt(2))+9/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(17/8)*c^(7/8)*sqrt(2))-9/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(17/8)*c^(7/8)*sqrt(2))],
[x^(3/2)/(a+c*x^4)^3,x,16,1/8*x^(5/2)/(a*(a+c*x^4)^2)+11/64*x^(5/2)/(a^2*(a+c*x^4))-33/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(19/8)*c^(5/8))-33/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(19/8)*c^(5/8))-33/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(19/8)*c^(5/8)*sqrt(2))+33/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(19/8)*c^(5/8)*sqrt(2))-33/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(19/8)*c^(5/8)*sqrt(2))+33/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(19/8)*c^(5/8)*sqrt(2))],
[sqrt(x)/(a+c*x^4)^3,x,16,1/8*x^(3/2)/(a*(a+c*x^4)^2)+13/64*x^(3/2)/(a^2*(a+c*x^4))+65/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(21/8)*c^(3/8))-65/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(21/8)*c^(3/8))+65/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(21/8)*c^(3/8)*sqrt(2))-65/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(21/8)*c^(3/8)*sqrt(2))-65/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(21/8)*c^(3/8)*sqrt(2))+65/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(21/8)*c^(3/8)*sqrt(2))],
[1/((a+c*x^4)^3*sqrt(x)),x,16,-105/256*arctan(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(23/8)*c^(1/8))-105/256*arctanh(c^(1/8)*sqrt(x)/(-a)^(1/8))/((-a)^(23/8)*c^(1/8))+105/256*arctan(1-c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(23/8)*c^(1/8)*sqrt(2))-105/256*arctan(1+c^(1/8)*sqrt(2)*sqrt(x)/(-a)^(1/8))/((-a)^(23/8)*c^(1/8)*sqrt(2))+105/512*log((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(23/8)*c^(1/8)*sqrt(2))-105/512*log((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*sqrt(2)*sqrt(x))/((-a)^(23/8)*c^(1/8)*sqrt(2))+1/8*sqrt(x)/(a*(a+c*x^4)^2)+15/64*sqrt(x)/(a^2*(a+c*x^4))],

# Integrands of the form x^m (a+b x^4)^(p/2)

# p>0
[x^11*sqrt(a+c*x^4),x,3,1/6*a^2*(a+c*x^4)^(3/2)/c^3-1/5*a*(a+c*x^4)^(5/2)/c^3+1/14*(a+c*x^4)^(7/2)/c^3],
[x^7*sqrt(a+c*x^4),x,3,-1/6*a*(a+c*x^4)^(3/2)/c^2+1/10*(a+c*x^4)^(5/2)/c^2],
[x^3*sqrt(a+c*x^4),x,1,1/6*(a+c*x^4)^(3/2)/c],
[sqrt(a+c*x^4)/x,x,4,-1/2*arctanh(sqrt(a+c*x^4)/sqrt(a))*sqrt(a)+1/2*sqrt(a+c*x^4)],
[sqrt(a+c*x^4)/x^5,x,4,-1/4*c*arctanh(sqrt(a+c*x^4)/sqrt(a))/sqrt(a)-1/4*sqrt(a+c*x^4)/x^4],
[sqrt(a+c*x^4)/x^9,x,5,1/16*c^2*arctanh(sqrt(a+c*x^4)/sqrt(a))/a^(3/2)-1/8*sqrt(a+c*x^4)/x^8-1/16*c*sqrt(a+c*x^4)/(a*x^4)],
[x^5*sqrt(a+c*x^4),x,5,-1/16*a^2*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/c^(3/2)+1/16*a*x^2*sqrt(a+c*x^4)/c+1/8*x^6*sqrt(a+c*x^4)],
[x*sqrt(a+c*x^4),x,4,1/4*a*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+1/4*x^2*sqrt(a+c*x^4)],
[sqrt(a+c*x^4)/x^3,x,4,1/2*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))*sqrt(c)-1/2*sqrt(a+c*x^4)/x^2],
[sqrt(a+c*x^4)/x^7,x,1,-1/6*(a+c*x^4)^(3/2)/(a*x^6)],
[sqrt(a+c*x^4)/x^11,x,2,-1/10*(a+c*x^4)^(3/2)/(a*x^10)+1/15*c*(a+c*x^4)^(3/2)/(a^2*x^6)],
[sqrt(a+c*x^4)/x^15,x,3,-1/14*(a+c*x^4)^(3/2)/(a*x^14)+2/35*c*(a+c*x^4)^(3/2)/(a^2*x^10)-4/105*c^2*(a+c*x^4)^(3/2)/(a^3*x^6)],
[x^4*sqrt(a+c*x^4),x,3,2/21*a*x*sqrt(a+c*x^4)/c+1/7*x^5*sqrt(a+c*x^4)-1/21*a^(7/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(5/4)*sqrt(a+c*x^4))],
[sqrt(a+c*x^4),x,2,1/3*x*sqrt(a+c*x^4)+1/3*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*sqrt(a+c*x^4))],
[sqrt(a+c*x^4)/x^4,x,2,-1/3*sqrt(a+c*x^4)/x^3+1/3*c^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*sqrt(a+c*x^4))],
[sqrt(a+c*x^4)/x^8,x,3,-1/7*sqrt(a+c*x^4)/x^7-2/21*c*sqrt(a+c*x^4)/(a*x^3)-1/21*c^(7/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(5/4)*sqrt(a+c*x^4))],
[x^2*sqrt(a+c*x^4),x,4,1/5*x^3*sqrt(a+c*x^4)+2/5*a*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-2/5*a^(5/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/5*a^(5/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[sqrt(a+c*x^4)/x^2,x,4,-sqrt(a+c*x^4)/x+2*x*sqrt(c)*sqrt(a+c*x^4)/(sqrt(a)+x^2*sqrt(c))-2*a^(1/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)+a^(1/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)],
[sqrt(a+c*x^4)/x^6,x,5,-1/5*sqrt(a+c*x^4)/x^5-2/5*c*sqrt(a+c*x^4)/(a*x)+2/5*c^(3/2)*x*sqrt(a+c*x^4)/(a*(sqrt(a)+x^2*sqrt(c)))-2/5*c^(5/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+c*x^4))+1/5*c^(5/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+c*x^4))],
[x^11*(a+c*x^4)^(3/2),x,3,1/10*a^2*(a+c*x^4)^(5/2)/c^3-1/7*a*(a+c*x^4)^(7/2)/c^3+1/18*(a+c*x^4)^(9/2)/c^3],
[x^7*(a+c*x^4)^(3/2),x,3,-1/10*a*(a+c*x^4)^(5/2)/c^2+1/14*(a+c*x^4)^(7/2)/c^2],
[x^3*(a+c*x^4)^(3/2),x,1,1/10*(a+c*x^4)^(5/2)/c],
[(a+c*x^4)^(3/2)/x,x,5,1/6*(a+c*x^4)^(3/2)-1/2*a^(3/2)*arctanh(sqrt(a+c*x^4)/sqrt(a))+1/2*a*sqrt(a+c*x^4)],
[(a+c*x^4)^(3/2)/x^5,x,5,-1/4*(a+c*x^4)^(3/2)/x^4-3/4*c*arctanh(sqrt(a+c*x^4)/sqrt(a))*sqrt(a)+3/4*c*sqrt(a+c*x^4)],
[(a+c*x^4)^(3/2)/x^9,x,5,-1/8*(a+c*x^4)^(3/2)/x^8-3/16*c^2*arctanh(sqrt(a+c*x^4)/sqrt(a))/sqrt(a)-3/16*c*sqrt(a+c*x^4)/x^4],
[x^5*(a+c*x^4)^(3/2),x,6,1/12*x^6*(a+c*x^4)^(3/2)-1/32*a^3*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/c^(3/2)+1/32*a^2*x^2*sqrt(a+c*x^4)/c+1/16*a*x^6*sqrt(a+c*x^4)],
[x*(a+c*x^4)^(3/2),x,5,1/8*x^2*(a+c*x^4)^(3/2)+3/16*a^2*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+3/16*a*x^2*sqrt(a+c*x^4)],
[(a+c*x^4)^(3/2)/x^3,x,5,-1/2*(a+c*x^4)^(3/2)/x^2+3/4*a*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))*sqrt(c)+3/4*c*x^2*sqrt(a+c*x^4)],
[(a+c*x^4)^(3/2)/x^7,x,5,-1/6*(a+c*x^4)^(3/2)/x^6+1/2*c^(3/2)*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))-1/2*c*sqrt(a+c*x^4)/x^2],
[(a+c*x^4)^(3/2)/x^11,x,1,-1/10*(a+c*x^4)^(5/2)/(a*x^10)],
[(a+c*x^4)^(3/2)/x^15,x,2,-1/14*(a+c*x^4)^(5/2)/(a*x^14)+1/35*c*(a+c*x^4)^(5/2)/(a^2*x^10)],
[(a+c*x^4)^(3/2)/x^19,x,3,-1/18*(a+c*x^4)^(5/2)/(a*x^18)+2/63*c*(a+c*x^4)^(5/2)/(a^2*x^14)-4/315*c^2*(a+c*x^4)^(5/2)/(a^3*x^10)],
[x^4*(a+c*x^4)^(3/2),x,4,1/11*x^5*(a+c*x^4)^(3/2)+4/77*a^2*x*sqrt(a+c*x^4)/c+6/77*a*x^5*sqrt(a+c*x^4)-2/77*a^(11/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(5/4)*sqrt(a+c*x^4))],
[(a+c*x^4)^(3/2),x,3,1/7*x*(a+c*x^4)^(3/2)+2/7*a*x*sqrt(a+c*x^4)+2/7*a^(7/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*sqrt(a+c*x^4))],
[(a+c*x^4)^(3/2)/x^4,x,3,-1/3*(a+c*x^4)^(3/2)/x^3+2/3*c*x*sqrt(a+c*x^4)+2/3*a^(3/4)*c^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)],
[(a+c*x^4)^(3/2)/x^8,x,3,-1/7*(a+c*x^4)^(3/2)/x^7-2/7*c*sqrt(a+c*x^4)/x^3+2/7*c^(7/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*sqrt(a+c*x^4))],
[x^2*(a+c*x^4)^(3/2),x,5,1/9*x^3*(a+c*x^4)^(3/2)+2/15*a*x^3*sqrt(a+c*x^4)+4/15*a^2*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-4/15*a^(9/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+2/15*a^(9/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[(a+c*x^4)^(3/2)/x^2,x,5,-(a+c*x^4)^(3/2)/x+6/5*c*x^3*sqrt(a+c*x^4)+12/5*a*x*sqrt(c)*sqrt(a+c*x^4)/(sqrt(a)+x^2*sqrt(c))-12/5*a^(5/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)+6/5*a^(5/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)],
[(a+c*x^4)^(3/2)/x^6,x,5,-1/5*(a+c*x^4)^(3/2)/x^5-6/5*c*sqrt(a+c*x^4)/x+12/5*c^(3/2)*x*sqrt(a+c*x^4)/(sqrt(a)+x^2*sqrt(c))-12/5*a^(1/4)*c^(5/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)+6/5*a^(1/4)*c^(5/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+c*x^4)],
[(1+x^4)^(3/2),x,3,1/7*x*(1+x^4)^(3/2)+2/7*x*sqrt(1+x^4)+2/7*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[(1-x^4)^(3/2),x,3,1/7*x*(1-x^4)^(3/2)+4/7*EllipticF(x,I)+2/7*x*sqrt(1-x^4)],
[x^7*sqrt(5+3*x^4),x,3,-5/54*(5+3*x^4)^(3/2)+1/90*(5+3*x^4)^(5/2)],
[x^3*sqrt(5+x^4),x,1,1/6*(5+x^4)^(3/2)],
[x*sqrt(3+2*x^4),x,3,3/4*arcsinh(x^2*sqrt(2/3))/sqrt(2)+1/4*x^2*sqrt(3+2*x^4)],
[x*sqrt(-2+x^4),x,4,-1/2*arctanh(x^2/sqrt(-2+x^4))+1/4*x^2*sqrt(-2+x^4)],
[(1+x^4)^(1/2),x,2,1/3*x*sqrt(1+x^4)+1/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[(1-x^4)^(1/2),x,2,2/3*EllipticF(x,I)+1/3*x*sqrt(1-x^4)],

# p<0
[x^11/sqrt(a+b*x^4),x,3,-1/3*a*(a+b*x^4)^(3/2)/b^3+1/10*(a+b*x^4)^(5/2)/b^3+1/2*a^2*sqrt(a+b*x^4)/b^3],
[x^7/sqrt(a+b*x^4),x,3,1/6*(a+b*x^4)^(3/2)/b^2-1/2*a*sqrt(a+b*x^4)/b^2],
[x^3/sqrt(a+b*x^4),x,1,1/2*sqrt(a+b*x^4)/b],
[1/(x*sqrt(a+b*x^4)),x,3,-1/2*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)],
[1/(x^5*sqrt(a+b*x^4)),x,4,1/4*b*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/4*sqrt(a+b*x^4)/(a*x^4)],
[x^5/sqrt(a+b*x^4),x,4,-1/4*a*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/4*x^2*sqrt(a+b*x^4)/b],
[x/sqrt(a+b*x^4),x,3,1/2*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)],
[1/(x^3*sqrt(a+b*x^4)),x,1,-1/2*sqrt(a+b*x^4)/(a*x^2)],
[1/(x^7*sqrt(a+b*x^4)),x,2,-1/6*sqrt(a+b*x^4)/(a*x^6)+1/3*b*sqrt(a+b*x^4)/(a^2*x^2)],
[1/(x^11*sqrt(a+b*x^4)),x,3,-1/10*sqrt(a+b*x^4)/(a*x^10)+2/15*b*sqrt(a+b*x^4)/(a^2*x^6)-4/15*b^2*sqrt(a+b*x^4)/(a^3*x^2)],
[x^8/sqrt(a+b*x^4),x,3,-5/21*a*x*sqrt(a+b*x^4)/b^2+1/7*x^5*sqrt(a+b*x^4)/b+5/42*a^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^4))],
[x^4/sqrt(a+b*x^4),x,2,1/3*x*sqrt(a+b*x^4)/b-1/6*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[1/sqrt(a+b*x^4),x,1,1/2*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*sqrt(a+b*x^4))],
[1/(x^4*sqrt(a+b*x^4)),x,2,-1/3*sqrt(a+b*x^4)/(a*x^3)-1/6*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[1/(x^8*sqrt(a+b*x^4)),x,3,-1/7*sqrt(a+b*x^4)/(a*x^7)+5/21*b*sqrt(a+b*x^4)/(a^2*x^3)+5/42*b^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(9/4)*sqrt(a+b*x^4))],
[x^10/sqrt(a+b*x^4),x,5,-7/45*a*x^3*sqrt(a+b*x^4)/b^2+1/9*x^7*sqrt(a+b*x^4)/b+7/15*a^2*x*sqrt(a+b*x^4)/(b^(5/2)*(sqrt(a)+x^2*sqrt(b)))-7/15*a^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^4))+7/30*a^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^4))],
[x^6/sqrt(a+b*x^4),x,4,1/5*x^3*sqrt(a+b*x^4)/b-3/5*a*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+3/5*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-3/10*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^2/sqrt(a+b*x^4),x,3,x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/2*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[1/(x^2*sqrt(a+b*x^4)),x,4,-sqrt(a+b*x^4)/(a*x)+x*sqrt(b)*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+1/2*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))],
[1/(x^6*sqrt(a+b*x^4)),x,5,-1/5*sqrt(a+b*x^4)/(a*x^5)+3/5*b*sqrt(a+b*x^4)/(a^2*x)-3/5*b^(3/2)*x*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))+3/5*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))-3/10*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))],
[x^11/sqrt(a-b*x^4),x,3,1/3*a*(a-b*x^4)^(3/2)/b^3-1/10*(a-b*x^4)^(5/2)/b^3-1/2*a^2*sqrt(a-b*x^4)/b^3],
[x^7/sqrt(a-b*x^4),x,3,1/6*(a-b*x^4)^(3/2)/b^2-1/2*a*sqrt(a-b*x^4)/b^2],
[x^3/sqrt(a-b*x^4),x,1,-1/2*sqrt(a-b*x^4)/b],
[1/(x*sqrt(a-b*x^4)),x,3,-1/2*arctanh(sqrt(a-b*x^4)/sqrt(a))/sqrt(a)],
[1/(x^5*sqrt(a-b*x^4)),x,4,-1/4*b*arctanh(sqrt(a-b*x^4)/sqrt(a))/a^(3/2)-1/4*sqrt(a-b*x^4)/(a*x^4)],
[x^5/sqrt(a-b*x^4),x,4,1/4*a*arctan(x^2*sqrt(b)/sqrt(a-b*x^4))/b^(3/2)-1/4*x^2*sqrt(a-b*x^4)/b],
[x/sqrt(a-b*x^4),x,3,1/2*arctan(x^2*sqrt(b)/sqrt(a-b*x^4))/sqrt(b)],
[1/(x^3*sqrt(a-b*x^4)),x,1,-1/2*sqrt(a-b*x^4)/(a*x^2)],
[1/(x^7*sqrt(a-b*x^4)),x,2,-1/6*sqrt(a-b*x^4)/(a*x^6)-1/3*b*sqrt(a-b*x^4)/(a^2*x^2)],
[1/(x^11*sqrt(a-b*x^4)),x,3,-1/10*sqrt(a-b*x^4)/(a*x^10)-2/15*b*sqrt(a-b*x^4)/(a^2*x^6)-4/15*b^2*sqrt(a-b*x^4)/(a^3*x^2)],
[x^8/sqrt(a-b*x^4),x,4,-5/21*a*x*sqrt(a-b*x^4)/b^2-1/7*x^5*sqrt(a-b*x^4)/b+5/21*a^(9/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(9/4)*sqrt(a-b*x^4))],
[x^4/sqrt(a-b*x^4),x,3,-1/3*x*sqrt(a-b*x^4)/b+1/3*a^(5/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(5/4)*sqrt(a-b*x^4))],
[1/sqrt(a-b*x^4),x,2,a^(1/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(1/4)*sqrt(a-b*x^4))],
[1/(x^4*sqrt(a-b*x^4)),x,3,-1/3*sqrt(a-b*x^4)/(a*x^3)+1/3*b^(3/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(a^(3/4)*sqrt(a-b*x^4))],
[1/(x^8*sqrt(a-b*x^4)),x,4,-1/7*sqrt(a-b*x^4)/(a*x^7)-5/21*b*sqrt(a-b*x^4)/(a^2*x^3)+5/21*b^(7/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(a^(7/4)*sqrt(a-b*x^4))],
[x^10/sqrt(a-b*x^4),x,8,-7/45*a*x^3*sqrt(a-b*x^4)/b^2-1/9*x^7*sqrt(a-b*x^4)/b+7/15*a^(11/4)*EllipticE(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(11/4)*sqrt(a-b*x^4))-7/15*a^(11/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(11/4)*sqrt(a-b*x^4))],
[x^6/sqrt(a-b*x^4),x,7,-1/5*x^3*sqrt(a-b*x^4)/b+3/5*a^(7/4)*EllipticE(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(7/4)*sqrt(a-b*x^4))-3/5*a^(7/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(7/4)*sqrt(a-b*x^4))],
[x^2/sqrt(a-b*x^4),x,6,a^(3/4)*EllipticE(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(3/4)*sqrt(a-b*x^4))-a^(3/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(3/4)*sqrt(a-b*x^4))],
[1/(x^2*sqrt(a-b*x^4)),x,7,-sqrt(a-b*x^4)/(a*x)-b^(1/4)*EllipticE(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(a^(1/4)*sqrt(a-b*x^4))+b^(1/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(a^(1/4)*sqrt(a-b*x^4))],
[1/(x^6*sqrt(a-b*x^4)),x,8,-1/5*sqrt(a-b*x^4)/(a*x^5)-3/5*b*sqrt(a-b*x^4)/(a^2*x)-3/5*b^(5/4)*EllipticE(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(a^(5/4)*sqrt(a-b*x^4))+3/5*b^(5/4)*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(a^(5/4)*sqrt(a-b*x^4))],
[x^11/(a+b*x^4)^(3/2),x,3,1/6*(a+b*x^4)^(3/2)/b^3-1/2*a^2/(b^3*sqrt(a+b*x^4))-a*sqrt(a+b*x^4)/b^3],
[x^7/(a+b*x^4)^(3/2),x,3,1/2*a/(b^2*sqrt(a+b*x^4))+1/2*sqrt(a+b*x^4)/b^2],
[x^3/(a+b*x^4)^(3/2),x,1,(-1/2)/(b*sqrt(a+b*x^4))],
[1/(x*(a+b*x^4)^(3/2)),x,4,-1/2*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)+1/2/(a*sqrt(a+b*x^4))],
[1/(x^5*(a+b*x^4)^(3/2)),x,5,3/4*b*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(5/2)-3/4*b/(a^2*sqrt(a+b*x^4))+(-1/4)/(a*x^4*sqrt(a+b*x^4))],
[x^9/(a+b*x^4)^(3/2),x,5,-3/4*a*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(5/2)-1/2*x^6/(b*sqrt(a+b*x^4))+3/4*x^2*sqrt(a+b*x^4)/b^2],
[x^5/(a+b*x^4)^(3/2),x,4,1/2*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)-1/2*x^2/(b*sqrt(a+b*x^4))],
[x/(a+b*x^4)^(3/2),x,1,1/2*x^2/(a*sqrt(a+b*x^4))],
[1/(x^3*(a+b*x^4)^(3/2)),x,2,(-1/2)/(a*x^2*sqrt(a+b*x^4))-b*x^2/(a^2*sqrt(a+b*x^4))],
[1/(x^7*(a+b*x^4)^(3/2)),x,3,(-1/6)/(a*x^6*sqrt(a+b*x^4))+2/3*b/(a^2*x^2*sqrt(a+b*x^4))+4/3*b^2*x^2/(a^3*sqrt(a+b*x^4))],
[x^12/(a+b*x^4)^(3/2),x,4,-1/2*x^9/(b*sqrt(a+b*x^4))-15/14*a*x*sqrt(a+b*x^4)/b^3+9/14*x^5*sqrt(a+b*x^4)/b^2+15/28*a^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(13/4)*sqrt(a+b*x^4))],
[x^8/(a+b*x^4)^(3/2),x,3,-1/2*x^5/(b*sqrt(a+b*x^4))+5/6*x*sqrt(a+b*x^4)/b^2-5/12*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^4))],
[x^4/(a+b*x^4)^(3/2),x,2,-1/2*x/(b*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(5/4)*sqrt(a+b*x^4))],
[1/(a+b*x^4)^(3/2),x,2,1/2*x/(a*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*b^(1/4)*sqrt(a+b*x^4))],
[1/(x^4*(a+b*x^4)^(3/2)),x,3,1/2/(a*x^3*sqrt(a+b*x^4))-5/6*sqrt(a+b*x^4)/(a^2*x^3)-5/12*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(9/4)*sqrt(a+b*x^4))],
[1/(x^8*(a+b*x^4)^(3/2)),x,4,1/2/(a*x^7*sqrt(a+b*x^4))-9/14*sqrt(a+b*x^4)/(a^2*x^7)+15/14*b*sqrt(a+b*x^4)/(a^3*x^3)+15/28*b^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(13/4)*sqrt(a+b*x^4))],
[x^14/(a+b*x^4)^(3/2),x,6,-1/2*x^11/(b*sqrt(a+b*x^4))-77/90*a*x^3*sqrt(a+b*x^4)/b^3+11/18*x^7*sqrt(a+b*x^4)/b^2+77/30*a^2*x*sqrt(a+b*x^4)/(b^(7/2)*(sqrt(a)+x^2*sqrt(b)))-77/30*a^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(15/4)*sqrt(a+b*x^4))+77/60*a^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(15/4)*sqrt(a+b*x^4))],
[x^10/(a+b*x^4)^(3/2),x,5,-1/2*x^7/(b*sqrt(a+b*x^4))+7/10*x^3*sqrt(a+b*x^4)/b^2-21/10*a*x*sqrt(a+b*x^4)/(b^(5/2)*(sqrt(a)+x^2*sqrt(b)))+21/10*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^4))-21/20*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^4))],
[x^6/(a+b*x^4)^(3/2),x,4,-1/2*x^3/(b*sqrt(a+b*x^4))+3/2*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))-3/2*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))+3/4*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^2/(a+b*x^4)^(3/2),x,4,1/2*x^3/(a*sqrt(a+b*x^4))-1/2*x*sqrt(a+b*x^4)/(a*sqrt(b)*(sqrt(a)+x^2*sqrt(b)))+1/2*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(3/4)*sqrt(a+b*x^4))-1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(3/4)*sqrt(a+b*x^4))],
[1/(x^2*(a+b*x^4)^(3/2)),x,5,1/2/(a*x*sqrt(a+b*x^4))-3/2*sqrt(a+b*x^4)/(a^2*x)+3/2*x*sqrt(b)*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))-3/2*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))+3/4*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))],
[1/(x^6*(a+b*x^4)^(3/2)),x,6,1/2/(a*x^5*sqrt(a+b*x^4))-7/10*sqrt(a+b*x^4)/(a^2*x^5)+21/10*b*sqrt(a+b*x^4)/(a^3*x)-21/10*b^(3/2)*x*sqrt(a+b*x^4)/(a^3*(sqrt(a)+x^2*sqrt(b)))+21/10*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(11/4)*sqrt(a+b*x^4))-21/20*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(11/4)*sqrt(a+b*x^4))],
[1/(a+b*x^4)^(5/2),x,3,1/6*x/(a*(a+b*x^4)^(3/2))+5/12*x/(a^2*sqrt(a+b*x^4))+5/24*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(9/4)*b^(1/4)*sqrt(a+b*x^4))],
[x^11/sqrt(1-x^4),x,3,1/3*(1-x^4)^(3/2)-1/10*(1-x^4)^(5/2)-1/2*sqrt(1-x^4)],
[x^7/sqrt(1-x^4),x,3,1/6*(1-x^4)^(3/2)-1/2*sqrt(1-x^4)],
[x^3/sqrt(1-x^4),x,1,-1/2*sqrt(1-x^4)],
[1/(x*sqrt(1-x^4)),x,3,-1/2*arctanh(sqrt(1-x^4))],
[1/(x^5*sqrt(1-x^4)),x,4,-1/4*arctanh(sqrt(1-x^4))-1/4*sqrt(1-x^4)/x^4],
[x^5/sqrt(1-x^4),x,3,1/4*arcsin(x^2)-1/4*x^2*sqrt(1-x^4)],
[x/sqrt(1-x^4),x,2,1/2*arcsin(x^2)],
[1/(x^3*sqrt(1-x^4)),x,1,-1/2*sqrt(1-x^4)/x^2],
[1/(x^7*sqrt(1-x^4)),x,2,-1/6*sqrt(1-x^4)/x^6-1/3*sqrt(1-x^4)/x^2],
[1/(x^11*sqrt(1-x^4)),x,3,-1/10*sqrt(1-x^4)/x^10-2/15*sqrt(1-x^4)/x^6-4/15*sqrt(1-x^4)/x^2],
[x^8/sqrt(1-x^4),x,3,5/21*EllipticF(x,I)-5/21*x*sqrt(1-x^4)-1/7*x^5*sqrt(1-x^4)],
[x^4/sqrt(1-x^4),x,2,1/3*EllipticF(x,I)-1/3*x*sqrt(1-x^4)],
[1/sqrt(1-x^4),x,1,EllipticF(x,I)],
[1/(x^4*sqrt(1-x^4)),x,2,1/3*EllipticF(x,I)-1/3*sqrt(1-x^4)/x^3],
[1/(x^8*sqrt(1-x^4)),x,3,5/21*EllipticF(x,I)-1/7*sqrt(1-x^4)/x^7-5/21*sqrt(1-x^4)/x^3],
[x^10/sqrt(1-x^4),x,6,7/15*EllipticE(x,I)-7/15*EllipticF(x,I)-7/45*x^3*sqrt(1-x^4)-1/9*x^7*sqrt(1-x^4)],
[x^6/sqrt(1-x^4),x,5,3/5*EllipticE(x,I)-3/5*EllipticF(x,I)-1/5*x^3*sqrt(1-x^4)],
[x^2/sqrt(1-x^4),x,4,EllipticE(x,I)-EllipticF(x,I)],
[1/(x^2*sqrt(1-x^4)),x,5,-EllipticE(x,I)+EllipticF(x,I)-sqrt(1-x^4)/x],
[1/(x^6*sqrt(1-x^4)),x,6,-3/5*EllipticE(x,I)+3/5*EllipticF(x,I)-1/5*sqrt(1-x^4)/x^5-3/5*sqrt(1-x^4)/x],
[x^11/(1-x^4)^(3/2),x,3,-1/6*(1-x^4)^(3/2)+1/2/sqrt(1-x^4)+sqrt(1-x^4)],
[x^7/(1-x^4)^(3/2),x,3,1/2/sqrt(1-x^4)+1/2*sqrt(1-x^4)],
[x^3/(1-x^4)^(3/2),x,1,1/2/sqrt(1-x^4)],
[1/(x*(1-x^4)^(3/2)),x,4,-1/2*arctanh(sqrt(1-x^4))+1/2/sqrt(1-x^4)],
[1/(x^5*(1-x^4)^(3/2)),x,5,-3/4*arctanh(sqrt(1-x^4))+3/4/sqrt(1-x^4)+(-1/4)/(x^4*sqrt(1-x^4))],
[x^9/(1-x^4)^(3/2),x,4,-3/4*arcsin(x^2)+1/2*x^6/sqrt(1-x^4)+3/4*x^2*sqrt(1-x^4)],
[x^5/(1-x^4)^(3/2),x,3,-1/2*arcsin(x^2)+1/2*x^2/sqrt(1-x^4)],
[x/(1-x^4)^(3/2),x,1,1/2*x^2/sqrt(1-x^4)],
[1/(x^3*(1-x^4)^(3/2)),x,2,(-1/2)/(x^2*sqrt(1-x^4))+x^2/sqrt(1-x^4)],
[1/(x^7*(1-x^4)^(3/2)),x,3,(-1/6)/(x^6*sqrt(1-x^4))+(-2/3)/(x^2*sqrt(1-x^4))+4/3*x^2/sqrt(1-x^4)],
[x^12/(1-x^4)^(3/2),x,4,-15/14*EllipticF(x,I)+1/2*x^9/sqrt(1-x^4)+15/14*x*sqrt(1-x^4)+9/14*x^5*sqrt(1-x^4)],
[x^8/(1-x^4)^(3/2),x,3,-5/6*EllipticF(x,I)+1/2*x^5/sqrt(1-x^4)+5/6*x*sqrt(1-x^4)],
[x^4/(1-x^4)^(3/2),x,2,-1/2*EllipticF(x,I)+1/2*x/sqrt(1-x^4)],
[1/(1-x^4)^(3/2),x,2,1/2*EllipticF(x,I)+1/2*x/sqrt(1-x^4)],
[1/(x^4*(1-x^4)^(3/2)),x,3,5/6*EllipticF(x,I)+1/2/(x^3*sqrt(1-x^4))-5/6*sqrt(1-x^4)/x^3],
[1/(x^8*(1-x^4)^(3/2)),x,4,15/14*EllipticF(x,I)+1/2/(x^7*sqrt(1-x^4))-9/14*sqrt(1-x^4)/x^7-15/14*sqrt(1-x^4)/x^3],
[x^14/(1-x^4)^(3/2),x,7,-77/30*EllipticE(x,I)+77/30*EllipticF(x,I)+1/2*x^11/sqrt(1-x^4)+77/90*x^3*sqrt(1-x^4)+11/18*x^7*sqrt(1-x^4)],
[x^10/(1-x^4)^(3/2),x,6,-21/10*EllipticE(x,I)+21/10*EllipticF(x,I)+1/2*x^7/sqrt(1-x^4)+7/10*x^3*sqrt(1-x^4)],
[x^6/(1-x^4)^(3/2),x,5,-3/2*EllipticE(x,I)+3/2*EllipticF(x,I)+1/2*x^3/sqrt(1-x^4)],
[x^2/(1-x^4)^(3/2),x,5,-1/2*EllipticE(x,I)+1/2*EllipticF(x,I)+1/2*x^3/sqrt(1-x^4)],
[1/(x^2*(1-x^4)^(3/2)),x,6,-3/2*EllipticE(x,I)+3/2*EllipticF(x,I)+1/2/(x*sqrt(1-x^4))-3/2*sqrt(1-x^4)/x],
[1/(x^6*(1-x^4)^(3/2)),x,7,-21/10*EllipticE(x,I)+21/10*EllipticF(x,I)+1/2/(x^5*sqrt(1-x^4))-7/10*sqrt(1-x^4)/x^5-21/10*sqrt(1-x^4)/x],
[1/(1-x^4)^(5/2),x,3,1/6*x/(1-x^4)^(3/2)+5/12*EllipticF(x,I)+5/12*x/sqrt(1-x^4)],
[x^11/sqrt(1+x^4),x,3,-1/3*(1+x^4)^(3/2)+1/10*(1+x^4)^(5/2)+1/2*sqrt(1+x^4)],
[x^7/sqrt(1+x^4),x,3,1/6*(1+x^4)^(3/2)-1/2*sqrt(1+x^4)],
[x^3/sqrt(1+x^4),x,1,1/2*sqrt(1+x^4)],
[1/(x*sqrt(1+x^4)),x,3,-1/2*arctanh(sqrt(1+x^4))],
[1/(x^5*sqrt(1+x^4)),x,4,1/4*arctanh(sqrt(1+x^4))-1/4*sqrt(1+x^4)/x^4],
[x^5/sqrt(1+x^4),x,3,-1/4*arcsinh(x^2)+1/4*x^2*sqrt(1+x^4)],
[x/sqrt(1+x^4),x,2,1/2*arcsinh(x^2)],
[1/(x^3*sqrt(1+x^4)),x,1,-1/2*sqrt(1+x^4)/x^2],
[1/(x^7*sqrt(1+x^4)),x,2,-1/6*sqrt(1+x^4)/x^6+1/3*sqrt(1+x^4)/x^2],
[1/(x^11*sqrt(1+x^4)),x,3,-1/10*sqrt(1+x^4)/x^10+2/15*sqrt(1+x^4)/x^6-4/15*sqrt(1+x^4)/x^2],
[x^8/sqrt(1+x^4),x,3,-5/21*x*sqrt(1+x^4)+1/7*x^5*sqrt(1+x^4)+5/42*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^4/sqrt(1+x^4),x,2,1/3*x*sqrt(1+x^4)-1/6*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/sqrt(1+x^4),x,1,1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^4*sqrt(1+x^4)),x,2,-1/3*sqrt(1+x^4)/x^3-1/6*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^8*sqrt(1+x^4)),x,3,-1/7*sqrt(1+x^4)/x^7+5/21*sqrt(1+x^4)/x^3+5/42*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^10/sqrt(1+x^4),x,5,-7/45*x^3*sqrt(1+x^4)+1/9*x^7*sqrt(1+x^4)+7/15*x*sqrt(1+x^4)/(1+x^2)-7/15*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)+7/30*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^6/sqrt(1+x^4),x,4,1/5*x^3*sqrt(1+x^4)-3/5*x*sqrt(1+x^4)/(1+x^2)+3/5*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)-3/10*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^2/sqrt(1+x^4),x,3,x*sqrt(1+x^4)/(1+x^2)-(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)+1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^2*sqrt(1+x^4)),x,4,-sqrt(1+x^4)/x+x*sqrt(1+x^4)/(1+x^2)-(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)+1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^6*sqrt(1+x^4)),x,5,-1/5*sqrt(1+x^4)/x^5+3/5*sqrt(1+x^4)/x-3/5*x*sqrt(1+x^4)/(1+x^2)+3/5*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)-3/10*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^11/(1+x^4)^(3/2),x,3,1/6*(1+x^4)^(3/2)+(-1/2)/sqrt(1+x^4)-sqrt(1+x^4)],
[x^7/(1+x^4)^(3/2),x,3,1/2/sqrt(1+x^4)+1/2*sqrt(1+x^4)],
[x^3/(1+x^4)^(3/2),x,1,(-1/2)/sqrt(1+x^4)],
[1/(x*(1+x^4)^(3/2)),x,4,-1/2*arctanh(sqrt(1+x^4))+1/2/sqrt(1+x^4)],
[1/(x^5*(1+x^4)^(3/2)),x,5,3/4*arctanh(sqrt(1+x^4))+(-3/4)/sqrt(1+x^4)+(-1/4)/(x^4*sqrt(1+x^4))],
[x^9/(1+x^4)^(3/2),x,4,-3/4*arcsinh(x^2)-1/2*x^6/sqrt(1+x^4)+3/4*x^2*sqrt(1+x^4)],
[x^5/(1+x^4)^(3/2),x,3,1/2*arcsinh(x^2)-1/2*x^2/sqrt(1+x^4)],
[x/(1+x^4)^(3/2),x,1,1/2*x^2/sqrt(1+x^4)],
[1/(x^3*(1+x^4)^(3/2)),x,2,(-1/2)/(x^2*sqrt(1+x^4))-x^2/sqrt(1+x^4)],
[1/(x^7*(1+x^4)^(3/2)),x,3,(-1/6)/(x^6*sqrt(1+x^4))+2/3/(x^2*sqrt(1+x^4))+4/3*x^2/sqrt(1+x^4)],
[x^12/(1+x^4)^(3/2),x,4,-1/2*x^9/sqrt(1+x^4)-15/14*x*sqrt(1+x^4)+9/14*x^5*sqrt(1+x^4)+15/28*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^8/(1+x^4)^(3/2),x,3,-1/2*x^5/sqrt(1+x^4)+5/6*x*sqrt(1+x^4)-5/12*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^4/(1+x^4)^(3/2),x,2,-1/2*x/sqrt(1+x^4)+1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(1+x^4)^(3/2),x,2,1/2*x/sqrt(1+x^4)+1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^4*(1+x^4)^(3/2)),x,3,1/2/(x^3*sqrt(1+x^4))-5/6*sqrt(1+x^4)/x^3-5/12*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^8*(1+x^4)^(3/2)),x,4,1/2/(x^7*sqrt(1+x^4))-9/14*sqrt(1+x^4)/x^7+15/14*sqrt(1+x^4)/x^3+15/28*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^14/(1+x^4)^(3/2),x,6,-1/2*x^11/sqrt(1+x^4)-77/90*x^3*sqrt(1+x^4)+11/18*x^7*sqrt(1+x^4)+77/30*x*sqrt(1+x^4)/(1+x^2)-77/30*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)+77/60*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^10/(1+x^4)^(3/2),x,5,-1/2*x^7/sqrt(1+x^4)+7/10*x^3*sqrt(1+x^4)-21/10*x*sqrt(1+x^4)/(1+x^2)+21/10*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)-21/20*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^6/(1+x^4)^(3/2),x,4,-1/2*x^3/sqrt(1+x^4)+3/2*x*sqrt(1+x^4)/(1+x^2)-3/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)+3/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^2/(1+x^4)^(3/2),x,4,1/2*x^3/sqrt(1+x^4)-1/2*x*sqrt(1+x^4)/(1+x^2)+1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)-1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^2*(1+x^4)^(3/2)),x,5,1/2/(x*sqrt(1+x^4))-3/2*sqrt(1+x^4)/x+3/2*x*sqrt(1+x^4)/(1+x^2)-3/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)+3/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(x^6*(1+x^4)^(3/2)),x,6,1/2/(x^5*sqrt(1+x^4))-7/10*sqrt(1+x^4)/x^5+21/10*sqrt(1+x^4)/x-21/10*x*sqrt(1+x^4)/(1+x^2)+21/10*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)-21/20*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[1/(1+x^4)^(5/2),x,3,1/6*x/(1+x^4)^(3/2)+5/12*x/sqrt(1+x^4)+5/24*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^7/sqrt(16-x^4),x,3,1/6*(16-x^4)^(3/2)-8*sqrt(16-x^4)],
[x^5/sqrt(16-x^4),x,3,4*arcsin(1/4*x^2)-1/4*x^2*sqrt(16-x^4)],
[x^3/sqrt(16-x^4),x,1,-1/2*sqrt(16-x^4)],
[x/sqrt(16-x^4),x,2,1/2*arcsin(1/4*x^2)],
[1/(x*sqrt(16-x^4)),x,3,-1/8*arctanh(1/4*sqrt(16-x^4))],
[1/(x^3*sqrt(16-x^4)),x,1,-1/32*sqrt(16-x^4)/x^2],
[1/(x^5*sqrt(16-x^4)),x,4,-1/256*arctanh(1/4*sqrt(16-x^4))-1/64*sqrt(16-x^4)/x^4],
[1/(x^7*sqrt(16-x^4)),x,2,-1/96*sqrt(16-x^4)/x^6-1/768*sqrt(16-x^4)/x^2],
[x^6/sqrt(16-x^4),x,6,96/5*EllipticE(1/2*x,I)-96/5*EllipticF(1/2*x,I)-1/5*x^3*sqrt(16-x^4)],
[x^4/sqrt(16-x^4),x,2,8/3*EllipticF(1/2*x,I)-1/3*x*sqrt(16-x^4)],
[x^2/sqrt(16-x^4),x,5,2*EllipticE(1/2*x,I)-2*EllipticF(1/2*x,I)],
[1/sqrt(16-x^4),x,1,1/2*EllipticF(1/2*x,I)],
[1/(x^2*sqrt(16-x^4)),x,6,-1/8*EllipticE(1/2*x,I)+1/8*EllipticF(1/2*x,I)-1/16*sqrt(16-x^4)/x],
[1/(x^4*sqrt(16-x^4)),x,2,1/96*EllipticF(1/2*x,I)-1/48*sqrt(16-x^4)/x^3],
[x/sqrt(-4+x^4),x,3,1/2*arctanh(x^2/sqrt(-4+x^4))],
[x/sqrt(4+x^4),x,2,1/2*arcsinh(1/2*x^2)],
[1/(x*sqrt(-1+x^4)),x,3,1/2*arctan(sqrt(-1+x^4))],
[x^4/sqrt(-1+x^4),x,2,1/3*EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))+1/3*x*sqrt(-1+x^4)],
[1/sqrt(-1+x^4),x,1,EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))],
[1/(x^4*sqrt(-1+x^4)),x,2,1/3*EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))+1/3*sqrt(-1+x^4)/x^3],
[x^6/sqrt(-1+x^4),x,4,3/5*x*(1+x^2)/sqrt(-1+x^4)+3/5*EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))-3/5*EllipticE(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(2)*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)+1/5*x^3*sqrt(-1+x^4)],
[x^2/sqrt(-1+x^4),x,3,x*(1+x^2)/sqrt(-1+x^4)+EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))-EllipticE(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(2)*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)],
[1/(x^2*sqrt(-1+x^4)),x,4,-x*(1+x^2)/sqrt(-1+x^4)-EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))+EllipticE(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(2)*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)+sqrt(-1+x^4)/x],
[x^2/sqrt(3-2*x^4),x,5,3^(1/4)*EllipticE((2/3)^(1/4)*x,I)/2^(3/4)-3^(1/4)*EllipticF((2/3)^(1/4)*x,I)/2^(3/4)],
[x^2/sqrt(3-b*x^4),x,4,3^(1/4)*EllipticE(b^(1/4)*x/3^(1/4),I)/b^(3/4)-3^(1/4)*EllipticF(b^(1/4)*x/3^(1/4),I)/b^(3/4)],

# Integrands of the form x^(m/2) (a+b x^4)^(p/2)

# Integrands of the form x^m (a+b x^4)^(p/3)
[x^7*(1+x^4)^(1/3),x,3,-3/16*(1+x^4)^(4/3)+3/28*(1+x^4)^(7/3)],
[x^3/(1+x^4)^(4/3),x,1,(-3/4)/(1+x^4)^(1/3)],
[x^3/(1+x^4)^(1/3),x,1,3/8*(1+x^4)^(2/3)],

# Integrands of the form x^m (a+b x^4)^(p/4)

# p>0
[x^19*(a+b*x^4)^(1/4),x,3,1/5*a^4*(a+b*x^4)^(5/4)/b^5-4/9*a^3*(a+b*x^4)^(9/4)/b^5+6/13*a^2*(a+b*x^4)^(13/4)/b^5-4/17*a*(a+b*x^4)^(17/4)/b^5+1/21*(a+b*x^4)^(21/4)/b^5],
[x^15*(a+b*x^4)^(1/4),x,3,-1/5*a^3*(a+b*x^4)^(5/4)/b^4+1/3*a^2*(a+b*x^4)^(9/4)/b^4-3/13*a*(a+b*x^4)^(13/4)/b^4+1/17*(a+b*x^4)^(17/4)/b^4],
[x^11*(a+b*x^4)^(1/4),x,3,1/5*a^2*(a+b*x^4)^(5/4)/b^3-2/9*a*(a+b*x^4)^(9/4)/b^3+1/13*(a+b*x^4)^(13/4)/b^3],
[x^7*(a+b*x^4)^(1/4),x,3,-1/5*a*(a+b*x^4)^(5/4)/b^2+1/9*(a+b*x^4)^(9/4)/b^2],
[x^3*(a+b*x^4)^(1/4),x,1,1/5*(a+b*x^4)^(5/4)/b],
[(a+b*x^4)^(1/4)/x,x,6,(a+b*x^4)^(1/4)-1/2*a^(1/4)*arctan((a+b*x^4)^(1/4)/a^(1/4))-1/2*a^(1/4)*arctanh((a+b*x^4)^(1/4)/a^(1/4))],
[(a+b*x^4)^(1/4)/x^5,x,6,-1/4*(a+b*x^4)^(1/4)/x^4-1/8*b*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(3/4)-1/8*b*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(3/4)],
[(a+b*x^4)^(1/4)/x^9,x,7,-1/8*(a+b*x^4)^(1/4)/x^8-1/32*b*(a+b*x^4)^(1/4)/(a*x^4)+3/64*b^2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(7/4)+3/64*b^2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(7/4)],
[x^9*(a+b*x^4)^(1/4),x,6,-2/77*a^2*x^2*(a+b*x^4)^(1/4)/b^2+1/77*a*x^6*(a+b*x^4)^(1/4)/b+1/11*x^10*(a+b*x^4)^(1/4)+4/77*a^(7/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(3/4))],
[x^5*(a+b*x^4)^(1/4),x,5,1/21*a*x^2*(a+b*x^4)^(1/4)/b+1/7*x^6*(a+b*x^4)^(1/4)-2/21*a^(5/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(3/4))],
[x*(a+b*x^4)^(1/4),x,4,1/3*x^2*(a+b*x^4)^(1/4)+1/3*a^(3/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(b))],
[(a+b*x^4)^(1/4)/x^3,x,4,-1/2*(a+b*x^4)^(1/4)/x^2+1/2*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(a+b*x^4)^(3/4)],
[(a+b*x^4)^(1/4)/x^7,x,5,-1/6*(a+b*x^4)^(1/4)/x^6-1/12*b*(a+b*x^4)^(1/4)/(a*x^2)-1/12*b^(3/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(a))],
[(a+b*x^4)^(1/4)/x^11,x,6,-1/10*(a+b*x^4)^(1/4)/x^10-1/60*b*(a+b*x^4)^(1/4)/(a*x^6)+1/24*b^2*(a+b*x^4)^(1/4)/(a^2*x^2)+1/24*b^(5/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(3/4))],
[x^6*(a+b*x^4)^(1/4),x,6,1/32*a*x^3*(a+b*x^4)^(1/4)/b+1/8*x^7*(a+b*x^4)^(1/4)+3/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(7/4)-3/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(7/4)],
[x^2*(a+b*x^4)^(1/4),x,5,1/4*x^3*(a+b*x^4)^(1/4)-1/8*a*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(3/4)+1/8*a*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(3/4)],
[(a+b*x^4)^(1/4)/x^2,x,5,-(a+b*x^4)^(1/4)/x-1/2*b^(1/4)*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))+1/2*b^(1/4)*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))],
[(a+b*x^4)^(1/4)/x^6,x,1,-1/5*(a+b*x^4)^(5/4)/(a*x^5)],
[(a+b*x^4)^(1/4)/x^10,x,2,-1/9*(a+b*x^4)^(5/4)/(a*x^9)+4/45*b*(a+b*x^4)^(5/4)/(a^2*x^5)],
[(a+b*x^4)^(1/4)/x^14,x,3,-1/13*(a+b*x^4)^(5/4)/(a*x^13)+8/117*b*(a+b*x^4)^(5/4)/(a^2*x^9)-32/585*b^2*(a+b*x^4)^(5/4)/(a^3*x^5)],
[(a+b*x^4)^(1/4)/x^18,x,4,-1/17*(a+b*x^4)^(5/4)/(a*x^17)+12/221*b*(a+b*x^4)^(5/4)/(a^2*x^13)-32/663*b^2*(a+b*x^4)^(5/4)/(a^3*x^9)+128/3315*b^3*(a+b*x^4)^(5/4)/(a^4*x^5)],
[x^12*(a+b*x^4)^(1/4),x,8,3/112*a^3*x*(a+b*x^4)^(1/4)/b^3-3/280*a^2*x^5*(a+b*x^4)^(1/4)/b^2+1/140*a*x^9*(a+b*x^4)^(1/4)/b+1/14*x^13*(a+b*x^4)^(1/4)+3/112*a^(7/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(3/4))],
[x^8*(a+b*x^4)^(1/4),x,7,-1/24*a^2*x*(a+b*x^4)^(1/4)/b^2+1/60*a*x^5*(a+b*x^4)^(1/4)/b+1/10*x^9*(a+b*x^4)^(1/4)-1/24*a^(5/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(3/4))],
[x^4*(a+b*x^4)^(1/4),x,6,1/12*a*x*(a+b*x^4)^(1/4)/b+1/6*x^5*(a+b*x^4)^(1/4)+1/12*a^(3/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(b))],
[(a+b*x^4)^(1/4),x,5,1/2*x*(a+b*x^4)^(1/4)-1/2*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(a+b*x^4)^(3/4)],
[(a+b*x^4)^(1/4)/x^4,x,5,-1/3*(a+b*x^4)^(1/4)/x^3-1/3*b^(3/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(a))],
[(a+b*x^4)^(1/4)/x^8,x,6,-1/7*(a+b*x^4)^(1/4)/x^7-1/21*b*(a+b*x^4)^(1/4)/(a*x^3)+2/21*b^(5/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(3/4))],
[(a+b*x^4)^(1/4)/x^12,x,7,-1/11*(a+b*x^4)^(1/4)/x^11-1/77*b*(a+b*x^4)^(1/4)/(a*x^7)+2/77*b^2*(a+b*x^4)^(1/4)/(a^2*x^3)-4/77*b^(7/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(3/4))],
[(a+b*x^4)^(1/4)/x^16,x,8,-1/15*(a+b*x^4)^(1/4)/x^15-1/165*b*(a+b*x^4)^(1/4)/(a*x^11)+2/231*b^2*(a+b*x^4)^(1/4)/(a^2*x^7)-4/231*b^3*(a+b*x^4)^(1/4)/(a^3*x^3)+8/231*b^(9/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a+b*x^4)^(3/4))],
[x^19*(a+b*x^4)^(3/4),x,3,1/7*a^4*(a+b*x^4)^(7/4)/b^5-4/11*a^3*(a+b*x^4)^(11/4)/b^5+2/5*a^2*(a+b*x^4)^(15/4)/b^5-4/19*a*(a+b*x^4)^(19/4)/b^5+1/23*(a+b*x^4)^(23/4)/b^5],
[x^15*(a+b*x^4)^(3/4),x,3,-1/7*a^3*(a+b*x^4)^(7/4)/b^4+3/11*a^2*(a+b*x^4)^(11/4)/b^4-1/5*a*(a+b*x^4)^(15/4)/b^4+1/19*(a+b*x^4)^(19/4)/b^4],
[x^11*(a+b*x^4)^(3/4),x,3,1/7*a^2*(a+b*x^4)^(7/4)/b^3-2/11*a*(a+b*x^4)^(11/4)/b^3+1/15*(a+b*x^4)^(15/4)/b^3],
[x^7*(a+b*x^4)^(3/4),x,3,-1/7*a*(a+b*x^4)^(7/4)/b^2+1/11*(a+b*x^4)^(11/4)/b^2],
[x^3*(a+b*x^4)^(3/4),x,1,1/7*(a+b*x^4)^(7/4)/b],
[(a+b*x^4)^(3/4)/x,x,6,1/3*(a+b*x^4)^(3/4)+1/2*a^(3/4)*arctan((a+b*x^4)^(1/4)/a^(1/4))-1/2*a^(3/4)*arctanh((a+b*x^4)^(1/4)/a^(1/4))],
[(a+b*x^4)^(3/4)/x^5,x,6,-1/4*(a+b*x^4)^(3/4)/x^4+3/8*b*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(1/4)-3/8*b*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(1/4)],
[(a+b*x^4)^(3/4)/x^9,x,7,-1/8*(a+b*x^4)^(3/4)/x^8-3/32*b*(a+b*x^4)^(3/4)/(a*x^4)-3/64*b^2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(5/4)+3/64*b^2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(5/4)],
[x^9*(a+b*x^4)^(3/4),x,7,4/65*a^3*x^2/(b^2*(a+b*x^4)^(1/4))-2/65*a^2*x^2*(a+b*x^4)^(3/4)/b^2+1/39*a*x^6*(a+b*x^4)^(3/4)/b+1/13*x^10*(a+b*x^4)^(3/4)-4/65*a^(7/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(1/4))],
[x^5*(a+b*x^4)^(3/4),x,6,-2/15*a^2*x^2/(b*(a+b*x^4)^(1/4))+1/15*a*x^2*(a+b*x^4)^(3/4)/b+1/9*x^6*(a+b*x^4)^(3/4)+2/15*a^(5/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(1/4))],
[x*(a+b*x^4)^(3/4),x,5,3/5*a*x^2/(a+b*x^4)^(1/4)+1/5*x^2*(a+b*x^4)^(3/4)-3/5*a^(3/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(1/4)*sqrt(b))],
[(a+b*x^4)^(3/4)/x^3,x,5,3/2*b*x^2/(a+b*x^4)^(1/4)-1/2*(a+b*x^4)^(3/4)/x^2-3/2*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(a+b*x^4)^(1/4)],
[(a+b*x^4)^(3/4)/x^7,x,6,1/4*b^2*x^2/(a*(a+b*x^4)^(1/4))-1/6*(a+b*x^4)^(3/4)/x^6-1/4*b*(a+b*x^4)^(3/4)/(a*x^2)-1/4*b^(3/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(1/4)*sqrt(a))],
[(a+b*x^4)^(3/4)/x^11,x,7,-3/40*b^3*x^2/(a^2*(a+b*x^4)^(1/4))-1/10*(a+b*x^4)^(3/4)/x^10-1/20*b*(a+b*x^4)^(3/4)/(a*x^6)+3/40*b^2*(a+b*x^4)^(3/4)/(a^2*x^2)+3/40*b^(5/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(1/4))],
[x^12*(a+b*x^4)^(3/4),x,8,45/2048*a^3*x*(a+b*x^4)^(3/4)/b^3-9/512*a^2*x^5*(a+b*x^4)^(3/4)/b^2+1/64*a*x^9*(a+b*x^4)^(3/4)/b+1/16*x^13*(a+b*x^4)^(3/4)-45/4096*a^4*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(13/4)-45/4096*a^4*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(13/4)],
[x^8*(a+b*x^4)^(3/4),x,7,-5/128*a^2*x*(a+b*x^4)^(3/4)/b^2+1/32*a*x^5*(a+b*x^4)^(3/4)/b+1/12*x^9*(a+b*x^4)^(3/4)+5/256*a^3*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(9/4)+5/256*a^3*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(9/4)],
[x^4*(a+b*x^4)^(3/4),x,6,3/32*a*x*(a+b*x^4)^(3/4)/b+1/8*x^5*(a+b*x^4)^(3/4)-3/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(5/4)-3/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(5/4)],
[(a+b*x^4)^(3/4),x,5,1/4*x*(a+b*x^4)^(3/4)+3/8*a*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(1/4)+3/8*a*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(1/4)],
[(a+b*x^4)^(3/4)/x^4,x,5,-1/3*(a+b*x^4)^(3/4)/x^3+1/2*b^(3/4)*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))+1/2*b^(3/4)*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))],
[(a+b*x^4)^(3/4)/x^8,x,1,-1/7*(a+b*x^4)^(7/4)/(a*x^7)],
[(a+b*x^4)^(3/4)/x^12,x,2,-1/11*(a+b*x^4)^(7/4)/(a*x^11)+4/77*b*(a+b*x^4)^(7/4)/(a^2*x^7)],
[(a+b*x^4)^(3/4)/x^16,x,3,-1/15*(a+b*x^4)^(7/4)/(a*x^15)+8/165*b*(a+b*x^4)^(7/4)/(a^2*x^11)-32/1155*b^2*(a+b*x^4)^(7/4)/(a^3*x^7)],
[(a+b*x^4)^(3/4)/x^20,x,4,-1/19*(a+b*x^4)^(7/4)/(a*x^19)+4/95*b*(a+b*x^4)^(7/4)/(a^2*x^15)-32/1045*b^2*(a+b*x^4)^(7/4)/(a^3*x^11)+128/7315*b^3*(a+b*x^4)^(7/4)/(a^4*x^7)],
[x^10*(a+b*x^4)^(3/4),x,8,3/80*a^3*x^3/(b^2*(a+b*x^4)^(1/4))-1/40*a^2*x^3*(a+b*x^4)^(3/4)/b^2+3/140*a*x^7*(a+b*x^4)^(3/4)/b+1/14*x^11*(a+b*x^4)^(3/4)+3/80*a^(7/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(1/4))],
[x^6*(a+b*x^4)^(3/4),x,7,-3/40*a^2*x^3/(b*(a+b*x^4)^(1/4))+1/20*a*x^3*(a+b*x^4)^(3/4)/b+1/10*x^7*(a+b*x^4)^(3/4)-3/40*a^(5/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(1/4))],
[x^2*(a+b*x^4)^(3/4),x,6,1/4*a*x^3/(a+b*x^4)^(1/4)+1/6*x^3*(a+b*x^4)^(3/4)+1/4*a^(3/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(1/4)*sqrt(b))],
[(a+b*x^4)^(3/4)/x^2,x,6,3/2*b*x^3/(a+b*x^4)^(1/4)-(a+b*x^4)^(3/4)/x+3/2*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(a+b*x^4)^(1/4)],
[(a+b*x^4)^(3/4)/x^6,x,6,-3/5*b/(x*(a+b*x^4)^(1/4))-1/5*(a+b*x^4)^(3/4)/x^5+3/5*b^(3/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(1/4)*sqrt(a))],
[(a+b*x^4)^(3/4)/x^10,x,7,2/15*b^2/(a*x*(a+b*x^4)^(1/4))-1/9*(a+b*x^4)^(3/4)/x^9-1/15*b*(a+b*x^4)^(3/4)/(a*x^5)-2/15*b^(5/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(1/4))],
[(a+b*x^4)^(3/4)/x^14,x,8,-4/65*b^3/(a^2*x*(a+b*x^4)^(1/4))-1/13*(a+b*x^4)^(3/4)/x^13-1/39*b*(a+b*x^4)^(3/4)/(a*x^9)+2/65*b^2*(a+b*x^4)^(3/4)/(a^2*x^5)+4/65*b^(7/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(1/4))],
[x^19*(a+b*x^4)^(5/4),x,3,1/9*a^4*(a+b*x^4)^(9/4)/b^5-4/13*a^3*(a+b*x^4)^(13/4)/b^5+6/17*a^2*(a+b*x^4)^(17/4)/b^5-4/21*a*(a+b*x^4)^(21/4)/b^5+1/25*(a+b*x^4)^(25/4)/b^5],
[x^15*(a+b*x^4)^(5/4),x,3,-1/9*a^3*(a+b*x^4)^(9/4)/b^4+3/13*a^2*(a+b*x^4)^(13/4)/b^4-3/17*a*(a+b*x^4)^(17/4)/b^4+1/21*(a+b*x^4)^(21/4)/b^4],
[x^11*(a+b*x^4)^(5/4),x,3,1/9*a^2*(a+b*x^4)^(9/4)/b^3-2/13*a*(a+b*x^4)^(13/4)/b^3+1/17*(a+b*x^4)^(17/4)/b^3],
[x^7*(a+b*x^4)^(5/4),x,3,-1/9*a*(a+b*x^4)^(9/4)/b^2+1/13*(a+b*x^4)^(13/4)/b^2],
[x^3*(a+b*x^4)^(5/4),x,1,1/9*(a+b*x^4)^(9/4)/b],
[(a+b*x^4)^(5/4)/x,x,7,a*(a+b*x^4)^(1/4)+1/5*(a+b*x^4)^(5/4)-1/2*a^(5/4)*arctan((a+b*x^4)^(1/4)/a^(1/4))-1/2*a^(5/4)*arctanh((a+b*x^4)^(1/4)/a^(1/4))],
[(a+b*x^4)^(5/4)/x^5,x,7,5/4*b*(a+b*x^4)^(1/4)-1/4*(a+b*x^4)^(5/4)/x^4-5/8*a^(1/4)*b*arctan((a+b*x^4)^(1/4)/a^(1/4))-5/8*a^(1/4)*b*arctanh((a+b*x^4)^(1/4)/a^(1/4))],
[(a+b*x^4)^(5/4)/x^9,x,7,-5/32*b*(a+b*x^4)^(1/4)/x^4-1/8*(a+b*x^4)^(5/4)/x^8-5/64*b^2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(3/4)-5/64*b^2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(3/4)],
[x^9*(a+b*x^4)^(5/4),x,7,-2/231*a^3*x^2*(a+b*x^4)^(1/4)/b^2+1/231*a^2*x^6*(a+b*x^4)^(1/4)/b+1/33*a*x^10*(a+b*x^4)^(1/4)+1/15*x^10*(a+b*x^4)^(5/4)+4/231*a^(9/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(3/4))],
[x^5*(a+b*x^4)^(5/4),x,6,5/231*a^2*x^2*(a+b*x^4)^(1/4)/b+5/77*a*x^6*(a+b*x^4)^(1/4)+1/11*x^6*(a+b*x^4)^(5/4)-10/231*a^(7/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(3/4))],
[x*(a+b*x^4)^(5/4),x,5,5/21*a*x^2*(a+b*x^4)^(1/4)+1/7*x^2*(a+b*x^4)^(5/4)+5/21*a^(5/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(b))],
[(a+b*x^4)^(5/4)/x^3,x,5,5/6*b*x^2*(a+b*x^4)^(1/4)-1/2*(a+b*x^4)^(5/4)/x^2+5/6*a^(3/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a+b*x^4)^(3/4)],
[(a+b*x^4)^(5/4)/x^7,x,5,-5/12*b*(a+b*x^4)^(1/4)/x^2-1/6*(a+b*x^4)^(5/4)/x^6+5/12*b^(3/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/(a+b*x^4)^(3/4)],
[(a+b*x^4)^(5/4)/x^11,x,6,-1/12*b*(a+b*x^4)^(1/4)/x^6-1/24*b^2*(a+b*x^4)^(1/4)/(a*x^2)-1/10*(a+b*x^4)^(5/4)/x^10-1/24*b^(5/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(a))],
[(a+b*x^4)^(5/4)/x^15,x,7,-1/28*b*(a+b*x^4)^(1/4)/x^10-1/168*b^2*(a+b*x^4)^(1/4)/(a*x^6)+5/336*b^3*(a+b*x^4)^(1/4)/(a^2*x^2)-1/14*(a+b*x^4)^(5/4)/x^14+5/336*b^(7/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(3/4))],
[x^10*(a+b*x^4)^(5/4),x,8,-35/6144*a^3*x^3*(a+b*x^4)^(1/4)/b^2+5/1536*a^2*x^7*(a+b*x^4)^(1/4)/b+5/192*a*x^11*(a+b*x^4)^(1/4)+1/16*x^11*(a+b*x^4)^(5/4)-35/4096*a^4*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(11/4)+35/4096*a^4*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(11/4)],
[x^6*(a+b*x^4)^(5/4),x,7,5/384*a^2*x^3*(a+b*x^4)^(1/4)/b+5/96*a*x^7*(a+b*x^4)^(1/4)+1/12*x^7*(a+b*x^4)^(5/4)+5/256*a^3*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(7/4)-5/256*a^3*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(7/4)],
[x^2*(a+b*x^4)^(5/4),x,6,5/32*a*x^3*(a+b*x^4)^(1/4)+1/8*x^3*(a+b*x^4)^(5/4)-5/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(3/4)+5/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(3/4)],
[(a+b*x^4)^(5/4)/x^2,x,6,5/4*b*x^3*(a+b*x^4)^(1/4)-(a+b*x^4)^(5/4)/x-5/8*a*b^(1/4)*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))+5/8*a*b^(1/4)*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))],
[(a+b*x^4)^(5/4)/x^6,x,6,-b*(a+b*x^4)^(1/4)/x-1/5*(a+b*x^4)^(5/4)/x^5-1/2*b^(5/4)*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))+1/2*b^(5/4)*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))],
[(a+b*x^4)^(5/4)/x^10,x,1,-1/9*(a+b*x^4)^(9/4)/(a*x^9)],
[(a+b*x^4)^(5/4)/x^14,x,2,-1/13*(a+b*x^4)^(9/4)/(a*x^13)+4/117*b*(a+b*x^4)^(9/4)/(a^2*x^9)],
[(a+b*x^4)^(5/4)/x^18,x,3,-1/17*(a+b*x^4)^(9/4)/(a*x^17)+8/221*b*(a+b*x^4)^(9/4)/(a^2*x^13)-32/1989*b^2*(a+b*x^4)^(9/4)/(a^3*x^9)],
[(a+b*x^4)^(5/4)/x^22,x,4,-1/21*(a+b*x^4)^(9/4)/(a*x^21)+4/119*b*(a+b*x^4)^(9/4)/(a^2*x^17)-32/1547*b^2*(a+b*x^4)^(9/4)/(a^3*x^13)+128/13923*b^3*(a+b*x^4)^(9/4)/(a^4*x^9)],
[x^12*(a+b*x^4)^(5/4),x,9,5/672*a^4*x*(a+b*x^4)^(1/4)/b^3-1/336*a^3*x^5*(a+b*x^4)^(1/4)/b^2+1/504*a^2*x^9*(a+b*x^4)^(1/4)/b+5/252*a*x^13*(a+b*x^4)^(1/4)+1/18*x^13*(a+b*x^4)^(5/4)+5/672*a^(9/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(3/4))],
[x^8*(a+b*x^4)^(5/4),x,8,-5/336*a^3*x*(a+b*x^4)^(1/4)/b^2+1/168*a^2*x^5*(a+b*x^4)^(1/4)/b+1/28*a*x^9*(a+b*x^4)^(1/4)+1/14*x^9*(a+b*x^4)^(5/4)-5/336*a^(7/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(3/4))],
[x^4*(a+b*x^4)^(5/4),x,7,1/24*a^2*x*(a+b*x^4)^(1/4)/b+1/12*a*x^5*(a+b*x^4)^(1/4)+1/10*x^5*(a+b*x^4)^(5/4)+1/24*a^(5/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(b))],
[(a+b*x^4)^(5/4),x,6,5/12*a*x*(a+b*x^4)^(1/4)+1/6*x*(a+b*x^4)^(5/4)-5/12*a^(3/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a+b*x^4)^(3/4)],
[(a+b*x^4)^(5/4)/x^4,x,6,5/6*b*x*(a+b*x^4)^(1/4)-1/3*(a+b*x^4)^(5/4)/x^3-5/6*b^(3/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/(a+b*x^4)^(3/4)],
[(a+b*x^4)^(5/4)/x^8,x,6,-5/21*b*(a+b*x^4)^(1/4)/x^3-1/7*(a+b*x^4)^(5/4)/x^7-5/21*b^(5/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(3/4)*sqrt(a))],
[(a+b*x^4)^(5/4)/x^12,x,7,-5/77*b*(a+b*x^4)^(1/4)/x^7-5/231*b^2*(a+b*x^4)^(1/4)/(a*x^3)-1/11*(a+b*x^4)^(5/4)/x^11+10/231*b^(7/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(3/4))],
[(a+b*x^4)^(5/4)/x^16,x,8,-1/33*b*(a+b*x^4)^(1/4)/x^11-1/231*b^2*(a+b*x^4)^(1/4)/(a*x^7)+2/231*b^3*(a+b*x^4)^(1/4)/(a^2*x^3)-1/15*(a+b*x^4)^(5/4)/x^15-4/231*b^(9/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(3/4))],
[(a+b*x^4)^(7/4),x,6,7/32*a*x*(a+b*x^4)^(3/4)+1/8*x*(a+b*x^4)^(7/4)+21/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(1/4)+21/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(1/4)],

# p<0
[x^19/(a+b*x^4)^(1/4),x,3,1/3*a^4*(a+b*x^4)^(3/4)/b^5-4/7*a^3*(a+b*x^4)^(7/4)/b^5+6/11*a^2*(a+b*x^4)^(11/4)/b^5-4/15*a*(a+b*x^4)^(15/4)/b^5+1/19*(a+b*x^4)^(19/4)/b^5],
[x^15/(a+b*x^4)^(1/4),x,3,-1/3*a^3*(a+b*x^4)^(3/4)/b^4+3/7*a^2*(a+b*x^4)^(7/4)/b^4-3/11*a*(a+b*x^4)^(11/4)/b^4+1/15*(a+b*x^4)^(15/4)/b^4],
[x^11/(a+b*x^4)^(1/4),x,3,1/3*a^2*(a+b*x^4)^(3/4)/b^3-2/7*a*(a+b*x^4)^(7/4)/b^3+1/11*(a+b*x^4)^(11/4)/b^3],
[x^7/(a+b*x^4)^(1/4),x,3,-1/3*a*(a+b*x^4)^(3/4)/b^2+1/7*(a+b*x^4)^(7/4)/b^2],
[x^3/(a+b*x^4)^(1/4),x,1,1/3*(a+b*x^4)^(3/4)/b],
[1/(x*(a+b*x^4)^(1/4)),x,5,1/2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(1/4)-1/2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(1/4)],
[1/(x^5*(a+b*x^4)^(1/4)),x,6,-1/4*(a+b*x^4)^(3/4)/(a*x^4)-1/8*b*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(5/4)+1/8*b*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(5/4)],
[1/(x^9*(a+b*x^4)^(1/4)),x,7,-1/8*(a+b*x^4)^(3/4)/(a*x^8)+5/32*b*(a+b*x^4)^(3/4)/(a^2*x^4)+5/64*b^2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(9/4)-5/64*b^2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(9/4)],
[x^13/(a+b*x^4)^(1/4),x,7,-8/39*a^3*x^2/(b^3*(a+b*x^4)^(1/4))+4/39*a^2*x^2*(a+b*x^4)^(3/4)/b^3-10/117*a*x^6*(a+b*x^4)^(3/4)/b^2+1/13*x^10*(a+b*x^4)^(3/4)/b+8/39*a^(7/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(7/2)*(a+b*x^4)^(1/4))],
[x^9/(a+b*x^4)^(1/4),x,6,4/15*a^2*x^2/(b^2*(a+b*x^4)^(1/4))-2/15*a*x^2*(a+b*x^4)^(3/4)/b^2+1/9*x^6*(a+b*x^4)^(3/4)/b-4/15*a^(5/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(1/4))],
[x^5/(a+b*x^4)^(1/4),x,5,-2/5*a*x^2/(b*(a+b*x^4)^(1/4))+1/5*x^2*(a+b*x^4)^(3/4)/b+2/5*a^(3/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(1/4))],
[x/(a+b*x^4)^(1/4),x,4,x^2/(a+b*x^4)^(1/4)-(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a+b*x^4)^(1/4)*sqrt(b))],
[1/(x^3*(a+b*x^4)^(1/4)),x,5,1/2*b*x^2/(a*(a+b*x^4)^(1/4))-1/2*(a+b*x^4)^(3/4)/(a*x^2)-1/2*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a+b*x^4)^(1/4)*sqrt(a))],
[1/(x^7*(a+b*x^4)^(1/4)),x,6,-1/4*b^2*x^2/(a^2*(a+b*x^4)^(1/4))-1/6*(a+b*x^4)^(3/4)/(a*x^6)+1/4*b*(a+b*x^4)^(3/4)/(a^2*x^2)+1/4*b^(3/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(1/4))],
[1/(x^11*(a+b*x^4)^(1/4)),x,7,7/40*b^3*x^2/(a^3*(a+b*x^4)^(1/4))-1/10*(a+b*x^4)^(3/4)/(a*x^10)+7/60*b*(a+b*x^4)^(3/4)/(a^2*x^6)-7/40*b^2*(a+b*x^4)^(3/4)/(a^3*x^2)-7/40*b^(5/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(1/4))],
[x^8/(a+b*x^4)^(1/4),x,6,-5/32*a*x*(a+b*x^4)^(3/4)/b^2+1/8*x^5*(a+b*x^4)^(3/4)/b+5/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(9/4)+5/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(9/4)],
[x^4/(a+b*x^4)^(1/4),x,5,1/4*x*(a+b*x^4)^(3/4)/b-1/8*a*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(5/4)-1/8*a*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(5/4)],
[1/(a+b*x^4)^(1/4),x,4,1/2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(1/4)+1/2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(1/4)],
[1/(x^4*(a+b*x^4)^(1/4)),x,1,-1/3*(a+b*x^4)^(3/4)/(a*x^3)],
[1/(x^8*(a+b*x^4)^(1/4)),x,2,-1/7*(a+b*x^4)^(3/4)/(a*x^7)+4/21*b*(a+b*x^4)^(3/4)/(a^2*x^3)],
[1/(x^12*(a+b*x^4)^(1/4)),x,3,-1/11*(a+b*x^4)^(3/4)/(a*x^11)+8/77*b*(a+b*x^4)^(3/4)/(a^2*x^7)-32/231*b^2*(a+b*x^4)^(3/4)/(a^3*x^3)],
[1/(x^16*(a+b*x^4)^(1/4)),x,4,-1/15*(a+b*x^4)^(3/4)/(a*x^15)+4/55*b*(a+b*x^4)^(3/4)/(a^2*x^11)-32/385*b^2*(a+b*x^4)^(3/4)/(a^3*x^7)+128/1155*b^3*(a+b*x^4)^(3/4)/(a^4*x^3)],
[1/(x^20*(a+b*x^4)^(1/4)),x,5,-1/19*(a+b*x^4)^(3/4)/(a*x^19)+16/285*b*(a+b*x^4)^(3/4)/(a^2*x^15)-64/1045*b^2*(a+b*x^4)^(3/4)/(a^3*x^11)+512/7315*b^3*(a+b*x^4)^(3/4)/(a^4*x^7)-2048/21945*b^4*(a+b*x^4)^(3/4)/(a^5*x^3)],
[x^10/(a+b*x^4)^(1/4),x,7,7/40*a^2*x^3/(b^2*(a+b*x^4)^(1/4))-7/60*a*x^3*(a+b*x^4)^(3/4)/b^2+1/10*x^7*(a+b*x^4)^(3/4)/b+7/40*a^(5/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(1/4))],
[x^6/(a+b*x^4)^(1/4),x,6,-1/4*a*x^3/(b*(a+b*x^4)^(1/4))+1/6*x^3*(a+b*x^4)^(3/4)/b-1/4*a^(3/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(1/4))],
[x^2/(a+b*x^4)^(1/4),x,5,1/2*x^3/(a+b*x^4)^(1/4)+1/2*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a+b*x^4)^(1/4)*sqrt(b))],
[1/(x^2*(a+b*x^4)^(1/4)),x,5,(-1)/(x*(a+b*x^4)^(1/4))+(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a+b*x^4)^(1/4)*sqrt(a))],
[1/(x^6*(a+b*x^4)^(1/4)),x,6,2/5*b/(a*x*(a+b*x^4)^(1/4))-1/5*(a+b*x^4)^(3/4)/(a*x^5)-2/5*b^(3/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(1/4))],
[1/(x^10*(a+b*x^4)^(1/4)),x,7,-4/15*b^2/(a^2*x*(a+b*x^4)^(1/4))-1/9*(a+b*x^4)^(3/4)/(a*x^9)+2/15*b*(a+b*x^4)^(3/4)/(a^2*x^5)+4/15*b^(5/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(1/4))],
[1/(x^14*(a+b*x^4)^(1/4)),x,8,8/39*b^3/(a^3*x*(a+b*x^4)^(1/4))-1/13*(a+b*x^4)^(3/4)/(a*x^13)+10/117*b*(a+b*x^4)^(3/4)/(a^2*x^9)-4/39*b^2*(a+b*x^4)^(3/4)/(a^3*x^5)-8/39*b^(7/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a+b*x^4)^(1/4))],
[x^19/(a+b*x^4)^(3/4),x,3,a^4*(a+b*x^4)^(1/4)/b^5-4/5*a^3*(a+b*x^4)^(5/4)/b^5+2/3*a^2*(a+b*x^4)^(9/4)/b^5-4/13*a*(a+b*x^4)^(13/4)/b^5+1/17*(a+b*x^4)^(17/4)/b^5],
[x^15/(a+b*x^4)^(3/4),x,3,-a^3*(a+b*x^4)^(1/4)/b^4+3/5*a^2*(a+b*x^4)^(5/4)/b^4-1/3*a*(a+b*x^4)^(9/4)/b^4+1/13*(a+b*x^4)^(13/4)/b^4],
[x^11/(a+b*x^4)^(3/4),x,3,a^2*(a+b*x^4)^(1/4)/b^3-2/5*a*(a+b*x^4)^(5/4)/b^3+1/9*(a+b*x^4)^(9/4)/b^3],
[x^7/(a+b*x^4)^(3/4),x,3,-a*(a+b*x^4)^(1/4)/b^2+1/5*(a+b*x^4)^(5/4)/b^2],
[x^3/(a+b*x^4)^(3/4),x,1,(a+b*x^4)^(1/4)/b],
[1/(x*(a+b*x^4)^(3/4)),x,5,-1/2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(3/4)-1/2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(3/4)],
[1/(x^5*(a+b*x^4)^(3/4)),x,6,-1/4*(a+b*x^4)^(1/4)/(a*x^4)+3/8*b*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(7/4)+3/8*b*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(7/4)],
[1/(x^9*(a+b*x^4)^(3/4)),x,7,-1/8*(a+b*x^4)^(1/4)/(a*x^8)+7/32*b*(a+b*x^4)^(1/4)/(a^2*x^4)-21/64*b^2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(11/4)-21/64*b^2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(11/4)],
[x^13/(a+b*x^4)^(3/4),x,6,20/77*a^2*x^2*(a+b*x^4)^(1/4)/b^3-10/77*a*x^6*(a+b*x^4)^(1/4)/b^2+1/11*x^10*(a+b*x^4)^(1/4)/b-40/77*a^(7/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(7/2)*(a+b*x^4)^(3/4))],
[x^9/(a+b*x^4)^(3/4),x,5,-2/7*a*x^2*(a+b*x^4)^(1/4)/b^2+1/7*x^6*(a+b*x^4)^(1/4)/b+4/7*a^(5/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(3/4))],
[x^5/(a+b*x^4)^(3/4),x,4,1/3*x^2*(a+b*x^4)^(1/4)/b-2/3*a^(3/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(3/4))],
[x/(a+b*x^4)^(3/4),x,3,(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a+b*x^4)^(3/4)*sqrt(b))],
[1/(x^3*(a+b*x^4)^(3/4)),x,4,-1/2*(a+b*x^4)^(1/4)/(a*x^2)-1/2*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a+b*x^4)^(3/4)*sqrt(a))],
[1/(x^7*(a+b*x^4)^(3/4)),x,5,-1/6*(a+b*x^4)^(1/4)/(a*x^6)+5/12*b*(a+b*x^4)^(1/4)/(a^2*x^2)+5/12*b^(3/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(3/4))],
[1/(x^11*(a+b*x^4)^(3/4)),x,6,-1/10*(a+b*x^4)^(1/4)/(a*x^10)+3/20*b*(a+b*x^4)^(1/4)/(a^2*x^6)-3/8*b^2*(a+b*x^4)^(1/4)/(a^3*x^2)-3/8*b^(5/2)*(1+b*x^4/a)^(3/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(3/4))],
[x^10/(a+b*x^4)^(3/4),x,6,-7/32*a*x^3*(a+b*x^4)^(1/4)/b^2+1/8*x^7*(a+b*x^4)^(1/4)/b-21/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(11/4)+21/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(11/4)],
[x^6/(a+b*x^4)^(3/4),x,5,1/4*x^3*(a+b*x^4)^(1/4)/b+3/8*a*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(7/4)-3/8*a*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(7/4)],
[x^2/(a+b*x^4)^(3/4),x,4,-1/2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(3/4)+1/2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(3/4)],
[1/(x^2*(a+b*x^4)^(3/4)),x,1,-(a+b*x^4)^(1/4)/(a*x)],
[1/(x^6*(a+b*x^4)^(3/4)),x,2,-1/5*(a+b*x^4)^(1/4)/(a*x^5)+4/5*b*(a+b*x^4)^(1/4)/(a^2*x)],
[1/(x^10*(a+b*x^4)^(3/4)),x,3,-1/9*(a+b*x^4)^(1/4)/(a*x^9)+8/45*b*(a+b*x^4)^(1/4)/(a^2*x^5)-32/45*b^2*(a+b*x^4)^(1/4)/(a^3*x)],
[1/(x^14*(a+b*x^4)^(3/4)),x,4,-1/13*(a+b*x^4)^(1/4)/(a*x^13)+4/39*b*(a+b*x^4)^(1/4)/(a^2*x^9)-32/195*b^2*(a+b*x^4)^(1/4)/(a^3*x^5)+128/195*b^3*(a+b*x^4)^(1/4)/(a^4*x)],
[x^12/(a+b*x^4)^(3/4),x,7,3/8*a^2*x*(a+b*x^4)^(1/4)/b^3-3/20*a*x^5*(a+b*x^4)^(1/4)/b^2+1/10*x^9*(a+b*x^4)^(1/4)/b+3/8*a^(5/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(3/4))],
[x^8/(a+b*x^4)^(3/4),x,6,-5/12*a*x*(a+b*x^4)^(1/4)/b^2+1/6*x^5*(a+b*x^4)^(1/4)/b-5/12*a^(3/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^4)^(3/4))],
[x^4/(a+b*x^4)^(3/4),x,5,1/2*x*(a+b*x^4)^(1/4)/b+1/2*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a+b*x^4)^(3/4)*sqrt(b))],
[1/(a+b*x^4)^(3/4),x,4,-(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a+b*x^4)^(3/4)*sqrt(a))],
[1/(x^4*(a+b*x^4)^(3/4)),x,5,-1/3*(a+b*x^4)^(1/4)/(a*x^3)+2/3*b^(3/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a+b*x^4)^(3/4))],
[1/(x^8*(a+b*x^4)^(3/4)),x,6,-1/7*(a+b*x^4)^(1/4)/(a*x^7)+2/7*b*(a+b*x^4)^(1/4)/(a^2*x^3)-4/7*b^(5/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(3/4))],
[1/(x^12*(a+b*x^4)^(3/4)),x,7,-1/11*(a+b*x^4)^(1/4)/(a*x^11)+10/77*b*(a+b*x^4)^(1/4)/(a^2*x^7)-20/77*b^2*(a+b*x^4)^(1/4)/(a^3*x^3)+40/77*b^(7/2)*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a+b*x^4)^(3/4))],
[x^19/(a+b*x^4)^(5/4),x,3,-a^4/(b^5*(a+b*x^4)^(1/4))-4/3*a^3*(a+b*x^4)^(3/4)/b^5+6/7*a^2*(a+b*x^4)^(7/4)/b^5-4/11*a*(a+b*x^4)^(11/4)/b^5+1/15*(a+b*x^4)^(15/4)/b^5],
[x^15/(a+b*x^4)^(5/4),x,3,a^3/(b^4*(a+b*x^4)^(1/4))+a^2*(a+b*x^4)^(3/4)/b^4-3/7*a*(a+b*x^4)^(7/4)/b^4+1/11*(a+b*x^4)^(11/4)/b^4],
[x^11/(a+b*x^4)^(5/4),x,3,-a^2/(b^3*(a+b*x^4)^(1/4))-2/3*a*(a+b*x^4)^(3/4)/b^3+1/7*(a+b*x^4)^(7/4)/b^3],
[x^7/(a+b*x^4)^(5/4),x,3,a/(b^2*(a+b*x^4)^(1/4))+1/3*(a+b*x^4)^(3/4)/b^2],
[x^3/(a+b*x^4)^(5/4),x,1,(-1)/(b*(a+b*x^4)^(1/4))],
[1/(x*(a+b*x^4)^(5/4)),x,6,1/(a*(a+b*x^4)^(1/4))+1/2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(5/4)-1/2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(5/4)],
[1/(x^5*(a+b*x^4)^(5/4)),x,7,-5/4*b/(a^2*(a+b*x^4)^(1/4))+(-1/4)/(a*x^4*(a+b*x^4)^(1/4))-5/8*b*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(9/4)+5/8*b*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(9/4)],
[1/(x^9*(a+b*x^4)^(5/4)),x,8,45/32*b^2/(a^3*(a+b*x^4)^(1/4))+(-1/8)/(a*x^8*(a+b*x^4)^(1/4))+9/32*b/(a^2*x^4*(a+b*x^4)^(1/4))+45/64*b^2*arctan((a+b*x^4)^(1/4)/a^(1/4))/a^(13/4)-45/64*b^2*arctanh((a+b*x^4)^(1/4)/a^(1/4))/a^(13/4)],
[x^13/(a+b*x^4)^(5/4),x,6,4/3*a^2*x^2/(b^3*(a+b*x^4)^(1/4))-2/9*a*x^6/(b^2*(a+b*x^4)^(1/4))+1/9*x^10/(b*(a+b*x^4)^(1/4))-8/3*a^(5/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(7/2)*(a+b*x^4)^(1/4))],
[x^9/(a+b*x^4)^(5/4),x,5,-6/5*a*x^2/(b^2*(a+b*x^4)^(1/4))+1/5*x^6/(b*(a+b*x^4)^(1/4))+12/5*a^(3/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(1/4))],
[x^5/(a+b*x^4)^(5/4),x,4,x^2/(b*(a+b*x^4)^(1/4))-2*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/(b^(3/2)*(a+b*x^4)^(1/4))],
[x/(a+b*x^4)^(5/4),x,3,(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(1/4)*sqrt(a)*sqrt(b))],
[1/(x^3*(a+b*x^4)^(5/4)),x,4,(-1/2)/(a*x^2*(a+b*x^4)^(1/4))-3/2*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(3/2)*(a+b*x^4)^(1/4))],
[1/(x^7*(a+b*x^4)^(5/4)),x,5,(-1/6)/(a*x^6*(a+b*x^4)^(1/4))+7/12*b/(a^2*x^2*(a+b*x^4)^(1/4))+7/4*b^(3/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(1/4))],
[1/(x^11*(a+b*x^4)^(5/4)),x,6,(-1/10)/(a*x^10*(a+b*x^4)^(1/4))+11/60*b/(a^2*x^6*(a+b*x^4)^(1/4))-77/120*b^2/(a^3*x^2*(a+b*x^4)^(1/4))-77/40*b^(5/2)*(1+b*x^4/a)^(1/4)*sqrt(cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a+b*x^4)^(1/4))],
[x^12/(a+b*x^4)^(5/4),x,7,-x^9/(b*(a+b*x^4)^(1/4))-45/32*a*x*(a+b*x^4)^(3/4)/b^3+9/8*x^5*(a+b*x^4)^(3/4)/b^2+45/64*a^2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(13/4)+45/64*a^2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(13/4)],
[x^8/(a+b*x^4)^(5/4),x,6,-x^5/(b*(a+b*x^4)^(1/4))+5/4*x*(a+b*x^4)^(3/4)/b^2-5/8*a*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(9/4)-5/8*a*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(9/4)],
[x^4/(a+b*x^4)^(5/4),x,5,-x/(b*(a+b*x^4)^(1/4))+1/2*arctan(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(5/4)+1/2*arctanh(b^(1/4)*x/(a+b*x^4)^(1/4))/b^(5/4)],
[1/(a+b*x^4)^(5/4),x,1,x/(a*(a+b*x^4)^(1/4))],
[1/(x^4*(a+b*x^4)^(5/4)),x,2,(-1/3)/(a*x^3*(a+b*x^4)^(1/4))-4/3*b*x/(a^2*(a+b*x^4)^(1/4))],
[1/(x^8*(a+b*x^4)^(5/4)),x,3,(-1/7)/(a*x^7*(a+b*x^4)^(1/4))+8/21*b/(a^2*x^3*(a+b*x^4)^(1/4))+32/21*b^2*x/(a^3*(a+b*x^4)^(1/4))],
[1/(x^12*(a+b*x^4)^(5/4)),x,4,(-1/11)/(a*x^11*(a+b*x^4)^(1/4))+12/77*b/(a^2*x^7*(a+b*x^4)^(1/4))-32/77*b^2/(a^3*x^3*(a+b*x^4)^(1/4))-128/77*b^3*x/(a^4*(a+b*x^4)^(1/4))],
[1/(x^16*(a+b*x^4)^(5/4)),x,5,(-1/15)/(a*x^15*(a+b*x^4)^(1/4))+16/165*b/(a^2*x^11*(a+b*x^4)^(1/4))-64/385*b^2/(a^3*x^7*(a+b*x^4)^(1/4))+512/1155*b^3/(a^4*x^3*(a+b*x^4)^(1/4))+2048/1155*b^4*x/(a^5*(a+b*x^4)^(1/4))],
[x^14/(a+b*x^4)^(5/4),x,7,77/120*a^2*x^3/(b^3*(a+b*x^4)^(1/4))-11/60*a*x^7/(b^2*(a+b*x^4)^(1/4))+1/10*x^11/(b*(a+b*x^4)^(1/4))+77/40*a^(5/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(7/2)*(a+b*x^4)^(1/4))],
[x^10/(a+b*x^4)^(5/4),x,6,-7/12*a*x^3/(b^2*(a+b*x^4)^(1/4))+1/6*x^7/(b*(a+b*x^4)^(1/4))-7/4*a^(3/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^4)^(1/4))],
[x^6/(a+b*x^4)^(5/4),x,5,1/2*x^3/(b*(a+b*x^4)^(1/4))+3/2*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/(b^(3/2)*(a+b*x^4)^(1/4))],
[x^2/(a+b*x^4)^(5/4),x,4,-(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a+b*x^4)^(1/4)*sqrt(a)*sqrt(b))],
[1/(x^2*(a+b*x^4)^(5/4)),x,5,(-1)/(a*x*(a+b*x^4)^(1/4))+2*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(3/2)*(a+b*x^4)^(1/4))],
[1/(x^6*(a+b*x^4)^(5/4)),x,6,(-1/5)/(a*x^5*(a+b*x^4)^(1/4))+6/5*b/(a^2*x*(a+b*x^4)^(1/4))-12/5*b^(3/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a+b*x^4)^(1/4))],
[1/(x^10*(a+b*x^4)^(5/4)),x,7,(-1/9)/(a*x^9*(a+b*x^4)^(1/4))+2/9*b/(a^2*x^5*(a+b*x^4)^(1/4))-4/3*b^2/(a^3*x*(a+b*x^4)^(1/4))+8/3*b^(5/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a+b*x^4)^(1/4))],
[1/(x^14*(a+b*x^4)^(5/4)),x,8,(-1/13)/(a*x^13*(a+b*x^4)^(1/4))+14/117*b/(a^2*x^9*(a+b*x^4)^(1/4))-28/117*b^2/(a^3*x^5*(a+b*x^4)^(1/4))+56/39*b^3/(a^4*x*(a+b*x^4)^(1/4))-112/39*b^(7/2)*(1+a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(9/2)*(a+b*x^4)^(1/4))],
[1/(a+b*x^4)^(7/4),x,5,1/3*x/(a*(a+b*x^4)^(3/4))-2/3*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(3/2)*(a+b*x^4)^(3/4))],
[1/(a+b*x^4)^(9/4),x,2,1/5*x/(a*(a+b*x^4)^(5/4))+4/5*x/(a^2*(a+b*x^4)^(1/4))],
[1/(a+b*x^4)^(11/4),x,6,1/7*x/(a*(a+b*x^4)^(7/4))+2/7*x/(a^2*(a+b*x^4)^(3/4))-4/7*(1+a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(5/2)*(a+b*x^4)^(3/4))],
[1/(a+b*x^4)^(13/4),x,3,1/9*x/(a*(a+b*x^4)^(9/4))+8/45*x/(a^2*(a+b*x^4)^(5/4))+32/45*x/(a^3*(a+b*x^4)^(1/4))],
[1/(a+b*x^4)^(17/4),x,4,1/13*x/(a*(a+b*x^4)^(13/4))+4/39*x/(a^2*(a+b*x^4)^(9/4))+32/195*x/(a^3*(a+b*x^4)^(5/4))+128/195*x/(a^4*(a+b*x^4)^(1/4))],

# Integrands of the form x^m (a-b x^4)^(p/4)

# p>0
[x^19*(a-b*x^4)^(1/4),x,3,-1/5*a^4*(a-b*x^4)^(5/4)/b^5+4/9*a^3*(a-b*x^4)^(9/4)/b^5-6/13*a^2*(a-b*x^4)^(13/4)/b^5+4/17*a*(a-b*x^4)^(17/4)/b^5-1/21*(a-b*x^4)^(21/4)/b^5],
[x^15*(a-b*x^4)^(1/4),x,3,-1/5*a^3*(a-b*x^4)^(5/4)/b^4+1/3*a^2*(a-b*x^4)^(9/4)/b^4-3/13*a*(a-b*x^4)^(13/4)/b^4+1/17*(a-b*x^4)^(17/4)/b^4],
[x^11*(a-b*x^4)^(1/4),x,3,-1/5*a^2*(a-b*x^4)^(5/4)/b^3+2/9*a*(a-b*x^4)^(9/4)/b^3-1/13*(a-b*x^4)^(13/4)/b^3],
[x^7*(a-b*x^4)^(1/4),x,3,-1/5*a*(a-b*x^4)^(5/4)/b^2+1/9*(a-b*x^4)^(9/4)/b^2],
[x^3*(a-b*x^4)^(1/4),x,1,-1/5*(a-b*x^4)^(5/4)/b],
[(a-b*x^4)^(1/4)/x,x,6,(a-b*x^4)^(1/4)-1/2*a^(1/4)*arctan((a-b*x^4)^(1/4)/a^(1/4))-1/2*a^(1/4)*arctanh((a-b*x^4)^(1/4)/a^(1/4))],
[(a-b*x^4)^(1/4)/x^5,x,6,-1/4*(a-b*x^4)^(1/4)/x^4+1/8*b*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(3/4)+1/8*b*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(3/4)],
[(a-b*x^4)^(1/4)/x^9,x,7,-1/8*(a-b*x^4)^(1/4)/x^8+1/32*b*(a-b*x^4)^(1/4)/(a*x^4)+3/64*b^2*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(7/4)+3/64*b^2*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(7/4)],
[x^9*(a-b*x^4)^(1/4),x,6,-2/77*a^2*x^2*(a-b*x^4)^(1/4)/b^2-1/77*a*x^6*(a-b*x^4)^(1/4)/b+1/11*x^10*(a-b*x^4)^(1/4)+4/77*a^(7/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a-b*x^4)^(3/4))],
[x^5*(a-b*x^4)^(1/4),x,5,-1/21*a*x^2*(a-b*x^4)^(1/4)/b+1/7*x^6*(a-b*x^4)^(1/4)+2/21*a^(5/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a-b*x^4)^(3/4))],
[x*(a-b*x^4)^(1/4),x,4,1/3*x^2*(a-b*x^4)^(1/4)+1/3*a^(3/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a-b*x^4)^(3/4)*sqrt(b))],
[(a-b*x^4)^(1/4)/x^3,x,4,-1/2*(a-b*x^4)^(1/4)/x^2-1/2*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(a-b*x^4)^(3/4)],
[(a-b*x^4)^(1/4)/x^7,x,5,-1/6*(a-b*x^4)^(1/4)/x^6+1/12*b*(a-b*x^4)^(1/4)/(a*x^2)-1/12*b^(3/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a-b*x^4)^(3/4)*sqrt(a))],
[(a-b*x^4)^(1/4)/x^11,x,6,-1/10*(a-b*x^4)^(1/4)/x^10+1/60*b*(a-b*x^4)^(1/4)/(a*x^6)+1/24*b^2*(a-b*x^4)^(1/4)/(a^2*x^2)-1/24*b^(5/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a-b*x^4)^(3/4))],
[x^6*(a-b*x^4)^(1/4),x,12,-1/32*a*x^3*(a-b*x^4)^(1/4)/b+1/8*x^7*(a-b*x^4)^(1/4)-3/64*a^2*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(7/4)*sqrt(2))+3/64*a^2*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(7/4)*sqrt(2))+3/128*a^2*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(7/4)*sqrt(2))-3/128*a^2*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(7/4)*sqrt(2))],
[x^2*(a-b*x^4)^(1/4),x,11,1/4*x^3*(a-b*x^4)^(1/4)-1/8*a*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(3/4)*sqrt(2))+1/8*a*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(3/4)*sqrt(2))+1/16*a*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(3/4)*sqrt(2))-1/16*a*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(3/4)*sqrt(2))],
[(a-b*x^4)^(1/4)/x^2,x,11,-(a-b*x^4)^(1/4)/x+1/2*b^(1/4)*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/sqrt(2)-1/2*b^(1/4)*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/sqrt(2)-1/4*b^(1/4)*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/sqrt(2)+1/4*b^(1/4)*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/sqrt(2)],
[(a-b*x^4)^(1/4)/x^6,x,1,-1/5*(a-b*x^4)^(5/4)/(a*x^5)],
[(a-b*x^4)^(1/4)/x^10,x,2,-1/9*(a-b*x^4)^(5/4)/(a*x^9)-4/45*b*(a-b*x^4)^(5/4)/(a^2*x^5)],
[(a-b*x^4)^(1/4)/x^14,x,3,-1/13*(a-b*x^4)^(5/4)/(a*x^13)-8/117*b*(a-b*x^4)^(5/4)/(a^2*x^9)-32/585*b^2*(a-b*x^4)^(5/4)/(a^3*x^5)],
[(a-b*x^4)^(1/4)/x^18,x,4,-1/17*(a-b*x^4)^(5/4)/(a*x^17)-12/221*b*(a-b*x^4)^(5/4)/(a^2*x^13)-32/663*b^2*(a-b*x^4)^(5/4)/(a^3*x^9)-128/3315*b^3*(a-b*x^4)^(5/4)/(a^4*x^5)],
[x^12*(a-b*x^4)^(1/4),x,8,-3/112*a^3*x*(a-b*x^4)^(1/4)/b^3-3/280*a^2*x^5*(a-b*x^4)^(1/4)/b^2-1/140*a*x^9*(a-b*x^4)^(1/4)/b+1/14*x^13*(a-b*x^4)^(1/4)-3/112*a^(7/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a-b*x^4)^(3/4))],
[x^8*(a-b*x^4)^(1/4),x,7,-1/24*a^2*x*(a-b*x^4)^(1/4)/b^2-1/60*a*x^5*(a-b*x^4)^(1/4)/b+1/10*x^9*(a-b*x^4)^(1/4)-1/24*a^(5/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a-b*x^4)^(3/4))],
[x^4*(a-b*x^4)^(1/4),x,6,-1/12*a*x*(a-b*x^4)^(1/4)/b+1/6*x^5*(a-b*x^4)^(1/4)-1/12*a^(3/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a-b*x^4)^(3/4)*sqrt(b))],
[(a-b*x^4)^(1/4),x,5,1/2*x*(a-b*x^4)^(1/4)-1/2*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(a-b*x^4)^(3/4)],
[(a-b*x^4)^(1/4)/x^4,x,5,-1/3*(a-b*x^4)^(1/4)/x^3+1/3*b^(3/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a-b*x^4)^(3/4)*sqrt(a))],
[(a-b*x^4)^(1/4)/x^8,x,6,-1/7*(a-b*x^4)^(1/4)/x^7+1/21*b*(a-b*x^4)^(1/4)/(a*x^3)+2/21*b^(5/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a-b*x^4)^(3/4))],
[(a-b*x^4)^(1/4)/x^12,x,7,-1/11*(a-b*x^4)^(1/4)/x^11+1/77*b*(a-b*x^4)^(1/4)/(a*x^7)+2/77*b^2*(a-b*x^4)^(1/4)/(a^2*x^3)+4/77*b^(7/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a-b*x^4)^(3/4))],
[(a-b*x^4)^(1/4)/x^16,x,8,-1/15*(a-b*x^4)^(1/4)/x^15+1/165*b*(a-b*x^4)^(1/4)/(a*x^11)+2/231*b^2*(a-b*x^4)^(1/4)/(a^2*x^7)+4/231*b^3*(a-b*x^4)^(1/4)/(a^3*x^3)+8/231*b^(9/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a-b*x^4)^(3/4))],

# p<0
[x^19/(a-b*x^4)^(1/4),x,3,-1/3*a^4*(a-b*x^4)^(3/4)/b^5+4/7*a^3*(a-b*x^4)^(7/4)/b^5-6/11*a^2*(a-b*x^4)^(11/4)/b^5+4/15*a*(a-b*x^4)^(15/4)/b^5-1/19*(a-b*x^4)^(19/4)/b^5],
[x^15/(a-b*x^4)^(1/4),x,3,-1/3*a^3*(a-b*x^4)^(3/4)/b^4+3/7*a^2*(a-b*x^4)^(7/4)/b^4-3/11*a*(a-b*x^4)^(11/4)/b^4+1/15*(a-b*x^4)^(15/4)/b^4],
[x^11/(a-b*x^4)^(1/4),x,3,-1/3*a^2*(a-b*x^4)^(3/4)/b^3+2/7*a*(a-b*x^4)^(7/4)/b^3-1/11*(a-b*x^4)^(11/4)/b^3],
[x^7/(a-b*x^4)^(1/4),x,3,-1/3*a*(a-b*x^4)^(3/4)/b^2+1/7*(a-b*x^4)^(7/4)/b^2],
[x^3/(a-b*x^4)^(1/4),x,1,-1/3*(a-b*x^4)^(3/4)/b],
[1/(x*(a-b*x^4)^(1/4)),x,5,1/2*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(1/4)-1/2*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(1/4)],
[1/(x^5*(a-b*x^4)^(1/4)),x,6,-1/4*(a-b*x^4)^(3/4)/(a*x^4)+1/8*b*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(5/4)-1/8*b*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(5/4)],
[1/(x^9*(a-b*x^4)^(1/4)),x,7,-1/8*(a-b*x^4)^(3/4)/(a*x^8)-5/32*b*(a-b*x^4)^(3/4)/(a^2*x^4)+5/64*b^2*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(9/4)-5/64*b^2*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(9/4)],
[x^13/(a-b*x^4)^(1/4),x,6,-4/39*a^2*x^2*(a-b*x^4)^(3/4)/b^3-10/117*a*x^6*(a-b*x^4)^(3/4)/b^2-1/13*x^10*(a-b*x^4)^(3/4)/b+8/39*a^(7/2)*(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(7/2)*(a-b*x^4)^(1/4))],
[x^9/(a-b*x^4)^(1/4),x,5,-2/15*a*x^2*(a-b*x^4)^(3/4)/b^2-1/9*x^6*(a-b*x^4)^(3/4)/b+4/15*a^(5/2)*(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a-b*x^4)^(1/4))],
[x^5/(a-b*x^4)^(1/4),x,4,-1/5*x^2*(a-b*x^4)^(3/4)/b+2/5*a^(3/2)*(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a-b*x^4)^(1/4))],
[x/(a-b*x^4)^(1/4),x,3,(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a-b*x^4)^(1/4)*sqrt(b))],
[1/(x^3*(a-b*x^4)^(1/4)),x,4,-1/2*(a-b*x^4)^(3/4)/(a*x^2)-1/2*(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a-b*x^4)^(1/4)*sqrt(a))],
[1/(x^7*(a-b*x^4)^(1/4)),x,5,-1/6*(a-b*x^4)^(3/4)/(a*x^6)-1/4*b*(a-b*x^4)^(3/4)/(a^2*x^2)-1/4*b^(3/2)*(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a-b*x^4)^(1/4))],
[1/(x^11*(a-b*x^4)^(1/4)),x,6,-1/10*(a-b*x^4)^(3/4)/(a*x^10)-7/60*b*(a-b*x^4)^(3/4)/(a^2*x^6)-7/40*b^2*(a-b*x^4)^(3/4)/(a^3*x^2)-7/40*b^(5/2)*(1-b*x^4/a)^(1/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a-b*x^4)^(1/4))],
[x^8/(a-b*x^4)^(1/4),x,12,-5/32*a*x*(a-b*x^4)^(3/4)/b^2-1/8*x^5*(a-b*x^4)^(3/4)/b-5/64*a^2*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(9/4)*sqrt(2))+5/64*a^2*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(9/4)*sqrt(2))-5/128*a^2*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(9/4)*sqrt(2))+5/128*a^2*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(9/4)*sqrt(2))],
[x^4/(a-b*x^4)^(1/4),x,11,-1/4*x*(a-b*x^4)^(3/4)/b-1/8*a*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(5/4)*sqrt(2))+1/8*a*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(5/4)*sqrt(2))-1/16*a*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(5/4)*sqrt(2))+1/16*a*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(5/4)*sqrt(2))],
[1/(a-b*x^4)^(1/4),x,10,-1/2*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(1/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(1/4)*sqrt(2))-1/4*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(1/4)*sqrt(2))+1/4*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(1/4)*sqrt(2))],
[1/(x^4*(a-b*x^4)^(1/4)),x,1,-1/3*(a-b*x^4)^(3/4)/(a*x^3)],
[1/(x^8*(a-b*x^4)^(1/4)),x,2,-1/7*(a-b*x^4)^(3/4)/(a*x^7)-4/21*b*(a-b*x^4)^(3/4)/(a^2*x^3)],
[1/(x^12*(a-b*x^4)^(1/4)),x,3,-1/11*(a-b*x^4)^(3/4)/(a*x^11)-8/77*b*(a-b*x^4)^(3/4)/(a^2*x^7)-32/231*b^2*(a-b*x^4)^(3/4)/(a^3*x^3)],
[1/(x^16*(a-b*x^4)^(1/4)),x,4,-1/15*(a-b*x^4)^(3/4)/(a*x^15)-4/55*b*(a-b*x^4)^(3/4)/(a^2*x^11)-32/385*b^2*(a-b*x^4)^(3/4)/(a^3*x^7)-128/1155*b^3*(a-b*x^4)^(3/4)/(a^4*x^3)],
[1/(x^20*(a-b*x^4)^(1/4)),x,5,-1/19*(a-b*x^4)^(3/4)/(a*x^19)-16/285*b*(a-b*x^4)^(3/4)/(a^2*x^15)-64/1045*b^2*(a-b*x^4)^(3/4)/(a^3*x^11)-512/7315*b^3*(a-b*x^4)^(3/4)/(a^4*x^7)-2048/21945*b^4*(a-b*x^4)^(3/4)/(a^5*x^3)],
[x^10/(a-b*x^4)^(1/4),x,7,-7/40*a^2*(a-b*x^4)^(3/4)/(b^3*x)-7/60*a*x^3*(a-b*x^4)^(3/4)/b^2-1/10*x^7*(a-b*x^4)^(3/4)/b+7/40*a^(5/2)*(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a-b*x^4)^(1/4))],
[x^6/(a-b*x^4)^(1/4),x,6,-1/4*a*(a-b*x^4)^(3/4)/(b^2*x)-1/6*x^3*(a-b*x^4)^(3/4)/b+1/4*a^(3/2)*(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a-b*x^4)^(1/4))],
[x^2/(a-b*x^4)^(1/4),x,5,-1/2*(a-b*x^4)^(3/4)/(b*x)+1/2*(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a-b*x^4)^(1/4)*sqrt(b))],
[1/(x^2*(a-b*x^4)^(1/4)),x,4,-(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a-b*x^4)^(1/4)*sqrt(a))],
[1/(x^6*(a-b*x^4)^(1/4)),x,5,-1/5*(a-b*x^4)^(3/4)/(a*x^5)-2/5*b^(3/2)*(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a-b*x^4)^(1/4))],
[1/(x^10*(a-b*x^4)^(1/4)),x,6,-1/9*(a-b*x^4)^(3/4)/(a*x^9)-2/15*b*(a-b*x^4)^(3/4)/(a^2*x^5)-4/15*b^(5/2)*(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a-b*x^4)^(1/4))],
[1/(x^14*(a-b*x^4)^(1/4)),x,7,-1/13*(a-b*x^4)^(3/4)/(a*x^13)-10/117*b*(a-b*x^4)^(3/4)/(a^2*x^9)-4/39*b^2*(a-b*x^4)^(3/4)/(a^3*x^5)-8/39*b^(7/2)*(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a-b*x^4)^(1/4))],
[x^19/(a-b*x^4)^(3/4),x,3,-a^4*(a-b*x^4)^(1/4)/b^5+4/5*a^3*(a-b*x^4)^(5/4)/b^5-2/3*a^2*(a-b*x^4)^(9/4)/b^5+4/13*a*(a-b*x^4)^(13/4)/b^5-1/17*(a-b*x^4)^(17/4)/b^5],
[x^15/(a-b*x^4)^(3/4),x,3,-a^3*(a-b*x^4)^(1/4)/b^4+3/5*a^2*(a-b*x^4)^(5/4)/b^4-1/3*a*(a-b*x^4)^(9/4)/b^4+1/13*(a-b*x^4)^(13/4)/b^4],
[x^11/(a-b*x^4)^(3/4),x,3,-a^2*(a-b*x^4)^(1/4)/b^3+2/5*a*(a-b*x^4)^(5/4)/b^3-1/9*(a-b*x^4)^(9/4)/b^3],
[x^7/(a-b*x^4)^(3/4),x,3,-a*(a-b*x^4)^(1/4)/b^2+1/5*(a-b*x^4)^(5/4)/b^2],
[x^3/(a-b*x^4)^(3/4),x,1,-(a-b*x^4)^(1/4)/b],
[1/(x*(a-b*x^4)^(3/4)),x,5,-1/2*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(3/4)-1/2*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(3/4)],
[1/(x^5*(a-b*x^4)^(3/4)),x,6,-1/4*(a-b*x^4)^(1/4)/(a*x^4)-3/8*b*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(7/4)-3/8*b*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(7/4)],
[1/(x^9*(a-b*x^4)^(3/4)),x,7,-1/8*(a-b*x^4)^(1/4)/(a*x^8)-7/32*b*(a-b*x^4)^(1/4)/(a^2*x^4)-21/64*b^2*arctan((a-b*x^4)^(1/4)/a^(1/4))/a^(11/4)-21/64*b^2*arctanh((a-b*x^4)^(1/4)/a^(1/4))/a^(11/4)],
[x^13/(a-b*x^4)^(3/4),x,6,-20/77*a^2*x^2*(a-b*x^4)^(1/4)/b^3-10/77*a*x^6*(a-b*x^4)^(1/4)/b^2-1/11*x^10*(a-b*x^4)^(1/4)/b+40/77*a^(7/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(7/2)*(a-b*x^4)^(3/4))],
[x^9/(a-b*x^4)^(3/4),x,5,-2/7*a*x^2*(a-b*x^4)^(1/4)/b^2-1/7*x^6*(a-b*x^4)^(1/4)/b+4/7*a^(5/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a-b*x^4)^(3/4))],
[x^5/(a-b*x^4)^(3/4),x,4,-1/3*x^2*(a-b*x^4)^(1/4)/b+2/3*a^(3/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a-b*x^4)^(3/4))],
[x/(a-b*x^4)^(3/4),x,3,(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a-b*x^4)^(3/4)*sqrt(b))],
[1/(x^3*(a-b*x^4)^(3/4)),x,4,-1/2*(a-b*x^4)^(1/4)/(a*x^2)+1/2*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a-b*x^4)^(3/4)*sqrt(a))],
[1/(x^7*(a-b*x^4)^(3/4)),x,5,-1/6*(a-b*x^4)^(1/4)/(a*x^6)-5/12*b*(a-b*x^4)^(1/4)/(a^2*x^2)+5/12*b^(3/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a-b*x^4)^(3/4))],
[1/(x^11*(a-b*x^4)^(3/4)),x,6,-1/10*(a-b*x^4)^(1/4)/(a*x^10)-3/20*b*(a-b*x^4)^(1/4)/(a^2*x^6)-3/8*b^2*(a-b*x^4)^(1/4)/(a^3*x^2)+3/8*b^(5/2)*(1-b*x^4/a)^(3/4)*sqrt(cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arcsin(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arcsin(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a-b*x^4)^(3/4))],
[x^10/(a-b*x^4)^(3/4),x,12,-7/32*a*x^3*(a-b*x^4)^(1/4)/b^2-1/8*x^7*(a-b*x^4)^(1/4)/b-21/64*a^2*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(11/4)*sqrt(2))+21/64*a^2*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(11/4)*sqrt(2))+21/128*a^2*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(11/4)*sqrt(2))-21/128*a^2*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(11/4)*sqrt(2))],
[x^6/(a-b*x^4)^(3/4),x,11,-1/4*x^3*(a-b*x^4)^(1/4)/b-3/8*a*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(7/4)*sqrt(2))+3/8*a*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(7/4)*sqrt(2))+3/16*a*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(7/4)*sqrt(2))-3/16*a*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(7/4)*sqrt(2))],
[x^2/(a-b*x^4)^(3/4),x,10,-1/2*arctan(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(3/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4))/(b^(3/4)*sqrt(2))+1/4*log(1-b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(3/4)*sqrt(2))-1/4*log(1+b^(1/4)*x*sqrt(2)/(a-b*x^4)^(1/4)+x^2*sqrt(b)/sqrt(a-b*x^4))/(b^(3/4)*sqrt(2))],
[1/(x^2*(a-b*x^4)^(3/4)),x,1,-(a-b*x^4)^(1/4)/(a*x)],
[1/(x^6*(a-b*x^4)^(3/4)),x,2,-1/5*(a-b*x^4)^(1/4)/(a*x^5)-4/5*b*(a-b*x^4)^(1/4)/(a^2*x)],
[1/(x^10*(a-b*x^4)^(3/4)),x,3,-1/9*(a-b*x^4)^(1/4)/(a*x^9)-8/45*b*(a-b*x^4)^(1/4)/(a^2*x^5)-32/45*b^2*(a-b*x^4)^(1/4)/(a^3*x)],
[1/(x^14*(a-b*x^4)^(3/4)),x,4,-1/13*(a-b*x^4)^(1/4)/(a*x^13)-4/39*b*(a-b*x^4)^(1/4)/(a^2*x^9)-32/195*b^2*(a-b*x^4)^(1/4)/(a^3*x^5)-128/195*b^3*(a-b*x^4)^(1/4)/(a^4*x)],
[x^12/(a-b*x^4)^(3/4),x,7,-3/8*a^2*x*(a-b*x^4)^(1/4)/b^3-3/20*a*x^5*(a-b*x^4)^(1/4)/b^2-1/10*x^9*(a-b*x^4)^(1/4)/b-3/8*a^(5/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a-b*x^4)^(3/4))],
[x^8/(a-b*x^4)^(3/4),x,6,-5/12*a*x*(a-b*x^4)^(1/4)/b^2-1/6*x^5*(a-b*x^4)^(1/4)/b-5/12*a^(3/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a-b*x^4)^(3/4))],
[x^4/(a-b*x^4)^(3/4),x,5,-1/2*x*(a-b*x^4)^(1/4)/b-1/2*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/((a-b*x^4)^(3/4)*sqrt(b))],
[1/(a-b*x^4)^(3/4),x,4,-(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((a-b*x^4)^(3/4)*sqrt(a))],
[1/(x^4*(a-b*x^4)^(3/4)),x,5,-1/3*(a-b*x^4)^(1/4)/(a*x^3)-2/3*b^(3/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*(a-b*x^4)^(3/4))],
[1/(x^8*(a-b*x^4)^(3/4)),x,6,-1/7*(a-b*x^4)^(1/4)/(a*x^7)-2/7*b*(a-b*x^4)^(1/4)/(a^2*x^3)-4/7*b^(5/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*(a-b*x^4)^(3/4))],
[1/(x^12*(a-b*x^4)^(3/4)),x,7,-1/11*(a-b*x^4)^(1/4)/(a*x^11)-10/77*b*(a-b*x^4)^(1/4)/(a^2*x^7)-20/77*b^2*(a-b*x^4)^(1/4)/(a^3*x^3)-40/77*b^(7/2)*(1-a/(b*x^4))^(3/4)*x^3*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*(a-b*x^4)^(3/4))],
[x^2/(a-b*x^4)^(5/4),x,5,1/(b*x*(a-b*x^4)^(1/4))-(1-a/(b*x^4))^(1/4)*x*sqrt(cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccsc(x^2*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccsc(x^2*sqrt(b)/sqrt(a))),sqrt(2))/((a-b*x^4)^(1/4)*sqrt(a)*sqrt(b))],

# Integrands of the form x^m (a+b x^4)^p with p symbolic
[x^7*(a+b*x^4)^p,x,3,-1/4*a*(a+b*x^4)^(1+p)/(b^2*(1+p))+1/4*(a+b*x^4)^(2+p)/(b^2*(2+p))],
[x^3*(a+b*x^4)^p,x,1,1/4*(a+b*x^4)^(1+p)/(b*(1+p))],
[(a+b*x^4)^p/x,x,2,-1/4*(a+b*x^4)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^4/a)/(a*(1+p))],

# Integrands of the form (c x)^m (a+b x^5)^p

# Integrands of the form x^m (a+b x^5)^p

# p>0

# p<0
[x^24/(a+b*x^5),x,3,-1/5*a^3*x^5/b^4+1/10*a^2*x^10/b^3-1/15*a*x^15/b^2+1/20*x^20/b+1/5*a^4*log(a+b*x^5)/b^5],
[x^19/(a+b*x^5),x,3,1/5*a^2*x^5/b^3-1/10*a*x^10/b^2+1/15*x^15/b-1/5*a^3*log(a+b*x^5)/b^4],
[x^14/(a+b*x^5),x,3,-1/5*a*x^5/b^2+1/10*x^10/b+1/5*a^2*log(a+b*x^5)/b^3],
[x^9/(a+b*x^5),x,3,1/5*x^5/b-1/5*a*log(a+b*x^5)/b^2],
[x^4/(a+b*x^5),x,1,1/5*log(a+b*x^5)/b],
[1/(x*(a+b*x^5)),x,4,log(x)/a-1/5*log(a+b*x^5)/a],
[1/(x^6*(a+b*x^5)),x,3,(-1/5)/(a*x^5)-b*log(x)/a^2+1/5*b*log(a+b*x^5)/a^2],
[1/(x^11*(a+b*x^5)),x,3,(-1/10)/(a*x^10)+1/5*b/(a^2*x^5)+b^2*log(x)/a^3-1/5*b^2*log(a+b*x^5)/a^3],
[1/(x^16*(a+b*x^5)),x,3,(-1/15)/(a*x^15)+1/10*b/(a^2*x^10)-1/5*b^2/(a^3*x^5)-b^3*log(x)/a^4+1/5*b^3*log(a+b*x^5)/a^4],
[1/(a+b*x^5),x,6,1/5*log(a^(1/5)+b^(1/5)*x)/(a^(4/5)*b^(1/5))-1/20*log(2*a^(2/5)+2*b^(2/5)*x^2-a^(1/5)*b^(1/5)*x*(1-sqrt(5)))*(1-sqrt(5))/(a^(4/5)*b^(1/5))-1/20*log(2*a^(2/5)+2*b^(2/5)*x^2-a^(1/5)*b^(1/5)*x*(1+sqrt(5)))*(1+sqrt(5))/(a^(4/5)*b^(1/5))-1/5*arctan((-4*b^(1/5)*x+a^(1/5)*(1+sqrt(5)))/(a^(1/5)*sqrt(2*(5-sqrt(5)))))*sqrt(1/2*(5-sqrt(5)))/(a^(4/5)*b^(1/5))-1/5*arctan((-4*b^(1/5)*x+a^(1/5)*(1-sqrt(5)))/(a^(1/5)*sqrt(2*(5+sqrt(5)))))*sqrt(1/2*(5+sqrt(5)))/(a^(4/5)*b^(1/5)),1/5*log(a^(1/5)+b^(1/5)*x)/(a^(4/5)*b^(1/5))-1/20*log(a^(2/5)+b^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(1-sqrt(5)))*(1-sqrt(5))/(a^(4/5)*b^(1/5))-1/20*log(a^(2/5)+b^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(1+sqrt(5)))*(1+sqrt(5))/(a^(4/5)*b^(1/5))-1/5*arctan(-b^(1/5)*x*sqrt(2/5*(5+sqrt(5)))/a^(1/5)+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5-sqrt(5)))/(a^(4/5)*b^(1/5))+1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*b^(1/5)*x*sqrt(2/(5+sqrt(5)))/a^(1/5))*sqrt(1/2*(5+sqrt(5)))/(a^(4/5)*b^(1/5))],
[x^24/(a+b*x^5)^2,x,3,3/5*a^2*x^5/b^4-1/5*a*x^10/b^3+1/15*x^15/b^2-1/5*a^4/(b^5*(a+b*x^5))-4/5*a^3*log(a+b*x^5)/b^5],
[x^19/(a+b*x^5)^2,x,3,-2/5*a*x^5/b^3+1/10*x^10/b^2+1/5*a^3/(b^4*(a+b*x^5))+3/5*a^2*log(a+b*x^5)/b^4],
[x^14/(a+b*x^5)^2,x,3,1/5*x^5/b^2-1/5*a^2/(b^3*(a+b*x^5))-2/5*a*log(a+b*x^5)/b^3],
[x^9/(a+b*x^5)^2,x,3,1/5*a/(b^2*(a+b*x^5))+1/5*log(a+b*x^5)/b^2],
[x^4/(a+b*x^5)^2,x,1,(-1/5)/(b*(a+b*x^5))],
[1/(x*(a+b*x^5)^2),x,3,1/5/(a*(a+b*x^5))+log(x)/a^2-1/5*log(a+b*x^5)/a^2],
[1/(x^6*(a+b*x^5)^2),x,3,(-1/5)/(a^2*x^5)-1/5*b/(a^2*(a+b*x^5))-2*b*log(x)/a^3+2/5*b*log(a+b*x^5)/a^3],
[1/(x^11*(a+b*x^5)^2),x,3,(-1/10)/(a^2*x^10)+2/5*b/(a^3*x^5)+1/5*b^2/(a^3*(a+b*x^5))+3*b^2*log(x)/a^4-3/5*b^2*log(a+b*x^5)/a^4],
[x^14/(2*b+b*x^5),x,3,-2/5*x^5/b+1/10*x^10/b+4/5*log(2+x^5)/b],
[x^9/(2*b+b*x^5),x,3,1/5*x^5/b-2/5*log(2+x^5)/b],
[x^4/(2*b+b*x^5),x,1,1/5*log(2+x^5)/b],
[1/(x*(2*b+b*x^5)),x,4,1/2*log(x)/b-1/10*log(2+x^5)/b],
[1/(x^6*(2*b+b*x^5)),x,3,(-1/10)/(b*x^5)-1/4*log(x)/b+1/20*log(2+x^5)/b],
[x^14/(3+b*x^5),x,3,-3/5*x^5/b^2+1/10*x^10/b+9/5*log(3+b*x^5)/b^3],
[x^9/(3+b*x^5),x,3,1/5*x^5/b-3/5*log(3+b*x^5)/b^2],
[x^4/(3+b*x^5),x,1,1/5*log(3+b*x^5)/b],
[1/(x*(3+b*x^5)),x,4,1/3*log(x)-1/15*log(3+b*x^5)],
[1/(x^6*(3+b*x^5)),x,3,(-1/15)/x^5-1/9*b*log(x)+1/45*b*log(3+b*x^5)],
[x^14/(1+x^5),x,3,-1/5*x^5+1/10*x^10+1/5*log(1+x^5)],
[x^9/(1+x^5),x,3,1/5*x^5-1/5*log(1+x^5)],
[x^4/(1+x^5),x,1,1/5*log(1+x^5)],
[1/(x*(1+x^5)),x,4,log(x)-1/5*log(1+x^5)],
[1/(x^6*(1+x^5)),x,3,(-1/5)/x^5-log(x)+1/5*log(1+x^5)],
[x^5/(1+x^5),x,7,x-1/5*log(1+x)+1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1-sqrt(5))+1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1+sqrt(5))+1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5-sqrt(5)))-1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[x^3/(1+x^5),x,6,-1/5*log(1+x)+1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1-sqrt(5))+1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1+sqrt(5))-1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5-sqrt(5)))+1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[x^2/(1+x^5),x,6,1/5*log(1+x)-1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1-sqrt(5))-1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1+sqrt(5))-1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))-1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[x/(1+x^5),x,6,-1/5*log(1+x)+1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1-sqrt(5))+1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1+sqrt(5))-1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))-1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[1/(1+x^5),x,6,1/5*log(1+x)-1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1-sqrt(5))-1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1+sqrt(5))-1/5*arctan(1/2*(1-4*x+sqrt(5))*sqrt(1/10*(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))-1/5*arctan((1-4*x-sqrt(5))/sqrt(2*(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5))),1/5*log(1+x)-1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1-sqrt(5))-1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1+sqrt(5))-1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5-sqrt(5)))+1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[1/(x^2*(1+x^5)),x,7,(-1)/x+1/5*log(1+x)-1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1-sqrt(5))-1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1+sqrt(5))+1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5-sqrt(5)))-1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[1/(x^3*(1+x^5)),x,7,(-1/2)/x^2-1/5*log(1+x)+1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1-sqrt(5))+1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1+sqrt(5))+1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))+1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],
[1/(x^4*(1+x^5)),x,7,(-1/3)/x^3+1/5*log(1+x)-1/20*log(1+x^2-1/2*x*(1+sqrt(5)))*(1-sqrt(5))-1/20*log(1+x^2-1/2*x*(1-sqrt(5)))*(1+sqrt(5))+1/5*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*x*sqrt(2/(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))+1/5*arctan(-x*sqrt(2/5*(5+sqrt(5)))+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],

# Integrands of the form x^(m/2) (a+b x^5)^p

# p>0

# p<0

# Integrands of the form x^m (a+b x^5)^(p/2)

# p>0

# p<0

# Integrands of the form x^(m/2) (a+b x^5)^(p/2)

# p>0

# p<0

#  Integrands of the form x^(n/2)/Sqrt[a+b*x^5] where n mod 10 = 0 
[x^(23/2)/sqrt(a+b*x^5),x,6,3/20*a^2*arctanh(x^(5/2)*sqrt(b)/sqrt(a+b*x^5))/b^(5/2)-3/20*a*x^(5/2)*sqrt(a+b*x^5)/b^2+1/10*x^(15/2)*sqrt(a+b*x^5)/b],
[x^(13/2)/sqrt(a+b*x^5),x,5,-1/5*a*arctanh(x^(5/2)*sqrt(b)/sqrt(a+b*x^5))/b^(3/2)+1/5*x^(5/2)*sqrt(a+b*x^5)/b],
[x^(3/2)/sqrt(a+b*x^5),x,4,2/5*arctanh(x^(5/2)*sqrt(b)/sqrt(a+b*x^5))/sqrt(b)],
[1/(x^(7/2)*sqrt(a+b*x^5)),x,1,-2/5*sqrt(a+b*x^5)/(a*x^(5/2))],
[1/(x^(17/2)*sqrt(a+b*x^5)),x,2,-2/15*sqrt(a+b*x^5)/(a*x^(15/2))+4/15*b*sqrt(a+b*x^5)/(a^2*x^(5/2))],

#  Integrands of the form x^(n/2)/Sqrt[1+x^5] where n mod 10 = 0 
[x^(23/2)/sqrt(1+x^5),x,5,3/20*arcsinh(x^(5/2))-3/20*x^(5/2)*sqrt(1+x^5)+1/10*x^(15/2)*sqrt(1+x^5)],
[x^(13/2)/sqrt(1+x^5),x,4,-1/5*arcsinh(x^(5/2))+1/5*x^(5/2)*sqrt(1+x^5)],
[x^(3/2)/sqrt(1+x^5),x,3,2/5*arcsinh(x^(5/2))],
[1/(x^(7/2)*sqrt(1+x^5)),x,1,-2/5*sqrt(1+x^5)/x^(5/2)],
[1/(x^(17/2)*sqrt(1+x^5)),x,2,-2/15*sqrt(1+x^5)/x^(15/2)+4/15*sqrt(1+x^5)/x^(5/2)],

# Integrands of the form (c x)^m (a+b x^6)^p

# Integrands of the form x^m (a+b x^6)^p

# p>0

# p<0
[x^8/(a+b*x^6),x,3,1/3*x^3/b-1/3*arctan(x^3*sqrt(b)/sqrt(a))*sqrt(a)/b^(3/2)],
[x^7/(a+b*x^6),x,8,1/2*x^2/b-1/6*a^(1/3)*log(a^(1/3)+b^(1/3)*x^2)/b^(4/3)+1/12*a^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/b^(4/3)+1/2*a^(1/3)*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(b^(4/3)*sqrt(3))],
[x^6/(a+b*x^6),x,11,x/b-1/3*a^(1/6)*arctan(b^(1/6)*x/a^(1/6))/b^(7/6)+1/6*a^(1/6)*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/b^(7/6)-1/6*a^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/b^(7/6)+1/4*a^(1/6)*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(b^(7/6)*sqrt(3))-1/4*a^(1/6)*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(b^(7/6)*sqrt(3))],
[x^5/(a+b*x^6),x,1,1/6*log(a+b*x^6)/b],
[x^4/(a+b*x^6),x,10,1/3*arctan(b^(1/6)*x/a^(1/6))/(a^(1/6)*b^(5/6))-1/6*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(1/6)*b^(5/6))+1/6*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(1/6)*b^(5/6))+1/4*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(1/6)*b^(5/6)*sqrt(3))-1/4*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(1/6)*b^(5/6)*sqrt(3))],
[x^3/(a+b*x^6),x,7,-1/6*log(a^(1/3)+b^(1/3)*x^2)/(a^(1/3)*b^(2/3))+1/12*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(1/3)*b^(2/3))-1/2*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*sqrt(3))],
[x^2/(a+b*x^6),x,2,1/3*arctan(x^3*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],
[x/(a+b*x^6),x,7,1/6*log(a^(1/3)+b^(1/3)*x^2)/(a^(2/3)*b^(1/3))-1/12*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(2/3)*b^(1/3))-1/2*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*sqrt(3))],
[1/(a+b*x^6),x,10,1/3*arctan(b^(1/6)*x/a^(1/6))/(a^(5/6)*b^(1/6))-1/6*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(5/6)*b^(1/6))+1/6*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(5/6)*b^(1/6))-1/4*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(5/6)*b^(1/6)*sqrt(3))+1/4*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(5/6)*b^(1/6)*sqrt(3))],
[1/(x*(a+b*x^6)),x,4,log(x)/a-1/6*log(a+b*x^6)/a],
[1/(x^2*(a+b*x^6)),x,11,(-1)/(a*x)-1/3*b^(1/6)*arctan(b^(1/6)*x/a^(1/6))/a^(7/6)+1/6*b^(1/6)*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/a^(7/6)-1/6*b^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/a^(7/6)-1/4*b^(1/6)*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(7/6)*sqrt(3))+1/4*b^(1/6)*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(7/6)*sqrt(3))],
[1/(x^3*(a+b*x^6)),x,8,(-1/2)/(a*x^2)+1/6*b^(1/3)*log(a^(1/3)+b^(1/3)*x^2)/a^(4/3)-1/12*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/a^(4/3)+1/2*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(4/3)*sqrt(3))],
[1/(x^4*(a+b*x^6)),x,3,(-1/3)/(a*x^3)-1/3*arctan(x^3*sqrt(b)/sqrt(a))*sqrt(b)/a^(3/2)],
[x^8/(a+b*x^6)^2,x,3,-1/6*x^3/(b*(a+b*x^6))+1/6*arctan(x^3*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))],
[x^7/(a+b*x^6)^2,x,8,-1/6*x^2/(b*(a+b*x^6))+1/18*log(a^(1/3)+b^(1/3)*x^2)/(a^(2/3)*b^(4/3))-1/36*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(2/3)*b^(4/3))-1/6*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(4/3)*sqrt(3))],
[x^6/(a+b*x^6)^2,x,11,-1/6*x/(b*(a+b*x^6))+1/18*arctan(b^(1/6)*x/a^(1/6))/(a^(5/6)*b^(7/6))-1/36*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(5/6)*b^(7/6))+1/36*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(5/6)*b^(7/6))-1/24*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(5/6)*b^(7/6)*sqrt(3))+1/24*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(5/6)*b^(7/6)*sqrt(3))],
[x^5/(a+b*x^6)^2,x,1,(-1/6)/(b*(a+b*x^6))],
[x^4/(a+b*x^6)^2,x,11,1/6*x^5/(a*(a+b*x^6))+1/18*arctan(b^(1/6)*x/a^(1/6))/(a^(7/6)*b^(5/6))-1/36*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(7/6)*b^(5/6))+1/36*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(7/6)*b^(5/6))+1/24*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(7/6)*b^(5/6)*sqrt(3))-1/24*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(7/6)*b^(5/6)*sqrt(3))],
[x^3/(a+b*x^6)^2,x,8,1/6*x^4/(a*(a+b*x^6))-1/18*log(a^(1/3)+b^(1/3)*x^2)/(a^(4/3)*b^(2/3))+1/36*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(4/3)*b^(2/3))-1/6*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(2/3)*sqrt(3))],
[x^2/(a+b*x^6)^2,x,3,1/6*x^3/(a*(a+b*x^6))+1/6*arctan(x^3*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))],
[x/(a+b*x^6)^2,x,8,1/6*x^2/(a*(a+b*x^6))+1/9*log(a^(1/3)+b^(1/3)*x^2)/(a^(5/3)*b^(1/3))-1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/(a^(5/3)*b^(1/3))-1/3*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(1/3)*sqrt(3))],
[1/(a+b*x^6)^2,x,11,1/6*x/(a*(a+b*x^6))+5/18*arctan(b^(1/6)*x/a^(1/6))/(a^(11/6)*b^(1/6))-5/36*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(11/6)*b^(1/6))+5/36*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/(a^(11/6)*b^(1/6))-5/24*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(11/6)*b^(1/6)*sqrt(3))+5/24*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(11/6)*b^(1/6)*sqrt(3))],
[1/(x*(a+b*x^6)^2),x,3,1/6/(a*(a+b*x^6))+log(x)/a^2-1/6*log(a+b*x^6)/a^2],
[1/(x^2*(a+b*x^6)^2),x,12,(-7/6)/(a^2*x)+1/6/(a*x*(a+b*x^6))-7/18*b^(1/6)*arctan(b^(1/6)*x/a^(1/6))/a^(13/6)+7/36*b^(1/6)*arctan((-2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/a^(13/6)-7/36*b^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*sqrt(3))/a^(1/6))/a^(13/6)-7/24*b^(1/6)*log(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(13/6)*sqrt(3))+7/24*b^(1/6)*log(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(13/6)*sqrt(3))],
[1/(x^3*(a+b*x^6)^2),x,9,(-2/3)/(a^2*x^2)+1/6/(a*x^2*(a+b*x^6))+2/9*b^(1/3)*log(a^(1/3)+b^(1/3)*x^2)/a^(7/3)-1/9*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/a^(7/3)+2/3*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*x^2)/(a^(1/3)*sqrt(3)))/(a^(7/3)*sqrt(3))],
[1/(x^4*(a+b*x^6)^2),x,4,(-1/2)/(a^2*x^3)+1/6/(a*x^3*(a+b*x^6))-1/2*arctan(x^3*sqrt(b)/sqrt(a))*sqrt(b)/a^(5/2)],
[x^8/(1-x^6),x,3,-1/3*x^3+1/3*arctanh(x^3)],
[x^7/(1-x^6),x,8,-1/2*x^2-1/6*log(1-x^2)+1/12*log(1+x^2+x^4)+1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[x^6/(1-x^6),x,11,-x+1/3*arctanh(x)+1/6*arctanh(x/(1+x^2))+1/2*arctan(x*sqrt(3)/(1-x^2))/sqrt(3),-x+1/3*arctanh(x)-1/12*log(1-x+x^2)+1/12*log(1+x+x^2)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^5/(1-x^6),x,1,-1/6*log(1-x^6)],
[x^4/(1-x^6),x,10,1/3*arctanh(x)+1/6*arctanh(x/(1+x^2))-1/2*arctan(x*sqrt(3)/(1-x^2))/sqrt(3),1/3*arctanh(x)-1/12*log(1-x+x^2)+1/12*log(1+x+x^2)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^3/(1-x^6),x,7,-1/6*log(1-x^2)+1/12*log(1+x^2+x^4)-1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[x^2/(1-x^6),x,2,1/3*arctanh(x^3)],
[x/(1-x^6),x,7,-1/6*log(1-x^2)+1/12*log(1+x^2+x^4)+1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[1/(1-x^6),x,10,1/3*arctanh(x)+1/6*arctanh(x/(1+x^2))+1/2*arctan(x*sqrt(3)/(1-x^2))/sqrt(3),1/3*arctanh(x)-1/12*log(1-x+x^2)+1/12*log(1+x+x^2)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[1/(x*(1-x^6)),x,4,log(x)-1/6*log(1-x^6)],
[1/(x^2*(1-x^6)),x,11,(-1)/x+1/3*arctanh(x)+1/6*arctanh(x/(1+x^2))-1/2*arctan(x*sqrt(3)/(1-x^2))/sqrt(3),(-1)/x+1/3*arctanh(x)-1/12*log(1-x+x^2)+1/12*log(1+x+x^2)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[1/(x^3*(1-x^6)),x,8,(-1/2)/x^2-1/6*log(1-x^2)+1/12*log(1+x^2+x^4)-1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[1/(x^4*(1-x^6)),x,3,(-1/3)/x^3+1/3*arctanh(x^3)],
[1/(x^5*(1-x^6)),x,8,(-1/4)/x^4-1/6*log(1-x^2)+1/12*log(1+x^2+x^4)+1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[1/(x^6*(1-x^6)),x,11,(-1/5)/x^5+1/3*arctanh(x)-1/12*log(1-x+x^2)+1/12*log(1+x+x^2)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[1/(x^7*(1-x^6)),x,3,(-1/6)/x^6+log(x)-1/6*log(1-x^6)],
[1/(x^8*(1-x^6)),x,12,(-1/7)/x^7+(-1)/x+1/3*arctanh(x)-1/12*log(1-x+x^2)+1/12*log(1+x+x^2)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^8/(1+x^6),x,3,1/3*x^3-1/3*arctan(x^3)],
[x^7/(1+x^6),x,8,1/2*x^2-1/6*log(1+x^2)+1/12*log(1-x^2+x^4)+1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)],
[x^6/(1+x^6),x,11,x-1/3*arctan(x)+1/6*arctan(-2*x+sqrt(3))-1/6*arctan(2*x+sqrt(3))+1/4*log(1+x^2-x*sqrt(3))/sqrt(3)-1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[x^5/(1+x^6),x,1,1/6*log(1+x^6)],
[x^4/(1+x^6),x,10,1/3*arctan(x)-1/6*arctan(-2*x+sqrt(3))+1/6*arctan(2*x+sqrt(3))+1/4*log(1+x^2-x*sqrt(3))/sqrt(3)-1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[x^3/(1+x^6),x,7,-1/6*log(1+x^2)+1/12*log(1-x^2+x^4)-1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)],
[x^2/(1+x^6),x,2,1/3*arctan(x^3)],
[x/(1+x^6),x,7,1/6*log(1+x^2)-1/12*log(1-x^2+x^4)-1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)],
[1/(1+x^6),x,10,1/3*arctan(x)-1/6*arctan(-2*x+sqrt(3))+1/6*arctan(2*x+sqrt(3))-1/4*log(1+x^2-x*sqrt(3))/sqrt(3)+1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(x*(1+x^6)),x,4,log(x)-1/6*log(1+x^6)],
[1/(x^2*(1+x^6)),x,11,(-1)/x-1/3*arctan(x)+1/6*arctan(-2*x+sqrt(3))-1/6*arctan(2*x+sqrt(3))-1/4*log(1+x^2-x*sqrt(3))/sqrt(3)+1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(x^3*(1+x^6)),x,8,(-1/2)/x^2+1/6*log(1+x^2)-1/12*log(1-x^2+x^4)+1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)],
[1/(x^4*(1+x^6)),x,3,(-1/3)/x^3-1/3*arctan(x^3)],
[1/(x^5*(1+x^6)),x,8,(-1/4)/x^4-1/6*log(1+x^2)+1/12*log(1-x^2+x^4)+1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)],
[1/(x^6*(1+x^6)),x,11,(-1/5)/x^5-1/3*arctan(x)+1/6*arctan(-2*x+sqrt(3))-1/6*arctan(2*x+sqrt(3))+1/4*log(1+x^2-x*sqrt(3))/sqrt(3)-1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(x^7*(1+x^6)),x,3,(-1/6)/x^6-log(x)+1/6*log(1+x^6)],
[1/(x^8*(1+x^6)),x,12,(-1/7)/x^7+1/x+1/3*arctan(x)-1/6*arctan(-2*x+sqrt(3))+1/6*arctan(2*x+sqrt(3))+1/4*log(1+x^2-x*sqrt(3))/sqrt(3)-1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(2-3*x^6),x,10,-1/2*arctan(-2^(5/6)*x/3^(1/3)+1/sqrt(3))/(2^(5/6)*3^(2/3))+1/2*arctan(2^(5/6)*x/3^(1/3)+1/sqrt(3))/(2^(5/6)*3^(2/3))+1/3*arctanh((3/2)^(1/6)*x)/(2^(5/6)*3^(1/6))-1/12*log(2^(1/3)-6^(1/6)*x+3^(1/3)*x^2)/(2^(5/6)*3^(1/6))+1/12*log(2^(1/3)+6^(1/6)*x+3^(1/3)*x^2)/(2^(5/6)*3^(1/6))],

# Integrands of the form x^(m/3) (a+b x^6)^p

# p>0

# p<0
[x^(1/3)/(1-x^6),x,8,1/3*arctan((x^(2/3)-cos(2/9*Pi))*csc(2/9*Pi))*cos(1/18*Pi)-1/6*log(1-x^(2/3))+1/12*log(1+x^(2/3)+x^(4/3))-1/6*cos(2/9*Pi)*log(1+x^(4/3)+2*x^(2/3)*cos(1/9*Pi))+1/6*cos(1/9*Pi)*log(1+x^(4/3)-2*x^(2/3)*sin(1/18*Pi))-1/6*log(1+x^(4/3)-2*x^(2/3)*cos(2/9*Pi))*sin(1/18*Pi)+1/3*arctan(sec(1/18*Pi)*(x^(2/3)-sin(1/18*Pi)))*sin(1/9*Pi)-1/3*arctan((x^(2/3)+cos(1/9*Pi))*csc(1/9*Pi))*sin(2/9*Pi)-1/2*arctan((1+2*x^(2/3))/sqrt(3))/sqrt(3)],

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

# p>0
[x^8*sqrt(-1+4*x^6),x,5,-1/192*arctanh(2*x^3/sqrt(-1+4*x^6))-1/96*x^3*sqrt(-1+4*x^6)+1/12*x^9*sqrt(-1+4*x^6)],
[x^5*sqrt(a^6-x^6),x,1,-1/9*(a^6-x^6)^(3/2)],
[x^2*sqrt(-2+x^6),x,4,-1/3*arctanh(x^3/sqrt(-2+x^6))+1/6*x^3*sqrt(-2+x^6)],

# p<0
[x^23/sqrt(2+x^6),x,3,4/3*(2+x^6)^(3/2)-2/5*(2+x^6)^(5/2)+1/21*(2+x^6)^(7/2)-8/3*sqrt(2+x^6)],
[x^17/sqrt(2+x^6),x,3,-4/9*(2+x^6)^(3/2)+1/15*(2+x^6)^(5/2)+4/3*sqrt(2+x^6)],
[x^11/sqrt(2+x^6),x,3,1/9*(2+x^6)^(3/2)-2/3*sqrt(2+x^6)],
[x^5/sqrt(2+x^6),x,1,1/3*sqrt(2+x^6)],
[1/(x*sqrt(2+x^6)),x,3,-1/3*arctanh(sqrt(2+x^6)/sqrt(2))/sqrt(2)],
[1/(x^7*sqrt(2+x^6)),x,4,1/12*arctanh(sqrt(2+x^6)/sqrt(2))/sqrt(2)-1/12*sqrt(2+x^6)/x^6],
[1/(x^13*sqrt(2+x^6)),x,5,-1/32*arctanh(sqrt(2+x^6)/sqrt(2))/sqrt(2)-1/24*sqrt(2+x^6)/x^12+1/32*sqrt(2+x^6)/x^6],
[x^14/sqrt(2+x^6),x,4,1/2*arcsinh(x^3/sqrt(2))-1/4*x^3*sqrt(2+x^6)+1/12*x^9*sqrt(2+x^6)],
[x^8/sqrt(2+x^6),x,3,-1/3*arcsinh(x^3/sqrt(2))+1/6*x^3*sqrt(2+x^6)],
[x^2/sqrt(2+x^6),x,2,1/3*arcsinh(x^3/sqrt(2))],
[1/(x^4*sqrt(2+x^6)),x,1,-1/6*sqrt(2+x^6)/x^3],
[1/(x^10*sqrt(2+x^6)),x,2,-1/18*sqrt(2+x^6)/x^9+1/18*sqrt(2+x^6)/x^3],
[1/(x^16*sqrt(2+x^6)),x,3,-1/30*sqrt(2+x^6)/x^15+1/45*sqrt(2+x^6)/x^9-1/45*sqrt(2+x^6)/x^3],
[x^7/sqrt(2+x^6),x,3,1/5*x^2*sqrt(2+x^6)-2/5*2^(5/6)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x/sqrt(2+x^6),x,2,(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/6)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[1/(x^5*sqrt(2+x^6)),x,3,-1/8*sqrt(2+x^6)/x^4-1/8*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/6)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^6/sqrt(2+x^6),x,2,1/4*x*sqrt(2+x^6)-1/4*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[1/sqrt(2+x^6),x,1,1/2*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[1/(x^6*sqrt(2+x^6)),x,2,-1/10*sqrt(2+x^6)/x^5-1/10*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[x^9/sqrt(2+x^6),x,5,1/7*x^4*sqrt(2+x^6)-8/7*sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))-8/7*2^(2/3)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))+4/7*2^(1/6)*3^(1/4)*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^3/sqrt(2+x^6),x,4,sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))+2^(2/3)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))-3^(1/4)*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[1/(x^3*sqrt(2+x^6)),x,5,-1/4*sqrt(2+x^6)/x^2+1/4*sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))+1/2*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))-1/4*3^(1/4)*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^10/sqrt(2+x^6),x,4,1/8*x^5*sqrt(2+x^6)-5/8*x*(1+sqrt(3))*sqrt(2+x^6)/(2^(1/3)+x^2*(1+sqrt(3)))+5/4*3^(1/4)*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))+5/8*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[x^4/sqrt(2+x^6),x,3,1/2*x*(1+sqrt(3))*sqrt(2+x^6)/(2^(1/3)+x^2*(1+sqrt(3)))-3^(1/4)*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))-1/2*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[1/(x^2*sqrt(2+x^6)),x,4,-1/2*sqrt(2+x^6)/x+1/2*x*(1+sqrt(3))*sqrt(2+x^6)/(2^(1/3)+x^2*(1+sqrt(3)))-3^(1/4)*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))-1/2*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[x^23/(2+x^6)^(3/2),x,3,-2/3*(2+x^6)^(3/2)+1/15*(2+x^6)^(5/2)+8/3/sqrt(2+x^6)+4*sqrt(2+x^6)],
[x^17/(2+x^6)^(3/2),x,3,1/9*(2+x^6)^(3/2)+(-4/3)/sqrt(2+x^6)-4/3*sqrt(2+x^6)],
[x^11/(2+x^6)^(3/2),x,3,2/3/sqrt(2+x^6)+1/3*sqrt(2+x^6)],
[x^5/(2+x^6)^(3/2),x,1,(-1/3)/sqrt(2+x^6)],
[1/(x*(2+x^6)^(3/2)),x,4,-1/6*arctanh(sqrt(2+x^6)/sqrt(2))/sqrt(2)+1/6/sqrt(2+x^6)],
[1/(x^7*(2+x^6)^(3/2)),x,5,1/8*arctanh(sqrt(2+x^6)/sqrt(2))/sqrt(2)+(-1/8)/sqrt(2+x^6)+(-1/12)/(x^6*sqrt(2+x^6))],
[1/(x^13*(2+x^6)^(3/2)),x,6,-5/64*arctanh(sqrt(2+x^6)/sqrt(2))/sqrt(2)+5/64/sqrt(2+x^6)+(-1/24)/(x^12*sqrt(2+x^6))+5/96/(x^6*sqrt(2+x^6))],
[x^14/(2+x^6)^(3/2),x,4,-arcsinh(x^3/sqrt(2))-1/3*x^9/sqrt(2+x^6)+1/2*x^3*sqrt(2+x^6)],
[x^8/(2+x^6)^(3/2),x,3,1/3*arcsinh(x^3/sqrt(2))-1/3*x^3/sqrt(2+x^6)],
[x^2/(2+x^6)^(3/2),x,1,1/6*x^3/sqrt(2+x^6)],
[1/(x^4*(2+x^6)^(3/2)),x,2,(-1/6)/(x^3*sqrt(2+x^6))-1/6*x^3/sqrt(2+x^6)],
[1/(x^10*(2+x^6)^(3/2)),x,3,(-1/18)/(x^9*sqrt(2+x^6))+1/9/(x^3*sqrt(2+x^6))+1/9*x^3/sqrt(2+x^6)],
[x^13/(2+x^6)^(3/2),x,4,-1/3*x^8/sqrt(2+x^6)+8/15*x^2*sqrt(2+x^6)-16/15*2^(5/6)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^7/(2+x^6)^(3/2),x,3,-1/3*x^2/sqrt(2+x^6)+1/3*2^(5/6)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x/(2+x^6)^(3/2),x,3,1/6*x^2/sqrt(2+x^6)+1/6*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/6)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[1/(x^5*(2+x^6)^(3/2)),x,4,1/6/(x^4*sqrt(2+x^6))-7/48*sqrt(2+x^6)/x^4-7/48*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/6)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^12/(2+x^6)^(3/2),x,3,-1/3*x^7/sqrt(2+x^6)+7/12*x*sqrt(2+x^6)-7/12*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[x^6/(2+x^6)^(3/2),x,2,-1/3*x/sqrt(2+x^6)+1/6*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[1/(2+x^6)^(3/2),x,2,1/6*x/sqrt(2+x^6)+1/6*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[1/(x^6*(2+x^6)^(3/2)),x,3,1/6/(x^5*sqrt(2+x^6))-2/15*sqrt(2+x^6)/x^5-1/15*2^(2/3)*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[x^15/(2+x^6)^(3/2),x,6,-1/3*x^10/sqrt(2+x^6)+10/21*x^4*sqrt(2+x^6)-80/21*sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))-80/21*2^(2/3)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))+40/7*2^(1/6)*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^9/(2+x^6)^(3/2),x,5,-1/3*x^4/sqrt(2+x^6)+4/3*sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))+4/3*2^(2/3)*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))-2*2^(1/6)*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^3/(2+x^6)^(3/2),x,5,1/6*x^4/sqrt(2+x^6)-1/6*sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))-1/3*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))+1/2*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*3^(3/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[1/(x^3*(2+x^6)^(3/2)),x,6,1/6/(x^2*sqrt(2+x^6))-5/24*sqrt(2+x^6)/x^2+5/24*sqrt(2+x^6)/(x^2+2^(1/3)*(1+sqrt(3)))+5/12*(2^(1/3)+x^2)*EllipticF((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(1/3)*3^(1/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))-5/8*(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-sqrt(3)))/(x^2+2^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+sqrt(3)))^2)/(2^(5/6)*3^(3/4)*sqrt(2+x^6)*sqrt((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+sqrt(3)))^2))],
[x^10/(2+x^6)^(3/2),x,4,-1/3*x^5/sqrt(2+x^6)+5/6*x*(1+sqrt(3))*sqrt(2+x^6)/(2^(1/3)+x^2*(1+sqrt(3)))-5*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(3/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))-5/6*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[x^4/(2+x^6)^(3/2),x,4,1/6*x^5/sqrt(2+x^6)-1/6*x*(1+sqrt(3))*sqrt(2+x^6)/(2^(1/3)+x^2*(1+sqrt(3)))+x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(3/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))+1/6*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],
[1/(x^2*(2+x^6)^(3/2)),x,5,1/6/(x*sqrt(2+x^6))-1/3*sqrt(2+x^6)/x+1/3*x*(1+sqrt(3))*sqrt(2+x^6)/(2^(1/3)+x^2*(1+sqrt(3)))-2^(1/3)*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(3^(3/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))-1/3*x*(2^(1/3)+x^2)*sqrt(cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)+x^2*(1-sqrt(3)))/(2^(1/3)+x^2*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+sqrt(3)))^2)/(2^(2/3)*3^(1/4)*sqrt(2+x^6)*sqrt(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+sqrt(3)))^2))],

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

# Integrands of the form (c x)^m (a+b x^7)^p

# Integrands of the form x^m (a+b x^7)^p

# p>0
[x^m*(a+b*x^7),x,2,a*x^(1+m)/(1+m)+b*x^(8+m)/(8+m)],
[x^8*(a+b*x^7),x,2,1/9*a*x^9+1/16*b*x^16],
[(a+b*x^7)/x^8,x,2,-1/7*a/x^7+b*log(x)],
[x^m*(a+b*x^7)^2,x,2,a^2*x^(1+m)/(1+m)+2*a*b*x^(8+m)/(8+m)+b^2*x^(15+m)/(15+m)],
[x^8*(a+b*x^7)^2,x,2,1/9*a^2*x^9+1/8*a*b*x^16+1/23*b^2*x^23],
[(a+b*x^7)^2/x^8,x,3,-1/7*a^2/x^7+1/7*b^2*x^7+2*a*b*log(x)],

# p<0
[x^m/(a+b*x^7),x,1,x^(1+m)*hypergeom([1,1/7*(1+m)],[1/7*(8+m)],-b*x^7/a)/(a*(1+m))],
[x^6/(a+b*x^7),x,1,1/7*log(a+b*x^7)/b],
[1/(a+b*x^7),x,6,2/7*arctan(b^(1/7)*x*sec(1/14*Pi)/a^(1/7)-tan(1/14*Pi))*cos(1/14*Pi)/(a^(6/7)*b^(1/7))+2/7*arctan(b^(1/7)*x*sec(3/14*Pi)/a^(1/7)+tan(3/14*Pi))*cos(3/14*Pi)/(a^(6/7)*b^(1/7))+1/7*log(a^(1/7)+b^(1/7)*x)/(a^(6/7)*b^(1/7))-1/7*cos(1/7*Pi)*log(a^(2/7)+b^(2/7)*x^2-2*a^(1/7)*b^(1/7)*x*cos(1/7*Pi))/(a^(6/7)*b^(1/7))-1/7*log(a^(2/7)+b^(2/7)*x^2-2*a^(1/7)*b^(1/7)*x*sin(1/14*Pi))*sin(1/14*Pi)/(a^(6/7)*b^(1/7))-2/7*arctan(cot(1/7*Pi)-b^(1/7)*x*csc(1/7*Pi)/a^(1/7))*sin(1/7*Pi)/(a^(6/7)*b^(1/7))+1/7*log(a^(2/7)+b^(2/7)*x^2+2*a^(1/7)*b^(1/7)*x*sin(3/14*Pi))*sin(3/14*Pi)/(a^(6/7)*b^(1/7))],
[1/(x*(a+b*x^7)),x,4,log(x)/a-1/7*log(a+b*x^7)/a],
[x^m/(a-b*x^7),x,1,x^(1+m)*hypergeom([1,1/7*(1+m)],[1/7*(8+m)],b*x^7/a)/(a*(1+m))],
[x^6/(a-b*x^7),x,1,-1/7*log(a-b*x^7)/b],
[1/(a-b*x^7),x,6,2/7*arctan(b^(1/7)*x*sec(1/14*Pi)/a^(1/7)+tan(1/14*Pi))*cos(1/14*Pi)/(a^(6/7)*b^(1/7))+2/7*arctan(b^(1/7)*x*sec(3/14*Pi)/a^(1/7)-tan(3/14*Pi))*cos(3/14*Pi)/(a^(6/7)*b^(1/7))-1/7*log(a^(1/7)-b^(1/7)*x)/(a^(6/7)*b^(1/7))+1/7*cos(1/7*Pi)*log(a^(2/7)+b^(2/7)*x^2+2*a^(1/7)*b^(1/7)*x*cos(1/7*Pi))/(a^(6/7)*b^(1/7))+1/7*log(a^(2/7)+b^(2/7)*x^2+2*a^(1/7)*b^(1/7)*x*sin(1/14*Pi))*sin(1/14*Pi)/(a^(6/7)*b^(1/7))+2/7*arctan(cot(1/7*Pi)+b^(1/7)*x*csc(1/7*Pi)/a^(1/7))*sin(1/7*Pi)/(a^(6/7)*b^(1/7))-1/7*log(a^(2/7)+b^(2/7)*x^2-2*a^(1/7)*b^(1/7)*x*sin(3/14*Pi))*sin(3/14*Pi)/(a^(6/7)*b^(1/7))],
[1/(x*(a-b*x^7)),x,4,log(x)/a-1/7*log(a-b*x^7)/a],
[1/(1-x^7),x,6,2/7*arctan(sec(1/14*Pi)*(x+sin(1/14*Pi)))*cos(1/14*Pi)+2/7*arctan(sec(3/14*Pi)*(x-sin(3/14*Pi)))*cos(3/14*Pi)-1/7*log(1-x)+1/7*cos(1/7*Pi)*log(1+x^2+2*x*cos(1/7*Pi))+1/7*log(1+x^2+2*x*sin(1/14*Pi))*sin(1/14*Pi)+2/7*arctan((x+cos(1/7*Pi))*csc(1/7*Pi))*sin(1/7*Pi)-1/7*log(1+x^2-2*x*sin(3/14*Pi))*sin(3/14*Pi)],
[1/(1+x^7),x,6,2/7*arctan(x*sec(1/14*Pi)-tan(1/14*Pi))*cos(1/14*Pi)+2/7*arctan(x*sec(3/14*Pi)+tan(3/14*Pi))*cos(3/14*Pi)+1/7*log(1+x)-1/7*cos(1/7*Pi)*log(1+x^2-2*x*cos(1/7*Pi))-1/7*log(1+x^2-2*x*sin(1/14*Pi))*sin(1/14*Pi)-2/7*arctan(cot(1/7*Pi)-x*csc(1/7*Pi))*sin(1/7*Pi)+1/7*log(1+x^2+2*x*sin(3/14*Pi))*sin(3/14*Pi)],

# Integrands of the form x^(m/2) (a+b x^7)^p

# Integrands of the form x^m (a+b x^7)^(p/2)

# Integrands of the form x^(m/2) (a+b x^7)^(p/2)

# Integrands of the form (c x)^m (a+b x^8)^p

# Integrands of the form x^m (a+b x^8)^p

# p>0

# p<0
[x^9/(a+b*x^8),x,11,1/2*x^2/b+1/4*a^(1/4)*arctan(1-b^(1/4)*x^2*sqrt(2)/a^(1/4))/(b^(5/4)*sqrt(2))-1/4*a^(1/4)*arctan(1+b^(1/4)*x^2*sqrt(2)/a^(1/4))/(b^(5/4)*sqrt(2))+1/8*a^(1/4)*log(-a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(b^(5/4)*sqrt(2))-1/8*a^(1/4)*log(a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(b^(5/4)*sqrt(2))],
[x^7/(a+b*x^8),x,1,1/8*log(a+b*x^8)/b],
[x^5/(a+b*x^8),x,10,-1/4*arctan(1-b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(1/4)*b^(3/4)*sqrt(2))+1/4*arctan(1+b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(1/4)*b^(3/4)*sqrt(2))+1/8*log(-a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(1/4)*b^(3/4)*sqrt(2))-1/8*log(a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(1/4)*b^(3/4)*sqrt(2))],
[x^3/(a+b*x^8),x,2,1/4*arctan(x^4*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],
[x/(a+b*x^8),x,10,-1/4*arctan(1-b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*sqrt(2))+1/4*arctan(1+b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*sqrt(2))-1/8*log(-a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(3/4)*b^(1/4)*sqrt(2))+1/8*log(a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(3/4)*b^(1/4)*sqrt(2))],
[1/(x*(a+b*x^8)),x,4,log(x)/a-1/8*log(a+b*x^8)/a],
[1/(x^3*(a+b*x^8)),x,11,(-1/2)/(a*x^2)+1/4*b^(1/4)*arctan(1-b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(5/4)*sqrt(2))-1/4*b^(1/4)*arctan(1+b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(5/4)*sqrt(2))-1/8*b^(1/4)*log(-a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(5/4)*sqrt(2))+1/8*b^(1/4)*log(a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(5/4)*sqrt(2))],
[1/(x^5*(a+b*x^8)),x,3,(-1/4)/(a*x^4)-1/4*arctan(x^4*sqrt(b)/sqrt(a))*sqrt(b)/a^(3/2)],
[1/(x^7*(a+b*x^8)),x,11,(-1/6)/(a*x^6)+1/4*b^(3/4)*arctan(1-b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(7/4)*sqrt(2))-1/4*b^(3/4)*arctan(1+b^(1/4)*x^2*sqrt(2)/a^(1/4))/(a^(7/4)*sqrt(2))+1/8*b^(3/4)*log(-a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(7/4)*sqrt(2))-1/8*b^(3/4)*log(a^(1/4)*b^(1/4)*x^2*sqrt(2)+sqrt(a)+x^4*sqrt(b))/(a^(7/4)*sqrt(2))],
[1/(x^9*(a+b*x^8)),x,3,(-1/8)/(a*x^8)-b*log(x)/a^2+1/8*b*log(a+b*x^8)/a^2],
[x^8/(a+b*x^8),x,14,x/b-1/4*(-a)^(1/8)*arctan(b^(1/8)*x/(-a)^(1/8))/b^(9/8)-1/4*(-a)^(1/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/b^(9/8)+1/4*(-a)^(1/8)*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/(b^(9/8)*sqrt(2))-1/4*(-a)^(1/8)*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/(b^(9/8)*sqrt(2))+1/8*(-a)^(1/8)*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/(b^(9/8)*sqrt(2))-1/8*(-a)^(1/8)*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/(b^(9/8)*sqrt(2))],
[x^6/(a+b*x^8),x,13,1/4*arctan(b^(1/8)*x/(-a)^(1/8))/((-a)^(1/8)*b^(7/8))-1/4*arctanh(b^(1/8)*x/(-a)^(1/8))/((-a)^(1/8)*b^(7/8))-1/4*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(1/8)*b^(7/8)*sqrt(2))+1/4*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(1/8)*b^(7/8)*sqrt(2))+1/8*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(1/8)*b^(7/8)*sqrt(2))-1/8*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(1/8)*b^(7/8)*sqrt(2))],
[x^4/(a+b*x^8),x,13,-1/4*arctan(b^(1/8)*x/(-a)^(1/8))/((-a)^(3/8)*b^(5/8))-1/4*arctanh(b^(1/8)*x/(-a)^(1/8))/((-a)^(3/8)*b^(5/8))-1/4*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(3/8)*b^(5/8)*sqrt(2))+1/4*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(3/8)*b^(5/8)*sqrt(2))-1/8*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(3/8)*b^(5/8)*sqrt(2))+1/8*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(3/8)*b^(5/8)*sqrt(2))],
[x^2/(a+b*x^8),x,13,1/4*arctan(b^(1/8)*x/(-a)^(1/8))/((-a)^(5/8)*b^(3/8))-1/4*arctanh(b^(1/8)*x/(-a)^(1/8))/((-a)^(5/8)*b^(3/8))+1/4*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(5/8)*b^(3/8)*sqrt(2))-1/4*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(5/8)*b^(3/8)*sqrt(2))-1/8*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(5/8)*b^(3/8)*sqrt(2))+1/8*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(5/8)*b^(3/8)*sqrt(2))],
[1/(a+b*x^8),x,13,-1/4*arctan(b^(1/8)*x/(-a)^(1/8))/((-a)^(7/8)*b^(1/8))-1/4*arctanh(b^(1/8)*x/(-a)^(1/8))/((-a)^(7/8)*b^(1/8))+1/4*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(7/8)*b^(1/8)*sqrt(2))-1/4*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(7/8)*b^(1/8)*sqrt(2))+1/8*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(7/8)*b^(1/8)*sqrt(2))-1/8*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(7/8)*b^(1/8)*sqrt(2))],
[1/(x^2*(a+b*x^8)),x,14,(-1)/(a*x)+1/4*b^(1/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(9/8)-1/4*b^(1/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)^(9/8)-1/4*b^(1/8)*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(9/8)*sqrt(2))+1/4*b^(1/8)*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(9/8)*sqrt(2))+1/8*b^(1/8)*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(9/8)*sqrt(2))-1/8*b^(1/8)*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(9/8)*sqrt(2))],
[1/(x^4*(a+b*x^8)),x,14,(-1/3)/(a*x^3)-1/4*b^(3/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(11/8)-1/4*b^(3/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)^(11/8)-1/4*b^(3/8)*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(11/8)*sqrt(2))+1/4*b^(3/8)*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(11/8)*sqrt(2))-1/8*b^(3/8)*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(11/8)*sqrt(2))+1/8*b^(3/8)*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(11/8)*sqrt(2))],
[1/(x^6*(a+b*x^8)),x,14,(-1/5)/(a*x^5)+1/4*b^(5/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(13/8)-1/4*b^(5/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)^(13/8)+1/4*b^(5/8)*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(13/8)*sqrt(2))-1/4*b^(5/8)*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(13/8)*sqrt(2))-1/8*b^(5/8)*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(13/8)*sqrt(2))+1/8*b^(5/8)*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(13/8)*sqrt(2))],
[1/(x^8*(a+b*x^8)),x,14,(-1/7)/(a*x^7)-1/4*b^(7/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(15/8)-1/4*b^(7/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)^(15/8)+1/4*b^(7/8)*arctan(1-b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(15/8)*sqrt(2))-1/4*b^(7/8)*arctan(1+b^(1/8)*x*sqrt(2)/(-a)^(1/8))/((-a)^(15/8)*sqrt(2))+1/8*b^(7/8)*log((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(15/8)*sqrt(2))-1/8*b^(7/8)*log((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(15/8)*sqrt(2))],
[1/(a-b*x^8),x,13,1/4*arctan(b^(1/8)*x/a^(1/8))/(a^(7/8)*b^(1/8))+1/4*arctanh(b^(1/8)*x/a^(1/8))/(a^(7/8)*b^(1/8))-1/4*arctan(1-b^(1/8)*x*sqrt(2)/a^(1/8))/(a^(7/8)*b^(1/8)*sqrt(2))+1/4*arctan(1+b^(1/8)*x*sqrt(2)/a^(1/8))/(a^(7/8)*b^(1/8)*sqrt(2))-1/8*log(a^(1/4)+b^(1/4)*x^2-a^(1/8)*b^(1/8)*x*sqrt(2))/(a^(7/8)*b^(1/8)*sqrt(2))+1/8*log(a^(1/4)+b^(1/4)*x^2+a^(1/8)*b^(1/8)*x*sqrt(2))/(a^(7/8)*b^(1/8)*sqrt(2))],
[x^9/(1-x^8),x,5,-1/2*x^2+1/4*arctan(x^2)+1/4*arctanh(x^2)],
[x^7/(1-x^8),x,1,-1/8*log(1-x^8)],
[x^5/(1-x^8),x,4,-1/4*arctan(x^2)+1/4*arctanh(x^2)],
[x^3/(1-x^8),x,2,1/4*arctanh(x^4)],
[x/(1-x^8),x,4,1/4*arctan(x^2)+1/4*arctanh(x^2)],
[1/(x*(1-x^8)),x,4,log(x)-1/8*log(1-x^8)],
[1/(x^3*(1-x^8)),x,5,(-1/2)/x^2-1/4*arctan(x^2)+1/4*arctanh(x^2)],
[1/(x^5*(1-x^8)),x,3,(-1/4)/x^4+1/4*arctanh(x^4)],
[1/(x^7*(1-x^8)),x,5,(-1/6)/x^6+1/4*arctan(x^2)+1/4*arctanh(x^2)],
[1/(x^9*(1-x^8)),x,3,(-1/8)/x^8+log(x)-1/8*log(1-x^8)],
[x^8/(1-x^8),x,14,-x+1/4*arctan(x)+1/4*arctanh(x)-1/4*arctan(1-x*sqrt(2))/sqrt(2)+1/4*arctan(1+x*sqrt(2))/sqrt(2)-1/8*log(1+x^2-x*sqrt(2))/sqrt(2)+1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^6/(1-x^8),x,13,-1/4*arctan(x)+1/4*arctanh(x)+1/4*arctan(1-x*sqrt(2))/sqrt(2)-1/4*arctan(1+x*sqrt(2))/sqrt(2)-1/8*log(1+x^2-x*sqrt(2))/sqrt(2)+1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^4/(1-x^8),x,13,1/4*arctan(x)+1/4*arctanh(x)+1/4*arctan(1-x*sqrt(2))/sqrt(2)-1/4*arctan(1+x*sqrt(2))/sqrt(2)+1/8*log(1+x^2-x*sqrt(2))/sqrt(2)-1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^2/(1-x^8),x,13,-1/4*arctan(x)+1/4*arctanh(x)-1/4*arctan(1-x*sqrt(2))/sqrt(2)+1/4*arctan(1+x*sqrt(2))/sqrt(2)+1/8*log(1+x^2-x*sqrt(2))/sqrt(2)-1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(1-x^8),x,13,1/4*arctan(x)+1/4*arctanh(x)-1/4*arctan(1-x*sqrt(2))/sqrt(2)+1/4*arctan(1+x*sqrt(2))/sqrt(2)-1/8*log(1+x^2-x*sqrt(2))/sqrt(2)+1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^2*(1-x^8)),x,14,(-1)/x-1/4*arctan(x)+1/4*arctanh(x)+1/4*arctan(1-x*sqrt(2))/sqrt(2)-1/4*arctan(1+x*sqrt(2))/sqrt(2)-1/8*log(1+x^2-x*sqrt(2))/sqrt(2)+1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^4*(1-x^8)),x,14,(-1/3)/x^3+1/4*arctan(x)+1/4*arctanh(x)+1/4*arctan(1-x*sqrt(2))/sqrt(2)-1/4*arctan(1+x*sqrt(2))/sqrt(2)+1/8*log(1+x^2-x*sqrt(2))/sqrt(2)-1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^6*(1-x^8)),x,14,(-1/5)/x^5-1/4*arctan(x)+1/4*arctanh(x)-1/4*arctan(1-x*sqrt(2))/sqrt(2)+1/4*arctan(1+x*sqrt(2))/sqrt(2)+1/8*log(1+x^2-x*sqrt(2))/sqrt(2)-1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^8*(1-x^8)),x,14,(-1/7)/x^7+1/4*arctan(x)+1/4*arctanh(x)-1/4*arctan(1-x*sqrt(2))/sqrt(2)+1/4*arctan(1+x*sqrt(2))/sqrt(2)-1/8*log(1+x^2-x*sqrt(2))/sqrt(2)+1/8*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^9/(1+x^8),x,11,1/2*x^2+1/4*arctan(1-x^2*sqrt(2))/sqrt(2)-1/4*arctan(1+x^2*sqrt(2))/sqrt(2)+1/8*log(1+x^4-x^2*sqrt(2))/sqrt(2)-1/8*log(1+x^4+x^2*sqrt(2))/sqrt(2)],
[x^7/(1+x^8),x,1,1/8*log(1+x^8)],
[x^5/(1+x^8),x,10,-1/4*arctan(1-x^2*sqrt(2))/sqrt(2)+1/4*arctan(1+x^2*sqrt(2))/sqrt(2)+1/8*log(1+x^4-x^2*sqrt(2))/sqrt(2)-1/8*log(1+x^4+x^2*sqrt(2))/sqrt(2)],
[x^3/(1+x^8),x,2,1/4*arctan(x^4)],
[x/(1+x^8),x,10,-1/4*arctan(1-x^2*sqrt(2))/sqrt(2)+1/4*arctan(1+x^2*sqrt(2))/sqrt(2)-1/8*log(1+x^4-x^2*sqrt(2))/sqrt(2)+1/8*log(1+x^4+x^2*sqrt(2))/sqrt(2)],
[1/(x*(1+x^8)),x,4,log(x)-1/8*log(1+x^8)],
[1/(x^3*(1+x^8)),x,11,(-1/2)/x^2+1/4*arctan(1-x^2*sqrt(2))/sqrt(2)-1/4*arctan(1+x^2*sqrt(2))/sqrt(2)-1/8*log(1+x^4-x^2*sqrt(2))/sqrt(2)+1/8*log(1+x^4+x^2*sqrt(2))/sqrt(2)],
[1/(x^5*(1+x^8)),x,3,(-1/4)/x^4-1/4*arctan(x^4)],
[1/(x^7*(1+x^8)),x,11,(-1/6)/x^6+1/4*arctan(1-x^2*sqrt(2))/sqrt(2)-1/4*arctan(1+x^2*sqrt(2))/sqrt(2)+1/8*log(1+x^4-x^2*sqrt(2))/sqrt(2)-1/8*log(1+x^4+x^2*sqrt(2))/sqrt(2)],
[1/(x^9*(1+x^8)),x,3,(-1/8)/x^8-log(x)+1/8*log(1+x^8)],
[x^8/(1+x^8),x,20,x+1/16*log(1+x^2-x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-1/16*log(1+x^2+x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/16*log(1+x^2-x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-1/16*log(1+x^2+x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[x^6/(1+x^8),x,21,1/16*log(1+x^2-x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-1/16*log(1+x^2+x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/16*log(1+x^2-x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-1/16*log(1+x^2+x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))+1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[x^4/(1+x^8),x,19,-1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/8*log(1+x^2-x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/8*log(1+x^2+x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))+1/8*log(1+x^2-x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/8*log(1+x^2+x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[x^2/(1+x^8),x,19,1/8*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2-sqrt(2))-1/8*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2-sqrt(2))+1/8*log(1+x^2-x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/8*log(1+x^2+x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/8*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2+sqrt(2))+1/8*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2+sqrt(2))-1/8*log(1+x^2-x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))+1/8*log(1+x^2+x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[1/(1+x^8),x,19,-1/16*log(1+x^2-x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/16*log(1+x^2+x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/16*log(1+x^2-x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/16*log(1+x^2+x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))+1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[1/(x^2*(1+x^8)),x,22,(-1)/x-1/16*log(1+x^2-x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/16*log(1+x^2+x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/16*log(1+x^2-x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/16*log(1+x^2+x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[1/(x^4*(1+x^8)),x,20,(-1/3)/x^3+1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/8*log(1+x^2-x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/8*log(1+x^2+x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))+1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/8*log(1+x^2-x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))+1/8*log(1+x^2+x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[1/(x^6*(1+x^8)),x,20,(-1/5)/x^5-1/8*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2-sqrt(2))+1/8*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2-sqrt(2))-1/8*log(1+x^2-x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/8*log(1+x^2+x*sqrt(2-sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/8*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2+sqrt(2))-1/8*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2+sqrt(2))+1/8*log(1+x^2-x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/8*log(1+x^2+x*sqrt(2+sqrt(2)))/sqrt(2*(2+sqrt(2)))],
[1/(x^8*(1+x^8)),x,20,(-1/7)/x^7+1/16*log(1+x^2-x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-1/16*log(1+x^2+x*sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/4*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))-1/4*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2*(2-sqrt(2)))+1/16*log(1+x^2-x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-1/16*log(1+x^2+x*sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/4*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))-1/4*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2*(2+sqrt(2)))],

# Integrands of the form x^(m/2) (a+b x^8)^p

# p>0

# p<0

# Integrands of the form x^m (a+b x^8)^(p/2)

# p>0
[x^3*sqrt(1+x^8),x,3,1/8*arcsinh(x^4)+1/8*x^4*sqrt(1+x^8)],
[x*sqrt(1+x^8),x,3,1/6*x^2*sqrt(1+x^8)+1/6*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[sqrt(1+x^8)/x,x,4,-1/4*arctanh(sqrt(1+x^8))+1/4*sqrt(1+x^8)],
[sqrt(1+x^8)/x^3,x,5,-1/2*sqrt(1+x^8)/x^2+x^2*sqrt(1+x^8)/(1+x^4)-(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticE(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)+1/2*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[x^3*sqrt(-2+x^8),x,4,-1/4*arctanh(x^4/sqrt(-2+x^8))+1/8*x^4*sqrt(-2+x^8)],

# p<0
[x^19/sqrt(1+x^8),x,4,3/32*arcsinh(x^4)-3/32*x^4*sqrt(1+x^8)+1/16*x^12*sqrt(1+x^8)],
[x^15/sqrt(1+x^8),x,3,1/12*(1+x^8)^(3/2)-1/4*sqrt(1+x^8)],
[x^11/sqrt(1+x^8),x,3,-1/8*arcsinh(x^4)+1/8*x^4*sqrt(1+x^8)],
[x^7/sqrt(1+x^8),x,1,1/4*sqrt(1+x^8)],
[x^3/sqrt(1+x^8),x,2,1/4*arcsinh(x^4)],
[1/(x*sqrt(1+x^8)),x,3,-1/4*arctanh(sqrt(1+x^8))],
[1/(x^5*sqrt(1+x^8)),x,1,-1/4*sqrt(1+x^8)/x^4],
[1/(x^9*sqrt(1+x^8)),x,4,1/8*arctanh(sqrt(1+x^8))-1/8*sqrt(1+x^8)/x^8],
[1/(x^13*sqrt(1+x^8)),x,2,-1/12*sqrt(1+x^8)/x^12+1/6*sqrt(1+x^8)/x^4],
[1/(x^17*sqrt(1+x^8)),x,5,-3/32*arctanh(sqrt(1+x^8))-1/16*sqrt(1+x^8)/x^16+3/32*sqrt(1+x^8)/x^8],
[x^13/sqrt(1+x^8),x,5,1/10*x^6*sqrt(1+x^8)-3/10*x^2*sqrt(1+x^8)/(1+x^4)+3/10*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticE(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)-3/20*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[x^9/sqrt(1+x^8),x,3,1/6*x^2*sqrt(1+x^8)-1/12*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[x^5/sqrt(1+x^8),x,4,1/2*x^2*sqrt(1+x^8)/(1+x^4)-1/2*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticE(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)+1/4*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[x/sqrt(1+x^8),x,2,1/4*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[1/(x^3*sqrt(1+x^8)),x,5,-1/2*sqrt(1+x^8)/x^2+1/2*x^2*sqrt(1+x^8)/(1+x^4)-1/2*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticE(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)+1/4*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[1/(x^7*sqrt(1+x^8)),x,3,-1/6*sqrt(1+x^8)/x^6-1/12*(1+x^4)*sqrt(cos(2*arctan(x^2))^2)/cos(2*arctan(x^2))*EllipticF(sin(2*arctan(x^2)),sqrt(1/2))*sqrt((1+x^8)/(1+x^4)^2)/sqrt(1+x^8)],
[x^10/sqrt(1+x^8),x,4,1/7*x^3*sqrt(1+x^8)+3/14*x^3*EllipticF(1/2*sqrt((2*x^2+sqrt(2)+x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt(-(1-x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1-x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))+3/14*x^3*EllipticF(1/2*sqrt((2*x^2-sqrt(2)-x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt((1+x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1+x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))],
[x^8/sqrt(1+x^8),x,4,1/5*x*sqrt(1+x^8)+1/10*x^3*EllipticF(1/2*sqrt((2*x^2+sqrt(2)+x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt(-(1-x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1-x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))-1/10*x^3*EllipticF(1/2*sqrt((2*x^2-sqrt(2)-x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt((1+x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1+x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))],
[x^6/sqrt(1+x^8),x,1,1/7*x^7*hypergeom([1/2,7/8],[15/8],-x^8)],
[x^4/sqrt(1+x^8),x,1,1/5*x^5*hypergeom([1/2,5/8],[13/8],-x^8)],
[x^2/sqrt(1+x^8),x,3,-1/2*x^3*EllipticF(1/2*sqrt((2*x^2+sqrt(2)+x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt(-(1-x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1-x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))-1/2*x^3*EllipticF(1/2*sqrt((2*x^2-sqrt(2)-x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt((1+x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1+x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))],
[1/sqrt(1+x^8),x,3,-1/2*x^3*EllipticF(1/2*sqrt((2*x^2+sqrt(2)+x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt(-(1-x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1-x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))+1/2*x^3*EllipticF(1/2*sqrt((2*x^2-sqrt(2)-x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt((1+x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1+x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))],
[1/(x^2*sqrt(1+x^8)),x,2,3/7*x^7*hypergeom([1/2,7/8],[15/8],-x^8)-sqrt(1+x^8)/x],
[1/(x^4*sqrt(1+x^8)),x,2,1/15*x^5*hypergeom([1/2,5/8],[13/8],-x^8)-1/3*sqrt(1+x^8)/x^3],
[1/(x^6*sqrt(1+x^8)),x,4,-1/5*sqrt(1+x^8)/x^5+1/10*x^3*EllipticF(1/2*sqrt((2*x^2+sqrt(2)+x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt(-(1-x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1-x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))+1/10*x^3*EllipticF(1/2*sqrt((2*x^2-sqrt(2)-x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt((1+x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1+x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))],
[1/(x^8*sqrt(1+x^8)),x,4,-1/7*sqrt(1+x^8)/x^7+3/14*x^3*EllipticF(1/2*sqrt((2*x^2+sqrt(2)+x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt(-(1-x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1-x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))-3/14*x^3*EllipticF(1/2*sqrt((2*x^2-sqrt(2)-x^4*sqrt(2))/x^2),sqrt(-2*(1-sqrt(2))))*sqrt((1+x^2)^2/x^2)*sqrt((-1-x^8)/x^4)/((1+x^2)*sqrt(1+x^8)*sqrt(2+sqrt(2)))],

# Integrands of the form x^(m/2) (a+b x^8)^(p/2)

# p>0

# p<0

# Integrands of the form (c x)^m (a+b x^10)^p

# Integrands of the form x^m (a+b x^10)^p

# p>0

# p<0
[1/(1-x^10),x,10,1/5*arctanh(x)+1/20*arctanh(1/2*x*(1-sqrt(5))/(1+x^2))*(1-sqrt(5))+1/20*arctanh(1/2*x*(1+sqrt(5))/(1+x^2))*(1+sqrt(5))+1/20*arctan(1/2*x*sqrt(10-2*sqrt(5))/(1-x^2))*sqrt(10-2*sqrt(5))+1/20*arctan(1/2*x*sqrt(10+2*sqrt(5))/(1-x^2))*sqrt(10+2*sqrt(5)),1/5*arctanh(x)-1/40*log(1+x^2-1/2*x*(1-sqrt(5)))*(1-sqrt(5))+1/40*log(1+x^2+1/2*x*(1-sqrt(5)))*(1-sqrt(5))-1/40*log(1+x^2-1/2*x*(1+sqrt(5)))*(1+sqrt(5))+1/40*log(1+x^2+1/2*x*(1+sqrt(5)))*(1+sqrt(5))-1/10*arctan(1/2*(1-4*x+sqrt(5))*sqrt(1/10*(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))+1/10*arctan(1/2*(1+4*x+sqrt(5))*sqrt(1/10*(5+sqrt(5))))*sqrt(1/2*(5-sqrt(5)))-1/10*arctan((1-4*x-sqrt(5))/sqrt(2*(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5)))+1/10*arctan((1+4*x-sqrt(5))/sqrt(2*(5+sqrt(5))))*sqrt(1/2*(5+sqrt(5)))],

# Integrands of the form x^(m/2) (a+b x^10)^p

# p>0

# p<0

# Integrands of the form x^m (a+b x^10)^(p/2)

# p>0

# p<0
[x^4/sqrt(1-x^10),x,2,1/5*arcsin(x^5)],
[x^4/sqrt(-2+x^10),x,3,1/5*arctanh(x^5/sqrt(-2+x^10))],

# Integrands of the form x^(m/2) (a+b x^10)^(p/2)

# p>0

# p<0

# Integrands of the form (c x)^m (a+b x^12)^p

# Integrands of the form x^m (a+b x^12)^p

# p>0

# p<0
[x^5/(9+x^12),x,2,1/18*arctan(1/3*x^6)],
[x^5/(9-x^12),x,2,1/18*arctanh(1/3*x^6)],

# Integrands of the form x^(m/2) (a+b x^12)^p

# p>0

# p<0

# Integrands of the form x^m (a+b x^12)^(p/2)

# p>0
[x^5*sqrt(9+x^12),x,3,3/4*arcsinh(1/3*x^6)+1/12*x^6*sqrt(9+x^12)],

# p<0

# Integrands of the form x^(m/2) (a+b x^12)^(p/2)

# p>0

# p<0

# Integrands of the form (c x)^m (a+b x^n)^p when n<0 is an integer

# Integrands of the form (c x)^m (a+b/x)^p

# Integrands of the form x^m (a+b/x)^p

# p>0
[(a+b/x)*x^6,x,2,1/6*b*x^6+1/7*a*x^7],
[(a+b/x)*x^5,x,2,1/5*b*x^5+1/6*a*x^6],
[(a+b/x)*x^4,x,2,1/4*b*x^4+1/5*a*x^5],
[(a+b/x)*x^3,x,2,1/3*b*x^3+1/4*a*x^4],
[(a+b/x)*x^2,x,2,1/2*b*x^2+1/3*a*x^3],
[(a+b/x)*x,x,2,b*x+1/2*a*x^2],
[a+b/x,x,1,a*x+b*log(x)],
[(a+b/x)/x,x,2,-b/x+a*log(x)],
[(a+b/x)/x^2,x,2,-1/2*b/x^2-a/x],
[(a+b/x)/x^3,x,2,-1/3*b/x^3-1/2*a/x^2],
[(a+b/x)/x^4,x,2,-1/4*b/x^4-1/3*a/x^3],
[(a+b/x)/x^5,x,2,-1/5*b/x^5-1/4*a/x^4],
[(a+b/x)/x^6,x,2,-1/6*b/x^6-1/5*a/x^5],
[(a+b/x)^2*x^5,x,3,1/4*b^2*x^4+2/5*a*b*x^5+1/6*a^2*x^6],
[(a+b/x)^2*x^4,x,3,1/3*b^2*x^3+1/2*a*b*x^4+1/5*a^2*x^5],
[(a+b/x)^2*x^3,x,3,1/2*b^2*x^2+2/3*a*b*x^3+1/4*a^2*x^4],
[(a+b/x)^2*x^2,x,2,1/3*(b+a*x)^3/a],
[(a+b/x)^2*x,x,3,2*a*b*x+1/2*a^2*x^2+b^2*log(x)],
[(a+b/x)^2,x,3,-b^2/x+a^2*x+2*a*b*log(x)],
[(a+b/x)^2/x,x,3,-1/2*b^2/x^2-2*a*b/x+a^2*log(x)],
[(a+b/x)^2/x^2,x,1,-1/3*(a+b/x)^3/b],
[(a+b/x)^2/x^3,x,3,-1/4*b^2/x^4-2/3*a*b/x^3-1/2*a^2/x^2],
[(a+b/x)^2/x^4,x,3,-1/5*b^2/x^5-1/2*a*b/x^4-1/3*a^2/x^3],
[(a+b/x)^2/x^5,x,3,-1/6*b^2/x^6-2/5*a*b/x^5-1/4*a^2/x^4],
[(a+b/x)^3*x^6,x,3,1/4*b^3*x^4+3/5*a*b^2*x^5+1/2*a^2*b*x^6+1/7*a^3*x^7],
[(a+b/x)^3*x^5,x,3,1/3*b^3*x^3+3/4*a*b^2*x^4+3/5*a^2*b*x^5+1/6*a^3*x^6],
[(a+b/x)^3*x^4,x,3,-1/4*b*(b+a*x)^4/a^2+1/5*(b+a*x)^5/a^2],
[(a+b/x)^3*x^3,x,2,1/4*(b+a*x)^4/a],
[(a+b/x)^3*x^2,x,3,3*a*b^2*x+3/2*a^2*b*x^2+1/3*a^3*x^3+b^3*log(x)],
[(a+b/x)^3*x,x,3,-b^3/x+3*a^2*b*x+1/2*a^3*x^2+3*a*b^2*log(x)],
[(a+b/x)^3,x,3,-1/2*b^3/x^2-3*a*b^2/x+a^3*x+3*a^2*b*log(x)],
[(a+b/x)^3/x,x,3,-1/3*b^3/x^3-3/2*a*b^2/x^2-3*a^2*b/x+a^3*log(x)],
[(a+b/x)^3/x^2,x,1,-1/4*(a+b/x)^4/b],
[(a+b/x)^3/x^3,x,3,-1/5*(b+a*x)^4/(b*x^5)+1/20*a*(b+a*x)^4/(b^2*x^4)],
[(a+b/x)^3/x^4,x,3,-1/6*b^3/x^6-3/5*a*b^2/x^5-3/4*a^2*b/x^4-1/3*a^3/x^3],
[(a+b/x)^3/x^5,x,3,-1/7*b^3/x^7-1/2*a*b^2/x^6-3/5*a^2*b/x^5-1/4*a^3/x^4],
[(a+b/x)^3/x^6,x,3,-1/8*b^3/x^8-3/7*a*b^2/x^7-1/2*a^2*b/x^6-1/5*a^3/x^5],
[(a+b/x)^8*x^16,x,3,1/9*b^8*x^9+4/5*a*b^7*x^10+28/11*a^2*b^6*x^11+14/3*a^3*b^5*x^12+70/13*a^4*b^4*x^13+4*a^5*b^3*x^14+28/15*a^6*b^2*x^15+1/2*a^7*b*x^16+1/17*a^8*x^17],
[(a+b/x)^8*x^15,x,3,1/8*b^8*x^8+8/9*a*b^7*x^9+14/5*a^2*b^6*x^10+56/11*a^3*b^5*x^11+35/6*a^4*b^4*x^12+56/13*a^5*b^3*x^13+2*a^6*b^2*x^14+8/15*a^7*b*x^15+1/16*a^8*x^16],
[(a+b/x)^8*x^13,x,3,-1/9*b^5*(b+a*x)^9/a^6+1/2*b^4*(b+a*x)^10/a^6-10/11*b^3*(b+a*x)^11/a^6+5/6*b^2*(b+a*x)^12/a^6-5/13*b*(b+a*x)^13/a^6+1/14*(b+a*x)^14/a^6],
[(a+b/x)^8*x^12,x,3,1/9*b^4*(b+a*x)^9/a^5-2/5*b^3*(b+a*x)^10/a^5+6/11*b^2*(b+a*x)^11/a^5-1/3*b*(b+a*x)^12/a^5+1/13*(b+a*x)^13/a^5],
[(a+b/x)^8*x^11,x,3,-1/9*b^3*(b+a*x)^9/a^4+3/10*b^2*(b+a*x)^10/a^4-3/11*b*(b+a*x)^11/a^4+1/12*(b+a*x)^12/a^4],
[(a+b/x)^8*x^10,x,3,1/9*b^2*(b+a*x)^9/a^3-1/5*b*(b+a*x)^10/a^3+1/11*(b+a*x)^11/a^3],
[(a+b/x)^8*x^9,x,3,-1/9*b*(b+a*x)^9/a^2+1/10*(b+a*x)^10/a^2],
[(a+b/x)^8*x^8,x,2,1/9*(b+a*x)^9/a],
[(a+b/x)^8*x^7,x,3,8*a*b^7*x+14*a^2*b^6*x^2+56/3*a^3*b^5*x^3+35/2*a^4*b^4*x^4+56/5*a^5*b^3*x^5+14/3*a^6*b^2*x^6+8/7*a^7*b*x^7+1/8*a^8*x^8+b^8*log(x)],
[(a+b/x)^8*x^6,x,3,-b^8/x+28*a^2*b^6*x+28*a^3*b^5*x^2+70/3*a^4*b^4*x^3+14*a^5*b^3*x^4+28/5*a^6*b^2*x^5+4/3*a^7*b*x^6+1/7*a^8*x^7+8*a*b^7*log(x)],
[(a+b/x)^8*x^5,x,3,-1/2*b^8/x^2-8*a*b^7/x+56*a^3*b^5*x+35*a^4*b^4*x^2+56/3*a^5*b^3*x^3+7*a^6*b^2*x^4+8/5*a^7*b*x^5+1/6*a^8*x^6+28*a^2*b^6*log(x)],
[(a+b/x)^8*x^4,x,3,-1/3*b^8/x^3-4*a*b^7/x^2-28*a^2*b^6/x+70*a^4*b^4*x+28*a^5*b^3*x^2+28/3*a^6*b^2*x^3+2*a^7*b*x^4+1/5*a^8*x^5+56*a^3*b^5*log(x)],
[(a+b/x)^8*x^3,x,3,-1/4*b^8/x^4-8/3*a*b^7/x^3-14*a^2*b^6/x^2-56*a^3*b^5/x+56*a^5*b^3*x+14*a^6*b^2*x^2+8/3*a^7*b*x^3+1/4*a^8*x^4+70*a^4*b^4*log(x)],
[(a+b/x)^8*x^2,x,3,-1/5*b^8/x^5-2*a*b^7/x^4-28/3*a^2*b^6/x^3-28*a^3*b^5/x^2-70*a^4*b^4/x+28*a^6*b^2*x+4*a^7*b*x^2+1/3*a^8*x^3+56*a^5*b^3*log(x)],
[(a+b/x)^8*x,x,3,-1/6*b^8/x^6-8/5*a*b^7/x^5-7*a^2*b^6/x^4-56/3*a^3*b^5/x^3-35*a^4*b^4/x^2-56*a^5*b^3/x+8*a^7*b*x+1/2*a^8*x^2+28*a^6*b^2*log(x)],
[(a+b/x)^8,x,3,-1/7*b^8/x^7-4/3*a*b^7/x^6-28/5*a^2*b^6/x^5-14*a^3*b^5/x^4-70/3*a^4*b^4/x^3-28*a^5*b^3/x^2-28*a^6*b^2/x+a^8*x+8*a^7*b*log(x)],
[(a+b/x)^8/x,x,3,-1/8*b^8/x^8-8/7*a*b^7/x^7-14/3*a^2*b^6/x^6-56/5*a^3*b^5/x^5-35/2*a^4*b^4/x^4-56/3*a^5*b^3/x^3-14*a^6*b^2/x^2-8*a^7*b/x+a^8*log(x)],
[(a+b/x)^8/x^2,x,1,-1/9*(a+b/x)^9/b],
[(a+b/x)^8/x^3,x,3,-1/10*(b+a*x)^9/(b*x^10)+1/90*a*(b+a*x)^9/(b^2*x^9)],
[(a+b/x)^8/x^4,x,4,-1/11*(b+a*x)^9/(b*x^11)+1/55*a*(b+a*x)^9/(b^2*x^10)-1/495*a^2*(b+a*x)^9/(b^3*x^9)],
[(a+b/x)^8/x^5,x,5,-1/12*(b+a*x)^9/(b*x^12)+1/44*a*(b+a*x)^9/(b^2*x^11)-1/220*a^2*(b+a*x)^9/(b^3*x^10)+1/1980*a^3*(b+a*x)^9/(b^4*x^9)],
[(a+b/x)^8/x^6,x,6,-1/13*(b+a*x)^9/(b*x^13)+1/39*a*(b+a*x)^9/(b^2*x^12)-1/143*a^2*(b+a*x)^9/(b^3*x^11)+1/715*a^3*(b+a*x)^9/(b^4*x^10)-1/6435*a^4*(b+a*x)^9/(b^5*x^9)],
[(a+b/x)^8/x^7,x,3,-1/14*b^8/x^14-8/13*a*b^7/x^13-7/3*a^2*b^6/x^12-56/11*a^3*b^5/x^11-7*a^4*b^4/x^10-56/9*a^5*b^3/x^9-7/2*a^6*b^2/x^8-8/7*a^7*b/x^7-1/6*a^8/x^6],
[(a+b/x)^8/x^8,x,3,-1/15*b^8/x^15-4/7*a*b^7/x^14-28/13*a^2*b^6/x^13-14/3*a^3*b^5/x^12-70/11*a^4*b^4/x^11-28/5*a^5*b^3/x^10-28/9*a^6*b^2/x^9-a^7*b/x^8-1/7*a^8/x^7],

# p<0
[x^4/(a+b/x),x,3,b^4*x/a^5-1/2*b^3*x^2/a^4+1/3*b^2*x^3/a^3-1/4*b*x^4/a^2+1/5*x^5/a-b^5*log(b+a*x)/a^6],
[x^3/(a+b/x),x,3,-b^3*x/a^4+1/2*b^2*x^2/a^3-1/3*b*x^3/a^2+1/4*x^4/a+b^4*log(b+a*x)/a^5],
[x^2/(a+b/x),x,3,b^2*x/a^3-1/2*b*x^2/a^2+1/3*x^3/a-b^3*log(b+a*x)/a^4],
[x/(a+b/x),x,3,-b*x/a^2+1/2*x^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/((a+b/x)*x),x,2,log(b+a*x)/a],
[1/((a+b/x)*x^2),x,1,-log(a+b/x)/b],
[1/((a+b/x)*x^3),x,3,(-1)/(b*x)-a*log(x)/b^2+a*log(b+a*x)/b^2],
[1/((a+b/x)*x^4),x,3,(-1/2)/(b*x^2)+a/(b^2*x)+a^2*log(x)/b^3-a^2*log(b+a*x)/b^3],
[1/((a+b/x)*x^5),x,3,(-1/3)/(b*x^3)+1/2*a/(b^2*x^2)-a^2/(b^3*x)-a^3*log(x)/b^4+a^3*log(b+a*x)/b^4],
[1/((a+b/x)*x^6),x,3,(-1/4)/(b*x^4)+1/3*a/(b^2*x^3)-1/2*a^2/(b^3*x^2)+a^3/(b^4*x)+a^4*log(x)/b^5-a^4*log(b+a*x)/b^5],
[1/((a+b/x)*x^7),x,3,(-1/5)/(b*x^5)+1/4*a/(b^2*x^4)-1/3*a^2/(b^3*x^3)+1/2*a^3/(b^4*x^2)-a^4/(b^5*x)-a^5*log(x)/b^6+a^5*log(b+a*x)/b^6],
[x^5/(a+b/x)^2,x,3,-6*b^5*x/a^7+5/2*b^4*x^2/a^6-4/3*b^3*x^3/a^5+3/4*b^2*x^4/a^4-2/5*b*x^5/a^3+1/6*x^6/a^2+b^7/(a^8*(b+a*x))+7*b^6*log(b+a*x)/a^8],
[x^4/(a+b/x)^2,x,3,5*b^4*x/a^6-2*b^3*x^2/a^5+b^2*x^3/a^4-1/2*b*x^4/a^3+1/5*x^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,3,-4*b^3*x/a^5+3/2*b^2*x^2/a^4-2/3*b*x^3/a^3+1/4*x^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,3,3*b^2*x/a^4-b*x^2/a^3+1/3*x^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,3,-2*b*x/a^3+1/2*x^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,4,2*x/a^2-x/(a*(a+b/x))-2*b*log(b+a*x)/a^3],
[1/((a+b/x)^2*x),x,3,b/(a^2*(b+a*x))+log(b+a*x)/a^2],
[1/((a+b/x)^2*x^2),x,1,1/(b*(a+b/x))],
[1/((a+b/x)^2*x^3),x,3,1/(b*(b+a*x))+log(x)/b^2-log(b+a*x)/b^2],
[1/((a+b/x)^2*x^4),x,3,(-1)/(b^2*x)-a/(b^2*(b+a*x))-2*a*log(x)/b^3+2*a*log(b+a*x)/b^3],
[1/((a+b/x)^2*x^5),x,3,(-1/2)/(b^2*x^2)+2*a/(b^3*x)+a^2/(b^3*(b+a*x))+3*a^2*log(x)/b^4-3*a^2*log(b+a*x)/b^4],
[1/((a+b/x)^2*x^6),x,3,(-1/3)/(b^2*x^3)+a/(b^3*x^2)-3*a^2/(b^4*x)-a^3/(b^4*(b+a*x))-4*a^3*log(x)/b^5+4*a^3*log(b+a*x)/b^5],
[1/((a+b/x)^2*x^7),x,3,(-1/4)/(b^2*x^4)+2/3*a/(b^3*x^3)-3/2*a^2/(b^4*x^2)+4*a^3/(b^5*x)+a^4/(b^5*(b+a*x))+5*a^4*log(x)/b^6-5*a^4*log(b+a*x)/b^6],
[1/((a+b/x)^2*x^8),x,3,(-1/5)/(b^2*x^5)+1/2*a/(b^3*x^4)-a^2/(b^4*x^3)+2*a^3/(b^5*x^2)-5*a^4/(b^6*x)-a^5/(b^6*(b+a*x))-6*a^5*log(x)/b^7+6*a^5*log(b+a*x)/b^7],
[x^4/(a+b/x)^3,x,3,15*b^4*x/a^7-5*b^3*x^2/a^6+2*b^2*x^3/a^5-3/4*b*x^4/a^4+1/5*x^5/a^3+1/2*b^7/(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,3,-10*b^3*x/a^6+3*b^2*x^2/a^5-b*x^3/a^4+1/4*x^4/a^3-1/2*b^6/(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,3,6*b^2*x/a^5-3/2*b*x^2/a^4+1/3*x^3/a^3+1/2*b^5/(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,3,-3*b*x/a^4+1/2*x^2/a^3-1/2*b^4/(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,5,3*x/a^3-1/2*x/(a*(a+b/x)^2)-3/2*x/(a^2*(a+b/x))-3*b*log(b+a*x)/a^4],
[1/((a+b/x)^3*x),x,3,-1/2*b^2/(a^3*(b+a*x)^2)+2*b/(a^3*(b+a*x))+log(b+a*x)/a^3],
[1/((a+b/x)^3*x^2),x,1,1/2/(b*(a+b/x)^2)],
[1/((a+b/x)^3*x^3),x,2,(-1/2)/(a*(b+a*x)^2)],
[1/((a+b/x)^3*x^4),x,3,1/2/(b*(b+a*x)^2)+1/(b^2*(b+a*x))+log(x)/b^3-log(b+a*x)/b^3],
[1/((a+b/x)^3*x^5),x,3,(-1)/(b^3*x)-1/2*a/(b^2*(b+a*x)^2)-2*a/(b^3*(b+a*x))-3*a*log(x)/b^4+3*a*log(b+a*x)/b^4],
[1/((a+b/x)^3*x^6),x,3,(-1/2)/(b^3*x^2)+3*a/(b^4*x)+1/2*a^2/(b^3*(b+a*x)^2)+3*a^2/(b^4*(b+a*x))+6*a^2*log(x)/b^5-6*a^2*log(b+a*x)/b^5],
[1/((a+b/x)^3*x^7),x,3,(-1/3)/(b^3*x^3)+3/2*a/(b^4*x^2)-6*a^2/(b^5*x)-1/2*a^3/(b^4*(b+a*x)^2)-4*a^3/(b^5*(b+a*x))-10*a^3*log(x)/b^6+10*a^3*log(b+a*x)/b^6],
[1/((a+b/x)^3*x^8),x,3,(-1/4)/(b^3*x^4)+a/(b^4*x^3)-3*a^2/(b^5*x^2)+10*a^3/(b^6*x)+1/2*a^4/(b^5*(b+a*x)^2)+5*a^4/(b^6*(b+a*x))+15*a^4*log(x)/b^7-15*a^4*log(b+a*x)/b^7],
[1/((a+b/x)^3*x^9),x,3,(-1/5)/(b^3*x^5)+3/4*a/(b^4*x^4)-2*a^2/(b^5*x^3)+5*a^3/(b^6*x^2)-15*a^4/(b^7*x)-1/2*a^5/(b^6*(b+a*x)^2)-6*a^5/(b^7*(b+a*x))-21*a^5*log(x)/b^8+21*a^5*log(b+a*x)/b^8],

# Integrands of the form x^(m/2) (a+b/x)^p

# p>0
[(a+b/x)*x^(5/2),x,2,2/5*b*x^(5/2)+2/7*a*x^(7/2)],
[(a+b/x)*x^(3/2),x,2,2/3*b*x^(3/2)+2/5*a*x^(5/2)],
[(a+b/x)*x^(1/2),x,2,2/3*a*x^(3/2)+2*b*sqrt(x)],
[(a+b/x)/x^(1/2),x,2,-2*b/sqrt(x)+2*a*sqrt(x)],
[(a+b/x)/x^(3/2),x,2,-2/3*b/x^(3/2)-2*a/sqrt(x)],
[(a+b/x)/x^(5/2),x,2,-2/5*b/x^(5/2)-2/3*a/x^(3/2)],
[(a+b/x)^2*x^(5/2),x,3,2/3*b^2*x^(3/2)+4/5*a*b*x^(5/2)+2/7*a^2*x^(7/2)],
[(a+b/x)^2*x^(3/2),x,3,4/3*a*b*x^(3/2)+2/5*a^2*x^(5/2)+2*b^2*sqrt(x)],
[(a+b/x)^2*x^(1/2),x,3,2/3*a^2*x^(3/2)-2*b^2/sqrt(x)+4*a*b*sqrt(x)],
[(a+b/x)^2/x^(1/2),x,3,-2/3*b^2/x^(3/2)-4*a*b/sqrt(x)+2*a^2*sqrt(x)],
[(a+b/x)^2/x^(3/2),x,3,-2/5*b^2/x^(5/2)-4/3*a*b/x^(3/2)-2*a^2/sqrt(x)],
[(a+b/x)^2/x^(5/2),x,3,-2/7*b^2/x^(7/2)-4/5*a*b/x^(5/2)-2/3*a^2/x^(3/2)],
[(a+b/x)^3*x^(5/2),x,3,2*a*b^2*x^(3/2)+6/5*a^2*b*x^(5/2)+2/7*a^3*x^(7/2)+2*b^3*sqrt(x)],
[(a+b/x)^3*x^(3/2),x,3,2*a^2*b*x^(3/2)+2/5*a^3*x^(5/2)-2*b^3/sqrt(x)+6*a*b^2*sqrt(x)],
[(a+b/x)^3*x^(1/2),x,3,-2/3*b^3/x^(3/2)+2/3*a^3*x^(3/2)-6*a*b^2/sqrt(x)+6*a^2*b*sqrt(x)],
[(a+b/x)^3/x^(1/2),x,3,-2/5*b^3/x^(5/2)-2*a*b^2/x^(3/2)-6*a^2*b/sqrt(x)+2*a^3*sqrt(x)],
[(a+b/x)^3/x^(3/2),x,3,-2/7*b^3/x^(7/2)-6/5*a*b^2/x^(5/2)-2*a^2*b/x^(3/2)-2*a^3/sqrt(x)],
[(a+b/x)^3/x^(5/2),x,3,-2/9*b^3/x^(9/2)-6/7*a*b^2/x^(7/2)-6/5*a^2*b/x^(5/2)-2/3*a^3/x^(3/2)],

# p<0
[x^(5/2)/(a+b/x),x,7,2/3*b^2*x^(3/2)/a^3-2/5*b*x^(5/2)/a^2+2/7*x^(7/2)/a+2*b^(7/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(9/2)-2*b^3*sqrt(x)/a^4],
[x^(3/2)/(a+b/x),x,6,-2/3*b*x^(3/2)/a^2+2/5*x^(5/2)/a-2*b^(5/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(7/2)+2*b^2*sqrt(x)/a^3],
[x^(1/2)/(a+b/x),x,5,2/3*x^(3/2)/a+2*b^(3/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(5/2)-2*b*sqrt(x)/a^2],
[1/((a+b/x)*x^(1/2)),x,4,-2*arctan(sqrt(a)*sqrt(x)/sqrt(b))*sqrt(b)/a^(3/2)+2*sqrt(x)/a],
[1/((a+b/x)*x^(3/2)),x,3,2*arctan(sqrt(a)*sqrt(x)/sqrt(b))/(sqrt(a)*sqrt(b))],
[1/((a+b/x)*x^(5/2)),x,4,-2*arctan(sqrt(a)*sqrt(x)/sqrt(b))*sqrt(a)/b^(3/2)+(-2)/(b*sqrt(x))],
[1/((a+b/x)*x^(7/2)),x,5,(-2/3)/(b*x^(3/2))+2*a^(3/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/b^(5/2)+2*a/(b^2*sqrt(x))],
[1/((a+b/x)*x^(9/2)),x,6,(-2/5)/(b*x^(5/2))+2/3*a/(b^2*x^(3/2))-2*a^(5/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/b^(7/2)-2*a^2/(b^3*sqrt(x))],
[x^(5/2)/(a+b/x)^2,x,8,3*b^2*x^(3/2)/a^4-9/5*b*x^(5/2)/a^3+9/7*x^(7/2)/a^2-x^(9/2)/(a*(b+a*x))+9*b^(7/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(11/2)-9*b^3*sqrt(x)/a^5],
[x^(3/2)/(a+b/x)^2,x,7,-7/3*b*x^(3/2)/a^3+7/5*x^(5/2)/a^2-x^(7/2)/(a*(b+a*x))-7*b^(5/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(9/2)+7*b^2*sqrt(x)/a^4],
[x^(1/2)/(a+b/x)^2,x,6,5/3*x^(3/2)/a^2-x^(5/2)/(a*(b+a*x))+5*b^(3/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(7/2)-5*b*sqrt(x)/a^3],
[1/((a+b/x)^2*x^(1/2)),x,5,-x^(3/2)/(a*(b+a*x))-3*arctan(sqrt(a)*sqrt(x)/sqrt(b))*sqrt(b)/a^(5/2)+3*sqrt(x)/a^2],
[1/((a+b/x)^2*x^(3/2)),x,4,arctan(sqrt(a)*sqrt(x)/sqrt(b))/(a^(3/2)*sqrt(b))-sqrt(x)/(a*(b+a*x))],
[1/((a+b/x)^2*x^(5/2)),x,4,arctan(sqrt(a)*sqrt(x)/sqrt(b))/(b^(3/2)*sqrt(a))+sqrt(x)/(b*(b+a*x))],
[1/((a+b/x)^2*x^(7/2)),x,5,-3*arctan(sqrt(a)*sqrt(x)/sqrt(b))*sqrt(a)/b^(5/2)+(-3)/(b^2*sqrt(x))+1/(b*(b+a*x)*sqrt(x))],
[1/((a+b/x)^2*x^(9/2)),x,6,(-5/3)/(b^2*x^(3/2))+1/(b*x^(3/2)*(b+a*x))+5*a^(3/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/b^(7/2)+5*a/(b^3*sqrt(x))],
[1/((a+b/x)^2*x^(11/2)),x,7,(-7/5)/(b^2*x^(5/2))+7/3*a/(b^3*x^(3/2))+1/(b*x^(5/2)*(b+a*x))-7*a^(5/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/b^(9/2)-7*a^2/(b^4*sqrt(x))],
[x^(3/2)/(a+b/x)^3,x,8,-21/4*b*x^(3/2)/a^4+63/20*x^(5/2)/a^3-1/2*x^(9/2)/(a*(b+a*x)^2)-9/4*x^(7/2)/(a^2*(b+a*x))-63/4*b^(5/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(11/2)+63/4*b^2*sqrt(x)/a^5],
[x^(1/2)/(a+b/x)^3,x,7,35/12*x^(3/2)/a^3-1/2*x^(7/2)/(a*(b+a*x)^2)-7/4*x^(5/2)/(a^2*(b+a*x))+35/4*b^(3/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/a^(9/2)-35/4*b*sqrt(x)/a^4],
[1/((a+b/x)^3*x^(1/2)),x,6,-1/2*x^(5/2)/(a*(b+a*x)^2)-5/4*x^(3/2)/(a^2*(b+a*x))-15/4*arctan(sqrt(a)*sqrt(x)/sqrt(b))*sqrt(b)/a^(7/2)+15/4*sqrt(x)/a^3],
[1/((a+b/x)^3*x^(3/2)),x,5,-1/2*x^(3/2)/(a*(b+a*x)^2)+3/4*arctan(sqrt(a)*sqrt(x)/sqrt(b))/(a^(5/2)*sqrt(b))-3/4*sqrt(x)/(a^2*(b+a*x))],
[1/((a+b/x)^3*x^(5/2)),x,5,1/4*arctan(sqrt(a)*sqrt(x)/sqrt(b))/(a^(3/2)*b^(3/2))-1/2*sqrt(x)/(a*(b+a*x)^2)+1/4*sqrt(x)/(a*b*(b+a*x))],
[1/((a+b/x)^3*x^(7/2)),x,5,3/4*arctan(sqrt(a)*sqrt(x)/sqrt(b))/(b^(5/2)*sqrt(a))+1/2*sqrt(x)/(b*(b+a*x)^2)+3/4*sqrt(x)/(b^2*(b+a*x))],
[1/((a+b/x)^3*x^(9/2)),x,6,-15/4*arctan(sqrt(a)*sqrt(x)/sqrt(b))*sqrt(a)/b^(7/2)+(-15/4)/(b^3*sqrt(x))+1/2/(b*(b+a*x)^2*sqrt(x))+5/4/(b^2*(b+a*x)*sqrt(x))],
[1/((a+b/x)^3*x^(11/2)),x,7,(-35/12)/(b^3*x^(3/2))+1/2/(b*x^(3/2)*(b+a*x)^2)+7/4/(b^2*x^(3/2)*(b+a*x))+35/4*a^(3/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/b^(9/2)+35/4*a/(b^4*sqrt(x))],
[1/((a+b/x)^3*x^(13/2)),x,8,(-63/20)/(b^3*x^(5/2))+21/4*a/(b^4*x^(3/2))+1/2/(b*x^(5/2)*(b+a*x)^2)+9/4/(b^2*x^(5/2)*(b+a*x))-63/4*a^(5/2)*arctan(sqrt(a)*sqrt(x)/sqrt(b))/b^(11/2)-63/4*a^2/(b^5*sqrt(x))],

# Integrands of the form x^m (a+b/x)^(p/2)

# p>0
[x^3*sqrt(a+b/x),x,7,-5/64*b^4*arctanh(sqrt(a+b/x)/sqrt(a))/a^(7/2)+5/64*b^3*x*sqrt(a+b/x)/a^3-5/96*b^2*x^2*sqrt(a+b/x)/a^2+1/24*b*x^3*sqrt(a+b/x)/a+1/4*x^4*sqrt(a+b/x)],
[x^2*sqrt(a+b/x),x,6,1/8*b^3*arctanh(sqrt(a+b/x)/sqrt(a))/a^(5/2)-1/8*b^2*x*sqrt(a+b/x)/a^2+1/12*b*x^2*sqrt(a+b/x)/a+1/3*x^3*sqrt(a+b/x)],
[x*sqrt(a+b/x),x,5,-1/4*b^2*arctanh(sqrt(a+b/x)/sqrt(a))/a^(3/2)+1/4*b*x*sqrt(a+b/x)/a+1/2*x^2*sqrt(a+b/x)],
[sqrt(a+b/x),x,4,b*arctanh(sqrt(a+b/x)/sqrt(a))/sqrt(a)+x*sqrt(a+b/x)],
[sqrt(a+b/x)/x,x,4,2*arctanh(sqrt(a+b/x)/sqrt(a))*sqrt(a)-2*sqrt(a+b/x)],
[sqrt(a+b/x)/x^2,x,1,-2/3*(a+b/x)^(3/2)/b],
[sqrt(a+b/x)/x^3,x,3,2/3*a*(a+b/x)^(3/2)/b^2-2/5*(a+b/x)^(5/2)/b^2],
[sqrt(a+b/x)/x^4,x,3,-2/3*a^2*(a+b/x)^(3/2)/b^3+4/5*a*(a+b/x)^(5/2)/b^3-2/7*(a+b/x)^(7/2)/b^3],
[sqrt(a+b/x)/x^5,x,3,2/3*a^3*(a+b/x)^(3/2)/b^4-6/5*a^2*(a+b/x)^(5/2)/b^4+6/7*a*(a+b/x)^(7/2)/b^4-2/9*(a+b/x)^(9/2)/b^4],
[sqrt(a+b/x)/x^6,x,3,-2/3*a^4*(a+b/x)^(3/2)/b^5+8/5*a^3*(a+b/x)^(5/2)/b^5-12/7*a^2*(a+b/x)^(7/2)/b^5+8/9*a*(a+b/x)^(9/2)/b^5-2/11*(a+b/x)^(11/2)/b^5],
[(a+b/x)^(3/2)*x^3,x,7,1/4*(a+b/x)^(3/2)*x^4+3/64*b^4*arctanh(sqrt(a+b/x)/sqrt(a))/a^(5/2)-3/64*b^3*x*sqrt(a+b/x)/a^2+1/32*b^2*x^2*sqrt(a+b/x)/a+1/8*b*x^3*sqrt(a+b/x)],
[(a+b/x)^(3/2)*x^2,x,6,1/3*(a+b/x)^(3/2)*x^3-1/8*b^3*arctanh(sqrt(a+b/x)/sqrt(a))/a^(3/2)+1/8*b^2*x*sqrt(a+b/x)/a+1/4*b*x^2*sqrt(a+b/x)],
[(a+b/x)^(3/2)*x,x,5,1/2*(a+b/x)^(3/2)*x^2+3/4*b^2*arctanh(sqrt(a+b/x)/sqrt(a))/sqrt(a)+3/4*b*x*sqrt(a+b/x)],
[(a+b/x)^(3/2),x,5,(a+b/x)^(3/2)*x+3*b*arctanh(sqrt(a+b/x)/sqrt(a))*sqrt(a)-3*b*sqrt(a+b/x)],
[(a+b/x)^(3/2)/x,x,5,-2/3*(a+b/x)^(3/2)+2*a^(3/2)*arctanh(sqrt(a+b/x)/sqrt(a))-2*a*sqrt(a+b/x)],
[(a+b/x)^(3/2)/x^2,x,1,-2/5*(a+b/x)^(5/2)/b],
[(a+b/x)^(3/2)/x^3,x,3,2/5*a*(a+b/x)^(5/2)/b^2-2/7*(a+b/x)^(7/2)/b^2],
[(a+b/x)^(3/2)/x^4,x,3,-2/5*a^2*(a+b/x)^(5/2)/b^3+4/7*a*(a+b/x)^(7/2)/b^3-2/9*(a+b/x)^(9/2)/b^3],
[(a+b/x)^(3/2)/x^5,x,3,2/5*a^3*(a+b/x)^(5/2)/b^4-6/7*a^2*(a+b/x)^(7/2)/b^4+2/3*a*(a+b/x)^(9/2)/b^4-2/11*(a+b/x)^(11/2)/b^4],
[(a+b/x)^(3/2)/x^6,x,3,-2/5*a^4*(a+b/x)^(5/2)/b^5+8/7*a^3*(a+b/x)^(7/2)/b^5-4/3*a^2*(a+b/x)^(9/2)/b^5+8/11*a*(a+b/x)^(11/2)/b^5-2/13*(a+b/x)^(13/2)/b^5],
[(a+b/x)^(3/2)/x^7,x,3,2/5*a^5*(a+b/x)^(5/2)/b^6-10/7*a^4*(a+b/x)^(7/2)/b^6+20/9*a^3*(a+b/x)^(9/2)/b^6-20/11*a^2*(a+b/x)^(11/2)/b^6+10/13*a*(a+b/x)^(13/2)/b^6-2/15*(a+b/x)^(15/2)/b^6],
[(a+b/x)^(5/2)*x^3,x,7,5/24*b*(a+b/x)^(3/2)*x^3+1/4*(a+b/x)^(5/2)*x^4-5/64*b^4*arctanh(sqrt(a+b/x)/sqrt(a))/a^(3/2)+5/64*b^3*x*sqrt(a+b/x)/a+5/32*b^2*x^2*sqrt(a+b/x)],
[(a+b/x)^(5/2)*x^2,x,6,5/12*b*(a+b/x)^(3/2)*x^2+1/3*(a+b/x)^(5/2)*x^3+5/8*b^3*arctanh(sqrt(a+b/x)/sqrt(a))/sqrt(a)+5/8*b^2*x*sqrt(a+b/x)],
[(a+b/x)^(5/2)*x,x,6,5/4*b*(a+b/x)^(3/2)*x+1/2*(a+b/x)^(5/2)*x^2+15/4*b^2*arctanh(sqrt(a+b/x)/sqrt(a))*sqrt(a)-15/4*b^2*sqrt(a+b/x)],
[(a+b/x)^(5/2),x,6,-5/3*b*(a+b/x)^(3/2)+(a+b/x)^(5/2)*x+5*a^(3/2)*b*arctanh(sqrt(a+b/x)/sqrt(a))-5*a*b*sqrt(a+b/x)],
[(a+b/x)^(5/2)/x,x,6,-2/3*a*(a+b/x)^(3/2)-2/5*(a+b/x)^(5/2)+2*a^(5/2)*arctanh(sqrt(a+b/x)/sqrt(a))-2*a^2*sqrt(a+b/x)],
[(a+b/x)^(5/2)/x^2,x,1,-2/7*(a+b/x)^(7/2)/b],
[(a+b/x)^(5/2)/x^3,x,3,2/7*a*(a+b/x)^(7/2)/b^2-2/9*(a+b/x)^(9/2)/b^2],
[(a+b/x)^(5/2)/x^4,x,3,-2/7*a^2*(a+b/x)^(7/2)/b^3+4/9*a*(a+b/x)^(9/2)/b^3-2/11*(a+b/x)^(11/2)/b^3],
[(a+b/x)^(5/2)/x^5,x,3,2/7*a^3*(a+b/x)^(7/2)/b^4-2/3*a^2*(a+b/x)^(9/2)/b^4+6/11*a*(a+b/x)^(11/2)/b^4-2/13*(a+b/x)^(13/2)/b^4],
[(a+b/x)^(5/2)/x^6,x,3,-2/7*a^4*(a+b/x)^(7/2)/b^5+8/9*a^3*(a+b/x)^(9/2)/b^5-12/11*a^2*(a+b/x)^(11/2)/b^5+8/13*a*(a+b/x)^(13/2)/b^5-2/15*(a+b/x)^(15/2)/b^5],

# p<0
[x^3/sqrt(a+b/x),x,7,35/64*b^4*arctanh(sqrt(a+b/x)/sqrt(a))/a^(9/2)-35/64*b^3*x*sqrt(a+b/x)/a^4+35/96*b^2*x^2*sqrt(a+b/x)/a^3-7/24*b*x^3*sqrt(a+b/x)/a^2+1/4*x^4*sqrt(a+b/x)/a],
[x^2/sqrt(a+b/x),x,6,-5/8*b^3*arctanh(sqrt(a+b/x)/sqrt(a))/a^(7/2)+5/8*b^2*x*sqrt(a+b/x)/a^3-5/12*b*x^2*sqrt(a+b/x)/a^2+1/3*x^3*sqrt(a+b/x)/a],
[x/sqrt(a+b/x),x,5,3/4*b^2*arctanh(sqrt(a+b/x)/sqrt(a))/a^(5/2)-3/4*b*x*sqrt(a+b/x)/a^2+1/2*x^2*sqrt(a+b/x)/a],
[1/sqrt(a+b/x),x,4,-b*arctanh(sqrt(a+b/x)/sqrt(a))/a^(3/2)+x*sqrt(a+b/x)/a],
[1/(x*sqrt(a+b/x)),x,3,2*arctanh(sqrt(a+b/x)/sqrt(a))/sqrt(a)],
[1/(x^2*sqrt(a+b/x)),x,1,-2*sqrt(a+b/x)/b],
[1/(x^3*sqrt(a+b/x)),x,3,-2/3*(a+b/x)^(3/2)/b^2+2*a*sqrt(a+b/x)/b^2],
[1/(x^4*sqrt(a+b/x)),x,3,4/3*a*(a+b/x)^(3/2)/b^3-2/5*(a+b/x)^(5/2)/b^3-2*a^2*sqrt(a+b/x)/b^3],
[1/(x^5*sqrt(a+b/x)),x,3,-2*a^2*(a+b/x)^(3/2)/b^4+6/5*a*(a+b/x)^(5/2)/b^4-2/7*(a+b/x)^(7/2)/b^4+2*a^3*sqrt(a+b/x)/b^4],
[1/(x^6*sqrt(a+b/x)),x,3,8/3*a^3*(a+b/x)^(3/2)/b^5-12/5*a^2*(a+b/x)^(5/2)/b^5+8/7*a*(a+b/x)^(7/2)/b^5-2/9*(a+b/x)^(9/2)/b^5-2*a^4*sqrt(a+b/x)/b^5],
[x^2/(a+b/x)^(3/2),x,7,-35/8*b^3*arctanh(sqrt(a+b/x)/sqrt(a))/a^(9/2)+35/8*b^3/(a^4*sqrt(a+b/x))+35/24*b^2*x/(a^3*sqrt(a+b/x))-7/12*b*x^2/(a^2*sqrt(a+b/x))+1/3*x^3/(a*sqrt(a+b/x))],
[x/(a+b/x)^(3/2),x,6,15/4*b^2*arctanh(sqrt(a+b/x)/sqrt(a))/a^(7/2)-15/4*b^2/(a^3*sqrt(a+b/x))-5/4*b*x/(a^2*sqrt(a+b/x))+1/2*x^2/(a*sqrt(a+b/x))],
[1/(a+b/x)^(3/2),x,5,-3*b*arctanh(sqrt(a+b/x)/sqrt(a))/a^(5/2)+3*b/(a^2*sqrt(a+b/x))+x/(a*sqrt(a+b/x))],
[1/((a+b/x)^(3/2)*x),x,4,2*arctanh(sqrt(a+b/x)/sqrt(a))/a^(3/2)+(-2)/(a*sqrt(a+b/x))],
[1/((a+b/x)^(3/2)*x^2),x,1,2/(b*sqrt(a+b/x))],
[1/((a+b/x)^(3/2)*x^3),x,3,-2*a/(b^2*sqrt(a+b/x))-2*sqrt(a+b/x)/b^2],
[1/((a+b/x)^(3/2)*x^4),x,3,-2/3*(a+b/x)^(3/2)/b^3+2*a^2/(b^3*sqrt(a+b/x))+4*a*sqrt(a+b/x)/b^3],
[1/((a+b/x)^(3/2)*x^5),x,3,2*a*(a+b/x)^(3/2)/b^4-2/5*(a+b/x)^(5/2)/b^4-2*a^3/(b^4*sqrt(a+b/x))-6*a^2*sqrt(a+b/x)/b^4],
[1/((a+b/x)^(3/2)*x^6),x,3,-4*a^2*(a+b/x)^(3/2)/b^5+8/5*a*(a+b/x)^(5/2)/b^5-2/7*(a+b/x)^(7/2)/b^5+2*a^4/(b^5*sqrt(a+b/x))+8*a^3*sqrt(a+b/x)/b^5],
[1/((a+b/x)^(3/2)*x^7),x,3,20/3*a^3*(a+b/x)^(3/2)/b^6-4*a^2*(a+b/x)^(5/2)/b^6+10/7*a*(a+b/x)^(7/2)/b^6-2/9*(a+b/x)^(9/2)/b^6-2*a^5/(b^6*sqrt(a+b/x))-10*a^4*sqrt(a+b/x)/b^6],
[x^2/(a+b/x)^(5/2),x,8,35/8*b^3/(a^4*(a+b/x)^(3/2))+21/8*b^2*x/(a^3*(a+b/x)^(3/2))-3/4*b*x^2/(a^2*(a+b/x)^(3/2))+1/3*x^3/(a*(a+b/x)^(3/2))-105/8*b^3*arctanh(sqrt(a+b/x)/sqrt(a))/a^(11/2)+105/8*b^3/(a^5*sqrt(a+b/x))],
[x/(a+b/x)^(5/2),x,7,-35/12*b^2/(a^3*(a+b/x)^(3/2))-7/4*b*x/(a^2*(a+b/x)^(3/2))+1/2*x^2/(a*(a+b/x)^(3/2))+35/4*b^2*arctanh(sqrt(a+b/x)/sqrt(a))/a^(9/2)-35/4*b^2/(a^4*sqrt(a+b/x))],
[1/(a+b/x)^(5/2),x,6,5/3*b/(a^2*(a+b/x)^(3/2))+x/(a*(a+b/x)^(3/2))-5*b*arctanh(sqrt(a+b/x)/sqrt(a))/a^(7/2)+5*b/(a^3*sqrt(a+b/x))],
[1/((a+b/x)^(5/2)*x),x,5,(-2/3)/(a*(a+b/x)^(3/2))+2*arctanh(sqrt(a+b/x)/sqrt(a))/a^(5/2)+(-2)/(a^2*sqrt(a+b/x))],
[1/((a+b/x)^(5/2)*x^2),x,1,2/3/(b*(a+b/x)^(3/2))],
[1/((a+b/x)^(5/2)*x^3),x,3,-2/3*a/(b^2*(a+b/x)^(3/2))+2/(b^2*sqrt(a+b/x))],
[1/((a+b/x)^(5/2)*x^4),x,3,2/3*a^2/(b^3*(a+b/x)^(3/2))-4*a/(b^3*sqrt(a+b/x))-2*sqrt(a+b/x)/b^3],
[1/((a+b/x)^(5/2)*x^5),x,3,-2/3*a^3/(b^4*(a+b/x)^(3/2))-2/3*(a+b/x)^(3/2)/b^4+6*a^2/(b^4*sqrt(a+b/x))+6*a*sqrt(a+b/x)/b^4],
[1/((a+b/x)^(5/2)*x^6),x,3,2/3*a^4/(b^5*(a+b/x)^(3/2))+8/3*a*(a+b/x)^(3/2)/b^5-2/5*(a+b/x)^(5/2)/b^5-8*a^3/(b^5*sqrt(a+b/x))-12*a^2*sqrt(a+b/x)/b^5],
[1/((a+b/x)^(5/2)*x^7),x,3,-2/3*a^5/(b^6*(a+b/x)^(3/2))-20/3*a^2*(a+b/x)^(3/2)/b^6+2*a*(a+b/x)^(5/2)/b^6-2/7*(a+b/x)^(7/2)/b^6+10*a^4/(b^6*sqrt(a+b/x))+20*a^3*sqrt(a+b/x)/b^6],

# Integrands of the form x^(m/2) (a+b/x)^(p/2)

# p>0
[(a+b/x)^(1/2)*x^(7/2),x,4,-32/315*b^3*(a+b/x)^(3/2)*x^(3/2)/a^4+16/105*b^2*(a+b/x)^(3/2)*x^(5/2)/a^3-4/21*b*(a+b/x)^(3/2)*x^(7/2)/a^2+2/9*(a+b/x)^(3/2)*x^(9/2)/a],
[(a+b/x)^(1/2)*x^(5/2),x,3,16/105*b^2*(a+b/x)^(3/2)*x^(3/2)/a^3-8/35*b*(a+b/x)^(3/2)*x^(5/2)/a^2+2/7*(a+b/x)^(3/2)*x^(7/2)/a],
[(a+b/x)^(1/2)*x^(3/2),x,2,-4/15*b*(a+b/x)^(3/2)*x^(3/2)/a^2+2/5*(a+b/x)^(3/2)*x^(5/2)/a],
[(a+b/x)^(1/2)*x^(1/2),x,1,2/3*(a+b/x)^(3/2)*x^(3/2)/a],
[(a+b/x)^(1/2)/x^(1/2),x,4,-2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))*sqrt(b)+2*sqrt(a+b/x)*sqrt(x)],
[(a+b/x)^(1/2)/x^(3/2),x,4,-a*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/sqrt(b)-sqrt(a+b/x)/sqrt(x)],
[(a+b/x)^(1/2)/x^(5/2),x,5,1/4*a^2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(3/2)-1/2*sqrt(a+b/x)/x^(3/2)-1/4*a*sqrt(a+b/x)/(b*sqrt(x))],
[(a+b/x)^(1/2)/x^(7/2),x,6,-1/8*a^3*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(5/2)-1/3*sqrt(a+b/x)/x^(5/2)-1/12*a*sqrt(a+b/x)/(b*x^(3/2))+1/8*a^2*sqrt(a+b/x)/(b^2*sqrt(x))],
[(a+b/x)^(3/2)*x^(9/2),x,4,-32/1155*b^3*(a+b/x)^(5/2)*x^(5/2)/a^4+16/231*b^2*(a+b/x)^(5/2)*x^(7/2)/a^3-4/33*b*(a+b/x)^(5/2)*x^(9/2)/a^2+2/11*(a+b/x)^(5/2)*x^(11/2)/a],
[(a+b/x)^(3/2)*x^(7/2),x,3,16/315*b^2*(a+b/x)^(5/2)*x^(5/2)/a^3-8/63*b*(a+b/x)^(5/2)*x^(7/2)/a^2+2/9*(a+b/x)^(5/2)*x^(9/2)/a],
[(a+b/x)^(3/2)*x^(5/2),x,2,-4/35*b*(a+b/x)^(5/2)*x^(5/2)/a^2+2/7*(a+b/x)^(5/2)*x^(7/2)/a],
[(a+b/x)^(3/2)*x^(3/2),x,1,2/5*(a+b/x)^(5/2)*x^(5/2)/a],
[(a+b/x)^(3/2)*x^(1/2),x,5,2/3*(a+b/x)^(3/2)*x^(3/2)-2*b^(3/2)*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))+2*b*sqrt(a+b/x)*sqrt(x)],
[(a+b/x)^(3/2)/x^(1/2),x,5,-3*a*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))*sqrt(b)-3*b*sqrt(a+b/x)/sqrt(x)+2*(a+b/x)^(3/2)*sqrt(x)],
[(a+b/x)^(3/2)/x^(3/2),x,5,-3/4*a^2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/sqrt(b)-1/2*(a+b/x)^(3/2)/sqrt(x)-3/4*a*sqrt(a+b/x)/sqrt(x)],
[(a+b/x)^(3/2)/x^(5/2),x,6,-1/3*(a+b/x)^(3/2)/x^(3/2)+1/8*a^3*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(3/2)-1/4*a*sqrt(a+b/x)/x^(3/2)-1/8*a^2*sqrt(a+b/x)/(b*sqrt(x))],
[(a+b/x)^(5/2)*x^(11/2),x,4,-32/3003*b^3*(a+b/x)^(7/2)*x^(7/2)/a^4+16/429*b^2*(a+b/x)^(7/2)*x^(9/2)/a^3-12/143*b*(a+b/x)^(7/2)*x^(11/2)/a^2+2/13*(a+b/x)^(7/2)*x^(13/2)/a],
[(a+b/x)^(5/2)*x^(9/2),x,3,16/693*b^2*(a+b/x)^(7/2)*x^(7/2)/a^3-8/99*b*(a+b/x)^(7/2)*x^(9/2)/a^2+2/11*(a+b/x)^(7/2)*x^(11/2)/a],
[(a+b/x)^(5/2)*x^(7/2),x,2,-4/63*b*(a+b/x)^(7/2)*x^(7/2)/a^2+2/9*(a+b/x)^(7/2)*x^(9/2)/a],
[(a+b/x)^(5/2)*x^(5/2),x,1,2/7*(a+b/x)^(7/2)*x^(7/2)/a],
[(a+b/x)^(5/2)*x^(3/2),x,6,2/3*b*(a+b/x)^(3/2)*x^(3/2)+2/5*(a+b/x)^(5/2)*x^(5/2)-2*b^(5/2)*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))+2*b^2*sqrt(a+b/x)*sqrt(x)],
[(a+b/x)^(5/2)*x^(1/2),x,6,2/3*(a+b/x)^(5/2)*x^(3/2)-5*a*b^(3/2)*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))-5*b^2*sqrt(a+b/x)/sqrt(x)+10/3*b*(a+b/x)^(3/2)*sqrt(x)],
[(a+b/x)^(5/2)/x^(1/2),x,6,-15/4*a^2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))*sqrt(b)-5/2*b*(a+b/x)^(3/2)/sqrt(x)-15/4*a*b*sqrt(a+b/x)/sqrt(x)+2*(a+b/x)^(5/2)*sqrt(x)],
[(a+b/x)^(5/2)/x^(3/2),x,6,-5/8*a^3*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/sqrt(b)-5/12*a*(a+b/x)^(3/2)/sqrt(x)-1/3*(a+b/x)^(5/2)/sqrt(x)-5/8*a^2*sqrt(a+b/x)/sqrt(x)],
[(a+b/x)^(5/2)/x^(5/2),x,7,-5/24*a*(a+b/x)^(3/2)/x^(3/2)-1/4*(a+b/x)^(5/2)/x^(3/2)+5/64*a^4*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(3/2)-5/32*a^2*sqrt(a+b/x)/x^(3/2)-5/64*a^3*sqrt(a+b/x)/(b*sqrt(x))],

# p<0
[x^(7/2)/(a+b/x)^(1/2),x,5,-128/315*b^3*x^(3/2)*sqrt(a+b/x)/a^4+32/105*b^2*x^(5/2)*sqrt(a+b/x)/a^3-16/63*b*x^(7/2)*sqrt(a+b/x)/a^2+2/9*x^(9/2)*sqrt(a+b/x)/a+256/315*b^4*sqrt(a+b/x)*sqrt(x)/a^5],
[x^(5/2)/(a+b/x)^(1/2),x,4,16/35*b^2*x^(3/2)*sqrt(a+b/x)/a^3-12/35*b*x^(5/2)*sqrt(a+b/x)/a^2+2/7*x^(7/2)*sqrt(a+b/x)/a-32/35*b^3*sqrt(a+b/x)*sqrt(x)/a^4],
[x^(3/2)/(a+b/x)^(1/2),x,3,-8/15*b*x^(3/2)*sqrt(a+b/x)/a^2+2/5*x^(5/2)*sqrt(a+b/x)/a+16/15*b^2*sqrt(a+b/x)*sqrt(x)/a^3],
[x^(1/2)/(a+b/x)^(1/2),x,2,2/3*x^(3/2)*sqrt(a+b/x)/a-4/3*b*sqrt(a+b/x)*sqrt(x)/a^2],
[1/((a+b/x)^(1/2)*x^(1/2)),x,1,2*sqrt(a+b/x)*sqrt(x)/a],
[1/((a+b/x)^(1/2)*x^(3/2)),x,3,-2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/sqrt(b)],
[1/((a+b/x)^(1/2)*x^(5/2)),x,4,a*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(3/2)-sqrt(a+b/x)/(b*sqrt(x))],
[1/((a+b/x)^(1/2)*x^(7/2)),x,5,-3/4*a^2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(5/2)-1/2*sqrt(a+b/x)/(b*x^(3/2))+3/4*a*sqrt(a+b/x)/(b^2*sqrt(x))],
[1/((a+b/x)^(1/2)*x^(9/2)),x,6,5/8*a^3*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(7/2)-1/3*sqrt(a+b/x)/(b*x^(5/2))+5/12*a*sqrt(a+b/x)/(b^2*x^(3/2))-5/8*a^2*sqrt(a+b/x)/(b^3*sqrt(x))],
[x^(5/2)/(a+b/x)^(3/2),x,5,32/35*b^2*x^(3/2)/(a^3*sqrt(a+b/x))-16/35*b*x^(5/2)/(a^2*sqrt(a+b/x))+2/7*x^(7/2)/(a*sqrt(a+b/x))-256/35*b^4/(a^5*sqrt(a+b/x)*sqrt(x))-128/35*b^3*sqrt(x)/(a^4*sqrt(a+b/x))],
[x^(3/2)/(a+b/x)^(3/2),x,4,-4/5*b*x^(3/2)/(a^2*sqrt(a+b/x))+2/5*x^(5/2)/(a*sqrt(a+b/x))+32/5*b^3/(a^4*sqrt(a+b/x)*sqrt(x))+16/5*b^2*sqrt(x)/(a^3*sqrt(a+b/x))],
[x^(1/2)/(a+b/x)^(3/2),x,3,2/3*x^(3/2)/(a*sqrt(a+b/x))-16/3*b^2/(a^3*sqrt(a+b/x)*sqrt(x))-8/3*b*sqrt(x)/(a^2*sqrt(a+b/x))],
[1/((a+b/x)^(3/2)*x^(1/2)),x,2,4*b/(a^2*sqrt(a+b/x)*sqrt(x))+2*sqrt(x)/(a*sqrt(a+b/x))],
[1/((a+b/x)^(3/2)*x^(3/2)),x,1,(-2)/(a*sqrt(a+b/x)*sqrt(x))],
[1/((a+b/x)^(3/2)*x^(5/2)),x,4,-2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(3/2)+2/(b*sqrt(a+b/x)*sqrt(x))],
[1/((a+b/x)^(3/2)*x^(7/2)),x,5,3*a*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(5/2)+2/(b*x^(3/2)*sqrt(a+b/x))-3*sqrt(a+b/x)/(b^2*sqrt(x))],
[1/((a+b/x)^(3/2)*x^(9/2)),x,6,-15/4*a^2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(7/2)+2/(b*x^(5/2)*sqrt(a+b/x))-5/2*sqrt(a+b/x)/(b^2*x^(3/2))+15/4*a*sqrt(a+b/x)/(b^3*sqrt(x))],
[1/((a+b/x)^(3/2)*x^(11/2)),x,7,35/8*a^3*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(9/2)+2/(b*x^(7/2)*sqrt(a+b/x))-7/3*sqrt(a+b/x)/(b^2*x^(5/2))+35/12*a*sqrt(a+b/x)/(b^3*x^(3/2))-35/8*a^2*sqrt(a+b/x)/(b^4*sqrt(x))],
[x^(5/2)/(a+b/x)^(5/2),x,6,-512/21*b^5/(a^6*(a+b/x)^(3/2)*x^(3/2))+32/21*b^2*x^(3/2)/(a^3*(a+b/x)^(3/2))-4/7*b*x^(5/2)/(a^2*(a+b/x)^(3/2))+2/7*x^(7/2)/(a*(a+b/x)^(3/2))-256/7*b^4/(a^5*(a+b/x)^(3/2)*sqrt(x))-64/7*b^3*sqrt(x)/(a^4*(a+b/x)^(3/2))],
[x^(3/2)/(a+b/x)^(5/2),x,5,256/15*b^4/(a^5*(a+b/x)^(3/2)*x^(3/2))-16/15*b*x^(3/2)/(a^2*(a+b/x)^(3/2))+2/5*x^(5/2)/(a*(a+b/x)^(3/2))+128/5*b^3/(a^4*(a+b/x)^(3/2)*sqrt(x))+32/5*b^2*sqrt(x)/(a^3*(a+b/x)^(3/2))],
[x^(1/2)/(a+b/x)^(5/2),x,4,-32/3*b^3/(a^4*(a+b/x)^(3/2)*x^(3/2))+2/3*x^(3/2)/(a*(a+b/x)^(3/2))-16*b^2/(a^3*(a+b/x)^(3/2)*sqrt(x))-4*b*sqrt(x)/(a^2*(a+b/x)^(3/2))],
[1/((a+b/x)^(5/2)*x^(1/2)),x,3,16/3*b^2/(a^3*(a+b/x)^(3/2)*x^(3/2))+8*b/(a^2*(a+b/x)^(3/2)*sqrt(x))+2*sqrt(x)/(a*(a+b/x)^(3/2))],
[1/((a+b/x)^(5/2)*x^(3/2)),x,2,-4/3*b/(a^2*(a+b/x)^(3/2)*x^(3/2))+(-2)/(a*(a+b/x)^(3/2)*sqrt(x))],
[1/((a+b/x)^(5/2)*x^(5/2)),x,1,(-2/3)/(a*(a+b/x)^(3/2)*x^(3/2))],
[1/((a+b/x)^(5/2)*x^(7/2)),x,5,2/3/(b*(a+b/x)^(3/2)*x^(3/2))-2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(5/2)+2/(b^2*sqrt(a+b/x)*sqrt(x))],
[1/((a+b/x)^(5/2)*x^(9/2)),x,6,2/3/(b*(a+b/x)^(3/2)*x^(5/2))+5*a*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(7/2)+10/3/(b^2*x^(3/2)*sqrt(a+b/x))-5*sqrt(a+b/x)/(b^3*sqrt(x))],
[1/((a+b/x)^(5/2)*x^(11/2)),x,7,2/3/(b*(a+b/x)^(3/2)*x^(7/2))-35/4*a^2*arctanh(sqrt(b)/(sqrt(a+b/x)*sqrt(x)))/b^(9/2)+14/3/(b^2*x^(5/2)*sqrt(a+b/x))-35/6*sqrt(a+b/x)/(b^3*x^(3/2))+35/4*a*sqrt(a+b/x)/(b^4*sqrt(x))],

# Integrands of the form (c x)^m (a+b/x^2)^p

# Integrands of the form x^m (a+b/x^2)^p

# p>0
[(a+b/x^2)*x^6,x,2,1/5*b*x^5+1/7*a*x^7],
[(a+b/x^2)*x^5,x,2,1/4*b*x^4+1/6*a*x^6],
[(a+b/x^2)*x^4,x,2,1/3*b*x^3+1/5*a*x^5],
[(a+b/x^2)*x^3,x,2,1/2*b*x^2+1/4*a*x^4],
[(a+b/x^2)*x^2,x,2,b*x+1/3*a*x^3],
[(a+b/x^2)*x,x,2,1/2*a*x^2+b*log(x)],
[a+b/x^2,x,1,-b/x+a*x],
[(a+b/x^2)/x,x,2,-1/2*b/x^2+a*log(x)],
[(a+b/x^2)/x^2,x,2,-1/3*b/x^3-a/x],
[(a+b/x^2)/x^3,x,2,-1/4*b/x^4-1/2*a/x^2],
[(a+b/x^2)/x^4,x,2,-1/5*b/x^5-1/3*a/x^3],
[(a+b/x^2)/x^5,x,2,-1/6*b/x^6-1/4*a/x^4],
[(a+b/x^2)/x^6,x,2,-1/7*b/x^7-1/5*a/x^5],
[(a+b/x^2)^2*x^6,x,3,1/3*b^2*x^3+2/5*a*b*x^5+1/7*a^2*x^7],
[(a+b/x^2)^2*x^5,x,2,1/6*(b+a*x^2)^3/a],
[(a+b/x^2)^2*x^4,x,3,b^2*x+2/3*a*b*x^3+1/5*a^2*x^5],
[(a+b/x^2)^2*x^3,x,4,a*b*x^2+1/4*a^2*x^4+b^2*log(x)],
[(a+b/x^2)^2*x^2,x,3,-b^2/x+2*a*b*x+1/3*a^2*x^3],
[(a+b/x^2)^2*x,x,4,-1/2*b^2/x^2+1/2*a^2*x^2+2*a*b*log(x)],
[(a+b/x^2)^2,x,3,-1/3*b^2/x^3-2*a*b/x+a^2*x],
[(a+b/x^2)^2/x,x,4,-1/4*b^2/x^4-a*b/x^2+a^2*log(x)],
[(a+b/x^2)^2/x^2,x,3,-1/5*b^2/x^5-2/3*a*b/x^3-a^2/x],
[(a+b/x^2)^2/x^3,x,1,-1/6*(a+b/x^2)^3/b],
[(a+b/x^2)^2/x^4,x,3,-1/7*b^2/x^7-2/5*a*b/x^5-1/3*a^2/x^3],
[(a+b/x^2)^2/x^5,x,4,-1/8*b^2/x^8-1/3*a*b/x^6-1/4*a^2/x^4],
[(a+b/x^2)^2/x^6,x,3,-1/9*b^2/x^9-2/7*a*b/x^7-1/5*a^2/x^5],
[(a+b/x^2)^3*x^6,x,3,b^3*x+a*b^2*x^3+3/5*a^2*b*x^5+1/7*a^3*x^7],
[(a+b/x^2)^3*x^5,x,4,3/2*a*b^2*x^2+3/4*a^2*b*x^4+1/6*a^3*x^6+b^3*log(x)],
[(a+b/x^2)^3*x^4,x,3,-b^3/x+3*a*b^2*x+a^2*b*x^3+1/5*a^3*x^5],
[(a+b/x^2)^3*x^3,x,4,-1/2*b^3/x^2+3/2*a^2*b*x^2+1/4*a^3*x^4+3*a*b^2*log(x)],
[(a+b/x^2)^3*x^2,x,3,-1/3*b^3/x^3-3*a*b^2/x+3*a^2*b*x+1/3*a^3*x^3],
[(a+b/x^2)^3*x,x,4,-1/4*b^3/x^4-3/2*a*b^2/x^2+1/2*a^3*x^2+3*a^2*b*log(x)],
[(a+b/x^2)^3,x,3,-1/5*b^3/x^5-a*b^2/x^3-3*a^2*b/x+a^3*x],
[(a+b/x^2)^3/x,x,4,-1/6*b^3/x^6-3/4*a*b^2/x^4-3/2*a^2*b/x^2+a^3*log(x)],
[(a+b/x^2)^3/x^2,x,3,-1/7*b^3/x^7-3/5*a*b^2/x^5-a^2*b/x^3-a^3/x],
[(a+b/x^2)^3/x^3,x,1,-1/8*(a+b/x^2)^4/b],
[(a+b/x^2)^3/x^4,x,3,-1/9*b^3/x^9-3/7*a*b^2/x^7-3/5*a^2*b/x^5-1/3*a^3/x^3],
[(a+b/x^2)^3/x^5,x,4,-1/10*(b+a*x^2)^4/(b*x^10)+1/40*a*(b+a*x^2)^4/(b^2*x^8)],
[(a+b/x^2)^3/x^6,x,3,-1/11*b^3/x^11-1/3*a*b^2/x^9-3/7*a^2*b/x^7-1/5*a^3/x^5],

# p<0
[x^6/(a+b/x^2),x,4,-b^3*x/a^4+1/3*b^2*x^3/a^3-1/5*b*x^5/a^2+1/7*x^7/a+b^(7/2)*arctan(x*sqrt(a)/sqrt(b))/a^(9/2)],
[x^5/(a+b/x^2),x,4,1/2*b^2*x^2/a^3-1/4*b*x^4/a^2+1/6*x^6/a-1/2*b^3*log(b+a*x^2)/a^4],
[x^4/(a+b/x^2),x,4,b^2*x/a^3-1/3*b*x^3/a^2+1/5*x^5/a-b^(5/2)*arctan(x*sqrt(a)/sqrt(b))/a^(7/2)],
[x^3/(a+b/x^2),x,4,-1/2*b*x^2/a^2+1/4*x^4/a+1/2*b^2*log(b+a*x^2)/a^3],
[x^2/(a+b/x^2),x,4,-b*x/a^2+1/3*x^3/a+b^(3/2)*arctan(x*sqrt(a)/sqrt(b))/a^(5/2)],
[x/(a+b/x^2),x,4,1/2*x^2/a-1/2*b*log(b+a*x^2)/a^2],
[1/(a+b/x^2),x,3,x/a-arctan(x*sqrt(a)/sqrt(b))*sqrt(b)/a^(3/2)],
[1/((a+b/x^2)*x),x,2,1/2*log(b+a*x^2)/a],
[1/((a+b/x^2)*x^2),x,2,arctan(x*sqrt(a)/sqrt(b))/(sqrt(a)*sqrt(b))],
[1/((a+b/x^2)*x^3),x,1,-1/2*log(a+b/x^2)/b],
[1/((a+b/x^2)*x^4),x,3,(-1)/(b*x)-arctan(x*sqrt(a)/sqrt(b))*sqrt(a)/b^(3/2)],
[1/((a+b/x^2)*x^5),x,4,(-1/2)/(b*x^2)-a*log(x)/b^2+1/2*a*log(b+a*x^2)/b^2],
[1/((a+b/x^2)*x^6),x,4,(-1/3)/(b*x^3)+a/(b^2*x)+a^(3/2)*arctan(x*sqrt(a)/sqrt(b))/b^(5/2)],
[1/((a+b/x^2)*x^7),x,4,(-1/4)/(b*x^4)+1/2*a/(b^2*x^2)+a^2*log(x)/b^3-1/2*a^2*log(b+a*x^2)/b^3],
[1/((a+b/x^2)*x^8),x,5,(-1/5)/(b*x^5)+1/3*a/(b^2*x^3)-a^2/(b^3*x)-a^(5/2)*arctan(x*sqrt(a)/sqrt(b))/b^(7/2)],
[x^6/(a+b/x^2)^2,x,5,-9/2*b^3*x/a^5+3/2*b^2*x^3/a^4-9/10*b*x^5/a^3+9/14*x^7/a^2-1/2*x^9/(a*(b+a*x^2))+9/2*b^(7/2)*arctan(x*sqrt(a)/sqrt(b))/a^(11/2)],
[x^5/(a+b/x^2)^2,x,4,3/2*b^2*x^2/a^4-1/2*b*x^4/a^3+1/6*x^6/a^2-1/2*b^4/(a^5*(b+a*x^2))-2*b^3*log(b+a*x^2)/a^5],
[x^4/(a+b/x^2)^2,x,5,7/2*b^2*x/a^4-7/6*b*x^3/a^3+7/10*x^5/a^2-1/2*x^7/(a*(b+a*x^2))-7/2*b^(5/2)*arctan(x*sqrt(a)/sqrt(b))/a^(9/2)],
[x^3/(a+b/x^2)^2,x,4,-b*x^2/a^3+1/4*x^4/a^2+1/2*b^3/(a^4*(b+a*x^2))+3/2*b^2*log(b+a*x^2)/a^4],
[x^2/(a+b/x^2)^2,x,5,-5/2*b*x/a^3+5/6*x^3/a^2-1/2*x^5/(a*(b+a*x^2))+5/2*b^(3/2)*arctan(x*sqrt(a)/sqrt(b))/a^(7/2)],
[x/(a+b/x^2)^2,x,4,1/2*x^2/a^2-1/2*b^2/(a^3*(b+a*x^2))-b*log(b+a*x^2)/a^3],
[1/(a+b/x^2)^2,x,4,3/2*x/a^2-1/2*x^3/(a*(b+a*x^2))-3/2*arctan(x*sqrt(a)/sqrt(b))*sqrt(b)/a^(5/2)],
[1/((a+b/x^2)^2*x),x,4,1/2*b/(a^2*(b+a*x^2))+1/2*log(b+a*x^2)/a^2],
[1/((a+b/x^2)^2*x^2),x,3,-1/2*x/(a*(b+a*x^2))+1/2*arctan(x*sqrt(a)/sqrt(b))/(a^(3/2)*sqrt(b))],
[1/((a+b/x^2)^2*x^3),x,1,1/2/(b*(a+b/x^2))],
[1/((a+b/x^2)^2*x^4),x,3,1/2*x/(b*(b+a*x^2))+1/2*arctan(x*sqrt(a)/sqrt(b))/(b^(3/2)*sqrt(a))],
[1/((a+b/x^2)^2*x^5),x,4,1/2/(b*(b+a*x^2))+log(x)/b^2-1/2*log(b+a*x^2)/b^2],
[1/((a+b/x^2)^2*x^6),x,4,(-3/2)/(b^2*x)+1/2/(b*x*(b+a*x^2))-3/2*arctan(x*sqrt(a)/sqrt(b))*sqrt(a)/b^(5/2)],
[1/((a+b/x^2)^2*x^7),x,4,(-1/2)/(b^2*x^2)-1/2*a/(b^2*(b+a*x^2))-2*a*log(x)/b^3+a*log(b+a*x^2)/b^3],
[1/((a+b/x^2)^2*x^8),x,5,(-5/6)/(b^2*x^3)+5/2*a/(b^3*x)+1/2/(b*x^3*(b+a*x^2))+5/2*a^(3/2)*arctan(x*sqrt(a)/sqrt(b))/b^(7/2)],
[1/((a+b/x^2)^2*x^9),x,4,(-1/4)/(b^2*x^4)+a/(b^3*x^2)+1/2*a^2/(b^3*(b+a*x^2))+3*a^2*log(x)/b^4-3/2*a^2*log(b+a*x^2)/b^4],
[x^5/(a+b/x^2)^3,x,4,3*b^2*x^2/a^5-3/4*b*x^4/a^4+1/6*x^6/a^3+1/4*b^5/(a^6*(b+a*x^2)^2)-5/2*b^4/(a^6*(b+a*x^2))-5*b^3*log(b+a*x^2)/a^6],
[x^4/(a+b/x^2)^3,x,6,63/8*b^2*x/a^5-21/8*b*x^3/a^4+63/40*x^5/a^3-1/4*x^9/(a*(b+a*x^2)^2)-9/8*x^7/(a^2*(b+a*x^2))-63/8*b^(5/2)*arctan(x*sqrt(a)/sqrt(b))/a^(11/2)],
[x^3/(a+b/x^2)^3,x,4,-3/2*b*x^2/a^4+1/4*x^4/a^3-1/4*b^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,6,-35/8*b*x/a^4+35/24*x^3/a^3-1/4*x^7/(a*(b+a*x^2)^2)-7/8*x^5/(a^2*(b+a*x^2))+35/8*b^(3/2)*arctan(x*sqrt(a)/sqrt(b))/a^(9/2)],
[x/(a+b/x^2)^3,x,4,1/2*x^2/a^3+1/4*b^3/(a^4*(b+a*x^2)^2)-3/2*b^2/(a^4*(b+a*x^2))-3/2*b*log(b+a*x^2)/a^4],
[1/(a+b/x^2)^3,x,5,15/8*x/a^3-1/4*x^5/(a*(b+a*x^2)^2)-5/8*x^3/(a^2*(b+a*x^2))-15/8*arctan(x*sqrt(a)/sqrt(b))*sqrt(b)/a^(7/2)],
[1/((a+b/x^2)^3*x),x,4,-1/4*b^2/(a^3*(b+a*x^2)^2)+b/(a^3*(b+a*x^2))+1/2*log(b+a*x^2)/a^3],
[1/((a+b/x^2)^3*x^2),x,4,-1/4*x^3/(a*(b+a*x^2)^2)-3/8*x/(a^2*(b+a*x^2))+3/8*arctan(x*sqrt(a)/sqrt(b))/(a^(5/2)*sqrt(b))],
[1/((a+b/x^2)^3*x^3),x,1,1/4/(b*(a+b/x^2)^2)],
[1/((a+b/x^2)^3*x^4),x,4,-1/4*x/(a*(b+a*x^2)^2)+1/8*x/(a*b*(b+a*x^2))+1/8*arctan(x*sqrt(a)/sqrt(b))/(a^(3/2)*b^(3/2))],
[1/((a+b/x^2)^3*x^5),x,2,(-1/4)/(a*(b+a*x^2)^2)],
[1/((a+b/x^2)^3*x^6),x,4,1/4*x/(b*(b+a*x^2)^2)+3/8*x/(b^2*(b+a*x^2))+3/8*arctan(x*sqrt(a)/sqrt(b))/(b^(5/2)*sqrt(a))],
[1/((a+b/x^2)^3*x^7),x,4,1/4/(b*(b+a*x^2)^2)+1/2/(b^2*(b+a*x^2))+log(x)/b^3-1/2*log(b+a*x^2)/b^3],
[1/((a+b/x^2)^3*x^8),x,5,(-15/8)/(b^3*x)+1/4/(b*x*(b+a*x^2)^2)+5/8/(b^2*x*(b+a*x^2))-15/8*arctan(x*sqrt(a)/sqrt(b))*sqrt(a)/b^(7/2)],
[1/((a+b/x^2)^3*x^9),x,4,(-1/2)/(b^3*x^2)-1/4*a/(b^2*(b+a*x^2)^2)-a/(b^3*(b+a*x^2))-3*a*log(x)/b^4+3/2*a*log(b+a*x^2)/b^4],
[1/((a+b/x^2)^3*x^10),x,6,(-35/24)/(b^3*x^3)+35/8*a/(b^4*x)+1/4/(b*x^3*(b+a*x^2)^2)+7/8/(b^2*x^3*(b+a*x^2))+35/8*a^(3/2)*arctan(x*sqrt(a)/sqrt(b))/b^(9/2)],
[1/((a+b/x^2)^3*x^11),x,4,(-1/4)/(b^3*x^4)+3/2*a/(b^4*x^2)+1/4*a^2/(b^3*(b+a*x^2)^2)+3/2*a^2/(b^4*(b+a*x^2))+6*a^2*log(x)/b^5-3*a^2*log(b+a*x^2)/b^5],

# Integrands of the form x^m (a+b/x^2)^(p/2)

# p>0
[(a+b/x^2)^(1/2)*x^3,x,5,-1/8*b^2*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(3/2)+1/8*b*x^2*sqrt(a+b/x^2)/a+1/4*x^4*sqrt(a+b/x^2)],
[(a+b/x^2)^(1/2)*x^2,x,1,1/3*(a+b/x^2)^(3/2)*x^3/a],
[(a+b/x^2)^(1/2)*x,x,4,1/2*b*arctanh(sqrt(a+b/x^2)/sqrt(a))/sqrt(a)+1/2*x^2*sqrt(a+b/x^2)],
[(a+b/x^2)^(1/2),x,4,-arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))*sqrt(b)+x*sqrt(a+b/x^2)],
[(a+b/x^2)^(1/2)/x,x,4,arctanh(sqrt(a+b/x^2)/sqrt(a))*sqrt(a)-sqrt(a+b/x^2)],
[(a+b/x^2)^(1/2)/x^2,x,4,-1/2*a*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/sqrt(b)-1/2*sqrt(a+b/x^2)/x],
[(a+b/x^2)^(1/2)/x^3,x,1,-1/3*(a+b/x^2)^(3/2)/b],
[(a+b/x^2)^(1/2)/x^4,x,5,1/8*a^2*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(3/2)-1/4*sqrt(a+b/x^2)/x^3-1/8*a*sqrt(a+b/x^2)/(b*x)],
[(a+b/x^2)^(3/2)*x^3,x,5,1/4*(a+b/x^2)^(3/2)*x^4+3/8*b^2*arctanh(sqrt(a+b/x^2)/sqrt(a))/sqrt(a)+3/8*b*x^2*sqrt(a+b/x^2)],
[(a+b/x^2)^(3/2)*x^2,x,5,1/3*(a+b/x^2)^(3/2)*x^3-b^(3/2)*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))+b*x*sqrt(a+b/x^2)],
[(a+b/x^2)^(3/2)*x,x,5,1/2*(a+b/x^2)^(3/2)*x^2+3/2*b*arctanh(sqrt(a+b/x^2)/sqrt(a))*sqrt(a)-3/2*b*sqrt(a+b/x^2)],
[(a+b/x^2)^(3/2),x,5,(a+b/x^2)^(3/2)*x-3/2*a*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))*sqrt(b)-3/2*b*sqrt(a+b/x^2)/x],
[(a+b/x^2)^(3/2)/x,x,5,-1/3*(a+b/x^2)^(3/2)+a^(3/2)*arctanh(sqrt(a+b/x^2)/sqrt(a))-a*sqrt(a+b/x^2)],
[(a+b/x^2)^(3/2)/x^2,x,5,-1/4*(a+b/x^2)^(3/2)/x-3/8*a^2*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/sqrt(b)-3/8*a*sqrt(a+b/x^2)/x],
[(a+b/x^2)^(3/2)/x^3,x,1,-1/5*(a+b/x^2)^(5/2)/b],
[(a+b/x^2)^(3/2)/x^4,x,6,-1/6*(a+b/x^2)^(3/2)/x^3+1/16*a^3*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(3/2)-1/8*a*sqrt(a+b/x^2)/x^3-1/16*a^2*sqrt(a+b/x^2)/(b*x)],
[(a+b/x^2)^(5/2)*x^3,x,6,5/8*b*(a+b/x^2)^(3/2)*x^2+1/4*(a+b/x^2)^(5/2)*x^4+15/8*b^2*arctanh(sqrt(a+b/x^2)/sqrt(a))*sqrt(a)-15/8*b^2*sqrt(a+b/x^2)],
[(a+b/x^2)^(5/2)*x^2,x,6,5/3*b*(a+b/x^2)^(3/2)*x+1/3*(a+b/x^2)^(5/2)*x^3-5/2*a*b^(3/2)*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))-5/2*b^2*sqrt(a+b/x^2)/x],
[(a+b/x^2)^(5/2)*x,x,6,-5/6*b*(a+b/x^2)^(3/2)+1/2*(a+b/x^2)^(5/2)*x^2+5/2*a^(3/2)*b*arctanh(sqrt(a+b/x^2)/sqrt(a))-5/2*a*b*sqrt(a+b/x^2)],
[(a+b/x^2)^(5/2),x,6,-5/4*b*(a+b/x^2)^(3/2)/x+(a+b/x^2)^(5/2)*x-15/8*a^2*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))*sqrt(b)-15/8*a*b*sqrt(a+b/x^2)/x],
[(a+b/x^2)^(5/2)/x,x,6,-1/3*a*(a+b/x^2)^(3/2)-1/5*(a+b/x^2)^(5/2)+a^(5/2)*arctanh(sqrt(a+b/x^2)/sqrt(a))-a^2*sqrt(a+b/x^2)],
[(a+b/x^2)^(5/2)/x^2,x,6,-5/24*a*(a+b/x^2)^(3/2)/x-1/6*(a+b/x^2)^(5/2)/x-5/16*a^3*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/sqrt(b)-5/16*a^2*sqrt(a+b/x^2)/x],
[(a+b/x^2)^(5/2)/x^3,x,1,-1/7*(a+b/x^2)^(7/2)/b],
[(a+b/x^2)^(5/2)/x^4,x,7,-5/48*a*(a+b/x^2)^(3/2)/x^3-1/8*(a+b/x^2)^(5/2)/x^3+5/128*a^4*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(3/2)-5/64*a^2*sqrt(a+b/x^2)/x^3-5/128*a^3*sqrt(a+b/x^2)/(b*x)],

# p<0
[x^3/(a+b/x^2)^(1/2),x,5,3/8*b^2*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(5/2)-3/8*b*x^2*sqrt(a+b/x^2)/a^2+1/4*x^4*sqrt(a+b/x^2)/a],
[x/(a+b/x^2)^(1/2),x,4,-1/2*b*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(3/2)+1/2*x^2*sqrt(a+b/x^2)/a],
[1/((a+b/x^2)^(1/2)*x),x,3,arctanh(sqrt(a+b/x^2)/sqrt(a))/sqrt(a)],
[1/((a+b/x^2)^(1/2)*x^3),x,1,-sqrt(a+b/x^2)/b],
[1/((a+b/x^2)^(1/2)*x^5),x,3,-1/3*(a+b/x^2)^(3/2)/b^2+a*sqrt(a+b/x^2)/b^2],
[1/((a+b/x^2)^(1/2)*x^7),x,3,2/3*a*(a+b/x^2)^(3/2)/b^3-1/5*(a+b/x^2)^(5/2)/b^3-a^2*sqrt(a+b/x^2)/b^3],
[1/((a+b/x^2)^(1/2)*x^9),x,3,-a^2*(a+b/x^2)^(3/2)/b^4+3/5*a*(a+b/x^2)^(5/2)/b^4-1/7*(a+b/x^2)^(7/2)/b^4+a^3*sqrt(a+b/x^2)/b^4],
[x^4/(a+b/x^2)^(1/2),x,3,8/15*b^2*x*sqrt(a+b/x^2)/a^3-4/15*b*x^3*sqrt(a+b/x^2)/a^2+1/5*x^5*sqrt(a+b/x^2)/a],
[x^2/(a+b/x^2)^(1/2),x,2,-2/3*b*x*sqrt(a+b/x^2)/a^2+1/3*x^3*sqrt(a+b/x^2)/a],
[1/(a+b/x^2)^(1/2),x,1,x*sqrt(a+b/x^2)/a],
[1/((a+b/x^2)^(1/2)*x^2),x,3,-arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/sqrt(b)],
[1/((a+b/x^2)^(1/2)*x^4),x,4,1/2*a*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(3/2)-1/2*sqrt(a+b/x^2)/(b*x)],
[1/(x*sqrt(-a+b/x^2)),x,3,-arctan(sqrt(-a+b/x^2)/sqrt(a))/sqrt(a)],
[1/(x^2*sqrt(2+b/x^2)),x,2,-arccsch(x*sqrt(2)/sqrt(b))/sqrt(b)],
[1/(x^2*sqrt(2-b/x^2)),x,2,-arccsc(x*sqrt(2)/sqrt(b))/sqrt(b)],
[x^3/(a+b/x^2)^(3/2),x,6,15/8*b^2*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(7/2)-15/8*b^2/(a^3*sqrt(a+b/x^2))-5/8*b*x^2/(a^2*sqrt(a+b/x^2))+1/4*x^4/(a*sqrt(a+b/x^2))],
[x/(a+b/x^2)^(3/2),x,5,-3/2*b*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(5/2)+3/2*b/(a^2*sqrt(a+b/x^2))+1/2*x^2/(a*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(3/2)*x),x,4,arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(3/2)+(-1)/(a*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(3/2)*x^3),x,1,1/(b*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(3/2)*x^5),x,3,-a/(b^2*sqrt(a+b/x^2))-sqrt(a+b/x^2)/b^2],
[1/((a+b/x^2)^(3/2)*x^7),x,3,-1/3*(a+b/x^2)^(3/2)/b^3+a^2/(b^3*sqrt(a+b/x^2))+2*a*sqrt(a+b/x^2)/b^3],
[1/((a+b/x^2)^(3/2)*x^9),x,3,a*(a+b/x^2)^(3/2)/b^4-1/5*(a+b/x^2)^(5/2)/b^4-a^3/(b^4*sqrt(a+b/x^2))-3*a^2*sqrt(a+b/x^2)/b^4],
[x^4/(a+b/x^2)^(3/2),x,4,-8/5*b^2*x/(a^3*sqrt(a+b/x^2))-2/5*b*x^3/(a^2*sqrt(a+b/x^2))+1/5*x^5/(a*sqrt(a+b/x^2))+16/5*b^2*x*sqrt(a+b/x^2)/a^4],
[x^2/(a+b/x^2)^(3/2),x,3,4/3*b*x/(a^2*sqrt(a+b/x^2))+1/3*x^3/(a*sqrt(a+b/x^2))-8/3*b*x*sqrt(a+b/x^2)/a^3],
[1/(a+b/x^2)^(3/2),x,2,-x/(a*sqrt(a+b/x^2))+2*x*sqrt(a+b/x^2)/a^2],
[1/((a+b/x^2)^(3/2)*x^2),x,1,(-1)/(a*x*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(3/2)*x^4),x,4,-arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(3/2)+1/(b*x*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(3/2)*x^6),x,5,3/2*a*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(5/2)+1/(b*x^3*sqrt(a+b/x^2))-3/2*sqrt(a+b/x^2)/(b^2*x)],
[1/((a+b/x^2)^(3/2)*x^8),x,6,-15/8*a^2*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(7/2)+1/(b*x^5*sqrt(a+b/x^2))-5/4*sqrt(a+b/x^2)/(b^2*x^3)+15/8*a*sqrt(a+b/x^2)/(b^3*x)],
[x^3/(a+b/x^2)^(5/2),x,7,-35/24*b^2/(a^3*(a+b/x^2)^(3/2))-7/8*b*x^2/(a^2*(a+b/x^2)^(3/2))+1/4*x^4/(a*(a+b/x^2)^(3/2))+35/8*b^2*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(9/2)-35/8*b^2/(a^4*sqrt(a+b/x^2))],
[x/(a+b/x^2)^(5/2),x,6,5/6*b/(a^2*(a+b/x^2)^(3/2))+1/2*x^2/(a*(a+b/x^2)^(3/2))-5/2*b*arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(7/2)+5/2*b/(a^3*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(5/2)*x),x,5,(-1/3)/(a*(a+b/x^2)^(3/2))+arctanh(sqrt(a+b/x^2)/sqrt(a))/a^(5/2)+(-1)/(a^2*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(5/2)*x^3),x,1,1/3/(b*(a+b/x^2)^(3/2))],
[1/((a+b/x^2)^(5/2)*x^5),x,3,-1/3*a/(b^2*(a+b/x^2)^(3/2))+1/(b^2*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(5/2)*x^7),x,3,1/3*a^2/(b^3*(a+b/x^2)^(3/2))-2*a/(b^3*sqrt(a+b/x^2))-sqrt(a+b/x^2)/b^3],
[1/((a+b/x^2)^(5/2)*x^9),x,3,-1/3*a^3/(b^4*(a+b/x^2)^(3/2))-1/3*(a+b/x^2)^(3/2)/b^4+3*a^2/(b^4*sqrt(a+b/x^2))+3*a*sqrt(a+b/x^2)/b^4],
[x^2/(a+b/x^2)^(5/2),x,4,2/3*b*x/(a^2*(a+b/x^2)^(3/2))+1/3*x^3/(a*(a+b/x^2)^(3/2))+8/3*b*x/(a^3*sqrt(a+b/x^2))-16/3*b*x*sqrt(a+b/x^2)/a^4],
[1/(a+b/x^2)^(5/2),x,3,-1/3*x/(a*(a+b/x^2)^(3/2))-4/3*x/(a^2*sqrt(a+b/x^2))+8/3*x*sqrt(a+b/x^2)/a^3],
[1/((a+b/x^2)^(5/2)*x^2),x,2,-2/3*b/(a^2*(a+b/x^2)^(3/2)*x^3)+(-1)/(a*(a+b/x^2)^(3/2)*x)],
[1/((a+b/x^2)^(5/2)*x^4),x,1,(-1/3)/(a*(a+b/x^2)^(3/2)*x^3)],
[1/((a+b/x^2)^(5/2)*x^6),x,5,1/3/(b*(a+b/x^2)^(3/2)*x^3)-arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(5/2)+1/(b^2*x*sqrt(a+b/x^2))],
[1/((a+b/x^2)^(5/2)*x^8),x,6,1/3/(b*(a+b/x^2)^(3/2)*x^5)+5/2*a*arctanh(sqrt(b)/(x*sqrt(a+b/x^2)))/b^(7/2)+5/3/(b^2*x^3*sqrt(a+b/x^2))-5/2*sqrt(a+b/x^2)/(b^3*x)],

# Integrands of the form x^m (a+b/x^2)^(p/3)
[(1+1/x^2)^(1/3)/x^3,x,1,-3/8*(1+1/x^2)^(4/3)],
[(1+1/x^2)^(5/3)/x^3,x,1,-3/16*(1+1/x^2)^(8/3)],

# Integrands of the form (c x)^m (a+b/x^2)^p with m symbolic
[(1+b/x^2)^(3/2)*(c*x)^m,x,2,(c*x)^(1+m)*hypergeom([-3/2,1/2*(-1-m)],[1/2*(1-m)],-b/x^2)/(c*(1+m))],
[(1+b/x^2)^(1/2)*(c*x)^m,x,2,(c*x)^(1+m)*hypergeom([-1/2,1/2*(-1-m)],[1/2*(1-m)],-b/x^2)/(c*(1+m))],
[(c*x)^m/(1+b/x^2)^(1/2),x,2,(c*x)^(1+m)*hypergeom([1/2,1/2*(-1-m)],[1/2*(1-m)],-b/x^2)/(c*(1+m))],
[(c*x)^m/(1+b/x^2)^(3/2),x,2,(c*x)^(1+m)*hypergeom([3/2,1/2*(-1-m)],[1/2*(1-m)],-b/x^2)/(c*(1+m))],

# Integrands of the form (c x)^m (a+b/x^2)^p with p symbolic
[(1+b/x^2)^p*(c*x)^m,x,2,(c*x)^(1+m)*hypergeom([1/2*(-1-m),-p],[1/2*(1-m)],-b/x^2)/(c*(1+m))],
[(a+b/x^2)^p*(c*x)^m,x,3,(a+b/x^2)^p*(c*x)^(1+m)*hypergeom([1/2*(-1-m),-p],[1/2*(1-m)],-b/(a*x^2))/(c*(1+m)*(1+b/(a*x^2))^p)],

# Integrands of the form (c x)^m (a+b/x^3)^p

# Integrands of the form x^m (a+b/x^3)^p

# p>0

# p<0
[x^5/(a+b/x^3),x,4,-1/3*b*x^3/a^2+1/6*x^6/a+1/3*b^2*log(b+a*x^3)/a^3],
[x^4/(a+b/x^3),x,9,-1/2*b*x^2/a^2+1/5*x^5/a-1/3*b^(5/3)*log(b^(1/3)+a^(1/3)*x)/a^(8/3)+1/6*b^(5/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(8/3)-b^(5/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(8/3)*sqrt(3))],
[x^3/(a+b/x^3),x,9,-b*x/a^2+1/4*x^4/a+1/3*b^(4/3)*log(b^(1/3)+a^(1/3)*x)/a^(7/3)-1/6*b^(4/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(7/3)-b^(4/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(7/3)*sqrt(3))],
[x^2/(a+b/x^3),x,4,1/3*x^3/a-1/3*b*log(b+a*x^3)/a^2],
[x/(a+b/x^3),x,8,1/2*x^2/a+1/3*b^(2/3)*log(b^(1/3)+a^(1/3)*x)/a^(5/3)-1/6*b^(2/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(5/3)+b^(2/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3))],
[1/(a+b/x^3),x,8,x/a-1/3*b^(1/3)*log(b^(1/3)+a^(1/3)*x)/a^(4/3)+1/6*b^(1/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(4/3)+b^(1/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(4/3)*sqrt(3))],
[1/((a+b/x^3)*x),x,2,1/3*log(b+a*x^3)/a],
[1/((a+b/x^3)*x^2),x,7,-1/3*log(b^(1/3)+a^(1/3)*x)/(a^(2/3)*b^(1/3))+1/6*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/(a^(2/3)*b^(1/3))-arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*sqrt(3))],
[1/((a+b/x^3)*x^3),x,7,1/3*log(b^(1/3)+a^(1/3)*x)/(a^(1/3)*b^(2/3))-1/6*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/(a^(1/3)*b^(2/3))-arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*sqrt(3))],
[1/((a+b/x^3)*x^4),x,1,-1/3*log(a+b/x^3)/b],
[1/((a+b/x^3)*x^5),x,8,(-1)/(b*x)+1/3*a^(1/3)*log(b^(1/3)+a^(1/3)*x)/b^(4/3)-1/6*a^(1/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/b^(4/3)+a^(1/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(b^(4/3)*sqrt(3))],
[1/((a+b/x^3)*x^6),x,8,(-1/2)/(b*x^2)-1/3*a^(2/3)*log(b^(1/3)+a^(1/3)*x)/b^(5/3)+1/6*a^(2/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/b^(5/3)+a^(2/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3))],
[1/((a+b/x^3)*x^7),x,4,(-1/3)/(b*x^3)-a*log(x)/b^2+1/3*a*log(b+a*x^3)/b^2],
[x^5/(a+b/x^3)^2,x,4,-2/3*b*x^3/a^3+1/6*x^6/a^2+1/3*b^3/(a^4*(b+a*x^3))+b^2*log(b+a*x^3)/a^4],
[x^4/(a+b/x^3)^2,x,10,-4/3*b*x^2/a^3+8/15*x^5/a^2-1/3*x^8/(a*(b+a*x^3))-8/9*b^(5/3)*log(b^(1/3)+a^(1/3)*x)/a^(11/3)+4/9*b^(5/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(11/3)-8/3*b^(5/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(11/3)*sqrt(3))],
[x^3/(a+b/x^3)^2,x,10,-7/3*b*x/a^3+7/12*x^4/a^2-1/3*x^7/(a*(b+a*x^3))+7/9*b^(4/3)*log(b^(1/3)+a^(1/3)*x)/a^(10/3)-7/18*b^(4/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(10/3)-7/3*b^(4/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(10/3)*sqrt(3))],
[x^2/(a+b/x^3)^2,x,4,1/3*x^3/a^2-1/3*b^2/(a^3*(b+a*x^3))-2/3*b*log(b+a*x^3)/a^3],
[x/(a+b/x^3)^2,x,9,5/6*x^2/a^2-1/3*x^5/(a*(b+a*x^3))+5/9*b^(2/3)*log(b^(1/3)+a^(1/3)*x)/a^(8/3)-5/18*b^(2/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(8/3)+5/3*b^(2/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(8/3)*sqrt(3))],
[1/(a+b/x^3)^2,x,9,4/3*x/a^2-1/3*x^4/(a*(b+a*x^3))-4/9*b^(1/3)*log(b^(1/3)+a^(1/3)*x)/a^(7/3)+2/9*b^(1/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/a^(7/3)+4/3*b^(1/3)*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(7/3)*sqrt(3))],
[1/((a+b/x^3)^2*x),x,4,1/3*b/(a^2*(b+a*x^3))+1/3*log(b+a*x^3)/a^2],
[1/((a+b/x^3)^2*x^2),x,8,-1/3*x^2/(a*(b+a*x^3))-2/9*log(b^(1/3)+a^(1/3)*x)/(a^(5/3)*b^(1/3))+1/9*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/(a^(5/3)*b^(1/3))-2/3*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(5/3)*b^(1/3)*sqrt(3))],
[1/((a+b/x^3)^2*x^3),x,8,-1/3*x/(a*(b+a*x^3))+1/9*log(b^(1/3)+a^(1/3)*x)/(a^(4/3)*b^(2/3))-1/18*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/(a^(4/3)*b^(2/3))-1/3*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(4/3)*b^(2/3)*sqrt(3))],
[1/((a+b/x^3)^2*x^4),x,1,1/3/(b*(a+b/x^3))],
[1/((a+b/x^3)^2*x^5),x,8,1/3*x^2/(b*(b+a*x^3))-1/9*log(b^(1/3)+a^(1/3)*x)/(a^(2/3)*b^(4/3))+1/18*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/(a^(2/3)*b^(4/3))-1/3*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(2/3)*b^(4/3)*sqrt(3))],
[1/((a+b/x^3)^2*x^6),x,8,1/3*x/(b*(b+a*x^3))+2/9*log(b^(1/3)+a^(1/3)*x)/(a^(1/3)*b^(5/3))-1/9*log(b^(2/3)-a^(1/3)*b^(1/3)*x+a^(2/3)*x^2)/(a^(1/3)*b^(5/3))-2/3*arctan((b^(1/3)-2*a^(1/3)*x)/(b^(1/3)*sqrt(3)))/(a^(1/3)*b^(5/3)*sqrt(3))],
[1/((a+b/x^3)^2*x^7),x,4,1/3/(b*(b+a*x^3))+log(x)/b^2-1/3*log(b+a*x^3)/b^2],

# Integrands of the form x^m (a+b/x^3)^(p/2)

# p>0
[x^5*sqrt(a+b/x^3),x,5,-1/12*b^2*arctanh(sqrt(a+b/x^3)/sqrt(a))/a^(3/2)+1/12*b*x^3*sqrt(a+b/x^3)/a+1/6*x^6*sqrt(a+b/x^3)],
[x^2*sqrt(a+b/x^3),x,4,1/3*b*arctanh(sqrt(a+b/x^3)/sqrt(a))/sqrt(a)+1/3*x^3*sqrt(a+b/x^3)],
[sqrt(a+b/x^3)/x,x,4,2/3*arctanh(sqrt(a+b/x^3)/sqrt(a))*sqrt(a)-2/3*sqrt(a+b/x^3)],
[sqrt(a+b/x^3)/x^4,x,1,-2/9*(a+b/x^3)^(3/2)/b],
[sqrt(a+b/x^3)/x^7,x,3,2/9*a*(a+b/x^3)^(3/2)/b^2-2/15*(a+b/x^3)^(5/2)/b^2],
[sqrt(a+b/x^3)/x^10,x,3,-2/9*a^2*(a+b/x^3)^(3/2)/b^3+4/15*a*(a+b/x^3)^(5/2)/b^3-2/21*(a+b/x^3)^(7/2)/b^3],
[sqrt(a+b/x^3)/x^13,x,3,2/9*a^3*(a+b/x^3)^(3/2)/b^4-2/5*a^2*(a+b/x^3)^(5/2)/b^4+2/7*a*(a+b/x^3)^(7/2)/b^4-2/27*(a+b/x^3)^(9/2)/b^4],
[x^7*sqrt(a+b/x^3),x,5,-21/320*b^2*x^2*sqrt(a+b/x^3)/a^2+3/80*b*x^5*sqrt(a+b/x^3)/a+1/8*x^8*sqrt(a+b/x^3)-7/320*3^(3/4)*b^(8/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^2*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4*sqrt(a+b/x^3),x,4,3/20*b*x^2*sqrt(a+b/x^3)/a+1/5*x^5*sqrt(a+b/x^3)+1/20*3^(3/4)*b^(5/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x*sqrt(a+b/x^3),x,3,1/2*x^2*sqrt(a+b/x^3)-1/2*3^(3/4)*b^(2/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3)/x^2,x,3,-2/5*sqrt(a+b/x^3)/x-2/5*3^(3/4)*a*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3)/x^5,x,4,-2/11*sqrt(a+b/x^3)/x^4-6/55*a*sqrt(a+b/x^3)/(b*x)+4/55*3^(3/4)*a^2*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(4/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3)/x^8,x,5,-2/17*sqrt(a+b/x^3)/x^7-6/187*a*sqrt(a+b/x^3)/(b*x^4)+48/935*a^2*sqrt(a+b/x^3)/(b^2*x)-32/935*3^(3/4)*a^3*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(7/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^6*sqrt(a+b/x^3),x,7,-15/112*b^2*x*sqrt(a+b/x^3)/a^2+3/56*b*x^4*sqrt(a+b/x^3)/a+1/7*x^7*sqrt(a+b/x^3)+15/112*b^(7/3)*sqrt(a+b/x^3)/(a^2*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+5/56*3^(3/4)*b^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(2)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-15/224*3^(1/4)*b^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3*sqrt(a+b/x^3),x,6,3/8*b*x*sqrt(a+b/x^3)/a+1/4*x^4*sqrt(a+b/x^3)-3/8*b^(4/3)*sqrt(a+b/x^3)/(a*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-1/4*3^(3/4)*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(2)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+3/16*3^(1/4)*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3),x,5,x*sqrt(a+b/x^3)-3*b^(1/3)*sqrt(a+b/x^3)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))-3^(3/4)*a^(1/3)*b^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+3/2*3^(1/4)*a^(1/3)*b^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3)/x^3,x,5,-2/7*sqrt(a+b/x^3)/x^2-6/7*a*sqrt(a+b/x^3)/(b^(2/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-2/7*3^(3/4)*a^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+3/7*3^(1/4)*a^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3)/x^6,x,6,-2/13*sqrt(a+b/x^3)/x^5-6/91*a*sqrt(a+b/x^3)/(b*x^2)+24/91*a^2*sqrt(a+b/x^3)/(b^(5/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+8/91*3^(3/4)*a^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-12/91*3^(1/4)*a^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+b/x^3)/x^9,x,7,-2/19*sqrt(a+b/x^3)/x^8-6/247*a*sqrt(a+b/x^3)/(b*x^5)+60/1729*a^2*sqrt(a+b/x^3)/(b^2*x^2)-240/1729*a^3*sqrt(a+b/x^3)/(b^(8/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-80/1729*3^(3/4)*a^(10/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+120/1729*3^(1/4)*a^(10/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b/x^3)^(3/2)*x^5,x,5,1/6*(a+b/x^3)^(3/2)*x^6+1/4*b^2*arctanh(sqrt(a+b/x^3)/sqrt(a))/sqrt(a)+1/4*b*x^3*sqrt(a+b/x^3)],
[(a+b/x^3)^(3/2)*x^2,x,5,1/3*(a+b/x^3)^(3/2)*x^3+b*arctanh(sqrt(a+b/x^3)/sqrt(a))*sqrt(a)-b*sqrt(a+b/x^3)],
[(a+b/x^3)^(3/2)/x,x,5,-2/9*(a+b/x^3)^(3/2)+2/3*a^(3/2)*arctanh(sqrt(a+b/x^3)/sqrt(a))-2/3*a*sqrt(a+b/x^3)],
[(a+b/x^3)^(3/2)/x^4,x,1,-2/15*(a+b/x^3)^(5/2)/b],
[(a+b/x^3)^(3/2)/x^7,x,3,2/15*a*(a+b/x^3)^(5/2)/b^2-2/21*(a+b/x^3)^(7/2)/b^2],
[(a+b/x^3)^(3/2)/x^10,x,3,-2/15*a^2*(a+b/x^3)^(5/2)/b^3+4/21*a*(a+b/x^3)^(7/2)/b^3-2/27*(a+b/x^3)^(9/2)/b^3],
[(a+b/x^3)^(3/2)/x^13,x,3,2/15*a^3*(a+b/x^3)^(5/2)/b^4-2/7*a^2*(a+b/x^3)^(7/2)/b^4+2/9*a*(a+b/x^3)^(9/2)/b^4-2/33*(a+b/x^3)^(11/2)/b^4],

# p<0
[x^5/sqrt(a+b/x^3),x,5,1/4*b^2*arctanh(sqrt(a+b/x^3)/sqrt(a))/a^(5/2)-1/4*b*x^3*sqrt(a+b/x^3)/a^2+1/6*x^6*sqrt(a+b/x^3)/a],
[x^2/sqrt(a+b/x^3),x,4,-1/3*b*arctanh(sqrt(a+b/x^3)/sqrt(a))/a^(3/2)+1/3*x^3*sqrt(a+b/x^3)/a],
[1/(x*sqrt(a+b/x^3)),x,3,2/3*arctanh(sqrt(a+b/x^3)/sqrt(a))/sqrt(a)],
[1/(x^4*sqrt(a+b/x^3)),x,1,-2/3*sqrt(a+b/x^3)/b],
[1/(x^7*sqrt(a+b/x^3)),x,3,-2/9*(a+b/x^3)^(3/2)/b^2+2/3*a*sqrt(a+b/x^3)/b^2],
[1/(x^10*sqrt(a+b/x^3)),x,3,4/9*a*(a+b/x^3)^(3/2)/b^3-2/15*(a+b/x^3)^(5/2)/b^3-2/3*a^2*sqrt(a+b/x^3)/b^3],
[1/(x^13*sqrt(a+b/x^3)),x,3,-2/3*a^2*(a+b/x^3)^(3/2)/b^4+2/5*a*(a+b/x^3)^(5/2)/b^4-2/21*(a+b/x^3)^(7/2)/b^4+2/3*a^3*sqrt(a+b/x^3)/b^4],
[x^7/sqrt(a+b/x^3),x,5,91/320*b^2*x^2*sqrt(a+b/x^3)/a^3-13/80*b*x^5*sqrt(a+b/x^3)/a^2+1/8*x^8*sqrt(a+b/x^3)/a+91/320*b^(8/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^3*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4/sqrt(a+b/x^3),x,4,-7/20*b*x^2*sqrt(a+b/x^3)/a^2+1/5*x^5*sqrt(a+b/x^3)/a-7/20*b^(5/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^2*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x/sqrt(a+b/x^3),x,3,1/2*x^2*sqrt(a+b/x^3)/a+1/2*b^(2/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^2*sqrt(a+b/x^3)),x,2,-2*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(1/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^5*sqrt(a+b/x^3)),x,3,-2/5*sqrt(a+b/x^3)/(b*x)+4/5*a*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(4/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^8*sqrt(a+b/x^3)),x,4,-2/11*sqrt(a+b/x^3)/(b*x^4)+16/55*a*sqrt(a+b/x^3)/(b^2*x)-32/55*a^2*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(7/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^6/sqrt(a+b/x^3),x,7,55/112*b^2*x*sqrt(a+b/x^3)/a^3-11/56*b*x^4*sqrt(a+b/x^3)/a^2+1/7*x^7*sqrt(a+b/x^3)/a-55/112*b^(7/3)*sqrt(a+b/x^3)/(a^3*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-55/56*b^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(8/3)*sqrt(2)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+55/224*3^(1/4)*b^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3/sqrt(a+b/x^3),x,6,-5/8*b*x*sqrt(a+b/x^3)/a^2+1/4*x^4*sqrt(a+b/x^3)/a+5/8*b^(4/3)*sqrt(a+b/x^3)/(a^2*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+5/4*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(5/3)*sqrt(2)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-5/16*3^(1/4)*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/sqrt(a+b/x^3),x,5,x*sqrt(a+b/x^3)/a-b^(1/3)*sqrt(a+b/x^3)/(a*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-b^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+1/2*3^(1/4)*b^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^3*sqrt(a+b/x^3)),x,4,-2*sqrt(a+b/x^3)/(b^(2/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-2*a^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^6*sqrt(a+b/x^3)),x,5,-2/7*sqrt(a+b/x^3)/(b*x^2)+8/7*a*sqrt(a+b/x^3)/(b^(5/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+8/7*a^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-4/7*3^(1/4)*a^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^9*sqrt(a+b/x^3)),x,6,-2/13*sqrt(a+b/x^3)/(b*x^5)+20/91*a*sqrt(a+b/x^3)/(b^2*x^2)-80/91*a^2*sqrt(a+b/x^3)/(b^(8/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-80/91*a^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+40/91*3^(1/4)*a^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(x^12*sqrt(a+b/x^3)),x,7,-2/19*sqrt(a+b/x^3)/(b*x^8)+32/247*a*sqrt(a+b/x^3)/(b^2*x^5)-320/1729*a^2*sqrt(a+b/x^3)/(b^3*x^2)+1280/1729*a^3*sqrt(a+b/x^3)/(b^(11/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+1280/1729*a^(10/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(11/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-640/1729*3^(1/4)*a^(10/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(b^(11/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^5/(a+b/x^3)^(3/2),x,6,5/4*b^2*arctanh(sqrt(a+b/x^3)/sqrt(a))/a^(7/2)-5/4*b^2/(a^3*sqrt(a+b/x^3))-5/12*b*x^3/(a^2*sqrt(a+b/x^3))+1/6*x^6/(a*sqrt(a+b/x^3))],
[x^2/(a+b/x^3)^(3/2),x,5,-b*arctanh(sqrt(a+b/x^3)/sqrt(a))/a^(5/2)+b/(a^2*sqrt(a+b/x^3))+1/3*x^3/(a*sqrt(a+b/x^3))],
[1/((a+b/x^3)^(3/2)*x),x,4,2/3*arctanh(sqrt(a+b/x^3)/sqrt(a))/a^(3/2)+(-2/3)/(a*sqrt(a+b/x^3))],
[1/((a+b/x^3)^(3/2)*x^4),x,1,2/3/(b*sqrt(a+b/x^3))],
[1/((a+b/x^3)^(3/2)*x^7),x,3,-2/3*a/(b^2*sqrt(a+b/x^3))-2/3*sqrt(a+b/x^3)/b^2],
[1/((a+b/x^3)^(3/2)*x^10),x,3,-2/9*(a+b/x^3)^(3/2)/b^3+2/3*a^2/(b^3*sqrt(a+b/x^3))+4/3*a*sqrt(a+b/x^3)/b^3],
[1/((a+b/x^3)^(3/2)*x^13),x,3,2/3*a*(a+b/x^3)^(3/2)/b^4-2/15*(a+b/x^3)^(5/2)/b^4-2/3*a^3/(b^4*sqrt(a+b/x^3))-2*a^2*sqrt(a+b/x^3)/b^4],
[x^7/(a+b/x^3)^(3/2),x,6,-2/3*x^8/(a*sqrt(a+b/x^3))+1729/960*b^2*x^2*sqrt(a+b/x^3)/a^4-247/240*b*x^5*sqrt(a+b/x^3)/a^3+19/24*x^8*sqrt(a+b/x^3)/a^2+1729/960*b^(8/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^4*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4/(a+b/x^3)^(3/2),x,5,-2/3*x^5/(a*sqrt(a+b/x^3))-91/60*b*x^2*sqrt(a+b/x^3)/a^3+13/15*x^5*sqrt(a+b/x^3)/a^2-91/60*b^(5/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^3*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x/(a+b/x^3)^(3/2),x,4,-2/3*x^2/(a*sqrt(a+b/x^3))+7/6*x^2*sqrt(a+b/x^3)/a^2+7/6*b^(2/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^2*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^2),x,3,(-2/3)/(a*x*sqrt(a+b/x^3))-2/3*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*b^(1/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^5),x,3,2/3/(b*x*sqrt(a+b/x^3))-4/3*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(4/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^8),x,4,2/3/(b*x^4*sqrt(a+b/x^3))-16/15*sqrt(a+b/x^3)/(b^2*x)+32/15*a*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(7/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^6/(a+b/x^3)^(3/2),x,8,-2/3*x^7/(a*sqrt(a+b/x^3))+935/336*b^2*x*sqrt(a+b/x^3)/a^4-187/168*b*x^4*sqrt(a+b/x^3)/a^3+17/21*x^7*sqrt(a+b/x^3)/a^2-935/336*b^(7/3)*sqrt(a+b/x^3)/(a^4*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-935/168*b^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(11/3)*sqrt(2)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+935/224*b^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(11/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3/(a+b/x^3)^(3/2),x,7,-2/3*x^4/(a*sqrt(a+b/x^3))-55/24*b*x*sqrt(a+b/x^3)/a^3+11/12*x^4*sqrt(a+b/x^3)/a^2+55/24*b^(4/3)*sqrt(a+b/x^3)/(a^3*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+55/12*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(8/3)*sqrt(2)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-55/16*b^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/(a+b/x^3)^(3/2),x,6,-2/3*x/(a*sqrt(a+b/x^3))+5/3*x*sqrt(a+b/x^3)/a^2-5/3*b^(1/3)*sqrt(a+b/x^3)/(a^2*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-5/3*b^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+5/2*b^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^3),x,5,(-2/3)/(a*x^2*sqrt(a+b/x^3))+2/3*sqrt(a+b/x^3)/(a*b^(2/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+2/3*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(2/3)*b^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(2/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^6),x,5,2/3/(b*x^2*sqrt(a+b/x^3))-8/3*sqrt(a+b/x^3)/(b^(5/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-8/3*a^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+4*a^(1/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(5/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^9),x,6,2/3/(b*x^5*sqrt(a+b/x^3))-20/21*sqrt(a+b/x^3)/(b^2*x^2)+80/21*a*sqrt(a+b/x^3)/(b^(8/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))+80/21*a^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))-40/7*a^(4/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(8/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((a+b/x^3)^(3/2)*x^12),x,7,2/3/(b*x^8*sqrt(a+b/x^3))-32/39*sqrt(a+b/x^3)/(b^2*x^5)+320/273*a*sqrt(a+b/x^3)/(b^3*x^2)-1280/273*a^2*sqrt(a+b/x^3)/(b^(11/3)*(b^(1/3)/x+a^(1/3)*(1+sqrt(3))))-1280/273*a^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(11/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))+640/91*a^(7/3)*(a^(1/3)+b^(1/3)/x)*EllipticE((b^(1/3)/x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)/x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(11/3)*sqrt(a+b/x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+sqrt(3)))^2))],

# Integrands of the form (c x)^m (a+b/x^4)^p

# Integrands of the form x^m (a+b/x^4)^p

# p>0

# p<0
[1/(a+b/x^4),x,11,x/a+1/2*b^(1/4)*arctan(1-a^(1/4)*x*sqrt(2)/b^(1/4))/(a^(5/4)*sqrt(2))-1/2*b^(1/4)*arctan(1+a^(1/4)*x*sqrt(2)/b^(1/4))/(a^(5/4)*sqrt(2))+1/4*b^(1/4)*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+x^2*sqrt(a)+sqrt(b))/(a^(5/4)*sqrt(2))-1/4*b^(1/4)*log(a^(1/4)*b^(1/4)*x*sqrt(2)+x^2*sqrt(a)+sqrt(b))/(a^(5/4)*sqrt(2))],

# Integrands of the form x^m (a+b/x^4)^(p/2)

# p>0
[x^3*sqrt(a+b/x^4),x,4,1/4*b*arctanh(sqrt(a+b/x^4)/sqrt(a))/sqrt(a)+1/4*x^4*sqrt(a+b/x^4)],
[x*sqrt(a+b/x^4),x,5,-1/2*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))*sqrt(b)+1/2*x^2*sqrt(a+b/x^4)],
[sqrt(a+b/x^4)/x,x,4,1/2*arctanh(sqrt(a+b/x^4)/sqrt(a))*sqrt(a)-1/2*sqrt(a+b/x^4)],
[sqrt(a+b/x^4)/x^3,x,5,-1/4*a*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))/sqrt(b)-1/4*sqrt(a+b/x^4)/x^2],
[x^2*sqrt(a+b/x^4),x,3,1/3*x^3*sqrt(a+b/x^4)-1/3*b^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(1/4)*sqrt(a+b/x^4))],
[sqrt(a+b/x^4),x,5,x*sqrt(a+b/x^4)-2*sqrt(b)*sqrt(a+b/x^4)/(x*(sqrt(a)+sqrt(b)/x^2))+2*a^(1/4)*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)-a^(1/4)*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)],
[sqrt(a+b/x^4)/x^2,x,3,-1/3*sqrt(a+b/x^4)/x-1/3*a^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(1/4)*sqrt(a+b/x^4))],
[sqrt(a+b/x^4)/x^4,x,5,-1/5*sqrt(a+b/x^4)/x^3-2/5*a*sqrt(a+b/x^4)/(x*sqrt(b)*(sqrt(a)+sqrt(b)/x^2))+2/5*a^(5/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))-1/5*a^(5/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))],
[(a+b/x^4)^(3/2)*x^3,x,5,1/4*(a+b/x^4)^(3/2)*x^4+3/4*b*arctanh(sqrt(a+b/x^4)/sqrt(a))*sqrt(a)-3/4*b*sqrt(a+b/x^4)],
[(a+b/x^4)^(3/2)*x,x,6,1/2*(a+b/x^4)^(3/2)*x^2-3/4*a*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))*sqrt(b)-3/4*b*sqrt(a+b/x^4)/x^2],
[(a+b/x^4)^(3/2)/x,x,5,-1/6*(a+b/x^4)^(3/2)+1/2*a^(3/2)*arctanh(sqrt(a+b/x^4)/sqrt(a))-1/2*a*sqrt(a+b/x^4)],
[(a+b/x^4)^(3/2)/x^3,x,6,-1/8*(a+b/x^4)^(3/2)/x^2-3/16*a^2*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))/sqrt(b)-3/16*a*sqrt(a+b/x^4)/x^2],
[(a+b/x^4)^(3/2)*x^2,x,4,1/3*(a+b/x^4)^(3/2)*x^3-2/3*b*sqrt(a+b/x^4)/x-2/3*a^(3/4)*b^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)],
[(a+b/x^4)^(3/2),x,6,(a+b/x^4)^(3/2)*x-6/5*b*sqrt(a+b/x^4)/x^3-12/5*a*sqrt(b)*sqrt(a+b/x^4)/(x*(sqrt(a)+sqrt(b)/x^2))+12/5*a^(5/4)*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)-6/5*a^(5/4)*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)],
[(a+b/x^4)^(3/2)/x^2,x,4,-1/7*(a+b/x^4)^(3/2)/x-2/7*a*sqrt(a+b/x^4)/x-2/7*a^(7/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(1/4)*sqrt(a+b/x^4))],
[(a+b/x^4)^(3/2)/x^4,x,6,-1/9*(a+b/x^4)^(3/2)/x^3-2/15*a*sqrt(a+b/x^4)/x^3-4/15*a^2*sqrt(a+b/x^4)/(x*sqrt(b)*(sqrt(a)+sqrt(b)/x^2))+4/15*a^(9/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))-2/15*a^(9/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))],
[(a+b/x^4)^(5/2)*x^3,x,6,-5/12*b*(a+b/x^4)^(3/2)+1/4*(a+b/x^4)^(5/2)*x^4+5/4*a^(3/2)*b*arctanh(sqrt(a+b/x^4)/sqrt(a))-5/4*a*b*sqrt(a+b/x^4)],
[(a+b/x^4)^(5/2)*x,x,7,-5/8*b*(a+b/x^4)^(3/2)/x^2+1/2*(a+b/x^4)^(5/2)*x^2-15/16*a^2*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))*sqrt(b)-15/16*a*b*sqrt(a+b/x^4)/x^2],
[(a+b/x^4)^(5/2)/x,x,6,-1/6*a*(a+b/x^4)^(3/2)-1/10*(a+b/x^4)^(5/2)+1/2*a^(5/2)*arctanh(sqrt(a+b/x^4)/sqrt(a))-1/2*a^2*sqrt(a+b/x^4)],
[(a+b/x^4)^(5/2)/x^3,x,7,-5/48*a*(a+b/x^4)^(3/2)/x^2-1/12*(a+b/x^4)^(5/2)/x^2-5/32*a^3*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))/sqrt(b)-5/32*a^2*sqrt(a+b/x^4)/x^2],
[(a+b/x^4)^(5/2)*x^2,x,5,-10/21*b*(a+b/x^4)^(3/2)/x+1/3*(a+b/x^4)^(5/2)*x^3-20/21*a*b*sqrt(a+b/x^4)/x-20/21*a^(7/4)*b^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)],
[(a+b/x^4)^(5/2),x,7,-10/9*b*(a+b/x^4)^(3/2)/x^3+(a+b/x^4)^(5/2)*x-4/3*a*b*sqrt(a+b/x^4)/x^3-8/3*a^2*sqrt(b)*sqrt(a+b/x^4)/(x*(sqrt(a)+sqrt(b)/x^2))+8/3*a^(9/4)*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)-4/3*a^(9/4)*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/sqrt(a+b/x^4)],
[(a+b/x^4)^(5/2)/x^2,x,5,-10/77*a*(a+b/x^4)^(3/2)/x-1/11*(a+b/x^4)^(5/2)/x-20/77*a^2*sqrt(a+b/x^4)/x-20/77*a^(11/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(1/4)*sqrt(a+b/x^4))],
[(a+b/x^4)^(5/2)/x^4,x,7,-10/117*a*(a+b/x^4)^(3/2)/x^3-1/13*(a+b/x^4)^(5/2)/x^3-4/39*a^2*sqrt(a+b/x^4)/x^3-8/39*a^3*sqrt(a+b/x^4)/(x*sqrt(b)*(sqrt(a)+sqrt(b)/x^2))+8/39*a^(13/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))-4/39*a^(13/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))],

# p<0
[x^3/sqrt(a+b/x^4),x,4,-1/4*b*arctanh(sqrt(a+b/x^4)/sqrt(a))/a^(3/2)+1/4*x^4*sqrt(a+b/x^4)/a],
[x/sqrt(a+b/x^4),x,1,1/2*x^2*sqrt(a+b/x^4)/a],
[1/(x*sqrt(a+b/x^4)),x,3,1/2*arctanh(sqrt(a+b/x^4)/sqrt(a))/sqrt(a)],
[1/(x^3*sqrt(a+b/x^4)),x,4,-1/2*arctanh(sqrt(b)/(x^2*sqrt(a+b/x^4)))/sqrt(b)],
[x^2/sqrt(a+b/x^4),x,3,1/3*x^3*sqrt(a+b/x^4)/a+1/6*b^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(5/4)*sqrt(a+b/x^4))],
[1/sqrt(a+b/x^4),x,5,x*sqrt(a+b/x^4)/a-sqrt(b)*sqrt(a+b/x^4)/(a*x*(sqrt(a)+sqrt(b)/x^2))+b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(3/4)*sqrt(a+b/x^4))-1/2*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(3/4)*sqrt(a+b/x^4))],
[1/(x^2*sqrt(a+b/x^4)),x,2,-1/2*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(1/4)*b^(1/4)*sqrt(a+b/x^4))],
[1/(x^4*sqrt(a+b/x^4)),x,4,-sqrt(a+b/x^4)/(x*sqrt(b)*(sqrt(a)+sqrt(b)/x^2))+a^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))-1/2*a^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(b^(3/4)*sqrt(a+b/x^4))],
[x^3/(a+b/x^4)^(3/2),x,5,-3/4*b*arctanh(sqrt(a+b/x^4)/sqrt(a))/a^(5/2)+3/4*b/(a^2*sqrt(a+b/x^4))+1/4*x^4/(a*sqrt(a+b/x^4))],
[x/(a+b/x^4)^(3/2),x,2,-1/2*x^2/(a*sqrt(a+b/x^4))+x^2*sqrt(a+b/x^4)/a^2],
[1/((a+b/x^4)^(3/2)*x),x,4,1/2*arctanh(sqrt(a+b/x^4)/sqrt(a))/a^(3/2)+(-1/2)/(a*sqrt(a+b/x^4))],
[1/((a+b/x^4)^(3/2)*x^3),x,1,(-1/2)/(a*x^2*sqrt(a+b/x^4))],
[x^2/(a+b/x^4)^(3/2),x,4,-1/2*x^3/(a*sqrt(a+b/x^4))+5/6*x^3*sqrt(a+b/x^4)/a^2+5/12*b^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(9/4)*sqrt(a+b/x^4))],
[1/(a+b/x^4)^(3/2),x,6,-1/2*x/(a*sqrt(a+b/x^4))+3/2*x*sqrt(a+b/x^4)/a^2-3/2*sqrt(b)*sqrt(a+b/x^4)/(a^2*x*(sqrt(a)+sqrt(b)/x^2))+3/2*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(7/4)*sqrt(a+b/x^4))-3/4*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(7/4)*sqrt(a+b/x^4))],
[1/((a+b/x^4)^(3/2)*x^2),x,3,(-1/2)/(a*x*sqrt(a+b/x^4))-1/4*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(5/4)*b^(1/4)*sqrt(a+b/x^4))],
[1/((a+b/x^4)^(3/2)*x^4),x,5,(-1/2)/(a*x^3*sqrt(a+b/x^4))+1/2*sqrt(a+b/x^4)/(a*x*sqrt(b)*(sqrt(a)+sqrt(b)/x^2))-1/2*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(3/4)*b^(3/4)*sqrt(a+b/x^4))+1/4*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(3/4)*b^(3/4)*sqrt(a+b/x^4))],
[x^3/(a+b/x^4)^(5/2),x,6,5/12*b/(a^2*(a+b/x^4)^(3/2))+1/4*x^4/(a*(a+b/x^4)^(3/2))-5/4*b*arctanh(sqrt(a+b/x^4)/sqrt(a))/a^(7/2)+5/4*b/(a^3*sqrt(a+b/x^4))],
[x/(a+b/x^4)^(5/2),x,3,-1/6*x^2/(a*(a+b/x^4)^(3/2))-2/3*x^2/(a^2*sqrt(a+b/x^4))+4/3*x^2*sqrt(a+b/x^4)/a^3],
[1/((a+b/x^4)^(5/2)*x),x,5,(-1/6)/(a*(a+b/x^4)^(3/2))+1/2*arctanh(sqrt(a+b/x^4)/sqrt(a))/a^(5/2)+(-1/2)/(a^2*sqrt(a+b/x^4))],
[1/((a+b/x^4)^(5/2)*x^3),x,2,-1/3*b/(a^2*(a+b/x^4)^(3/2)*x^6)+(-1/2)/(a*(a+b/x^4)^(3/2)*x^2)],
[x^2/(a+b/x^4)^(5/2),x,5,-1/6*x^3/(a*(a+b/x^4)^(3/2))-3/4*x^3/(a^2*sqrt(a+b/x^4))+5/4*x^3*sqrt(a+b/x^4)/a^3+5/8*b^(3/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(13/4)*sqrt(a+b/x^4))],
[1/(a+b/x^4)^(5/2),x,7,-1/6*x/(a*(a+b/x^4)^(3/2))-7/12*x/(a^2*sqrt(a+b/x^4))+7/4*x*sqrt(a+b/x^4)/a^3-7/4*sqrt(b)*sqrt(a+b/x^4)/(a^3*x*(sqrt(a)+sqrt(b)/x^2))+7/4*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(11/4)*sqrt(a+b/x^4))-7/8*b^(1/4)*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(11/4)*sqrt(a+b/x^4))],
[1/((a+b/x^4)^(5/2)*x^2),x,4,(-1/6)/(a*(a+b/x^4)^(3/2)*x)+(-5/12)/(a^2*x*sqrt(a+b/x^4))-5/24*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(9/4)*b^(1/4)*sqrt(a+b/x^4))],
[1/((a+b/x^4)^(5/2)*x^4),x,6,(-1/6)/(a*(a+b/x^4)^(3/2)*x^3)+(-1/4)/(a^2*x^3*sqrt(a+b/x^4))+1/4*sqrt(a+b/x^4)/(a^2*x*sqrt(b)*(sqrt(a)+sqrt(b)/x^2))-1/4*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticE(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(7/4)*b^(3/4)*sqrt(a+b/x^4))+1/8*sqrt(cos(2*arccot(a^(1/4)*x/b^(1/4)))^2)/cos(2*arccot(a^(1/4)*x/b^(1/4)))*EllipticF(sin(2*arccot(a^(1/4)*x/b^(1/4))),sqrt(1/2))*(sqrt(a)+sqrt(b)/x^2)*sqrt((a+b/x^4)/(sqrt(a)+sqrt(b)/x^2)^2)/(a^(7/4)*b^(3/4)*sqrt(a+b/x^4))],

# Integrands of the form (c x)^m (a+b/x^5)^p

# Integrands of the form x^m (a+b/x^5)^p

# p>0

# p<0
[1/(a+b/x^5),x,8,x/a-1/5*b^(1/5)*log(b^(1/5)+a^(1/5)*x)/a^(6/5)+1/20*b^(1/5)*log(b^(2/5)+a^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(1-sqrt(5)))*(1-sqrt(5))/a^(6/5)+1/20*b^(1/5)*log(b^(2/5)+a^(2/5)*x^2-1/2*a^(1/5)*b^(1/5)*x*(1+sqrt(5)))*(1+sqrt(5))/a^(6/5)+1/5*b^(1/5)*arctan(-a^(1/5)*x*sqrt(2/5*(5+sqrt(5)))/b^(1/5)+sqrt(1/5*(5+2*sqrt(5))))*sqrt(1/2*(5-sqrt(5)))/a^(6/5)-1/5*b^(1/5)*arctan(sqrt(1/5*(5-2*sqrt(5)))+2*a^(1/5)*x*sqrt(2/(5+sqrt(5)))/b^(1/5))*sqrt(1/2*(5+sqrt(5)))/a^(6/5)],

# Integrands of the form x^m (a+b/x^5)^(p/2)

# p>0

# p<0
[1/(x*sqrt(a+b/x^5)),x,3,2/5*arctanh(sqrt(a+b/x^5)/sqrt(a))/sqrt(a)],
[1/(x*sqrt(-a+b/x^5)),x,3,-2/5*arctan(sqrt(-a+b/x^5)/sqrt(a))/sqrt(a)],

# Integrands of the form (c x)^m (a+b/x^6)^p

# Integrands of the form x^m (a+b/x^6)^p

# p>0

# p<0
[1/(a+b/x^6),x,12,x/a-1/3*b^(1/6)*arctan(a^(1/6)*x/b^(1/6))/a^(7/6)+1/6*b^(1/6)*arctan((-2*a^(1/6)*x+b^(1/6)*sqrt(3))/b^(1/6))/a^(7/6)-1/6*b^(1/6)*arctan((2*a^(1/6)*x+b^(1/6)*sqrt(3))/b^(1/6))/a^(7/6)+1/4*b^(1/6)*log(b^(1/3)+a^(1/3)*x^2-a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(7/6)*sqrt(3))-1/4*b^(1/6)*log(b^(1/3)+a^(1/3)*x^2+a^(1/6)*b^(1/6)*x*sqrt(3))/(a^(7/6)*sqrt(3))],

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

# Integrands of the form (c x)^m (a+b/x^8)^p

# Integrands of the form x^m (a+b/x^8)^p

# p>0

# p<0
[1/(a+b/x^8),x,15,x/a+1/4*b^(1/8)*arctan((-a)^(1/8)*x/b^(1/8))/(-a)^(9/8)+1/4*b^(1/8)*arctanh((-a)^(1/8)*x/b^(1/8))/(-a)^(9/8)-1/4*b^(1/8)*arctan(1-(-a)^(1/8)*x*sqrt(2)/b^(1/8))/((-a)^(9/8)*sqrt(2))+1/4*b^(1/8)*arctan(1+(-a)^(1/8)*x*sqrt(2)/b^(1/8))/((-a)^(9/8)*sqrt(2))-1/8*b^(1/8)*log(b^(1/4)+(-a)^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(9/8)*sqrt(2))+1/8*b^(1/8)*log(b^(1/4)+(-a)^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*sqrt(2))/((-a)^(9/8)*sqrt(2))],

# Integrands of the form x^m (a+b/x^8)^(p/2)

# p>0

# p<0

# Integrands of the form (c x)^m (a+b x^n)^p when n>0 is a fraction

# Integrands of the form (c x)^m (a+b x^(1/2))^p

# Integrands of the form x^m (a+b Sqrt[x])^p

# p>0
[x^4*(a+b*sqrt(x)),x,2,1/5*a*x^5+2/11*b*x^(11/2)],
[x^3*(a+b*sqrt(x)),x,2,1/4*a*x^4+2/9*b*x^(9/2)],
[x^2*(a+b*sqrt(x)),x,2,1/3*a*x^3+2/7*b*x^(7/2)],
[x*(a+b*sqrt(x)),x,2,1/2*a*x^2+2/5*b*x^(5/2)],
[a+b*sqrt(x),x,1,a*x+2/3*b*x^(3/2)],
[(a+b*sqrt(x))/x,x,2,a*log(x)+2*b*sqrt(x)],
[(a+b*sqrt(x))/x^2,x,2,-a/x-2*b/sqrt(x)],
[(a+b*sqrt(x))/x^3,x,2,-1/2*a/x^2-2/3*b/x^(3/2)],
[(a+b*sqrt(x))/x^4,x,2,-1/3*a/x^3-2/5*b/x^(5/2)],
[x^4*(a+b*sqrt(x))^2,x,3,1/5*a^2*x^5+4/11*a*b*x^(11/2)+1/6*b^2*x^6],
[x^3*(a+b*sqrt(x))^2,x,3,1/4*a^2*x^4+4/9*a*b*x^(9/2)+1/5*b^2*x^5],
[x^2*(a+b*sqrt(x))^2,x,3,1/3*a^2*x^3+4/7*a*b*x^(7/2)+1/4*b^2*x^4],
[x*(a+b*sqrt(x))^2,x,3,1/2*a^2*x^2+4/5*a*b*x^(5/2)+1/3*b^2*x^3],
[(a+b*sqrt(x))^2,x,3,a^2*x+4/3*a*b*x^(3/2)+1/2*b^2*x^2],
[(a+b*sqrt(x))^2/x,x,3,b^2*x+a^2*log(x)+4*a*b*sqrt(x)],
[(a+b*sqrt(x))^2/x^2,x,3,-a^2/x+b^2*log(x)-4*a*b/sqrt(x)],
[(a+b*sqrt(x))^2/x^3,x,3,-1/2*a^2/x^2-4/3*a*b/x^(3/2)-b^2/x],
[(a+b*sqrt(x))^2/x^4,x,3,-1/3*a^2/x^3-4/5*a*b/x^(5/2)-1/2*b^2/x^2],
[(a+b*sqrt(x))^2/x^5,x,3,-1/4*a^2/x^4-4/7*a*b/x^(7/2)-1/3*b^2/x^3],
[x^4*(a+b*sqrt(x))^3,x,3,1/5*a^3*x^5+6/11*a^2*b*x^(11/2)+1/2*a*b^2*x^6+2/13*b^3*x^(13/2)],
[x^3*(a+b*sqrt(x))^3,x,3,1/4*a^3*x^4+2/3*a^2*b*x^(9/2)+3/5*a*b^2*x^5+2/11*b^3*x^(11/2)],
[x^2*(a+b*sqrt(x))^3,x,3,1/3*a^3*x^3+6/7*a^2*b*x^(7/2)+3/4*a*b^2*x^4+2/9*b^3*x^(9/2)],
[x*(a+b*sqrt(x))^3,x,3,1/2*a^3*x^2+6/5*a^2*b*x^(5/2)+a*b^2*x^3+2/7*b^3*x^(7/2)],
[(a+b*sqrt(x))^3,x,3,-1/2*a*(a+b*sqrt(x))^4/b^2+2/5*(a+b*sqrt(x))^5/b^2],
[(a+b*sqrt(x))^3/x,x,3,3*a*b^2*x+2/3*b^3*x^(3/2)+a^3*log(x)+6*a^2*b*sqrt(x)],
[(a+b*sqrt(x))^3/x^2,x,3,-a^3/x+3*a*b^2*log(x)-6*a^2*b/sqrt(x)+2*b^3*sqrt(x)],
[(a+b*sqrt(x))^3/x^3,x,1,-1/2*(a+b*sqrt(x))^4/(a*x^2)],
[(a+b*sqrt(x))^3/x^4,x,3,-1/3*a^3/x^3-6/5*a^2*b/x^(5/2)-3/2*a*b^2/x^2-2/3*b^3/x^(3/2)],
[(a+b*sqrt(x))^3/x^5,x,3,-1/4*a^3/x^4-6/7*a^2*b/x^(7/2)-a*b^2/x^3-2/5*b^3/x^(5/2)],
[(a+b*sqrt(x))^3/x^6,x,3,-1/5*a^3/x^5-2/3*a^2*b/x^(9/2)-3/4*a*b^2/x^4-2/7*b^3/x^(7/2)],
[x^4*(a+b*sqrt(x))^5,x,3,1/5*a^5*x^5+10/11*a^4*b*x^(11/2)+5/3*a^3*b^2*x^6+20/13*a^2*b^3*x^(13/2)+5/7*a*b^4*x^7+2/15*b^5*x^(15/2)],
[x^3*(a+b*sqrt(x))^5,x,3,1/4*a^5*x^4+10/9*a^4*b*x^(9/2)+2*a^3*b^2*x^5+20/11*a^2*b^3*x^(11/2)+5/6*a*b^4*x^6+2/13*b^5*x^(13/2)],
[x^2*(a+b*sqrt(x))^5,x,3,1/3*a^5*x^3+10/7*a^4*b*x^(7/2)+5/2*a^3*b^2*x^4+20/9*a^2*b^3*x^(9/2)+a*b^4*x^5+2/11*b^5*x^(11/2)],
[x*(a+b*sqrt(x))^5,x,3,-1/3*a^3*(a+b*sqrt(x))^6/b^4+6/7*a^2*(a+b*sqrt(x))^7/b^4-3/4*a*(a+b*sqrt(x))^8/b^4+2/9*(a+b*sqrt(x))^9/b^4],
[(a+b*sqrt(x))^5,x,3,-1/3*a*(a+b*sqrt(x))^6/b^2+2/7*(a+b*sqrt(x))^7/b^2],
[(a+b*sqrt(x))^5/x,x,3,10*a^3*b^2*x+20/3*a^2*b^3*x^(3/2)+5/2*a*b^4*x^2+2/5*b^5*x^(5/2)+a^5*log(x)+10*a^4*b*sqrt(x)],
[(a+b*sqrt(x))^5/x^2,x,3,-a^5/x+5*a*b^4*x+2/3*b^5*x^(3/2)+10*a^3*b^2*log(x)-10*a^4*b/sqrt(x)+20*a^2*b^3*sqrt(x)],
[(a+b*sqrt(x))^5/x^3,x,3,-1/2*a^5/x^2-10/3*a^4*b/x^(3/2)-10*a^3*b^2/x+5*a*b^4*log(x)-20*a^2*b^3/sqrt(x)+2*b^5*sqrt(x)],
[(a+b*sqrt(x))^5/x^4,x,1,-1/3*(a+b*sqrt(x))^6/(a*x^3)],
[(a+b*sqrt(x))^5/x^5,x,4,-1/4*(a+b*sqrt(x))^6/(a*x^4)+1/14*b*(a+b*sqrt(x))^6/(a^2*x^(7/2))-1/84*b^2*(a+b*sqrt(x))^6/(a^3*x^3)],
[(a+b*sqrt(x))^5/x^6,x,3,-1/5*a^5/x^5-10/9*a^4*b/x^(9/2)-5/2*a^3*b^2/x^4-20/7*a^2*b^3/x^(7/2)-5/3*a*b^4/x^3-2/5*b^5/x^(5/2)],
[(a+b*sqrt(x))^5/x^7,x,3,-1/6*a^5/x^6-10/11*a^4*b/x^(11/2)-2*a^3*b^2/x^5-20/9*a^2*b^3/x^(9/2)-5/4*a*b^4/x^4-2/7*b^5/x^(7/2)],
[x^4*(a+b*sqrt(x))^10,x,3,1/5*a^10*x^5+20/11*a^9*b*x^(11/2)+15/2*a^8*b^2*x^6+240/13*a^7*b^3*x^(13/2)+30*a^6*b^4*x^7+168/5*a^5*b^5*x^(15/2)+105/4*a^4*b^6*x^8+240/17*a^3*b^7*x^(17/2)+5*a^2*b^8*x^9+20/19*a*b^9*x^(19/2)+1/10*b^10*x^10],
[x^3*(a+b*sqrt(x))^10,x,3,-2/11*a^7*(a+b*sqrt(x))^11/b^8+7/6*a^6*(a+b*sqrt(x))^12/b^8-42/13*a^5*(a+b*sqrt(x))^13/b^8+5*a^4*(a+b*sqrt(x))^14/b^8-14/3*a^3*(a+b*sqrt(x))^15/b^8+21/8*a^2*(a+b*sqrt(x))^16/b^8-14/17*a*(a+b*sqrt(x))^17/b^8+1/9*(a+b*sqrt(x))^18/b^8],
[x^2*(a+b*sqrt(x))^10,x,3,-2/11*a^5*(a+b*sqrt(x))^11/b^6+5/6*a^4*(a+b*sqrt(x))^12/b^6-20/13*a^3*(a+b*sqrt(x))^13/b^6+10/7*a^2*(a+b*sqrt(x))^14/b^6-2/3*a*(a+b*sqrt(x))^15/b^6+1/8*(a+b*sqrt(x))^16/b^6],
[x*(a+b*sqrt(x))^10,x,3,-2/11*a^3*(a+b*sqrt(x))^11/b^4+1/2*a^2*(a+b*sqrt(x))^12/b^4-6/13*a*(a+b*sqrt(x))^13/b^4+1/7*(a+b*sqrt(x))^14/b^4],
[(a+b*sqrt(x))^10,x,3,-2/11*a*(a+b*sqrt(x))^11/b^2+1/6*(a+b*sqrt(x))^12/b^2],
[(a+b*sqrt(x))^10/x,x,3,45*a^8*b^2*x+80*a^7*b^3*x^(3/2)+105*a^6*b^4*x^2+504/5*a^5*b^5*x^(5/2)+70*a^4*b^6*x^3+240/7*a^3*b^7*x^(7/2)+45/4*a^2*b^8*x^4+20/9*a*b^9*x^(9/2)+1/5*b^10*x^5+a^10*log(x)+20*a^9*b*sqrt(x)],
[(a+b*sqrt(x))^10/x^2,x,3,-a^10/x+210*a^6*b^4*x+168*a^5*b^5*x^(3/2)+105*a^4*b^6*x^2+48*a^3*b^7*x^(5/2)+15*a^2*b^8*x^3+20/7*a*b^9*x^(7/2)+1/4*b^10*x^4+45*a^8*b^2*log(x)-20*a^9*b/sqrt(x)+240*a^7*b^3*sqrt(x)],
[(a+b*sqrt(x))^10/x^3,x,3,-1/2*a^10/x^2-20/3*a^9*b/x^(3/2)-45*a^8*b^2/x+210*a^4*b^6*x+80*a^3*b^7*x^(3/2)+45/2*a^2*b^8*x^2+4*a*b^9*x^(5/2)+1/3*b^10*x^3+210*a^6*b^4*log(x)-240*a^7*b^3/sqrt(x)+504*a^5*b^5*sqrt(x)],
[(a+b*sqrt(x))^10/x^4,x,3,-1/3*a^10/x^3-4*a^9*b/x^(5/2)-45/2*a^8*b^2/x^2-80*a^7*b^3/x^(3/2)-210*a^6*b^4/x+45*a^2*b^8*x+20/3*a*b^9*x^(3/2)+1/2*b^10*x^2+210*a^4*b^6*log(x)-504*a^5*b^5/sqrt(x)+240*a^3*b^7*sqrt(x)],
[(a+b*sqrt(x))^10/x^5,x,3,-1/4*a^10/x^4-20/7*a^9*b/x^(7/2)-15*a^8*b^2/x^3-48*a^7*b^3/x^(5/2)-105*a^6*b^4/x^2-168*a^5*b^5/x^(3/2)-210*a^4*b^6/x+b^10*x+45*a^2*b^8*log(x)-240*a^3*b^7/sqrt(x)+20*a*b^9*sqrt(x)],
[(a+b*sqrt(x))^10/x^6,x,3,-1/5*a^10/x^5-20/9*a^9*b/x^(9/2)-45/4*a^8*b^2/x^4-240/7*a^7*b^3/x^(7/2)-70*a^6*b^4/x^3-504/5*a^5*b^5/x^(5/2)-105*a^4*b^6/x^2-80*a^3*b^7/x^(3/2)-45*a^2*b^8/x+b^10*log(x)-20*a*b^9/sqrt(x)],
[(a+b*sqrt(x))^10/x^7,x,3,-1/6*(a+b*sqrt(x))^11/(a*x^6)+1/66*b*(a+b*sqrt(x))^11/(a^2*x^(11/2))],
[(a+b*sqrt(x))^10/x^8,x,5,-1/7*(a+b*sqrt(x))^11/(a*x^7)+3/91*b*(a+b*sqrt(x))^11/(a^2*x^(13/2))-1/182*b^2*(a+b*sqrt(x))^11/(a^3*x^6)+1/2002*b^3*(a+b*sqrt(x))^11/(a^4*x^(11/2))],
[(a+b*sqrt(x))^10/x^9,x,7,-1/8*(a+b*sqrt(x))^11/(a*x^8)+1/24*b*(a+b*sqrt(x))^11/(a^2*x^(15/2))-1/84*b^2*(a+b*sqrt(x))^11/(a^3*x^7)+1/364*b^3*(a+b*sqrt(x))^11/(a^4*x^(13/2))-1/2184*b^4*(a+b*sqrt(x))^11/(a^5*x^6)+1/24024*b^5*(a+b*sqrt(x))^11/(a^6*x^(11/2))],
[(a+b*sqrt(x))^10/x^10,x,3,-1/9*a^10/x^9-20/17*a^9*b/x^(17/2)-45/8*a^8*b^2/x^8-16*a^7*b^3/x^(15/2)-30*a^6*b^4/x^7-504/13*a^5*b^5/x^(13/2)-35*a^4*b^6/x^6-240/11*a^3*b^7/x^(11/2)-9*a^2*b^8/x^5-20/9*a*b^9/x^(9/2)-1/4*b^10/x^4],
[(a+b*sqrt(x))^10/x^11,x,3,-1/10*a^10/x^10-20/19*a^9*b/x^(19/2)-5*a^8*b^2/x^9-240/17*a^7*b^3/x^(17/2)-105/4*a^6*b^4/x^8-168/5*a^5*b^5/x^(15/2)-30*a^4*b^6/x^7-240/13*a^3*b^7/x^(13/2)-15/2*a^2*b^8/x^6-20/11*a*b^9/x^(11/2)-1/5*b^10/x^5],
[x^5*(a+b*sqrt(x))^15,x,3,-1/8*a^11*(a+b*sqrt(x))^16/b^12+22/17*a^10*(a+b*sqrt(x))^17/b^12-55/9*a^9*(a+b*sqrt(x))^18/b^12+330/19*a^8*(a+b*sqrt(x))^19/b^12-33*a^7*(a+b*sqrt(x))^20/b^12+44*a^6*(a+b*sqrt(x))^21/b^12-42*a^5*(a+b*sqrt(x))^22/b^12+660/23*a^4*(a+b*sqrt(x))^23/b^12-55/4*a^3*(a+b*sqrt(x))^24/b^12+22/5*a^2*(a+b*sqrt(x))^25/b^12-11/13*a*(a+b*sqrt(x))^26/b^12+2/27*(a+b*sqrt(x))^27/b^12],
[x^4*(a+b*sqrt(x))^15,x,3,-1/8*a^9*(a+b*sqrt(x))^16/b^10+18/17*a^8*(a+b*sqrt(x))^17/b^10-4*a^7*(a+b*sqrt(x))^18/b^10+168/19*a^6*(a+b*sqrt(x))^19/b^10-63/5*a^5*(a+b*sqrt(x))^20/b^10+12*a^4*(a+b*sqrt(x))^21/b^10-84/11*a^3*(a+b*sqrt(x))^22/b^10+72/23*a^2*(a+b*sqrt(x))^23/b^10-3/4*a*(a+b*sqrt(x))^24/b^10+2/25*(a+b*sqrt(x))^25/b^10],
[x^3*(a+b*sqrt(x))^15,x,3,-1/8*a^7*(a+b*sqrt(x))^16/b^8+14/17*a^6*(a+b*sqrt(x))^17/b^8-7/3*a^5*(a+b*sqrt(x))^18/b^8+70/19*a^4*(a+b*sqrt(x))^19/b^8-7/2*a^3*(a+b*sqrt(x))^20/b^8+2*a^2*(a+b*sqrt(x))^21/b^8-7/11*a*(a+b*sqrt(x))^22/b^8+2/23*(a+b*sqrt(x))^23/b^8],
[x^2*(a+b*sqrt(x))^15,x,3,-1/8*a^5*(a+b*sqrt(x))^16/b^6+10/17*a^4*(a+b*sqrt(x))^17/b^6-10/9*a^3*(a+b*sqrt(x))^18/b^6+20/19*a^2*(a+b*sqrt(x))^19/b^6-1/2*a*(a+b*sqrt(x))^20/b^6+2/21*(a+b*sqrt(x))^21/b^6],
[x*(a+b*sqrt(x))^15,x,3,-1/8*a^3*(a+b*sqrt(x))^16/b^4+6/17*a^2*(a+b*sqrt(x))^17/b^4-1/3*a*(a+b*sqrt(x))^18/b^4+2/19*(a+b*sqrt(x))^19/b^4],
[(a+b*sqrt(x))^15,x,3,-1/8*a*(a+b*sqrt(x))^16/b^2+2/17*(a+b*sqrt(x))^17/b^2],
[(a+b*sqrt(x))^15/x,x,3,105*a^13*b^2*x+910/3*a^12*b^3*x^(3/2)+1365/2*a^11*b^4*x^2+6006/5*a^10*b^5*x^(5/2)+5005/3*a^9*b^6*x^3+12870/7*a^8*b^7*x^(7/2)+6435/4*a^7*b^8*x^4+10010/9*a^6*b^9*x^(9/2)+3003/5*a^5*b^10*x^5+2730/11*a^4*b^11*x^(11/2)+455/6*a^3*b^12*x^6+210/13*a^2*b^13*x^(13/2)+15/7*a*b^14*x^7+2/15*b^15*x^(15/2)+a^15*log(x)+30*a^14*b*sqrt(x)],
[(a+b*sqrt(x))^15/x^2,x,3,-a^15/x+1365*a^11*b^4*x+2002*a^10*b^5*x^(3/2)+5005/2*a^9*b^6*x^2+2574*a^8*b^7*x^(5/2)+2145*a^7*b^8*x^3+1430*a^6*b^9*x^(7/2)+3003/4*a^5*b^10*x^4+910/3*a^4*b^11*x^(9/2)+91*a^3*b^12*x^5+210/11*a^2*b^13*x^(11/2)+5/2*a*b^14*x^6+2/13*b^15*x^(13/2)+105*a^13*b^2*log(x)-30*a^14*b/sqrt(x)+910*a^12*b^3*sqrt(x)],
[(a+b*sqrt(x))^15/x^3,x,3,-1/2*a^15/x^2-10*a^14*b/x^(3/2)-105*a^13*b^2/x+5005*a^9*b^6*x+4290*a^8*b^7*x^(3/2)+6435/2*a^7*b^8*x^2+2002*a^6*b^9*x^(5/2)+1001*a^5*b^10*x^3+390*a^4*b^11*x^(7/2)+455/4*a^3*b^12*x^4+70/3*a^2*b^13*x^(9/2)+3*a*b^14*x^5+2/11*b^15*x^(11/2)+1365*a^11*b^4*log(x)-910*a^12*b^3/sqrt(x)+6006*a^10*b^5*sqrt(x)],
[(a+b*sqrt(x))^15/x^4,x,3,-1/3*a^15/x^3-6*a^14*b/x^(5/2)-105/2*a^13*b^2/x^2-910/3*a^12*b^3/x^(3/2)-1365*a^11*b^4/x+6435*a^7*b^8*x+10010/3*a^6*b^9*x^(3/2)+3003/2*a^5*b^10*x^2+546*a^4*b^11*x^(5/2)+455/3*a^3*b^12*x^3+30*a^2*b^13*x^(7/2)+15/4*a*b^14*x^4+2/9*b^15*x^(9/2)+5005*a^9*b^6*log(x)-6006*a^10*b^5/sqrt(x)+12870*a^8*b^7*sqrt(x)],
[(a+b*sqrt(x))^15/x^6,x,3,-1/5*a^15/x^5-10/3*a^14*b/x^(9/2)-105/4*a^13*b^2/x^4-130*a^12*b^3/x^(7/2)-455*a^11*b^4/x^3-6006/5*a^10*b^5/x^(5/2)-5005/2*a^9*b^6/x^2-4290*a^8*b^7/x^(3/2)-6435*a^7*b^8/x+455*a^3*b^12*x+70*a^2*b^13*x^(3/2)+15/2*a*b^14*x^2+2/5*b^15*x^(5/2)+3003*a^5*b^10*log(x)-10010*a^6*b^9/sqrt(x)+2730*a^4*b^11*sqrt(x)],
[(a+b*sqrt(x))^15/x^7,x,3,-1/6*a^15/x^6-30/11*a^14*b/x^(11/2)-21*a^13*b^2/x^5-910/9*a^12*b^3/x^(9/2)-1365/4*a^11*b^4/x^4-858*a^10*b^5/x^(7/2)-5005/3*a^9*b^6/x^3-2574*a^8*b^7/x^(5/2)-6435/2*a^7*b^8/x^2-10010/3*a^6*b^9/x^(3/2)-3003*a^5*b^10/x+15*a*b^14*x+2/3*b^15*x^(3/2)+455*a^3*b^12*log(x)-2730*a^4*b^11/sqrt(x)+210*a^2*b^13*sqrt(x)],
[(a+b*sqrt(x))^15/x^8,x,3,-1/7*a^15/x^7-30/13*a^14*b/x^(13/2)-35/2*a^13*b^2/x^6-910/11*a^12*b^3/x^(11/2)-273*a^11*b^4/x^5-2002/3*a^10*b^5/x^(9/2)-5005/4*a^9*b^6/x^4-12870/7*a^8*b^7/x^(7/2)-2145*a^7*b^8/x^3-2002*a^6*b^9/x^(5/2)-3003/2*a^5*b^10/x^2-910*a^4*b^11/x^(3/2)-455*a^3*b^12/x+15*a*b^14*log(x)-210*a^2*b^13/sqrt(x)+2*b^15*sqrt(x)],
[(a+b*sqrt(x))^15/x^9,x,1,-1/8*(a+b*sqrt(x))^16/(a*x^8)],
[(a+b*sqrt(x))^15/x^10,x,4,-1/9*(a+b*sqrt(x))^16/(a*x^9)+2/153*b*(a+b*sqrt(x))^16/(a^2*x^(17/2))-1/1224*b^2*(a+b*sqrt(x))^16/(a^3*x^8)],
[(a+b*sqrt(x))^15/x^11,x,6,-1/10*(a+b*sqrt(x))^16/(a*x^10)+2/95*b*(a+b*sqrt(x))^16/(a^2*x^(19/2))-1/285*b^2*(a+b*sqrt(x))^16/(a^3*x^9)+2/4845*b^3*(a+b*sqrt(x))^16/(a^4*x^(17/2))-1/38760*b^4*(a+b*sqrt(x))^16/(a^5*x^8)],
[(a+b*sqrt(x))^15/x^12,x,8,-1/11*(a+b*sqrt(x))^16/(a*x^11)+2/77*b*(a+b*sqrt(x))^16/(a^2*x^(21/2))-1/154*b^2*(a+b*sqrt(x))^16/(a^3*x^10)+2/1463*b^3*(a+b*sqrt(x))^16/(a^4*x^(19/2))-1/4389*b^4*(a+b*sqrt(x))^16/(a^5*x^9)+2/74613*b^5*(a+b*sqrt(x))^16/(a^6*x^(17/2))-1/596904*b^6*(a+b*sqrt(x))^16/(a^7*x^8)],
[(a+b*sqrt(x))^15/x^13,x,10,-1/12*(a+b*sqrt(x))^16/(a*x^12)+2/69*b*(a+b*sqrt(x))^16/(a^2*x^(23/2))-7/759*b^2*(a+b*sqrt(x))^16/(a^3*x^11)+2/759*b^3*(a+b*sqrt(x))^16/(a^4*x^(21/2))-1/1518*b^4*(a+b*sqrt(x))^16/(a^5*x^10)+2/14421*b^5*(a+b*sqrt(x))^16/(a^6*x^(19/2))-1/43263*b^6*(a+b*sqrt(x))^16/(a^7*x^9)+2/735471*b^7*(a+b*sqrt(x))^16/(a^8*x^(17/2))-1/5883768*b^8*(a+b*sqrt(x))^16/(a^9*x^8)],
[(a+b*sqrt(x))^15/x^14,x,12,-1/13*(a+b*sqrt(x))^16/(a*x^13)+2/65*b*(a+b*sqrt(x))^16/(a^2*x^(25/2))-3/260*b^2*(a+b*sqrt(x))^16/(a^3*x^12)+6/1495*b^3*(a+b*sqrt(x))^16/(a^4*x^(23/2))-21/16445*b^4*(a+b*sqrt(x))^16/(a^5*x^11)+6/16445*b^5*(a+b*sqrt(x))^16/(a^6*x^(21/2))-3/32890*b^6*(a+b*sqrt(x))^16/(a^7*x^10)+6/312455*b^7*(a+b*sqrt(x))^16/(a^8*x^(19/2))-1/312455*b^8*(a+b*sqrt(x))^16/(a^9*x^9)+2/5311735*b^9*(a+b*sqrt(x))^16/(a^10*x^(17/2))-1/42493880*b^10*(a+b*sqrt(x))^16/(a^11*x^8)],
[(a+b*sqrt(x))^15/x^15,x,3,-1/14*a^15/x^14-10/9*a^14*b/x^(27/2)-105/13*a^13*b^2/x^13-182/5*a^12*b^3/x^(25/2)-455/4*a^11*b^4/x^12-6006/23*a^10*b^5/x^(23/2)-455*a^9*b^6/x^11-4290/7*a^8*b^7/x^(21/2)-1287/2*a^7*b^8/x^10-10010/19*a^6*b^9/x^(19/2)-1001/3*a^5*b^10/x^9-2730/17*a^4*b^11/x^(17/2)-455/8*a^3*b^12/x^8-14*a^2*b^13/x^(15/2)-15/7*a*b^14/x^7-2/13*b^15/x^(13/2)],
[(a+b*sqrt(x))^15/x^16,x,3,-1/15*a^15/x^15-30/29*a^14*b/x^(29/2)-15/2*a^13*b^2/x^14-910/27*a^12*b^3/x^(27/2)-105*a^11*b^4/x^13-6006/25*a^10*b^5/x^(25/2)-5005/12*a^9*b^6/x^12-12870/23*a^8*b^7/x^(23/2)-585*a^7*b^8/x^11-1430/3*a^6*b^9/x^(21/2)-3003/10*a^5*b^10/x^10-2730/19*a^4*b^11/x^(19/2)-455/9*a^3*b^12/x^9-210/17*a^2*b^13/x^(17/2)-15/8*a*b^14/x^8-2/15*b^15/x^(15/2)],
[(a+b*sqrt(x))^15/x^17,x,3,-1/16*a^15/x^16-30/31*a^14*b/x^(31/2)-7*a^13*b^2/x^15-910/29*a^12*b^3/x^(29/2)-195/2*a^11*b^4/x^14-2002/9*a^10*b^5/x^(27/2)-385*a^9*b^6/x^13-2574/5*a^8*b^7/x^(25/2)-2145/4*a^7*b^8/x^12-10010/23*a^6*b^9/x^(23/2)-273*a^5*b^10/x^11-130*a^4*b^11/x^(21/2)-91/2*a^3*b^12/x^10-210/19*a^2*b^13/x^(19/2)-5/3*a*b^14/x^9-2/17*b^15/x^(17/2)],

# p<0
[x^3/(a+b*sqrt(x)),x,3,-a^5*x/b^6+2/3*a^4*x^(3/2)/b^5-1/2*a^3*x^2/b^4+2/5*a^2*x^(5/2)/b^3-1/3*a*x^3/b^2+2/7*x^(7/2)/b-2*a^7*log(a+b*sqrt(x))/b^8+2*a^6*sqrt(x)/b^7],
[x^2/(a+b*sqrt(x)),x,3,-a^3*x/b^4+2/3*a^2*x^(3/2)/b^3-1/2*a*x^2/b^2+2/5*x^(5/2)/b-2*a^5*log(a+b*sqrt(x))/b^6+2*a^4*sqrt(x)/b^5],
[x/(a+b*sqrt(x)),x,3,-a*x/b^2+2/3*x^(3/2)/b-2*a^3*log(a+b*sqrt(x))/b^4+2*a^2*sqrt(x)/b^3],
[1/(a+b*sqrt(x)),x,3,-2*a*log(a+b*sqrt(x))/b^2+2*sqrt(x)/b],
[1/(x*(a+b*sqrt(x))),x,4,log(x)/a-2*log(a+b*sqrt(x))/a],
[1/(x^2*(a+b*sqrt(x))),x,3,(-1)/(a*x)+b^2*log(x)/a^3-2*b^2*log(a+b*sqrt(x))/a^3+2*b/(a^2*sqrt(x))],
[1/(x^3*(a+b*sqrt(x))),x,3,(-1/2)/(a*x^2)+2/3*b/(a^2*x^(3/2))-b^2/(a^3*x)+b^4*log(x)/a^5-2*b^4*log(a+b*sqrt(x))/a^5+2*b^3/(a^4*sqrt(x))],
[1/(x^4*(a+b*sqrt(x))),x,3,(-1/3)/(a*x^3)+2/5*b/(a^2*x^(5/2))-1/2*b^2/(a^3*x^2)+2/3*b^3/(a^4*x^(3/2))-b^4/(a^5*x)+b^6*log(x)/a^7-2*b^6*log(a+b*sqrt(x))/a^7+2*b^5/(a^6*sqrt(x))],
[x^3/(a+b*sqrt(x))^2,x,3,5*a^4*x/b^6-8/3*a^3*x^(3/2)/b^5+3/2*a^2*x^2/b^4-4/5*a*x^(5/2)/b^3+1/3*x^3/b^2+14*a^6*log(a+b*sqrt(x))/b^8-12*a^5*sqrt(x)/b^7+2*a^7/(b^8*(a+b*sqrt(x)))],
[x^2/(a+b*sqrt(x))^2,x,3,3*a^2*x/b^4-4/3*a*x^(3/2)/b^3+1/2*x^2/b^2+10*a^4*log(a+b*sqrt(x))/b^6-8*a^3*sqrt(x)/b^5+2*a^5/(b^6*(a+b*sqrt(x)))],
[x/(a+b*sqrt(x))^2,x,3,x/b^2+6*a^2*log(a+b*sqrt(x))/b^4-4*a*sqrt(x)/b^3+2*a^3/(b^4*(a+b*sqrt(x)))],
[1/(a+b*sqrt(x))^2,x,3,2*log(a+b*sqrt(x))/b^2+2*a/(b^2*(a+b*sqrt(x)))],
[1/(x*(a+b*sqrt(x))^2),x,3,log(x)/a^2-2*log(a+b*sqrt(x))/a^2+2/(a*(a+b*sqrt(x)))],
[1/(x^2*(a+b*sqrt(x))^2),x,3,(-1)/(a^2*x)+3*b^2*log(x)/a^4-6*b^2*log(a+b*sqrt(x))/a^4+4*b/(a^3*sqrt(x))+2*b^2/(a^3*(a+b*sqrt(x)))],
[1/(x^3*(a+b*sqrt(x))^2),x,3,(-1/2)/(a^2*x^2)+4/3*b/(a^3*x^(3/2))-3*b^2/(a^4*x)+5*b^4*log(x)/a^6-10*b^4*log(a+b*sqrt(x))/a^6+8*b^3/(a^5*sqrt(x))+2*b^4/(a^5*(a+b*sqrt(x)))],
[1/(x^4*(a+b*sqrt(x))^2),x,3,(-1/3)/(a^2*x^3)+4/5*b/(a^3*x^(5/2))-3/2*b^2/(a^4*x^2)+8/3*b^3/(a^5*x^(3/2))-5*b^4/(a^6*x)+7*b^6*log(x)/a^8-14*b^6*log(a+b*sqrt(x))/a^8+12*b^5/(a^7*sqrt(x))+2*b^6/(a^7*(a+b*sqrt(x)))],
[x^3/(a+b*sqrt(x))^3,x,3,-10*a^3*x/b^6+4*a^2*x^(3/2)/b^5-3/2*a*x^2/b^4+2/5*x^(5/2)/b^3-42*a^5*log(a+b*sqrt(x))/b^8+30*a^4*sqrt(x)/b^7+a^7/(b^8*(a+b*sqrt(x))^2)-14*a^6/(b^8*(a+b*sqrt(x)))],
[x^2/(a+b*sqrt(x))^3,x,3,-3*a*x/b^4+2/3*x^(3/2)/b^3-20*a^3*log(a+b*sqrt(x))/b^6+12*a^2*sqrt(x)/b^5+a^5/(b^6*(a+b*sqrt(x))^2)-10*a^4/(b^6*(a+b*sqrt(x)))],
[x/(a+b*sqrt(x))^3,x,3,-6*a*log(a+b*sqrt(x))/b^4+2*sqrt(x)/b^3+a^3/(b^4*(a+b*sqrt(x))^2)-6*a^2/(b^4*(a+b*sqrt(x)))],
[1/(a+b*sqrt(x))^3,x,2,x/(a*(a+b*sqrt(x))^2)],
[1/(x*(a+b*sqrt(x))^3),x,3,log(x)/a^3-2*log(a+b*sqrt(x))/a^3+1/(a*(a+b*sqrt(x))^2)+2/(a^2*(a+b*sqrt(x)))],
[1/(x^2*(a+b*sqrt(x))^3),x,3,(-1)/(a^3*x)+6*b^2*log(x)/a^5-12*b^2*log(a+b*sqrt(x))/a^5+6*b/(a^4*sqrt(x))+b^2/(a^3*(a+b*sqrt(x))^2)+6*b^2/(a^4*(a+b*sqrt(x)))],
[1/(x^3*(a+b*sqrt(x))^3),x,3,(-1/2)/(a^3*x^2)+2*b/(a^4*x^(3/2))-6*b^2/(a^5*x)+15*b^4*log(x)/a^7-30*b^4*log(a+b*sqrt(x))/a^7+20*b^3/(a^6*sqrt(x))+b^4/(a^5*(a+b*sqrt(x))^2)+10*b^4/(a^6*(a+b*sqrt(x)))],
[1/(x^4*(a+b*sqrt(x))^3),x,3,(-1/3)/(a^3*x^3)+6/5*b/(a^4*x^(5/2))-3*b^2/(a^5*x^2)+20/3*b^3/(a^6*x^(3/2))-15*b^4/(a^7*x)+28*b^6*log(x)/a^9-56*b^6*log(a+b*sqrt(x))/a^9+42*b^5/(a^8*sqrt(x))+b^6/(a^7*(a+b*sqrt(x))^2)+14*b^6/(a^8*(a+b*sqrt(x)))],
[x^4/(a+b*sqrt(x))^5,x,3,-35*a^3*x/b^8+10*a^2*x^(3/2)/b^7-5/2*a*x^2/b^6+2/5*x^(5/2)/b^5-252*a^5*log(a+b*sqrt(x))/b^10+140*a^4*sqrt(x)/b^9+1/2*a^9/(b^10*(a+b*sqrt(x))^4)-6*a^8/(b^10*(a+b*sqrt(x))^3)+36*a^7/(b^10*(a+b*sqrt(x))^2)-168*a^6/(b^10*(a+b*sqrt(x)))],
[x^3/(a+b*sqrt(x))^5,x,3,-5*a*x/b^6+2/3*x^(3/2)/b^5-70*a^3*log(a+b*sqrt(x))/b^8+30*a^2*sqrt(x)/b^7+1/2*a^7/(b^8*(a+b*sqrt(x))^4)-14/3*a^6/(b^8*(a+b*sqrt(x))^3)+21*a^5/(b^8*(a+b*sqrt(x))^2)-70*a^4/(b^8*(a+b*sqrt(x)))],
[x^2/(a+b*sqrt(x))^5,x,3,-10*a*log(a+b*sqrt(x))/b^6+2*sqrt(x)/b^5+1/2*a^5/(b^6*(a+b*sqrt(x))^4)-10/3*a^4/(b^6*(a+b*sqrt(x))^3)+10*a^3/(b^6*(a+b*sqrt(x))^2)-20*a^2/(b^6*(a+b*sqrt(x)))],
[x/(a+b*sqrt(x))^5,x,1,1/2*x^2/(a*(a+b*sqrt(x))^4)],
[1/(a+b*sqrt(x))^5,x,3,1/2*a/(b^2*(a+b*sqrt(x))^4)+(-2/3)/(b^2*(a+b*sqrt(x))^3)],
[1/(x*(a+b*sqrt(x))^5),x,3,log(x)/a^5-2*log(a+b*sqrt(x))/a^5+1/2/(a*(a+b*sqrt(x))^4)+2/3/(a^2*(a+b*sqrt(x))^3)+1/(a^3*(a+b*sqrt(x))^2)+2/(a^4*(a+b*sqrt(x)))],
[1/(x^2*(a+b*sqrt(x))^5),x,3,(-1)/(a^5*x)+15*b^2*log(x)/a^7-30*b^2*log(a+b*sqrt(x))/a^7+10*b/(a^6*sqrt(x))+1/2*b^2/(a^3*(a+b*sqrt(x))^4)+2*b^2/(a^4*(a+b*sqrt(x))^3)+6*b^2/(a^5*(a+b*sqrt(x))^2)+20*b^2/(a^6*(a+b*sqrt(x)))],
[1/(x^3*(a+b*sqrt(x))^5),x,3,(-1/2)/(a^5*x^2)+10/3*b/(a^6*x^(3/2))-15*b^2/(a^7*x)+70*b^4*log(x)/a^9-140*b^4*log(a+b*sqrt(x))/a^9+70*b^3/(a^8*sqrt(x))+1/2*b^4/(a^5*(a+b*sqrt(x))^4)+10/3*b^4/(a^6*(a+b*sqrt(x))^3)+15*b^4/(a^7*(a+b*sqrt(x))^2)+70*b^4/(a^8*(a+b*sqrt(x)))],
[x^5/(a+b*sqrt(x))^8,x,3,36*a^2*x/b^10-16/3*a*x^(3/2)/b^9+1/2*x^2/b^8+660*a^4*log(a+b*sqrt(x))/b^12-240*a^3*sqrt(x)/b^11+2/7*a^11/(b^12*(a+b*sqrt(x))^7)-11/3*a^10/(b^12*(a+b*sqrt(x))^6)+22*a^9/(b^12*(a+b*sqrt(x))^5)-165/2*a^8/(b^12*(a+b*sqrt(x))^4)+220*a^7/(b^12*(a+b*sqrt(x))^3)-462*a^6/(b^12*(a+b*sqrt(x))^2)+924*a^5/(b^12*(a+b*sqrt(x)))],
[x^4/(a+b*sqrt(x))^8,x,3,x/b^8+72*a^2*log(a+b*sqrt(x))/b^10-16*a*sqrt(x)/b^9+2/7*a^9/(b^10*(a+b*sqrt(x))^7)-3*a^8/(b^10*(a+b*sqrt(x))^6)+72/5*a^7/(b^10*(a+b*sqrt(x))^5)-42*a^6/(b^10*(a+b*sqrt(x))^4)+84*a^5/(b^10*(a+b*sqrt(x))^3)-126*a^4/(b^10*(a+b*sqrt(x))^2)+168*a^3/(b^10*(a+b*sqrt(x)))],
[x^3/(a+b*sqrt(x))^8,x,3,2*log(a+b*sqrt(x))/b^8+2/7*a^7/(b^8*(a+b*sqrt(x))^7)-7/3*a^6/(b^8*(a+b*sqrt(x))^6)+42/5*a^5/(b^8*(a+b*sqrt(x))^5)-35/2*a^4/(b^8*(a+b*sqrt(x))^4)+70/3*a^3/(b^8*(a+b*sqrt(x))^3)-21*a^2/(b^8*(a+b*sqrt(x))^2)+14*a/(b^8*(a+b*sqrt(x)))],
[x^2/(a+b*sqrt(x))^8,x,3,2/7*x^3/(a*(a+b*sqrt(x))^7)+1/21*x^3/(a^2*(a+b*sqrt(x))^6)],
[x/(a+b*sqrt(x))^8,x,3,2/7*a^3/(b^4*(a+b*sqrt(x))^7)-a^2/(b^4*(a+b*sqrt(x))^6)+6/5*a/(b^4*(a+b*sqrt(x))^5)+(-1/2)/(b^4*(a+b*sqrt(x))^4)],
[1/(a+b*sqrt(x))^8,x,3,2/7*a/(b^2*(a+b*sqrt(x))^7)+(-1/3)/(b^2*(a+b*sqrt(x))^6)],
[1/(x*(a+b*sqrt(x))^8),x,3,log(x)/a^8-2*log(a+b*sqrt(x))/a^8+2/7/(a*(a+b*sqrt(x))^7)+1/3/(a^2*(a+b*sqrt(x))^6)+2/5/(a^3*(a+b*sqrt(x))^5)+1/2/(a^4*(a+b*sqrt(x))^4)+2/3/(a^5*(a+b*sqrt(x))^3)+1/(a^6*(a+b*sqrt(x))^2)+2/(a^7*(a+b*sqrt(x)))],
[1/(x^2*(a+b*sqrt(x))^8),x,3,(-1)/(a^8*x)+36*b^2*log(x)/a^10-72*b^2*log(a+b*sqrt(x))/a^10+16*b/(a^9*sqrt(x))+2/7*b^2/(a^3*(a+b*sqrt(x))^7)+b^2/(a^4*(a+b*sqrt(x))^6)+12/5*b^2/(a^5*(a+b*sqrt(x))^5)+5*b^2/(a^6*(a+b*sqrt(x))^4)+10*b^2/(a^7*(a+b*sqrt(x))^3)+21*b^2/(a^8*(a+b*sqrt(x))^2)+56*b^2/(a^9*(a+b*sqrt(x)))],
[1/(x^3*(a+b*sqrt(x))^8),x,3,(-1/2)/(a^8*x^2)+16/3*b/(a^9*x^(3/2))-36*b^2/(a^10*x)+330*b^4*log(x)/a^12-660*b^4*log(a+b*sqrt(x))/a^12+240*b^3/(a^11*sqrt(x))+2/7*b^4/(a^5*(a+b*sqrt(x))^7)+5/3*b^4/(a^6*(a+b*sqrt(x))^6)+6*b^4/(a^7*(a+b*sqrt(x))^5)+35/2*b^4/(a^8*(a+b*sqrt(x))^4)+140/3*b^4/(a^9*(a+b*sqrt(x))^3)+126*b^4/(a^10*(a+b*sqrt(x))^2)+420*b^4/(a^11*(a+b*sqrt(x)))],
[1/(x*(2+b*sqrt(x))),x,4,1/2*log(x)-log(2+b*sqrt(x))],

# Integrands of the form x^m (a+b Sqrt[x])^(p/2)

# p>0
[x^2*sqrt(a+b*sqrt(x)),x,3,-4/3*a^5*(a+b*sqrt(x))^(3/2)/b^6+4*a^4*(a+b*sqrt(x))^(5/2)/b^6-40/7*a^3*(a+b*sqrt(x))^(7/2)/b^6+40/9*a^2*(a+b*sqrt(x))^(9/2)/b^6-20/11*a*(a+b*sqrt(x))^(11/2)/b^6+4/13*(a+b*sqrt(x))^(13/2)/b^6],
[x*sqrt(a+b*sqrt(x)),x,3,-4/3*a^3*(a+b*sqrt(x))^(3/2)/b^4+12/5*a^2*(a+b*sqrt(x))^(5/2)/b^4-12/7*a*(a+b*sqrt(x))^(7/2)/b^4+4/9*(a+b*sqrt(x))^(9/2)/b^4],
[sqrt(a+b*sqrt(x)),x,3,-4/3*a*(a+b*sqrt(x))^(3/2)/b^2+4/5*(a+b*sqrt(x))^(5/2)/b^2],
[sqrt(a+b*sqrt(x))/x,x,4,-4*arctanh(sqrt(a+b*sqrt(x))/sqrt(a))*sqrt(a)+4*sqrt(a+b*sqrt(x))],
[sqrt(a+b*sqrt(x))/x^2,x,5,1/2*b^2*arctanh(sqrt(a+b*sqrt(x))/sqrt(a))/a^(3/2)-sqrt(a+b*sqrt(x))/x-1/2*b*sqrt(a+b*sqrt(x))/(a*sqrt(x))],
[sqrt(a+b*sqrt(x))/x^3,x,7,5/32*b^4*arctanh(sqrt(a+b*sqrt(x))/sqrt(a))/a^(7/2)-1/2*sqrt(a+b*sqrt(x))/x^2-1/12*b*sqrt(a+b*sqrt(x))/(a*x^(3/2))+5/48*b^2*sqrt(a+b*sqrt(x))/(a^2*x)-5/32*b^3*sqrt(a+b*sqrt(x))/(a^3*sqrt(x))],

# p<0
[x^2/sqrt(a+b*sqrt(x)),x,3,20/3*a^4*(a+b*sqrt(x))^(3/2)/b^6-8*a^3*(a+b*sqrt(x))^(5/2)/b^6+40/7*a^2*(a+b*sqrt(x))^(7/2)/b^6-20/9*a*(a+b*sqrt(x))^(9/2)/b^6+4/11*(a+b*sqrt(x))^(11/2)/b^6-4*a^5*sqrt(a+b*sqrt(x))/b^6],
[x/sqrt(a+b*sqrt(x)),x,3,4*a^2*(a+b*sqrt(x))^(3/2)/b^4-12/5*a*(a+b*sqrt(x))^(5/2)/b^4+4/7*(a+b*sqrt(x))^(7/2)/b^4-4*a^3*sqrt(a+b*sqrt(x))/b^4],
[1/sqrt(a+b*sqrt(x)),x,3,4/3*(a+b*sqrt(x))^(3/2)/b^2-4*a*sqrt(a+b*sqrt(x))/b^2],
[1/(x*sqrt(a+b*sqrt(x))),x,3,-4*arctanh(sqrt(a+b*sqrt(x))/sqrt(a))/sqrt(a)],
[1/(x^2*sqrt(a+b*sqrt(x))),x,5,-3/2*b^2*arctanh(sqrt(a+b*sqrt(x))/sqrt(a))/a^(5/2)-sqrt(a+b*sqrt(x))/(a*x)+3/2*b*sqrt(a+b*sqrt(x))/(a^2*sqrt(x))],
[1/(x^3*sqrt(a+b*sqrt(x))),x,7,-35/32*b^4*arctanh(sqrt(a+b*sqrt(x))/sqrt(a))/a^(9/2)-1/2*sqrt(a+b*sqrt(x))/(a*x^2)+7/12*b*sqrt(a+b*sqrt(x))/(a^2*x^(3/2))-35/48*b^2*sqrt(a+b*sqrt(x))/(a^3*x)+35/32*b^3*sqrt(a+b*sqrt(x))/(a^4*sqrt(x))],

# Integrands of the form x^(m/2) (a+b Sqrt[x])^p

# p>0
[sqrt(x)*(a+b*sqrt(x))^n,x,3,2*a^2*(a+b*sqrt(x))^(1+n)/(b^3*(1+n))-4*a*(a+b*sqrt(x))^(2+n)/(b^3*(2+n))+2*(a+b*sqrt(x))^(3+n)/(b^3*(3+n))],
[(a+b*sqrt(x))^n/sqrt(x),x,1,2*(a+b*sqrt(x))^(1+n)/(b*(1+n))],
[(1+sqrt(x))/sqrt(x),x,2,x+2*sqrt(x)],
[(1+sqrt(x))^2/sqrt(x),x,1,2/3*(1+sqrt(x))^3],
[(1+sqrt(x))^3/sqrt(x),x,1,1/2*(1+sqrt(x))^4],

# p<0
[sqrt(x)/(1+sqrt(x)),x,3,x+2*log(1+sqrt(x))-2*sqrt(x)],
[1/(sqrt(x)*(1+sqrt(x))),x,1,2*log(1+sqrt(x))],
[1/(sqrt(x)*(1+sqrt(x))^2),x,1,(-2)/(1+sqrt(x))],
[1/(sqrt(x)*(1+sqrt(x))^3),x,1,(-1)/(1+sqrt(x))^2],

# Integrands of the form x^(m/2) (a+b Sqrt[x])^(p/2)

# p>0
[sqrt(x)*sqrt(1+sqrt(x)),x,3,4/3*(1+sqrt(x))^(3/2)-8/5*(1+sqrt(x))^(5/2)+4/7*(1+sqrt(x))^(7/2)],
[sqrt(1+sqrt(x))/sqrt(x),x,1,4/3*(1+sqrt(x))^(3/2)],

# p<0

# Integrands of the form x^(m/3) (a+b Sqrt[x])^p

# p>0

# p<0
[x^(1/3)/(1+sqrt(x)),x,7,-3*x^(1/3)+6/5*x^(5/6)-3*log(1+x^(1/6))+log(1+sqrt(x))-2*arctan((1-2*x^(1/6))/sqrt(3))*sqrt(3)],

# Integrands of the form x^m (a+b Sqrt[x])^p with m symbolic
[x^m*(a+b*sqrt(x))^4,x,2,a^4*x^(1+m)/(1+m)+8*a^3*b*x^(3/2+m)/(3+2*m)+6*a^2*b^2*x^(2+m)/(2+m)+8*a*b^3*x^(5/2+m)/(5+2*m)+b^4*x^(3+m)/(3+m)],
[x^m*(a+b*sqrt(x))^3,x,2,a^3*x^(1+m)/(1+m)+6*a^2*b*x^(3/2+m)/(3+2*m)+3*a*b^2*x^(2+m)/(2+m)+2*b^3*x^(5/2+m)/(5+2*m)],
[x^m*(a+b*sqrt(x))^2,x,2,a^2*x^(1+m)/(1+m)+4*a*b*x^(3/2+m)/(3+2*m)+b^2*x^(2+m)/(2+m)],
[x^m*(a+b*sqrt(x)),x,2,a*x^(1+m)/(1+m)+2*b*x^(3/2+m)/(3+2*m)],
[x^m/(a+b*sqrt(x)),x,2,x^(1+m)*hypergeom([1,2*(1+m)],[3+2*m],-b*sqrt(x)/a)/(a*(1+m))],
[x^m/(a+b*sqrt(x))^2,x,2,x^(1+m)*hypergeom([2,2*(1+m)],[3+2*m],-b*sqrt(x)/a)/(a^2*(1+m))],

# Integrands of the form x^m (a+b Sqrt[x])^p with p symbolic
[x^m*(a+b*sqrt(x))^p,x,3,-2*x^(1+m)*hypergeom([1,3+2*m+p],[2+p],(a+b*sqrt(x))/a)*(a+b*sqrt(x))^(1+p)/(a*(1+p)),x^(1+m)*hypergeom([2*(1+m),-p],[3+2*m],-b*sqrt(x)/a)*(a+b*sqrt(x))^p/((1+m)*(1+b*sqrt(x)/a)^p)],
[x^3*(a+b*sqrt(x))^p,x,3,-2*a^7*(a+b*sqrt(x))^(1+p)/(b^8*(1+p))+14*a^6*(a+b*sqrt(x))^(2+p)/(b^8*(2+p))-42*a^5*(a+b*sqrt(x))^(3+p)/(b^8*(3+p))+70*a^4*(a+b*sqrt(x))^(4+p)/(b^8*(4+p))-70*a^3*(a+b*sqrt(x))^(5+p)/(b^8*(5+p))+42*a^2*(a+b*sqrt(x))^(6+p)/(b^8*(6+p))-14*a*(a+b*sqrt(x))^(7+p)/(b^8*(7+p))+2*(a+b*sqrt(x))^(8+p)/(b^8*(8+p))],
[x^2*(a+b*sqrt(x))^p,x,3,-2*a^5*(a+b*sqrt(x))^(1+p)/(b^6*(1+p))+10*a^4*(a+b*sqrt(x))^(2+p)/(b^6*(2+p))-20*a^3*(a+b*sqrt(x))^(3+p)/(b^6*(3+p))+20*a^2*(a+b*sqrt(x))^(4+p)/(b^6*(4+p))-10*a*(a+b*sqrt(x))^(5+p)/(b^6*(5+p))+2*(a+b*sqrt(x))^(6+p)/(b^6*(6+p))],
[x*(a+b*sqrt(x))^p,x,3,-2*a^3*(a+b*sqrt(x))^(1+p)/(b^4*(1+p))+6*a^2*(a+b*sqrt(x))^(2+p)/(b^4*(2+p))-6*a*(a+b*sqrt(x))^(3+p)/(b^4*(3+p))+2*(a+b*sqrt(x))^(4+p)/(b^4*(4+p))],
[(a+b*sqrt(x))^p,x,3,-2*a*(a+b*sqrt(x))^(1+p)/(b^2*(1+p))+2*(a+b*sqrt(x))^(2+p)/(b^2*(2+p))],
[(a+b*sqrt(x))^p/x,x,2,-2*hypergeom([1,1+p],[2+p],1+b*sqrt(x)/a)*(a+b*sqrt(x))^(1+p)/(a*(1+p))],
[(a+b*sqrt(x))^p/x^2,x,2,-2*b^2*hypergeom([3,1+p],[2+p],1+b*sqrt(x)/a)*(a+b*sqrt(x))^(1+p)/(a^3*(1+p))],

# Integrands of the form (c x)^m (a+b x^(3/2))^p

# Integrands of the form x^(m/2) (a+b x^(3/2))^p

# p>0

# p<0
[sqrt(x)/(1+x^(3/2)),x,1,2/3*log(1+x^(3/2))],

# Integrands of the form x^m (a+b x^(3/2))^(p/3)

# p>0

# p<0
[x^3/(a+b*x^(3/2))^(2/3),x,4,-5/9*a*x*(a+b*x^(3/2))^(1/3)/b^2+1/3*x^(5/2)*(a+b*x^(3/2))^(1/3)/b-5/9*a^2*log(-(a+b*x^(3/2))^(1/3)+b^(1/3)*sqrt(x))/b^(8/3)-10/9*a^2*arctan((1+2*b^(1/3)*sqrt(x)/(a+b*x^(3/2))^(1/3))/sqrt(3))/(b^(8/3)*sqrt(3))],
[1/(a+b*x^(3/2))^(2/3),x,2,-log(-(a+b*x^(3/2))^(1/3)+b^(1/3)*sqrt(x))/b^(2/3)-2*arctan((1+2*b^(1/3)*sqrt(x)/(a+b*x^(3/2))^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[1/(x^3*(a+b*x^(3/2))^(2/3)),x,2,-1/2*(a+b*x^(3/2))^(1/3)/(a*x^2)+3/2*b*(a+b*x^(3/2))^(1/3)/(a^2*sqrt(x))],
[1/(x^6*(a+b*x^(3/2))^(2/3)),x,4,-1/5*(a+b*x^(3/2))^(1/3)/(a*x^5)+9/35*b*(a+b*x^(3/2))^(1/3)/(a^2*x^(7/2))-27/70*b^2*(a+b*x^(3/2))^(1/3)/(a^3*x^2)+81/70*b^3*(a+b*x^(3/2))^(1/3)/(a^4*sqrt(x))],
[1/(x^9*(a+b*x^(3/2))^(2/3)),x,6,-1/8*(a+b*x^(3/2))^(1/3)/(a*x^8)+15/104*b*(a+b*x^(3/2))^(1/3)/(a^2*x^(13/2))-9/52*b^2*(a+b*x^(3/2))^(1/3)/(a^3*x^5)+81/364*b^3*(a+b*x^(3/2))^(1/3)/(a^4*x^(7/2))-243/728*b^4*(a+b*x^(3/2))^(1/3)/(a^5*x^2)+729/728*b^5*(a+b*x^(3/2))^(1/3)/(a^6*sqrt(x))],
[x^8/(a+b*x^(3/2))^(2/3),x,3,-2*a^5*(a+b*x^(3/2))^(1/3)/b^6+5/2*a^4*(a+b*x^(3/2))^(4/3)/b^6-20/7*a^3*(a+b*x^(3/2))^(7/3)/b^6+2*a^2*(a+b*x^(3/2))^(10/3)/b^6-10/13*a*(a+b*x^(3/2))^(13/3)/b^6+1/8*(a+b*x^(3/2))^(16/3)/b^6],
[x^5/(a+b*x^(3/2))^(2/3),x,3,-2*a^3*(a+b*x^(3/2))^(1/3)/b^4+3/2*a^2*(a+b*x^(3/2))^(4/3)/b^4-6/7*a*(a+b*x^(3/2))^(7/3)/b^4+1/5*(a+b*x^(3/2))^(10/3)/b^4],
[x^2/(a+b*x^(3/2))^(2/3),x,3,-2*a*(a+b*x^(3/2))^(1/3)/b^2+1/2*(a+b*x^(3/2))^(4/3)/b^2],
[1/(x*(a+b*x^(3/2))^(2/3)),x,5,-1/2*log(x)/a^(2/3)+log(a^(1/3)-(a+b*x^(3/2))^(1/3))/a^(2/3)-2*arctan((a^(1/3)+2*(a+b*x^(3/2))^(1/3))/(a^(1/3)*sqrt(3)))/(a^(2/3)*sqrt(3))],
[1/(x^4*(a+b*x^(3/2))^(2/3)),x,7,-1/3*(a+b*x^(3/2))^(1/3)/(a*x^3)+5/9*b*(a+b*x^(3/2))^(1/3)/(a^2*x^(3/2))-5/18*b^2*log(x)/a^(8/3)+5/9*b^2*log(a^(1/3)-(a+b*x^(3/2))^(1/3))/a^(8/3)-10/9*b^2*arctan((a^(1/3)+2*(a+b*x^(3/2))^(1/3))/(a^(1/3)*sqrt(3)))/(a^(8/3)*sqrt(3))],
[x^4/(a+b*x^(3/2))^(2/3),x,3,1/5*x^5*(a+b*x^(3/2))^(1/3)*hypergeom([1,11/3],[13/3],-b*x^(3/2)/a)/a,1/5*x^5*(1+b*x^(3/2)/a)^(2/3)*hypergeom([2/3,10/3],[13/3],-b*x^(3/2)/a)/(a+b*x^(3/2))^(2/3)],
[x/(a+b*x^(3/2))^(2/3),x,3,1/2*x^2*(a+b*x^(3/2))^(1/3)*hypergeom([1,5/3],[7/3],-b*x^(3/2)/a)/a,1/2*x^2*(1+b*x^(3/2)/a)^(2/3)*hypergeom([2/3,4/3],[7/3],-b*x^(3/2)/a)/(a+b*x^(3/2))^(2/3)],
[1/(x^2*(a+b*x^(3/2))^(2/3)),x,3,-(a+b*x^(3/2))^(1/3)*hypergeom([-1/3,1],[1/3],-b*x^(3/2)/a)/(a*x),-(1+b*x^(3/2)/a)^(2/3)*hypergeom([-2/3,2/3],[1/3],-b*x^(3/2)/a)/(x*(a+b*x^(3/2))^(2/3))],
[1/(x^5*(a+b*x^(3/2))^(2/3)),x,3,-1/4*(1+b*x^(3/2)/a)^(2/3)*hypergeom([-8/3,2/3],[-5/3],-b*x^(3/2)/a)/(x^4*(a+b*x^(3/2))^(2/3))],

# Integrands of the form (c x)^m (a+b x^(1/3))^p

# Integrands of the form x^m (a+b x^(1/3))^p

# p>0
[(a+b*x^(1/3))*x^4,x,2,1/5*a*x^5+3/16*b*x^(16/3)],
[(a+b*x^(1/3))*x^3,x,2,1/4*a*x^4+3/13*b*x^(13/3)],
[(a+b*x^(1/3))*x^2,x,2,1/3*a*x^3+3/10*b*x^(10/3)],
[(a+b*x^(1/3))*x,x,2,1/2*a*x^2+3/7*b*x^(7/3)],
[a+b*x^(1/3),x,1,a*x+3/4*b*x^(4/3)],
[(a+b*x^(1/3))/x,x,2,3*b*x^(1/3)+a*log(x)],
[(a+b*x^(1/3))/x^2,x,2,-a/x-3/2*b/x^(2/3)],
[(a+b*x^(1/3))/x^3,x,2,-1/2*a/x^2-3/5*b/x^(5/3)],
[(a+b*x^(1/3))/x^4,x,2,-1/3*a/x^3-3/8*b/x^(8/3)],
[(a+b*x^(1/3))^2*x^4,x,3,1/5*a^2*x^5+3/8*a*b*x^(16/3)+3/17*b^2*x^(17/3)],
[(a+b*x^(1/3))^2*x^3,x,3,1/4*a^2*x^4+6/13*a*b*x^(13/3)+3/14*b^2*x^(14/3)],
[(a+b*x^(1/3))^2*x^2,x,3,1/3*a^2*x^3+3/5*a*b*x^(10/3)+3/11*b^2*x^(11/3)],
[(a+b*x^(1/3))^2*x,x,3,1/2*a^2*x^2+6/7*a*b*x^(7/3)+3/8*b^2*x^(8/3)],
[(a+b*x^(1/3))^2,x,3,a^2*x+3/2*a*b*x^(4/3)+3/5*b^2*x^(5/3)],
[(a+b*x^(1/3))^2/x,x,3,6*a*b*x^(1/3)+3/2*b^2*x^(2/3)+a^2*log(x)],
[(a+b*x^(1/3))^2/x^2,x,1,-(a+b*x^(1/3))^3/(a*x)],
[(a+b*x^(1/3))^2/x^3,x,3,-1/2*a^2/x^2-6/5*a*b/x^(5/3)-3/4*b^2/x^(4/3)],
[(a+b*x^(1/3))^2/x^4,x,3,-1/3*a^2/x^3-3/4*a*b/x^(8/3)-3/7*b^2/x^(7/3)],
[(a+b*x^(1/3))^3*x^4,x,3,1/5*a^3*x^5+9/16*a^2*b*x^(16/3)+9/17*a*b^2*x^(17/3)+1/6*b^3*x^6],
[(a+b*x^(1/3))^3*x^3,x,3,1/4*a^3*x^4+9/13*a^2*b*x^(13/3)+9/14*a*b^2*x^(14/3)+1/5*b^3*x^5],
[(a+b*x^(1/3))^3*x^2,x,3,1/3*a^3*x^3+9/10*a^2*b*x^(10/3)+9/11*a*b^2*x^(11/3)+1/4*b^3*x^4],
[(a+b*x^(1/3))^3*x,x,3,1/2*a^3*x^2+9/7*a^2*b*x^(7/3)+9/8*a*b^2*x^(8/3)+1/3*b^3*x^3],
[(a+b*x^(1/3))^3,x,3,a^3*x+9/4*a^2*b*x^(4/3)+9/5*a*b^2*x^(5/3)+1/2*b^3*x^2],
[(a+b*x^(1/3))^3/x,x,3,9*a^2*b*x^(1/3)+9/2*a*b^2*x^(2/3)+b^3*x+a^3*log(x)],
[(a+b*x^(1/3))^3/x^2,x,3,-a^3/x-9/2*a^2*b/x^(2/3)-9*a*b^2/x^(1/3)+b^3*log(x)],
[(a+b*x^(1/3))^3/x^3,x,3,-1/2*a^3/x^2-9/5*a^2*b/x^(5/3)-9/4*a*b^2/x^(4/3)-b^3/x],
[(a+b*x^(1/3))^3/x^4,x,3,-1/3*a^3/x^3-9/8*a^2*b/x^(8/3)-9/7*a*b^2/x^(7/3)-1/2*b^3/x^2],
[(a+b*x^(1/3))^5*x^4,x,3,1/5*a^5*x^5+15/16*a^4*b*x^(16/3)+30/17*a^3*b^2*x^(17/3)+5/3*a^2*b^3*x^6+15/19*a*b^4*x^(19/3)+3/20*b^5*x^(20/3)],
[(a+b*x^(1/3))^5*x^3,x,3,1/4*a^5*x^4+15/13*a^4*b*x^(13/3)+15/7*a^3*b^2*x^(14/3)+2*a^2*b^3*x^5+15/16*a*b^4*x^(16/3)+3/17*b^5*x^(17/3)],
[(a+b*x^(1/3))^5*x^2,x,3,1/3*a^5*x^3+3/2*a^4*b*x^(10/3)+30/11*a^3*b^2*x^(11/3)+5/2*a^2*b^3*x^4+15/13*a*b^4*x^(13/3)+3/14*b^5*x^(14/3)],
[(a+b*x^(1/3))^5*x,x,3,1/2*a^5*x^2+15/7*a^4*b*x^(7/3)+15/4*a^3*b^2*x^(8/3)+10/3*a^2*b^3*x^3+3/2*a*b^4*x^(10/3)+3/11*b^5*x^(11/3)],
[(a+b*x^(1/3))^5,x,3,1/2*a^2*(a+b*x^(1/3))^6/b^3-6/7*a*(a+b*x^(1/3))^7/b^3+3/8*(a+b*x^(1/3))^8/b^3],
[(a+b*x^(1/3))^5/x,x,3,15*a^4*b*x^(1/3)+15*a^3*b^2*x^(2/3)+10*a^2*b^3*x+15/4*a*b^4*x^(4/3)+3/5*b^5*x^(5/3)+a^5*log(x)],
[(a+b*x^(1/3))^5/x^2,x,3,-a^5/x-15/2*a^4*b/x^(2/3)-30*a^3*b^2/x^(1/3)+15*a*b^4*x^(1/3)+3/2*b^5*x^(2/3)+10*a^2*b^3*log(x)],
[(a+b*x^(1/3))^5/x^3,x,1,-1/2*(a+b*x^(1/3))^6/(a*x^2)],
[(a+b*x^(1/3))^5/x^4,x,3,-1/3*a^5/x^3-15/8*a^4*b/x^(8/3)-30/7*a^3*b^2/x^(7/3)-5*a^2*b^3/x^2-3*a*b^4/x^(5/3)-3/4*b^5/x^(4/3)],
[(a+b*x^(1/3))^5/x^5,x,3,-1/4*a^5/x^4-15/11*a^4*b/x^(11/3)-3*a^3*b^2/x^(10/3)-10/3*a^2*b^3/x^3-15/8*a*b^4/x^(8/3)-3/7*b^5/x^(7/3)],
[(a+b*x^(1/3))^5/x^6,x,3,-1/5*a^5/x^5-15/14*a^4*b/x^(14/3)-30/13*a^3*b^2/x^(13/3)-5/2*a^2*b^3/x^4-15/11*a*b^4/x^(11/3)-3/10*b^5/x^(10/3)],
[(a+b*x^(1/3))^5/x^7,x,3,-1/6*a^5/x^6-15/17*a^4*b/x^(17/3)-15/8*a^3*b^2/x^(16/3)-2*a^2*b^3/x^5-15/14*a*b^4/x^(14/3)-3/13*b^5/x^(13/3)],
[(a+b*x^(1/3))^10*x^4,x,3,1/5*a^10*x^5+15/8*a^9*b*x^(16/3)+135/17*a^8*b^2*x^(17/3)+20*a^7*b^3*x^6+630/19*a^6*b^4*x^(19/3)+189/5*a^5*b^5*x^(20/3)+30*a^4*b^6*x^7+180/11*a^3*b^7*x^(22/3)+135/23*a^2*b^8*x^(23/3)+5/4*a*b^9*x^8+3/25*b^10*x^(25/3)],
[(a+b*x^(1/3))^10*x^3,x,3,1/4*a^10*x^4+30/13*a^9*b*x^(13/3)+135/14*a^8*b^2*x^(14/3)+24*a^7*b^3*x^5+315/8*a^6*b^4*x^(16/3)+756/17*a^5*b^5*x^(17/3)+35*a^4*b^6*x^6+360/19*a^3*b^7*x^(19/3)+27/4*a^2*b^8*x^(20/3)+10/7*a*b^9*x^7+3/22*b^10*x^(22/3)],
[(a+b*x^(1/3))^10*x^2,x,3,3/11*a^8*(a+b*x^(1/3))^11/b^9-2*a^7*(a+b*x^(1/3))^12/b^9+84/13*a^6*(a+b*x^(1/3))^13/b^9-12*a^5*(a+b*x^(1/3))^14/b^9+14*a^4*(a+b*x^(1/3))^15/b^9-21/2*a^3*(a+b*x^(1/3))^16/b^9+84/17*a^2*(a+b*x^(1/3))^17/b^9-4/3*a*(a+b*x^(1/3))^18/b^9+3/19*(a+b*x^(1/3))^19/b^9],
[(a+b*x^(1/3))^10*x,x,3,-3/11*a^5*(a+b*x^(1/3))^11/b^6+5/4*a^4*(a+b*x^(1/3))^12/b^6-30/13*a^3*(a+b*x^(1/3))^13/b^6+15/7*a^2*(a+b*x^(1/3))^14/b^6-a*(a+b*x^(1/3))^15/b^6+3/16*(a+b*x^(1/3))^16/b^6],
[(a+b*x^(1/3))^10,x,3,3/11*a^2*(a+b*x^(1/3))^11/b^3-1/2*a*(a+b*x^(1/3))^12/b^3+3/13*(a+b*x^(1/3))^13/b^3],
[(a+b*x^(1/3))^10/x,x,3,30*a^9*b*x^(1/3)+135/2*a^8*b^2*x^(2/3)+120*a^7*b^3*x+315/2*a^6*b^4*x^(4/3)+756/5*a^5*b^5*x^(5/3)+105*a^4*b^6*x^2+360/7*a^3*b^7*x^(7/3)+135/8*a^2*b^8*x^(8/3)+10/3*a*b^9*x^3+3/10*b^10*x^(10/3)+a^10*log(x)],
[(a+b*x^(1/3))^10/x^2,x,3,-a^10/x-15*a^9*b/x^(2/3)-135*a^8*b^2/x^(1/3)+630*a^6*b^4*x^(1/3)+378*a^5*b^5*x^(2/3)+210*a^4*b^6*x+90*a^3*b^7*x^(4/3)+27*a^2*b^8*x^(5/3)+5*a*b^9*x^2+3/7*b^10*x^(7/3)+120*a^7*b^3*log(x)],
[(a+b*x^(1/3))^10/x^3,x,3,-1/2*a^10/x^2-6*a^9*b/x^(5/3)-135/4*a^8*b^2/x^(4/3)-120*a^7*b^3/x-315*a^6*b^4/x^(2/3)-756*a^5*b^5/x^(1/3)+360*a^3*b^7*x^(1/3)+135/2*a^2*b^8*x^(2/3)+10*a*b^9*x+3/4*b^10*x^(4/3)+210*a^4*b^6*log(x)],
[(a+b*x^(1/3))^10/x^4,x,3,-1/3*a^10/x^3-15/4*a^9*b/x^(8/3)-135/7*a^8*b^2/x^(7/3)-60*a^7*b^3/x^2-126*a^6*b^4/x^(5/3)-189*a^5*b^5/x^(4/3)-210*a^4*b^6/x-180*a^3*b^7/x^(2/3)-135*a^2*b^8/x^(1/3)+3*b^10*x^(1/3)+10*a*b^9*log(x)],
[(a+b*x^(1/3))^10/x^5,x,3,-1/4*(a+b*x^(1/3))^11/(a*x^4)+1/44*b*(a+b*x^(1/3))^11/(a^2*x^(11/3))],
[(a+b*x^(1/3))^10/x^6,x,6,-1/5*(a+b*x^(1/3))^11/(a*x^5)+2/35*b*(a+b*x^(1/3))^11/(a^2*x^(14/3))-6/455*b^2*(a+b*x^(1/3))^11/(a^3*x^(13/3))+1/455*b^3*(a+b*x^(1/3))^11/(a^4*x^4)-1/5005*b^4*(a+b*x^(1/3))^11/(a^5*x^(11/3))],
[(a+b*x^(1/3))^10/x^7,x,3,-1/6*a^10/x^6-30/17*a^9*b/x^(17/3)-135/16*a^8*b^2/x^(16/3)-24*a^7*b^3/x^5-45*a^6*b^4/x^(14/3)-756/13*a^5*b^5/x^(13/3)-105/2*a^4*b^6/x^4-360/11*a^3*b^7/x^(11/3)-27/2*a^2*b^8/x^(10/3)-10/3*a*b^9/x^3-3/8*b^10/x^(8/3)],
[(a+b*x^(1/3))^10/x^8,x,3,-1/7*a^10/x^7-3/2*a^9*b/x^(20/3)-135/19*a^8*b^2/x^(19/3)-20*a^7*b^3/x^6-630/17*a^6*b^4/x^(17/3)-189/4*a^5*b^5/x^(16/3)-42*a^4*b^6/x^5-180/7*a^3*b^7/x^(14/3)-135/13*a^2*b^8/x^(13/3)-5/2*a*b^9/x^4-3/11*b^10/x^(11/3)],
[(a+b*x^(1/3))^10/x^9,x,3,-1/8*a^10/x^8-30/23*a^9*b/x^(23/3)-135/22*a^8*b^2/x^(22/3)-120/7*a^7*b^3/x^7-63/2*a^6*b^4/x^(20/3)-756/19*a^5*b^5/x^(19/3)-35*a^4*b^6/x^6-360/17*a^3*b^7/x^(17/3)-135/16*a^2*b^8/x^(16/3)-2*a*b^9/x^5-3/14*b^10/x^(14/3)],
[(a+b*x^(1/3))^10/x^10,x,3,-1/9*a^10/x^9-15/13*a^9*b/x^(26/3)-27/5*a^8*b^2/x^(25/3)-15*a^7*b^3/x^8-630/23*a^6*b^4/x^(23/3)-378/11*a^5*b^5/x^(22/3)-30*a^4*b^6/x^7-18*a^3*b^7/x^(20/3)-135/19*a^2*b^8/x^(19/3)-5/3*a*b^9/x^6-3/17*b^10/x^(17/3)],
[(a+b*x^(1/3))^15*x^5,x,3,1/6*a^15*x^6+45/19*a^14*b*x^(19/3)+63/4*a^13*b^2*x^(20/3)+65*a^12*b^3*x^7+4095/22*a^11*b^4*x^(22/3)+9009/23*a^10*b^5*x^(23/3)+5005/8*a^9*b^6*x^8+3861/5*a^8*b^7*x^(25/3)+1485/2*a^7*b^8*x^(26/3)+5005/9*a^6*b^9*x^9+1287/4*a^5*b^10*x^(28/3)+4095/29*a^4*b^11*x^(29/3)+91/2*a^3*b^12*x^10+315/31*a^2*b^13*x^(31/3)+45/32*a*b^14*x^(32/3)+1/11*b^15*x^11],
[(a+b*x^(1/3))^15*x^4,x,3,1/5*a^15*x^5+45/16*a^14*b*x^(16/3)+315/17*a^13*b^2*x^(17/3)+455/6*a^12*b^3*x^6+4095/19*a^11*b^4*x^(19/3)+9009/20*a^10*b^5*x^(20/3)+715*a^9*b^6*x^7+1755/2*a^8*b^7*x^(22/3)+19305/23*a^7*b^8*x^(23/3)+5005/8*a^6*b^9*x^8+9009/25*a^5*b^10*x^(25/3)+315/2*a^4*b^11*x^(26/3)+455/9*a^3*b^12*x^9+45/4*a^2*b^13*x^(28/3)+45/29*a*b^14*x^(29/3)+1/10*b^15*x^10],
[(a+b*x^(1/3))^15*x^3,x,3,-3/16*a^11*(a+b*x^(1/3))^16/b^12+33/17*a^10*(a+b*x^(1/3))^17/b^12-55/6*a^9*(a+b*x^(1/3))^18/b^12+495/19*a^8*(a+b*x^(1/3))^19/b^12-99/2*a^7*(a+b*x^(1/3))^20/b^12+66*a^6*(a+b*x^(1/3))^21/b^12-63*a^5*(a+b*x^(1/3))^22/b^12+990/23*a^4*(a+b*x^(1/3))^23/b^12-165/8*a^3*(a+b*x^(1/3))^24/b^12+33/5*a^2*(a+b*x^(1/3))^25/b^12-33/26*a*(a+b*x^(1/3))^26/b^12+1/9*(a+b*x^(1/3))^27/b^12],
[(a+b*x^(1/3))^15*x^2,x,3,3/16*a^8*(a+b*x^(1/3))^16/b^9-24/17*a^7*(a+b*x^(1/3))^17/b^9+14/3*a^6*(a+b*x^(1/3))^18/b^9-168/19*a^5*(a+b*x^(1/3))^19/b^9+21/2*a^4*(a+b*x^(1/3))^20/b^9-8*a^3*(a+b*x^(1/3))^21/b^9+42/11*a^2*(a+b*x^(1/3))^22/b^9-24/23*a*(a+b*x^(1/3))^23/b^9+1/8*(a+b*x^(1/3))^24/b^9],
[(a+b*x^(1/3))^15*x,x,3,-3/16*a^5*(a+b*x^(1/3))^16/b^6+15/17*a^4*(a+b*x^(1/3))^17/b^6-5/3*a^3*(a+b*x^(1/3))^18/b^6+30/19*a^2*(a+b*x^(1/3))^19/b^6-3/4*a*(a+b*x^(1/3))^20/b^6+1/7*(a+b*x^(1/3))^21/b^6],
[(a+b*x^(1/3))^15,x,3,3/16*a^2*(a+b*x^(1/3))^16/b^3-6/17*a*(a+b*x^(1/3))^17/b^3+1/6*(a+b*x^(1/3))^18/b^3],
[(a+b*x^(1/3))^15/x,x,3,45*a^14*b*x^(1/3)+315/2*a^13*b^2*x^(2/3)+455*a^12*b^3*x+4095/4*a^11*b^4*x^(4/3)+9009/5*a^10*b^5*x^(5/3)+5005/2*a^9*b^6*x^2+19305/7*a^8*b^7*x^(7/3)+19305/8*a^7*b^8*x^(8/3)+5005/3*a^6*b^9*x^3+9009/10*a^5*b^10*x^(10/3)+4095/11*a^4*b^11*x^(11/3)+455/4*a^3*b^12*x^4+315/13*a^2*b^13*x^(13/3)+45/14*a*b^14*x^(14/3)+1/5*b^15*x^5+a^15*log(x)],
[(a+b*x^(1/3))^15/x^2,x,3,-a^15/x-45/2*a^14*b/x^(2/3)-315*a^13*b^2/x^(1/3)+4095*a^11*b^4*x^(1/3)+9009/2*a^10*b^5*x^(2/3)+5005*a^9*b^6*x+19305/4*a^8*b^7*x^(4/3)+3861*a^7*b^8*x^(5/3)+5005/2*a^6*b^9*x^2+1287*a^5*b^10*x^(7/3)+4095/8*a^4*b^11*x^(8/3)+455/3*a^3*b^12*x^3+63/2*a^2*b^13*x^(10/3)+45/11*a*b^14*x^(11/3)+1/4*b^15*x^4+455*a^12*b^3*log(x)],
[(a+b*x^(1/3))^15/x^3,x,3,-1/2*a^15/x^2-9*a^14*b/x^(5/3)-315/4*a^13*b^2/x^(4/3)-455*a^12*b^3/x-4095/2*a^11*b^4/x^(2/3)-9009*a^10*b^5/x^(1/3)+19305*a^8*b^7*x^(1/3)+19305/2*a^7*b^8*x^(2/3)+5005*a^6*b^9*x+9009/4*a^5*b^10*x^(4/3)+819*a^4*b^11*x^(5/3)+455/2*a^3*b^12*x^2+45*a^2*b^13*x^(7/3)+45/8*a*b^14*x^(8/3)+1/3*b^15*x^3+5005*a^9*b^6*log(x)],
[(a+b*x^(1/3))^15/x^4,x,3,-1/3*a^15/x^3-45/8*a^14*b/x^(8/3)-45*a^13*b^2/x^(7/3)-455/2*a^12*b^3/x^2-819*a^11*b^4/x^(5/3)-9009/4*a^10*b^5/x^(4/3)-5005*a^9*b^6/x-19305/2*a^8*b^7/x^(2/3)-19305*a^7*b^8/x^(1/3)+9009*a^5*b^10*x^(1/3)+4095/2*a^4*b^11*x^(2/3)+455*a^3*b^12*x+315/4*a^2*b^13*x^(4/3)+9*a*b^14*x^(5/3)+1/2*b^15*x^2+5005*a^6*b^9*log(x)],
[(a+b*x^(1/3))^15/x^6,x,3,-1/5*a^15/x^5-45/14*a^14*b/x^(14/3)-315/13*a^13*b^2/x^(13/3)-455/4*a^12*b^3/x^4-4095/11*a^11*b^4/x^(11/3)-9009/10*a^10*b^5/x^(10/3)-5005/3*a^9*b^6/x^3-19305/8*a^8*b^7/x^(8/3)-19305/7*a^7*b^8/x^(7/3)-5005/2*a^6*b^9/x^2-9009/5*a^5*b^10/x^(5/3)-4095/4*a^4*b^11/x^(4/3)-455*a^3*b^12/x-315/2*a^2*b^13/x^(2/3)-45*a*b^14/x^(1/3)+b^15*log(x)],
[(a+b*x^(1/3))^15/x^7,x,4,-1/6*(a+b*x^(1/3))^16/(a*x^6)+1/51*b*(a+b*x^(1/3))^16/(a^2*x^(17/3))-1/816*b^2*(a+b*x^(1/3))^16/(a^3*x^(16/3))],
[(a+b*x^(1/3))^15/x^8,x,7,-1/7*(a+b*x^(1/3))^16/(a*x^7)+1/28*b*(a+b*x^(1/3))^16/(a^2*x^(20/3))-1/133*b^2*(a+b*x^(1/3))^16/(a^3*x^(19/3))+1/798*b^3*(a+b*x^(1/3))^16/(a^4*x^6)-1/6783*b^4*(a+b*x^(1/3))^16/(a^5*x^(17/3))+1/108528*b^5*(a+b*x^(1/3))^16/(a^6*x^(16/3))],
[(a+b*x^(1/3))^15/x^9,x,10,-1/8*(a+b*x^(1/3))^16/(a*x^8)+1/23*b*(a+b*x^(1/3))^16/(a^2*x^(23/3))-7/506*b^2*(a+b*x^(1/3))^16/(a^3*x^(22/3))+1/253*b^3*(a+b*x^(1/3))^16/(a^4*x^7)-1/1012*b^4*(a+b*x^(1/3))^16/(a^5*x^(20/3))+1/4807*b^5*(a+b*x^(1/3))^16/(a^6*x^(19/3))-1/28842*b^6*(a+b*x^(1/3))^16/(a^7*x^6)+1/245157*b^7*(a+b*x^(1/3))^16/(a^8*x^(17/3))-1/3922512*b^8*(a+b*x^(1/3))^16/(a^9*x^(16/3))],
[(a+b*x^(1/3))^15/x^10,x,3,-1/9*a^15/x^9-45/26*a^14*b/x^(26/3)-63/5*a^13*b^2/x^(25/3)-455/8*a^12*b^3/x^8-4095/23*a^11*b^4/x^(23/3)-819/2*a^10*b^5/x^(22/3)-715*a^9*b^6/x^7-3861/4*a^8*b^7/x^(20/3)-19305/19*a^7*b^8/x^(19/3)-5005/6*a^6*b^9/x^6-9009/17*a^5*b^10/x^(17/3)-4095/16*a^4*b^11/x^(16/3)-91*a^3*b^12/x^5-45/2*a^2*b^13/x^(14/3)-45/13*a*b^14/x^(13/3)-1/4*b^15/x^4],
[(a+b*x^(1/3))^15/x^11,x,3,-1/10*a^15/x^10-45/29*a^14*b/x^(29/3)-45/4*a^13*b^2/x^(28/3)-455/9*a^12*b^3/x^9-315/2*a^11*b^4/x^(26/3)-9009/25*a^10*b^5/x^(25/3)-5005/8*a^9*b^6/x^8-19305/23*a^8*b^7/x^(23/3)-1755/2*a^7*b^8/x^(22/3)-715*a^6*b^9/x^7-9009/20*a^5*b^10/x^(20/3)-4095/19*a^4*b^11/x^(19/3)-455/6*a^3*b^12/x^6-315/17*a^2*b^13/x^(17/3)-45/16*a*b^14/x^(16/3)-1/5*b^15/x^5],
[(a+b*x^(1/3))^15/x^12,x,3,-1/11*a^15/x^11-45/32*a^14*b/x^(32/3)-315/31*a^13*b^2/x^(31/3)-91/2*a^12*b^3/x^10-4095/29*a^11*b^4/x^(29/3)-1287/4*a^10*b^5/x^(28/3)-5005/9*a^9*b^6/x^9-1485/2*a^8*b^7/x^(26/3)-3861/5*a^7*b^8/x^(25/3)-5005/8*a^6*b^9/x^8-9009/23*a^5*b^10/x^(23/3)-4095/22*a^4*b^11/x^(22/3)-65*a^3*b^12/x^7-63/4*a^2*b^13/x^(20/3)-45/19*a*b^14/x^(19/3)-1/6*b^15/x^6],

# p<0
[x^3/(a+b*x^(1/3)),x,3,3*a^10*x^(1/3)/b^11-3/2*a^9*x^(2/3)/b^10+a^8*x/b^9-3/4*a^7*x^(4/3)/b^8+3/5*a^6*x^(5/3)/b^7-1/2*a^5*x^2/b^6+3/7*a^4*x^(7/3)/b^5-3/8*a^3*x^(8/3)/b^4+1/3*a^2*x^3/b^3-3/10*a*x^(10/3)/b^2+3/11*x^(11/3)/b-3*a^11*log(a+b*x^(1/3))/b^12],
[x^2/(a+b*x^(1/3)),x,3,-3*a^7*x^(1/3)/b^8+3/2*a^6*x^(2/3)/b^7-a^5*x/b^6+3/4*a^4*x^(4/3)/b^5-3/5*a^3*x^(5/3)/b^4+1/2*a^2*x^2/b^3-3/7*a*x^(7/3)/b^2+3/8*x^(8/3)/b+3*a^8*log(a+b*x^(1/3))/b^9],
[x/(a+b*x^(1/3)),x,3,3*a^4*x^(1/3)/b^5-3/2*a^3*x^(2/3)/b^4+a^2*x/b^3-3/4*a*x^(4/3)/b^2+3/5*x^(5/3)/b-3*a^5*log(a+b*x^(1/3))/b^6],
[1/(a+b*x^(1/3)),x,3,-3*a*x^(1/3)/b^2+3/2*x^(2/3)/b+3*a^2*log(a+b*x^(1/3))/b^3],
[1/((a+b*x^(1/3))*x),x,4,-3*log(a+b*x^(1/3))/a+log(x)/a],
[1/((a+b*x^(1/3))*x^2),x,3,(-1)/(a*x)+3/2*b/(a^2*x^(2/3))-3*b^2/(a^3*x^(1/3))+3*b^3*log(a+b*x^(1/3))/a^4-b^3*log(x)/a^4],
[1/((a+b*x^(1/3))*x^3),x,3,(-1/2)/(a*x^2)+3/5*b/(a^2*x^(5/3))-3/4*b^2/(a^3*x^(4/3))+b^3/(a^4*x)-3/2*b^4/(a^5*x^(2/3))+3*b^5/(a^6*x^(1/3))-3*b^6*log(a+b*x^(1/3))/a^7+b^6*log(x)/a^7],
[1/((a+b*x^(1/3))*x^4),x,3,(-1/3)/(a*x^3)+3/8*b/(a^2*x^(8/3))-3/7*b^2/(a^3*x^(7/3))+1/2*b^3/(a^4*x^2)-3/5*b^4/(a^5*x^(5/3))+3/4*b^5/(a^6*x^(4/3))-b^6/(a^7*x)+3/2*b^7/(a^8*x^(2/3))-3*b^8/(a^9*x^(1/3))+3*b^9*log(a+b*x^(1/3))/a^10-b^9*log(x)/a^10],
[1/((2+b*x^(1/3))*x),x,4,-3/2*log(2+b*x^(1/3))+1/2*log(x)],
[x^3/(a+b*x^(1/3))^2,x,3,3*a^11/(b^12*(a+b*x^(1/3)))-30*a^9*x^(1/3)/b^11+27/2*a^8*x^(2/3)/b^10-8*a^7*x/b^9+21/4*a^6*x^(4/3)/b^8-18/5*a^5*x^(5/3)/b^7+5/2*a^4*x^2/b^6-12/7*a^3*x^(7/3)/b^5+9/8*a^2*x^(8/3)/b^4-2/3*a*x^3/b^3+3/10*x^(10/3)/b^2+33*a^10*log(a+b*x^(1/3))/b^12],
[x^2/(a+b*x^(1/3))^2,x,3,-3*a^8/(b^9*(a+b*x^(1/3)))+21*a^6*x^(1/3)/b^8-9*a^5*x^(2/3)/b^7+5*a^4*x/b^6-3*a^3*x^(4/3)/b^5+9/5*a^2*x^(5/3)/b^4-a*x^2/b^3+3/7*x^(7/3)/b^2-24*a^7*log(a+b*x^(1/3))/b^9],
[x/(a+b*x^(1/3))^2,x,3,3*a^5/(b^6*(a+b*x^(1/3)))-12*a^3*x^(1/3)/b^5+9/2*a^2*x^(2/3)/b^4-2*a*x/b^3+3/4*x^(4/3)/b^2+15*a^4*log(a+b*x^(1/3))/b^6],
[1/(a+b*x^(1/3))^2,x,3,-3*a^2/(b^3*(a+b*x^(1/3)))+3*x^(1/3)/b^2-6*a*log(a+b*x^(1/3))/b^3],
[1/((a+b*x^(1/3))^2*x),x,3,3/(a*(a+b*x^(1/3)))-3*log(a+b*x^(1/3))/a^2+log(x)/a^2],
[1/((a+b*x^(1/3))^2*x^2),x,3,-3*b^3/(a^4*(a+b*x^(1/3)))+(-1)/(a^2*x)+3*b/(a^3*x^(2/3))-9*b^2/(a^4*x^(1/3))+12*b^3*log(a+b*x^(1/3))/a^5-4*b^3*log(x)/a^5],
[1/((a+b*x^(1/3))^2*x^3),x,3,3*b^6/(a^7*(a+b*x^(1/3)))+(-1/2)/(a^2*x^2)+6/5*b/(a^3*x^(5/3))-9/4*b^2/(a^4*x^(4/3))+4*b^3/(a^5*x)-15/2*b^4/(a^6*x^(2/3))+18*b^5/(a^7*x^(1/3))-21*b^6*log(a+b*x^(1/3))/a^8+7*b^6*log(x)/a^8],
[1/((a+b*x^(1/3))^2*x^4),x,3,-3*b^9/(a^10*(a+b*x^(1/3)))+(-1/3)/(a^2*x^3)+3/4*b/(a^3*x^(8/3))-9/7*b^2/(a^4*x^(7/3))+2*b^3/(a^5*x^2)-3*b^4/(a^6*x^(5/3))+9/2*b^5/(a^7*x^(4/3))-7*b^6/(a^8*x)+12*b^7/(a^9*x^(2/3))-27*b^8/(a^10*x^(1/3))+30*b^9*log(a+b*x^(1/3))/a^11-10*b^9*log(x)/a^11],
[x^3/(a+b*x^(1/3))^3,x,3,3/2*a^11/(b^12*(a+b*x^(1/3))^2)-33*a^10/(b^12*(a+b*x^(1/3)))+135*a^8*x^(1/3)/b^11-54*a^7*x^(2/3)/b^10+28*a^6*x/b^9-63/4*a^5*x^(4/3)/b^8+9*a^4*x^(5/3)/b^7-5*a^3*x^2/b^6+18/7*a^2*x^(7/3)/b^5-9/8*a*x^(8/3)/b^4+1/3*x^3/b^3-165*a^9*log(a+b*x^(1/3))/b^12],
[x^2/(a+b*x^(1/3))^3,x,3,-3/2*a^8/(b^9*(a+b*x^(1/3))^2)+24*a^7/(b^9*(a+b*x^(1/3)))-63*a^5*x^(1/3)/b^8+45/2*a^4*x^(2/3)/b^7-10*a^3*x/b^6+9/2*a^2*x^(4/3)/b^5-9/5*a*x^(5/3)/b^4+1/2*x^2/b^3+84*a^6*log(a+b*x^(1/3))/b^9],
[x/(a+b*x^(1/3))^3,x,3,3/2*a^5/(b^6*(a+b*x^(1/3))^2)-15*a^4/(b^6*(a+b*x^(1/3)))+18*a^2*x^(1/3)/b^5-9/2*a*x^(2/3)/b^4+x/b^3-30*a^3*log(a+b*x^(1/3))/b^6],
[1/(a+b*x^(1/3))^3,x,3,-3/2*a^2/(b^3*(a+b*x^(1/3))^2)+6*a/(b^3*(a+b*x^(1/3)))+3*log(a+b*x^(1/3))/b^3],
[1/((a+b*x^(1/3))^3*x),x,3,3/2/(a*(a+b*x^(1/3))^2)+3/(a^2*(a+b*x^(1/3)))-3*log(a+b*x^(1/3))/a^3+log(x)/a^3],
[1/((a+b*x^(1/3))^3*x^2),x,3,-3/2*b^3/(a^4*(a+b*x^(1/3))^2)-12*b^3/(a^5*(a+b*x^(1/3)))+(-1)/(a^3*x)+9/2*b/(a^4*x^(2/3))-18*b^2/(a^5*x^(1/3))+30*b^3*log(a+b*x^(1/3))/a^6-10*b^3*log(x)/a^6],
[1/((a+b*x^(1/3))^3*x^3),x,3,3/2*b^6/(a^7*(a+b*x^(1/3))^2)+21*b^6/(a^8*(a+b*x^(1/3)))+(-1/2)/(a^3*x^2)+9/5*b/(a^4*x^(5/3))-9/2*b^2/(a^5*x^(4/3))+10*b^3/(a^6*x)-45/2*b^4/(a^7*x^(2/3))+63*b^5/(a^8*x^(1/3))-84*b^6*log(a+b*x^(1/3))/a^9+28*b^6*log(x)/a^9],
[1/((a+b*x^(1/3))^3*x^4),x,3,-3/2*b^9/(a^10*(a+b*x^(1/3))^2)-30*b^9/(a^11*(a+b*x^(1/3)))+(-1/3)/(a^3*x^3)+9/8*b/(a^4*x^(8/3))-18/7*b^2/(a^5*x^(7/3))+5*b^3/(a^6*x^2)-9*b^4/(a^7*x^(5/3))+63/4*b^5/(a^8*x^(4/3))-28*b^6/(a^9*x)+54*b^7/(a^10*x^(2/3))-135*b^8/(a^11*x^(1/3))+165*b^9*log(a+b*x^(1/3))/a^12-55*b^9*log(x)/a^12],

# Integrands of the form x^m (a+b x^(1/3))^(p/2)

# p>0

# p<0
[1/sqrt(1+x^(1/3)),x,3,-4*(1+x^(1/3))^(3/2)+6/5*(1+x^(1/3))^(5/2)+6*sqrt(1+x^(1/3))],

# Integrands of the form x^(m/2) (a+b x^(1/3))^p

# p>0

# p<0
[1/((1+x^(1/3))*x^(3/2)),x,5,6/x^(1/6)+6*arctan(x^(1/6))+(-2)/sqrt(x)],

# Integrands of the form x^(m/3) (a+b x^(1/3))^p

# p>0

# p<0
[x^(2/3)/(1+x^(1/3)),x,3,-3*x^(1/3)+3/2*x^(2/3)-x+3/4*x^(4/3)+3*log(1+x^(1/3))],

# Integrands of the form (c x)^m (a+b x^(2/3))^p

# Integrands of the form x^m (a+b x^(2/3))^p

# p>0

# p<0
[1/(1+x^(2/3)),x,3,3*x^(1/3)-3*arctan(x^(1/3))],

# Integrands of the form x^(m/3) (a+b x^(2/3))^p

# p>0

# p<0
[1/((1+x^(2/3))*x^(1/3)),x,1,3/2*log(1+x^(2/3))],
[1/((1+x^(2/3))*x^(2/3)),x,2,3*arctan(x^(1/3))],

# Integrands of the form x^(m/3) (a+b x^(2/3))^(p/2)

# p>0
[sqrt(-1+x^(2/3))/x^(1/3),x,1,(-1+x^(2/3))^(3/2)],
[(1+x^(2/3))^(3/2)/x^(1/3),x,1,3/5*(1+x^(2/3))^(5/2)],

# p<0

# Integrands of the form x^(m/2) (a+b x^(2/3))^p

# p>0

# p<0
[sqrt(x)/(1+x^(2/3)),x,13,-6*x^(1/6)+6/5*x^(5/6)-3*arctan(1-x^(1/6)*sqrt(2))/sqrt(2)+3*arctan(1+x^(1/6)*sqrt(2))/sqrt(2)-3/2*log(1+x^(1/3)-x^(1/6)*sqrt(2))/sqrt(2)+3/2*log(1+x^(1/3)+x^(1/6)*sqrt(2))/sqrt(2)],

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

# Integrands of the form (c x)^m (a+b x^n)^p when n<0 is a fraction

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

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

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

# p>0
[sqrt(3+(-1)/sqrt(x)),x,5,-1/6*arctanh(sqrt(3+(-1)/sqrt(x))/sqrt(3))/sqrt(3)+x*sqrt(3+(-1)/sqrt(x))-1/6*sqrt(x)*sqrt(3+(-1)/sqrt(x))],

# p<0
[1/sqrt(1+1/sqrt(x)),x,5,3/2*arctanh(sqrt(1+1/sqrt(x)))+x*sqrt(1+1/sqrt(x))-3/2*sqrt(x)*sqrt(1+1/sqrt(x))],

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

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

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

# Integrands of the form (c x)^m (a+b/x^(n/3))^p

# Integrands of the form x^m (a+b/x^(n/3))^p

# p>0
[(a+b/x^(1/3))*x^4,x,2,3/14*b*x^(14/3)+1/5*a*x^5],
[(a+b/x^(1/3))*x^3,x,2,3/11*b*x^(11/3)+1/4*a*x^4],
[(a+b/x^(1/3))*x^2,x,2,3/8*b*x^(8/3)+1/3*a*x^3],
[(a+b/x^(1/3))*x,x,2,3/5*b*x^(5/3)+1/2*a*x^2],
[a+b/x^(1/3),x,1,3/2*b*x^(2/3)+a*x],
[(a+b/x^(1/3))/x,x,2,-3*b/x^(1/3)+a*log(x)],
[(a+b/x^(1/3))/x^2,x,2,-3/4*b/x^(4/3)-a/x],
[(a+b/x^(1/3))/x^3,x,2,-3/7*b/x^(7/3)-1/2*a/x^2],
[(a+b/x^(1/3))/x^4,x,2,-3/10*b/x^(10/3)-1/3*a/x^3],
[(a+b/x^(1/3))^2*x^4,x,4,3/13*b^2*x^(13/3)+3/7*a*b*x^(14/3)+1/5*a^2*x^5],
[(a+b/x^(1/3))^2*x^3,x,4,3/10*b^2*x^(10/3)+6/11*a*b*x^(11/3)+1/4*a^2*x^4],
[(a+b/x^(1/3))^2*x^2,x,4,3/7*b^2*x^(7/3)+3/4*a*b*x^(8/3)+1/3*a^2*x^3],
[(a+b/x^(1/3))^2*x,x,4,3/4*b^2*x^(4/3)+6/5*a*b*x^(5/3)+1/2*a^2*x^2],
[(a+b/x^(1/3))^2,x,2,(a+b/x^(1/3))^3*x/a],
[(a+b/x^(1/3))^2/x,x,4,-3/2*b^2/x^(2/3)-6*a*b/x^(1/3)+a^2*log(x)],
[(a+b/x^(1/3))^2/x^2,x,4,-3/5*b^2/x^(5/3)-3/2*a*b/x^(4/3)-a^2/x],
[(a+b/x^(1/3))^2/x^3,x,4,-3/8*b^2/x^(8/3)-6/7*a*b/x^(7/3)-1/2*a^2/x^2],
[(a+b/x^(1/3))^2/x^4,x,4,-3/11*b^2/x^(11/3)-3/5*a*b/x^(10/3)-1/3*a^2/x^3],
[(a+b/x^(1/3))^3*x^4,x,4,1/4*b^3*x^4+9/13*a*b^2*x^(13/3)+9/14*a^2*b*x^(14/3)+1/5*a^3*x^5],
[(a+b/x^(1/3))^3*x^3,x,4,1/3*b^3*x^3+9/10*a*b^2*x^(10/3)+9/11*a^2*b*x^(11/3)+1/4*a^3*x^4],
[(a+b/x^(1/3))^3*x^2,x,4,1/2*b^3*x^2+9/7*a*b^2*x^(7/3)+9/8*a^2*b*x^(8/3)+1/3*a^3*x^3],
[(a+b/x^(1/3))^3*x,x,4,b^3*x+9/4*a*b^2*x^(4/3)+9/5*a^2*b*x^(5/3)+1/2*a^3*x^2],
[(a+b/x^(1/3))^3,x,3,9*a*b^2*x^(1/3)+9/2*a^2*b*x^(2/3)+a^3*x+b^3*log(x)],
[(a+b/x^(1/3))^3/x,x,4,-b^3/x-9/2*a*b^2/x^(2/3)-9*a^2*b/x^(1/3)+a^3*log(x)],
[(a+b/x^(1/3))^3/x^2,x,4,-1/2*b^3/x^2-9/5*a*b^2/x^(5/3)-9/4*a^2*b/x^(4/3)-a^3/x],
[(a+b/x^(1/3))^3/x^3,x,4,-1/3*b^3/x^3-9/8*a*b^2/x^(8/3)-9/7*a^2*b/x^(7/3)-1/2*a^3/x^2],
[(a+b/x^(1/3))^3/x^4,x,4,-1/4*b^3/x^4-9/11*a*b^2/x^(11/3)-9/10*a^2*b/x^(10/3)-1/3*a^3/x^3],

# p<0
[x^2/(a+b/x^(1/3)),x,4,3*b^8*x^(1/3)/a^9-3/2*b^7*x^(2/3)/a^8+b^6*x/a^7-3/4*b^5*x^(4/3)/a^6+3/5*b^4*x^(5/3)/a^5-1/2*b^3*x^2/a^4+3/7*b^2*x^(7/3)/a^3-3/8*b*x^(8/3)/a^2+1/3*x^3/a-3*b^9*log(b+a*x^(1/3))/a^10],
[x/(a+b/x^(1/3)),x,4,-3*b^5*x^(1/3)/a^6+3/2*b^4*x^(2/3)/a^5-b^3*x/a^4+3/4*b^2*x^(4/3)/a^3-3/5*b*x^(5/3)/a^2+1/2*x^2/a+3*b^6*log(b+a*x^(1/3))/a^7],
[1/(a+b/x^(1/3)),x,3,3*b^2*x^(1/3)/a^3-3/2*b*x^(2/3)/a^2+x/a-3*b^3*log(a+b/x^(1/3))/a^4-b^3*log(x)/a^4],
[1/((a+b/x^(1/3))*x),x,2,3*log(b+a*x^(1/3))/a],
[1/((a+b/x^(1/3))*x^2),x,4,(-3/2)/(b*x^(2/3))+3*a/(b^2*x^(1/3))-3*a^2*log(b+a*x^(1/3))/b^3+a^2*log(x)/b^3],
[1/((a+b/x^(1/3))*x^3),x,4,(-3/5)/(b*x^(5/3))+3/4*a/(b^2*x^(4/3))-a^2/(b^3*x)+3/2*a^3/(b^4*x^(2/3))-3*a^4/(b^5*x^(1/3))+3*a^5*log(b+a*x^(1/3))/b^6-a^5*log(x)/b^6],
[1/((a+b/x^(1/3))*x^4),x,4,(-3/8)/(b*x^(8/3))+3/7*a/(b^2*x^(7/3))-1/2*a^2/(b^3*x^2)+3/5*a^3/(b^4*x^(5/3))-3/4*a^4/(b^5*x^(4/3))+a^5/(b^6*x)-3/2*a^6/(b^7*x^(2/3))+3*a^7/(b^8*x^(1/3))-3*a^8*log(b+a*x^(1/3))/b^9+a^8*log(x)/b^9],
[1/((a+b/x^(1/3))*x^5),x,4,(-3/11)/(b*x^(11/3))+3/10*a/(b^2*x^(10/3))-1/3*a^2/(b^3*x^3)+3/8*a^3/(b^4*x^(8/3))-3/7*a^4/(b^5*x^(7/3))+1/2*a^5/(b^6*x^2)-3/5*a^6/(b^7*x^(5/3))+3/4*a^7/(b^8*x^(4/3))-a^8/(b^9*x)+3/2*a^9/(b^10*x^(2/3))-3*a^10/(b^11*x^(1/3))+3*a^11*log(b+a*x^(1/3))/b^12-a^11*log(x)/b^12],
[x^2/(a+b/x^(1/3))^2,x,4,-3*b^10/(a^11*(b+a*x^(1/3)))+27*b^8*x^(1/3)/a^10-12*b^7*x^(2/3)/a^9+7*b^6*x/a^8-9/2*b^5*x^(4/3)/a^7+3*b^4*x^(5/3)/a^6-2*b^3*x^2/a^5+9/7*b^2*x^(7/3)/a^4-3/4*b*x^(8/3)/a^3+1/3*x^3/a^2-30*b^9*log(b+a*x^(1/3))/a^11],
[x/(a+b/x^(1/3))^2,x,4,3*b^7/(a^8*(b+a*x^(1/3)))-18*b^5*x^(1/3)/a^7+15/2*b^4*x^(2/3)/a^6-4*b^3*x/a^5+9/4*b^2*x^(4/3)/a^4-6/5*b*x^(5/3)/a^3+1/2*x^2/a^2+21*b^6*log(b+a*x^(1/3))/a^8],
[1/(a+b/x^(1/3))^2,x,3,3*b^3/(a^4*(a+b/x^(1/3)))+9*b^2*x^(1/3)/a^4-3*b*x^(2/3)/a^3+x/a^2-12*b^3*log(a+b/x^(1/3))/a^5-4*b^3*log(x)/a^5],
[1/((a+b/x^(1/3))^2*x),x,4,3*b/(a^2*(b+a*x^(1/3)))+3*log(b+a*x^(1/3))/a^2],
[1/((a+b/x^(1/3))^2*x^2),x,4,-3*a/(b^2*(b+a*x^(1/3)))+(-3)/(b^2*x^(1/3))+6*a*log(b+a*x^(1/3))/b^3-2*a*log(x)/b^3],
[1/((a+b/x^(1/3))^2*x^3),x,4,3*a^4/(b^5*(b+a*x^(1/3)))+(-3/4)/(b^2*x^(4/3))+2*a/(b^3*x)-9/2*a^2/(b^4*x^(2/3))+12*a^3/(b^5*x^(1/3))-15*a^4*log(b+a*x^(1/3))/b^6+5*a^4*log(x)/b^6],
[1/((a+b/x^(1/3))^2*x^4),x,4,-3*a^7/(b^8*(b+a*x^(1/3)))+(-3/7)/(b^2*x^(7/3))+a/(b^3*x^2)-9/5*a^2/(b^4*x^(5/3))+3*a^3/(b^5*x^(4/3))-5*a^4/(b^6*x)+9*a^5/(b^7*x^(2/3))-21*a^6/(b^8*x^(1/3))+24*a^7*log(b+a*x^(1/3))/b^9-8*a^7*log(x)/b^9],
[1/((a+b/x^(1/3))^2*x^5),x,4,3*a^10/(b^11*(b+a*x^(1/3)))+(-3/10)/(b^2*x^(10/3))+2/3*a/(b^3*x^3)-9/8*a^2/(b^4*x^(8/3))+12/7*a^3/(b^5*x^(7/3))-5/2*a^4/(b^6*x^2)+18/5*a^5/(b^7*x^(5/3))-21/4*a^6/(b^8*x^(4/3))+8*a^7/(b^9*x)-27/2*a^8/(b^10*x^(2/3))+30*a^9/(b^11*x^(1/3))-33*a^10*log(b+a*x^(1/3))/b^12+11*a^10*log(x)/b^12],
[x^2/(a+b/x^(1/3))^3,x,4,3/2*b^11/(a^12*(b+a*x^(1/3))^2)-33*b^10/(a^12*(b+a*x^(1/3)))+135*b^8*x^(1/3)/a^11-54*b^7*x^(2/3)/a^10+28*b^6*x/a^9-63/4*b^5*x^(4/3)/a^8+9*b^4*x^(5/3)/a^7-5*b^3*x^2/a^6+18/7*b^2*x^(7/3)/a^5-9/8*b*x^(8/3)/a^4+1/3*x^3/a^3-165*b^9*log(b+a*x^(1/3))/a^12],
[x/(a+b/x^(1/3))^3,x,4,-3/2*b^8/(a^9*(b+a*x^(1/3))^2)+24*b^7/(a^9*(b+a*x^(1/3)))-63*b^5*x^(1/3)/a^8+45/2*b^4*x^(2/3)/a^7-10*b^3*x/a^6+9/2*b^2*x^(4/3)/a^5-9/5*b*x^(5/3)/a^4+1/2*x^2/a^3+84*b^6*log(b+a*x^(1/3))/a^9],
[1/(a+b/x^(1/3))^3,x,3,3/2*b^3/(a^4*(a+b/x^(1/3))^2)+12*b^3/(a^5*(a+b/x^(1/3)))+18*b^2*x^(1/3)/a^5-9/2*b*x^(2/3)/a^4+x/a^3-30*b^3*log(a+b/x^(1/3))/a^6-10*b^3*log(x)/a^6],
[1/((a+b/x^(1/3))^3*x),x,4,-3/2*b^2/(a^3*(b+a*x^(1/3))^2)+6*b/(a^3*(b+a*x^(1/3)))+3*log(b+a*x^(1/3))/a^3],
[1/((a+b/x^(1/3))^3*x^2),x,4,3/2/(b*(b+a*x^(1/3))^2)+3/(b^2*(b+a*x^(1/3)))-3*log(b+a*x^(1/3))/b^3+log(x)/b^3],
[1/((a+b/x^(1/3))^3*x^3),x,4,-3/2*a^3/(b^4*(b+a*x^(1/3))^2)-12*a^3/(b^5*(b+a*x^(1/3)))+(-1)/(b^3*x)+9/2*a/(b^4*x^(2/3))-18*a^2/(b^5*x^(1/3))+30*a^3*log(b+a*x^(1/3))/b^6-10*a^3*log(x)/b^6],
[1/((a+b/x^(1/3))^3*x^4),x,4,3/2*a^6/(b^7*(b+a*x^(1/3))^2)+21*a^6/(b^8*(b+a*x^(1/3)))+(-1/2)/(b^3*x^2)+9/5*a/(b^4*x^(5/3))-9/2*a^2/(b^5*x^(4/3))+10*a^3/(b^6*x)-45/2*a^4/(b^7*x^(2/3))+63*a^5/(b^8*x^(1/3))-84*a^6*log(b+a*x^(1/3))/b^9+28*a^6*log(x)/b^9],
[1/((a+b/x^(1/3))^3*x^5),x,4,-3/2*a^9/(b^10*(b+a*x^(1/3))^2)-30*a^9/(b^11*(b+a*x^(1/3)))+(-1/3)/(b^3*x^3)+9/8*a/(b^4*x^(8/3))-18/7*a^2/(b^5*x^(7/3))+5*a^3/(b^6*x^2)-9*a^4/(b^7*x^(5/3))+63/4*a^5/(b^8*x^(4/3))-28*a^6/(b^9*x)+54*a^7/(b^10*x^(2/3))-135*a^8/(b^11*x^(1/3))+165*a^9*log(b+a*x^(1/3))/b^12-55*a^9*log(x)/b^12],
[1/(1+b/x^(1/3)),x,3,3*b^2*x^(1/3)-3/2*b*x^(2/3)+x-3*b^3*log(1+b/x^(1/3))-b^3*log(x)],

# Integrands of the form x^m (a+b/x^(n/3))^(p/2)

# Integrands of the form x^(m/2) (a+b/x^(n/3))^p

# Integrands of the form x^(m/2) (a+b/x^(n/3))^(p/2)

# Integrands of the form x^(m/3) (a+b/x^(n/3))^(p/3)
[x^(2/3)*(1+x^(5/3))^(2/3),x,1,9/25*(1+x^(5/3))^(5/3)],

# Integrands of the form x^(m/3) (a+b/x^(n/3))^(p/7)
[x^(7/3)*(a^(10/3)-x^(10/3))^(19/7),x,1,-21/260*(a^(10/3)-x^(10/3))^(26/7)],

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

# Integrands of the form (c x)^m (a+b x^n)^p when n symbolic

# Integrands of the form x^m (a+b x^n)^p

# Integrands of the form x^m (a+b x^n)^p

# p>0
[x^3*(a+b*x^n),x,2,1/4*a*x^4+b*x^(4+n)/(4+n)],
[x^2*(a+b*x^n),x,2,1/3*a*x^3+b*x^(3+n)/(3+n)],
[x*(a+b*x^n),x,2,1/2*a*x^2+b*x^(2+n)/(2+n)],
[a+b*x^n,x,1,a*x+b*x^(1+n)/(1+n)],
[(a+b*x^n)/x,x,2,b*x^n/n+a*log(x)],
[(a+b*x^n)/x^2,x,2,-a/x-b*x^(-1+n)/(1-n)],
[(a+b*x^n)/x^3,x,2,-1/2*a/x^2-b*x^(-2+n)/(2-n)],
[x^3*(a+b*x^n)^2,x,2,1/4*a^2*x^4+1/2*b^2*x^(2*(2+n))/(2+n)+2*a*b*x^(4+n)/(4+n)],
[x^2*(a+b*x^n)^2,x,2,1/3*a^2*x^3+2*a*b*x^(3+n)/(3+n)+b^2*x^(3+2*n)/(3+2*n)],
[x*(a+b*x^n)^2,x,2,1/2*a^2*x^2+1/2*b^2*x^(2*(1+n))/(1+n)+2*a*b*x^(2+n)/(2+n)],
[(a+b*x^n)^2,x,2,a^2*x+2*a*b*x^(1+n)/(1+n)+b^2*x^(1+2*n)/(1+2*n)],
[(a+b*x^n)^2/x,x,3,2*a*b*x^n/n+1/2*b^2*x^(2*n)/n+a^2*log(x)],
[(a+b*x^n)^2/x^2,x,2,-a^2/x-2*a*b*x^(-1+n)/(1-n)-b^2*x^(-1+2*n)/(1-2*n)],
[(a+b*x^n)^2/x^3,x,2,-1/2*a^2/x^2-1/2*b^2/((1-n)*x^(2*(1-n)))-2*a*b*x^(-2+n)/(2-n)],
[x^3*(a+b*x^n)^3,x,2,1/4*a^3*x^4+3/2*a*b^2*x^(2*(2+n))/(2+n)+3*a^2*b*x^(4+n)/(4+n)+b^3*x^(4+3*n)/(4+3*n)],
[x^2*(a+b*x^n)^3,x,2,1/3*a^3*x^3+1/3*b^3*x^(3*(1+n))/(1+n)+3*a^2*b*x^(3+n)/(3+n)+3*a*b^2*x^(3+2*n)/(3+2*n)],
[x*(a+b*x^n)^3,x,2,1/2*a^3*x^2+3/2*a*b^2*x^(2*(1+n))/(1+n)+3*a^2*b*x^(2+n)/(2+n)+b^3*x^(2+3*n)/(2+3*n)],
[(a+b*x^n)^3,x,2,a^3*x+3*a^2*b*x^(1+n)/(1+n)+3*a*b^2*x^(1+2*n)/(1+2*n)+b^3*x^(1+3*n)/(1+3*n)],
[(a+b*x^n)^3/x,x,3,3*a^2*b*x^n/n+3/2*a*b^2*x^(2*n)/n+1/3*b^3*x^(3*n)/n+a^3*log(x)],
[(a+b*x^n)^3/x^2,x,2,-a^3/x-3*a^2*b*x^(-1+n)/(1-n)-3*a*b^2*x^(-1+2*n)/(1-2*n)-b^3*x^(-1+3*n)/(1-3*n)],
[(a+b*x^n)^3/x^3,x,2,-1/2*a^3/x^2-3/2*a*b^2/((1-n)*x^(2*(1-n)))-3*a^2*b*x^(-2+n)/(2-n)-b^3*x^(-2+3*n)/(2-3*n)],

# p<0
[x/(a+b*x^n),x,1,1/2*x^2*hypergeom([1,2/n],[(2+n)/n],-b*x^n/a)/a],
[1/(a+b*x^n),x,1,x*hypergeom([1,1/n],[1+1/n],-b*x^n/a)/a],
[1/(x*(a+b*x^n)),x,4,log(x)/a-log(a+b*x^n)/(a*n)],
[1/(x^2*(a+b*x^n)),x,1,-hypergeom([1,(-1)/n],[(-1+n)/n],-b*x^n/a)/(a*x)],
[1/(x^3*(a+b*x^n)),x,1,-1/2*hypergeom([1,(-2)/n],[(-2+n)/n],-b*x^n/a)/(a*x^2)],
[x/(a+b*x^n)^2,x,1,1/2*x^2*hypergeom([2,2/n],[(2+n)/n],-b*x^n/a)/a^2],
[1/(a+b*x^n)^2,x,1,x*hypergeom([2,1/n],[1+1/n],-b*x^n/a)/a^2],
[1/(x*(a+b*x^n)^2),x,3,1/(a*n*(a+b*x^n))+log(x)/a^2-log(a+b*x^n)/(a^2*n)],
[1/(x^2*(a+b*x^n)^2),x,1,-hypergeom([2,(-1)/n],[(-1+n)/n],-b*x^n/a)/(a^2*x)],
[1/(x^3*(a+b*x^n)^2),x,1,-1/2*hypergeom([2,(-2)/n],[(-2+n)/n],-b*x^n/a)/(a^2*x^2)],
[x/(a+b*x^n)^3,x,1,1/2*x^2*hypergeom([3,2/n],[(2+n)/n],-b*x^n/a)/a^3],
[1/(a+b*x^n)^3,x,1,x*hypergeom([3,1/n],[1+1/n],-b*x^n/a)/a^3],
[1/(x*(a+b*x^n)^3),x,3,1/2/(a*n*(a+b*x^n)^2)+1/(a^2*n*(a+b*x^n))+log(x)/a^3-log(a+b*x^n)/(a^3*n)],
[1/(x^2*(a+b*x^n)^3),x,1,-hypergeom([3,(-1)/n],[(-1+n)/n],-b*x^n/a)/(a^3*x)],
[1/(x^3*(a+b*x^n)^3),x,1,-1/2*hypergeom([3,(-2)/n],[(-2+n)/n],-b*x^n/a)/(a^3*x^2)],

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

# p>0
[x*(a+b*x^n)^(1/2),x,2,1/2*x^2*(a+b*x^n)^(3/2)*hypergeom([1,3/2+2/n],[(2+n)/n],-b*x^n/a)/a,1/2*x^2*hypergeom([-1/2,2/n],[(2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/sqrt(1+b*x^n/a)],
[(a+b*x^n)^(1/2),x,2,x*(a+b*x^n)^(3/2)*hypergeom([1,3/2+1/n],[1+1/n],-b*x^n/a)/a,x*hypergeom([-1/2,1/n],[1+1/n],-b*x^n/a)*sqrt(a+b*x^n)/sqrt(1+b*x^n/a)],
[(a+b*x^n)^(1/2)/x,x,4,-2*arctanh(sqrt(a+b*x^n)/sqrt(a))*sqrt(a)/n+2*sqrt(a+b*x^n)/n],
[(a+b*x^n)^(1/2)/x^2,x,2,-(a+b*x^n)^(3/2)*hypergeom([1,3/2+(-1)/n],[(-1+n)/n],-b*x^n/a)/(a*x),-hypergeom([-1/2,(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(x*sqrt(1+b*x^n/a))],
[(a+b*x^n)^(1/2)/x^3,x,2,-1/2*(a+b*x^n)^(3/2)*hypergeom([1,3/2+(-2)/n],[(-2+n)/n],-b*x^n/a)/(a*x^2),-1/2*hypergeom([-1/2,(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(x^2*sqrt(1+b*x^n/a))],
[x*(a+b*x^n)^(3/2),x,2,1/2*x^2*(a+b*x^n)^(5/2)*hypergeom([1,5/2+2/n],[(2+n)/n],-b*x^n/a)/a,1/2*a*x^2*hypergeom([-3/2,2/n],[(2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/sqrt(1+b*x^n/a)],
[(a+b*x^n)^(3/2),x,2,x*(a+b*x^n)^(5/2)*hypergeom([1,5/2+1/n],[1+1/n],-b*x^n/a)/a,a*x*hypergeom([-3/2,1/n],[1+1/n],-b*x^n/a)*sqrt(a+b*x^n)/sqrt(1+b*x^n/a)],
[(a+b*x^n)^(3/2)/x,x,5,2/3*(a+b*x^n)^(3/2)/n-2*a^(3/2)*arctanh(sqrt(a+b*x^n)/sqrt(a))/n+2*a*sqrt(a+b*x^n)/n],
[(a+b*x^n)^(3/2)/x^2,x,2,-(a+b*x^n)^(5/2)*hypergeom([1,5/2+(-1)/n],[(-1+n)/n],-b*x^n/a)/(a*x),-a*hypergeom([-3/2,(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(x*sqrt(1+b*x^n/a))],
[(a+b*x^n)^(3/2)/x^3,x,2,-1/2*(a+b*x^n)^(5/2)*hypergeom([1,5/2+(-2)/n],[(-2+n)/n],-b*x^n/a)/(a*x^2),-1/2*a*hypergeom([-3/2,(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(x^2*sqrt(1+b*x^n/a))],
[x*(a+b*x^n)^(5/2),x,2,1/2*x^2*(a+b*x^n)^(7/2)*hypergeom([1,7/2+2/n],[(2+n)/n],-b*x^n/a)/a,1/2*a^2*x^2*hypergeom([-5/2,2/n],[(2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/sqrt(1+b*x^n/a)],
[(a+b*x^n)^(5/2),x,2,x*(a+b*x^n)^(7/2)*hypergeom([1,7/2+1/n],[1+1/n],-b*x^n/a)/a,a^2*x*hypergeom([-5/2,1/n],[1+1/n],-b*x^n/a)*sqrt(a+b*x^n)/sqrt(1+b*x^n/a)],
[(a+b*x^n)^(5/2)/x,x,6,2/3*a*(a+b*x^n)^(3/2)/n+2/5*(a+b*x^n)^(5/2)/n-2*a^(5/2)*arctanh(sqrt(a+b*x^n)/sqrt(a))/n+2*a^2*sqrt(a+b*x^n)/n],
[(a+b*x^n)^(5/2)/x^2,x,2,-(a+b*x^n)^(7/2)*hypergeom([1,7/2+(-1)/n],[(-1+n)/n],-b*x^n/a)/(a*x),-a^2*hypergeom([-5/2,(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(x*sqrt(1+b*x^n/a))],
[(a+b*x^n)^(5/2)/x^3,x,2,-1/2*(a+b*x^n)^(7/2)*hypergeom([1,7/2+(-2)/n],[(-2+n)/n],-b*x^n/a)/(a*x^2),-1/2*a^2*hypergeom([-5/2,(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(x^2*sqrt(1+b*x^n/a))],

# p<0
[x/(a+b*x^n)^(1/2),x,2,1/2*x^2*hypergeom([1,1/2+2/n],[(2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/a,1/2*x^2*hypergeom([1/2,2/n],[(2+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/sqrt(a+b*x^n)],
[1/(a+b*x^n)^(1/2),x,2,x*hypergeom([1,1/2+1/n],[1+1/n],-b*x^n/a)*sqrt(a+b*x^n)/a,x*hypergeom([1/2,1/n],[1+1/n],-b*x^n/a)*sqrt(1+b*x^n/a)/sqrt(a+b*x^n)],
[1/(x*(a+b*x^n)^(1/2)),x,3,-2*arctanh(sqrt(a+b*x^n)/sqrt(a))/(n*sqrt(a))],
[1/(x^2*(a+b*x^n)^(1/2)),x,2,-hypergeom([1,1/2+(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*x),-hypergeom([1/2,(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(x*sqrt(a+b*x^n))],
[1/(x^3*(a+b*x^n)^(1/2)),x,2,-1/2*hypergeom([1,1/2+(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*x^2),-1/2*hypergeom([1/2,(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(x^2*sqrt(a+b*x^n))],
[x/(a+b*x^n)^(3/2),x,2,1/2*x^2*hypergeom([1,-1/2+2/n],[(2+n)/n],-b*x^n/a)/(a*sqrt(a+b*x^n)),1/2*x^2*hypergeom([3/2,2/n],[(2+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a*sqrt(a+b*x^n))],
[1/(a+b*x^n)^(3/2),x,2,x*hypergeom([1,-1/2+1/n],[1+1/n],-b*x^n/a)/(a*sqrt(a+b*x^n)),x*hypergeom([3/2,1/n],[1+1/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a*sqrt(a+b*x^n))],
[1/(x*(a+b*x^n)^(3/2)),x,4,-2*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(3/2)*n)+2/(a*n*sqrt(a+b*x^n))],
[1/(x^2*(a+b*x^n)^(3/2)),x,2,-hypergeom([1,-1/2+(-1)/n],[(-1+n)/n],-b*x^n/a)/(a*x*sqrt(a+b*x^n)),-hypergeom([3/2,(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a*x*sqrt(a+b*x^n))],
[1/(x^3*(a+b*x^n)^(3/2)),x,2,-1/2*hypergeom([1,-1/2+(-2)/n],[(-2+n)/n],-b*x^n/a)/(a*x^2*sqrt(a+b*x^n)),-1/2*hypergeom([3/2,(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a*x^2*sqrt(a+b*x^n))],
[x/(a+b*x^n)^(5/2),x,2,1/2*x^2*hypergeom([1,-3/2+2/n],[(2+n)/n],-b*x^n/a)/(a*(a+b*x^n)^(3/2)),1/2*x^2*hypergeom([5/2,2/n],[(2+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a^2*sqrt(a+b*x^n))],
[1/(a+b*x^n)^(5/2),x,2,x*hypergeom([1,-3/2+1/n],[1+1/n],-b*x^n/a)/(a*(a+b*x^n)^(3/2)),x*hypergeom([5/2,1/n],[1+1/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a^2*sqrt(a+b*x^n))],
[1/(x*(a+b*x^n)^(5/2)),x,5,2/3/(a*n*(a+b*x^n)^(3/2))-2*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(5/2)*n)+2/(a^2*n*sqrt(a+b*x^n))],
[1/(x^2*(a+b*x^n)^(5/2)),x,2,-hypergeom([1,-3/2+(-1)/n],[(-1+n)/n],-b*x^n/a)/(a*x*(a+b*x^n)^(3/2)),-hypergeom([5/2,(-1)/n],[(-1+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a^2*x*sqrt(a+b*x^n))],
[1/(x^3*(a+b*x^n)^(5/2)),x,2,-1/2*hypergeom([1,-3/2+(-2)/n],[(-2+n)/n],-b*x^n/a)/(a*x^2*(a+b*x^n)^(3/2)),-1/2*hypergeom([5/2,(-2)/n],[(-2+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a^2*x^2*sqrt(a+b*x^n))],

# Integrands of the form x^m (a+b x^n)^(p/3)

# p>0
[(a+b*x^n)^(1/3)/x,x,6,3*(a+b*x^n)^(1/3)/n-1/2*a^(1/3)*log(x)+3/2*a^(1/3)*log(a^(1/3)-(a+b*x^n)^(1/3))/n-a^(1/3)*arctan((a^(1/3)+2*(a+b*x^n)^(1/3))/(a^(1/3)*sqrt(3)))*sqrt(3)/n],

# p<0

# Integrands of the form x^(k n-1) (a+b x^n)^p

# p>0
[x^(-1+4*n)*(a+b*x^n),x,2,1/4*a*x^(4*n)/n+1/5*b*x^(5*n)/n],
[x^(-1+3*n)*(a+b*x^n),x,2,1/3*a*x^(3*n)/n+1/4*b*x^(4*n)/n],
[x^(-1+2*n)*(a+b*x^n),x,2,1/2*a*x^(2*n)/n+1/3*b*x^(3*n)/n],
[x^(-1+n)*(a+b*x^n),x,2,a*x^n/n+1/2*b*x^(2*n)/n],
[(a+b*x^n)/x,x,2,b*x^n/n+a*log(x)],
[x^(-1-n)*(a+b*x^n),x,2,-a/(n*x^n)+b*log(x)],
[x^(-1-2*n)*(a+b*x^n),x,2,-1/2*a/(n*x^(2*n))-b/(n*x^n)],
[x^(-1-3*n)*(a+b*x^n),x,2,-1/3*a/(n*x^(3*n))-1/2*b/(n*x^(2*n))],
[x^(-1-4*n)*(a+b*x^n),x,2,-1/4*a/(n*x^(4*n))-1/3*b/(n*x^(3*n))],
[x^(-1-5*n)*(a+b*x^n),x,2,-1/5*a/(n*x^(5*n))-1/4*b/(n*x^(4*n))],
[x^(-1+4*n)*(a+b*x^n)^2,x,3,1/4*a^2*x^(4*n)/n+2/5*a*b*x^(5*n)/n+1/6*b^2*x^(6*n)/n],
[x^(-1+3*n)*(a+b*x^n)^2,x,3,1/3*a^2*x^(3*n)/n+1/2*a*b*x^(4*n)/n+1/5*b^2*x^(5*n)/n],
[x^(-1+2*n)*(a+b*x^n)^2,x,3,1/2*a^2*x^(2*n)/n+2/3*a*b*x^(3*n)/n+1/4*b^2*x^(4*n)/n],
[x^(-1+n)*(a+b*x^n)^2,x,1,1/3*(a+b*x^n)^3/(b*n)],
[(a+b*x^n)^2/x,x,3,2*a*b*x^n/n+1/2*b^2*x^(2*n)/n+a^2*log(x)],
[x^(-1-n)*(a+b*x^n)^2,x,3,-a^2/(n*x^n)+b^2*x^n/n+2*a*b*log(x)],
[x^(-1-2*n)*(a+b*x^n)^2,x,3,-1/2*a^2/(n*x^(2*n))-2*a*b/(n*x^n)+b^2*log(x)],
[x^(-1-3*n)*(a+b*x^n)^2,x,1,-1/3*(a+b*x^n)^3/(a*n*x^(3*n))],
[x^(-1-4*n)*(a+b*x^n)^2,x,3,-1/4*a^2/(n*x^(4*n))-2/3*a*b/(n*x^(3*n))-1/2*b^2/(n*x^(2*n))],
[x^(-1-5*n)*(a+b*x^n)^2,x,3,-1/5*a^2/(n*x^(5*n))-1/2*a*b/(n*x^(4*n))-1/3*b^2/(n*x^(3*n))],
[x^(-1-6*n)*(a+b*x^n)^2,x,3,-1/6*a^2/(n*x^(6*n))-2/5*a*b/(n*x^(5*n))-1/4*b^2/(n*x^(4*n))],
[x^(-1+4*n)*(a+b*x^n)^3,x,3,1/4*a^3*x^(4*n)/n+3/5*a^2*b*x^(5*n)/n+1/2*a*b^2*x^(6*n)/n+1/7*b^3*x^(7*n)/n],
[x^(-1+3*n)*(a+b*x^n)^3,x,3,1/3*a^3*x^(3*n)/n+3/4*a^2*b*x^(4*n)/n+3/5*a*b^2*x^(5*n)/n+1/6*b^3*x^(6*n)/n],
[x^(-1+2*n)*(a+b*x^n)^3,x,3,-1/4*a*(a+b*x^n)^4/(b^2*n)+1/5*(a+b*x^n)^5/(b^2*n)],
[x^(-1+n)*(a+b*x^n)^3,x,1,1/4*(a+b*x^n)^4/(b*n)],
[(a+b*x^n)^3/x,x,3,3*a^2*b*x^n/n+3/2*a*b^2*x^(2*n)/n+1/3*b^3*x^(3*n)/n+a^3*log(x)],
[x^(-1-n)*(a+b*x^n)^3,x,3,-a^3/(n*x^n)+3*a*b^2*x^n/n+1/2*b^3*x^(2*n)/n+3*a^2*b*log(x)],
[x^(-1-2*n)*(a+b*x^n)^3,x,3,-1/2*a^3/(n*x^(2*n))-3*a^2*b/(n*x^n)+b^3*x^n/n+3*a*b^2*log(x)],
[x^(-1-3*n)*(a+b*x^n)^3,x,3,-1/3*a^3/(n*x^(3*n))-3/2*a^2*b/(n*x^(2*n))-3*a*b^2/(n*x^n)+b^3*log(x)],
[x^(-1-4*n)*(a+b*x^n)^3,x,1,-1/4*(a+b*x^n)^4/(a*n*x^(4*n))],
[x^(-1-5*n)*(a+b*x^n)^3,x,3,-1/5*(a+b*x^n)^4/(a*n*x^(5*n))+1/20*b*(a+b*x^n)^4/(a^2*n*x^(4*n))],
[x^(-1-6*n)*(a+b*x^n)^3,x,3,-1/6*a^3/(n*x^(6*n))-3/5*a^2*b/(n*x^(5*n))-3/4*a*b^2/(n*x^(4*n))-1/3*b^3/(n*x^(3*n))],
[x^(-1-7*n)*(a+b*x^n)^3,x,3,-1/7*a^3/(n*x^(7*n))-1/2*a^2*b/(n*x^(6*n))-3/5*a*b^2/(n*x^(5*n))-1/4*b^3/(n*x^(4*n))],
[x^(-1+4*n)*(a+b*x^n)^5,x,3,-1/6*a^3*(a+b*x^n)^6/(b^4*n)+3/7*a^2*(a+b*x^n)^7/(b^4*n)-3/8*a*(a+b*x^n)^8/(b^4*n)+1/9*(a+b*x^n)^9/(b^4*n)],
[x^(-1+3*n)*(a+b*x^n)^5,x,3,1/6*a^2*(a+b*x^n)^6/(b^3*n)-2/7*a*(a+b*x^n)^7/(b^3*n)+1/8*(a+b*x^n)^8/(b^3*n)],
[x^(-1+2*n)*(a+b*x^n)^5,x,3,-1/6*a*(a+b*x^n)^6/(b^2*n)+1/7*(a+b*x^n)^7/(b^2*n)],
[x^(-1+n)*(a+b*x^n)^5,x,1,1/6*(a+b*x^n)^6/(b*n)],
[(a+b*x^n)^5/x,x,3,5*a^4*b*x^n/n+5*a^3*b^2*x^(2*n)/n+10/3*a^2*b^3*x^(3*n)/n+5/4*a*b^4*x^(4*n)/n+1/5*b^5*x^(5*n)/n+a^5*log(x)],
[x^(-1-n)*(a+b*x^n)^5,x,3,-a^5/(n*x^n)+10*a^3*b^2*x^n/n+5*a^2*b^3*x^(2*n)/n+5/3*a*b^4*x^(3*n)/n+1/4*b^5*x^(4*n)/n+5*a^4*b*log(x)],
[x^(-1-2*n)*(a+b*x^n)^5,x,3,-1/2*a^5/(n*x^(2*n))-5*a^4*b/(n*x^n)+10*a^2*b^3*x^n/n+5/2*a*b^4*x^(2*n)/n+1/3*b^5*x^(3*n)/n+10*a^3*b^2*log(x)],
[x^(-1-3*n)*(a+b*x^n)^5,x,3,-1/3*a^5/(n*x^(3*n))-5/2*a^4*b/(n*x^(2*n))-10*a^3*b^2/(n*x^n)+5*a*b^4*x^n/n+1/2*b^5*x^(2*n)/n+10*a^2*b^3*log(x)],
[x^(-1-4*n)*(a+b*x^n)^5,x,3,-1/4*a^5/(n*x^(4*n))-5/3*a^4*b/(n*x^(3*n))-5*a^3*b^2/(n*x^(2*n))-10*a^2*b^3/(n*x^n)+b^5*x^n/n+5*a*b^4*log(x)],
[x^(-1-5*n)*(a+b*x^n)^5,x,3,-1/5*a^5/(n*x^(5*n))-5/4*a^4*b/(n*x^(4*n))-10/3*a^3*b^2/(n*x^(3*n))-5*a^2*b^3/(n*x^(2*n))-5*a*b^4/(n*x^n)+b^5*log(x)],
[x^(-1-6*n)*(a+b*x^n)^5,x,1,-1/6*(a+b*x^n)^6/(a*n*x^(6*n))],
[x^(-1-7*n)*(a+b*x^n)^5,x,3,-1/7*(a+b*x^n)^6/(a*n*x^(7*n))+1/42*b*(a+b*x^n)^6/(a^2*n*x^(6*n))],
[x^(-1-8*n)*(a+b*x^n)^5,x,4,-1/8*(a+b*x^n)^6/(a*n*x^(8*n))+1/28*b*(a+b*x^n)^6/(a^2*n*x^(7*n))-1/168*b^2*(a+b*x^n)^6/(a^3*n*x^(6*n))],
[x^(-1-9*n)*(a+b*x^n)^5,x,3,-1/9*a^5/(n*x^(9*n))-5/8*a^4*b/(n*x^(8*n))-10/7*a^3*b^2/(n*x^(7*n))-5/3*a^2*b^3/(n*x^(6*n))-a*b^4/(n*x^(5*n))-1/4*b^5/(n*x^(4*n))],
[x^(-1-10*n)*(a+b*x^n)^5,x,3,-1/10*a^5/(n*x^(10*n))-5/9*a^4*b/(n*x^(9*n))-5/4*a^3*b^2/(n*x^(8*n))-10/7*a^2*b^3/(n*x^(7*n))-5/6*a*b^4/(n*x^(6*n))-1/5*b^5/(n*x^(5*n))],
[x^(-1+9*n)*(a+b*x^n)^8,x,3,1/9*a^8*x^(9*n)/n+4/5*a^7*b*x^(10*n)/n+28/11*a^6*b^2*x^(11*n)/n+14/3*a^5*b^3*x^(12*n)/n+70/13*a^4*b^4*x^(13*n)/n+4*a^3*b^5*x^(14*n)/n+28/15*a^2*b^6*x^(15*n)/n+1/2*a*b^7*x^(16*n)/n+1/17*b^8*x^(17*n)/n],
[x^(-1+8*n)*(a+b*x^n)^8,x,3,1/8*a^8*x^(8*n)/n+8/9*a^7*b*x^(9*n)/n+14/5*a^6*b^2*x^(10*n)/n+56/11*a^5*b^3*x^(11*n)/n+35/6*a^4*b^4*x^(12*n)/n+56/13*a^3*b^5*x^(13*n)/n+2*a^2*b^6*x^(14*n)/n+8/15*a*b^7*x^(15*n)/n+1/16*b^8*x^(16*n)/n],
[x^(-1+7*n)*(a+b*x^n)^8,x,3,1/9*a^6*(a+b*x^n)^9/(b^7*n)-3/5*a^5*(a+b*x^n)^10/(b^7*n)+15/11*a^4*(a+b*x^n)^11/(b^7*n)-5/3*a^3*(a+b*x^n)^12/(b^7*n)+15/13*a^2*(a+b*x^n)^13/(b^7*n)-3/7*a*(a+b*x^n)^14/(b^7*n)+1/15*(a+b*x^n)^15/(b^7*n)],
[x^(-1+6*n)*(a+b*x^n)^8,x,3,-1/9*a^5*(a+b*x^n)^9/(b^6*n)+1/2*a^4*(a+b*x^n)^10/(b^6*n)-10/11*a^3*(a+b*x^n)^11/(b^6*n)+5/6*a^2*(a+b*x^n)^12/(b^6*n)-5/13*a*(a+b*x^n)^13/(b^6*n)+1/14*(a+b*x^n)^14/(b^6*n)],
[x^(-1+5*n)*(a+b*x^n)^8,x,3,1/9*a^4*(a+b*x^n)^9/(b^5*n)-2/5*a^3*(a+b*x^n)^10/(b^5*n)+6/11*a^2*(a+b*x^n)^11/(b^5*n)-1/3*a*(a+b*x^n)^12/(b^5*n)+1/13*(a+b*x^n)^13/(b^5*n)],
[x^(-1+4*n)*(a+b*x^n)^8,x,3,-1/9*a^3*(a+b*x^n)^9/(b^4*n)+3/10*a^2*(a+b*x^n)^10/(b^4*n)-3/11*a*(a+b*x^n)^11/(b^4*n)+1/12*(a+b*x^n)^12/(b^4*n)],
[x^(-1+3*n)*(a+b*x^n)^8,x,3,1/9*a^2*(a+b*x^n)^9/(b^3*n)-1/5*a*(a+b*x^n)^10/(b^3*n)+1/11*(a+b*x^n)^11/(b^3*n)],
[x^(-1+2*n)*(a+b*x^n)^8,x,3,-1/9*a*(a+b*x^n)^9/(b^2*n)+1/10*(a+b*x^n)^10/(b^2*n)],
[x^(-1+n)*(a+b*x^n)^8,x,1,1/9*(a+b*x^n)^9/(b*n)],
[(a+b*x^n)^8/x,x,3,8*a^7*b*x^n/n+14*a^6*b^2*x^(2*n)/n+56/3*a^5*b^3*x^(3*n)/n+35/2*a^4*b^4*x^(4*n)/n+56/5*a^3*b^5*x^(5*n)/n+14/3*a^2*b^6*x^(6*n)/n+8/7*a*b^7*x^(7*n)/n+1/8*b^8*x^(8*n)/n+a^8*log(x)],
[x^(-1-n)*(a+b*x^n)^8,x,3,-a^8/(n*x^n)+28*a^6*b^2*x^n/n+28*a^5*b^3*x^(2*n)/n+70/3*a^4*b^4*x^(3*n)/n+14*a^3*b^5*x^(4*n)/n+28/5*a^2*b^6*x^(5*n)/n+4/3*a*b^7*x^(6*n)/n+1/7*b^8*x^(7*n)/n+8*a^7*b*log(x)],
[x^(-1-2*n)*(a+b*x^n)^8,x,3,-1/2*a^8/(n*x^(2*n))-8*a^7*b/(n*x^n)+56*a^5*b^3*x^n/n+35*a^4*b^4*x^(2*n)/n+56/3*a^3*b^5*x^(3*n)/n+7*a^2*b^6*x^(4*n)/n+8/5*a*b^7*x^(5*n)/n+1/6*b^8*x^(6*n)/n+28*a^6*b^2*log(x)],
[x^(-1-3*n)*(a+b*x^n)^8,x,3,-1/3*a^8/(n*x^(3*n))-4*a^7*b/(n*x^(2*n))-28*a^6*b^2/(n*x^n)+70*a^4*b^4*x^n/n+28*a^3*b^5*x^(2*n)/n+28/3*a^2*b^6*x^(3*n)/n+2*a*b^7*x^(4*n)/n+1/5*b^8*x^(5*n)/n+56*a^5*b^3*log(x)],
[x^(-1-4*n)*(a+b*x^n)^8,x,3,-1/4*a^8/(n*x^(4*n))-8/3*a^7*b/(n*x^(3*n))-14*a^6*b^2/(n*x^(2*n))-56*a^5*b^3/(n*x^n)+56*a^3*b^5*x^n/n+14*a^2*b^6*x^(2*n)/n+8/3*a*b^7*x^(3*n)/n+1/4*b^8*x^(4*n)/n+70*a^4*b^4*log(x)],
[x^(-1-5*n)*(a+b*x^n)^8,x,3,-1/5*a^8/(n*x^(5*n))-2*a^7*b/(n*x^(4*n))-28/3*a^6*b^2/(n*x^(3*n))-28*a^5*b^3/(n*x^(2*n))-70*a^4*b^4/(n*x^n)+28*a^2*b^6*x^n/n+4*a*b^7*x^(2*n)/n+1/3*b^8*x^(3*n)/n+56*a^3*b^5*log(x)],
[x^(-1-6*n)*(a+b*x^n)^8,x,3,-1/6*a^8/(n*x^(6*n))-8/5*a^7*b/(n*x^(5*n))-7*a^6*b^2/(n*x^(4*n))-56/3*a^5*b^3/(n*x^(3*n))-35*a^4*b^4/(n*x^(2*n))-56*a^3*b^5/(n*x^n)+8*a*b^7*x^n/n+1/2*b^8*x^(2*n)/n+28*a^2*b^6*log(x)],
[x^(-1-7*n)*(a+b*x^n)^8,x,3,-1/7*a^8/(n*x^(7*n))-4/3*a^7*b/(n*x^(6*n))-28/5*a^6*b^2/(n*x^(5*n))-14*a^5*b^3/(n*x^(4*n))-70/3*a^4*b^4/(n*x^(3*n))-28*a^3*b^5/(n*x^(2*n))-28*a^2*b^6/(n*x^n)+b^8*x^n/n+8*a*b^7*log(x)],
[x^(-1-8*n)*(a+b*x^n)^8,x,3,-1/8*a^8/(n*x^(8*n))-8/7*a^7*b/(n*x^(7*n))-14/3*a^6*b^2/(n*x^(6*n))-56/5*a^5*b^3/(n*x^(5*n))-35/2*a^4*b^4/(n*x^(4*n))-56/3*a^3*b^5/(n*x^(3*n))-14*a^2*b^6/(n*x^(2*n))-8*a*b^7/(n*x^n)+b^8*log(x)],
[x^(-1-9*n)*(a+b*x^n)^8,x,1,-1/9*(a+b*x^n)^9/(a*n*x^(9*n))],
[x^(-1-10*n)*(a+b*x^n)^8,x,3,-1/10*(a+b*x^n)^9/(a*n*x^(10*n))+1/90*b*(a+b*x^n)^9/(a^2*n*x^(9*n))],
[x^(-1-11*n)*(a+b*x^n)^8,x,4,-1/11*(a+b*x^n)^9/(a*n*x^(11*n))+1/55*b*(a+b*x^n)^9/(a^2*n*x^(10*n))-1/495*b^2*(a+b*x^n)^9/(a^3*n*x^(9*n))],
[x^(-1-12*n)*(a+b*x^n)^8,x,5,-1/12*(a+b*x^n)^9/(a*n*x^(12*n))+1/44*b*(a+b*x^n)^9/(a^2*n*x^(11*n))-1/220*b^2*(a+b*x^n)^9/(a^3*n*x^(10*n))+1/1980*b^3*(a+b*x^n)^9/(a^4*n*x^(9*n))],
[x^(-1-13*n)*(a+b*x^n)^8,x,6,-1/13*(a+b*x^n)^9/(a*n*x^(13*n))+1/39*b*(a+b*x^n)^9/(a^2*n*x^(12*n))-1/143*b^2*(a+b*x^n)^9/(a^3*n*x^(11*n))+1/715*b^3*(a+b*x^n)^9/(a^4*n*x^(10*n))-1/6435*b^4*(a+b*x^n)^9/(a^5*n*x^(9*n))],
[x^(-1-14*n)*(a+b*x^n)^8,x,3,-1/14*a^8/(n*x^(14*n))-8/13*a^7*b/(n*x^(13*n))-7/3*a^6*b^2/(n*x^(12*n))-56/11*a^5*b^3/(n*x^(11*n))-7*a^4*b^4/(n*x^(10*n))-56/9*a^3*b^5/(n*x^(9*n))-7/2*a^2*b^6/(n*x^(8*n))-8/7*a*b^7/(n*x^(7*n))-1/6*b^8/(n*x^(6*n))],
[x^(-1-15*n)*(a+b*x^n)^8,x,3,-1/15*a^8/(n*x^(15*n))-4/7*a^7*b/(n*x^(14*n))-28/13*a^6*b^2/(n*x^(13*n))-14/3*a^5*b^3/(n*x^(12*n))-70/11*a^4*b^4/(n*x^(11*n))-28/5*a^3*b^5/(n*x^(10*n))-28/9*a^2*b^6/(n*x^(9*n))-a*b^7/(n*x^(8*n))-1/7*b^8/(n*x^(7*n))],
[x^(-1+n)*(a+b*x^n)^16,x,1,1/17*(a+b*x^n)^17/(b*n)],
[x^12*(a+b*x^13)^12,x,1,1/169*(a+b*x^13)^13/b],
[x^24*(a+b*x^25)^12,x,1,1/325*(a+b*x^25)^13/b],
[x^36*(a+b*x^37)^12,x,1,1/481*(a+b*x^37)^13/b],
[x^(12*m)*(a+b*x^(1+12*m))^12,x,1,1/13*(a+b*x^(1+12*m))^13/(b*(1+12*m))],

#  Need to detect that degree of nomomial simplifies to 12*m so integrand is not be expanded! 
[x^(12+12*(-1+m))*(a+b*x^(1+12*m))^12,x,1,1/13*(a+b*x^(1+12*m))^13/(b*(1+12*m))],

# p<0
[x^(-1+5*n)/(a+b*x^n),x,3,-a^3*x^n/(b^4*n)+1/2*a^2*x^(2*n)/(b^3*n)-1/3*a*x^(3*n)/(b^2*n)+1/4*x^(4*n)/(b*n)+a^4*log(a+b*x^n)/(b^5*n)],
[x^(-1+4*n)/(a+b*x^n),x,3,a^2*x^n/(b^3*n)-1/2*a*x^(2*n)/(b^2*n)+1/3*x^(3*n)/(b*n)-a^3*log(a+b*x^n)/(b^4*n)],
[x^(-1+3*n)/(a+b*x^n),x,3,-a*x^n/(b^2*n)+1/2*x^(2*n)/(b*n)+a^2*log(a+b*x^n)/(b^3*n)],
[x^(-1+2*n)/(a+b*x^n),x,3,x^n/(b*n)-a*log(a+b*x^n)/(b^2*n)],
[x^(-1+n)/(a+b*x^n),x,1,log(a+b*x^n)/(b*n)],
[1/(x*(a+b*x^n)),x,4,log(x)/a-log(a+b*x^n)/(a*n)],
[x^(-1-n)/(a+b*x^n),x,3,(-1)/(a*n*x^n)-b*log(x)/a^2+b*log(a+b*x^n)/(a^2*n)],
[x^(-1-2*n)/(a+b*x^n),x,3,(-1/2)/(a*n*x^(2*n))+b/(a^2*n*x^n)+b^2*log(x)/a^3-b^2*log(a+b*x^n)/(a^3*n)],
[x^(-1-3*n)/(a+b*x^n),x,3,(-1/3)/(a*n*x^(3*n))+1/2*b/(a^2*n*x^(2*n))-b^2/(a^3*n*x^n)-b^3*log(x)/a^4+b^3*log(a+b*x^n)/(a^4*n)],

#  Note: Requires simplification of the exponent of the monomial. 
[x^(4+5*(-1+n))/(a+b*x^n),x,3,-a^3*x^n/(b^4*n)+1/2*a^2*x^(2*n)/(b^3*n)-1/3*a*x^(3*n)/(b^2*n)+1/4*x^(4*n)/(b*n)+a^4*log(a+b*x^n)/(b^5*n)],
[x^(3+4*(-1+n))/(a+b*x^n),x,3,a^2*x^n/(b^3*n)-1/2*a*x^(2*n)/(b^2*n)+1/3*x^(3*n)/(b*n)-a^3*log(a+b*x^n)/(b^4*n)],
[x^(2+3*(-1+n))/(a+b*x^n),x,3,-a*x^n/(b^2*n)+1/2*x^(2*n)/(b*n)+a^2*log(a+b*x^n)/(b^3*n)],
[x^(1+2*(-1+n))/(a+b*x^n),x,3,x^n/(b*n)-a*log(a+b*x^n)/(b^2*n)],
[x^(-1+n)/(a+b*x^n),x,1,log(a+b*x^n)/(b*n)],
[1/(x*(a+b*x^n)),x,4,log(x)/a-log(a+b*x^n)/(a*n)],
[x^(-1-n)/(a+b*x^n),x,3,(-1)/(a*n*x^n)-b*log(x)/a^2+b*log(a+b*x^n)/(a^2*n)],
[x^(-3-2*(-1+n))/(a+b*x^n),x,3,(-1/2)/(a*n*x^(2*n))+b/(a^2*n*x^n)+b^2*log(x)/a^3-b^2*log(a+b*x^n)/(a^3*n)],
[x^(-4-3*(-1+n))/(a+b*x^n),x,3,(-1/3)/(a*n*x^(3*n))+1/2*b/(a^2*n*x^(2*n))-b^2/(a^3*n*x^n)-b^3*log(x)/a^4+b^3*log(a+b*x^n)/(a^4*n)],
[x^(-1+5*n)/(2+b*x^n),x,3,-8*x^n/(b^4*n)+2*x^(2*n)/(b^3*n)-2/3*x^(3*n)/(b^2*n)+1/4*x^(4*n)/(b*n)+16*log(2+b*x^n)/(b^5*n)],
[x^(-1+4*n)/(2+b*x^n),x,3,4*x^n/(b^3*n)-x^(2*n)/(b^2*n)+1/3*x^(3*n)/(b*n)-8*log(2+b*x^n)/(b^4*n)],
[x^(-1+3*n)/(2+b*x^n),x,3,-2*x^n/(b^2*n)+1/2*x^(2*n)/(b*n)+4*log(2+b*x^n)/(b^3*n)],
[x^(-1+2*n)/(2+b*x^n),x,3,x^n/(b*n)-2*log(2+b*x^n)/(b^2*n)],
[x^(-1+n)/(2+b*x^n),x,1,log(2+b*x^n)/(b*n)],
[1/(x*(2+b*x^n)),x,4,1/2*log(x)-1/2*log(2+b*x^n)/n],
[x^(-1-n)/(2+b*x^n),x,3,(-1/2)/(n*x^n)-1/4*b*log(x)+1/4*b*log(2+b*x^n)/n],
[x^(-1-2*n)/(2+b*x^n),x,3,(-1/4)/(n*x^(2*n))+1/4*b/(n*x^n)+1/8*b^2*log(x)-1/8*b^2*log(2+b*x^n)/n],
[x^(-1-3*n)/(2+b*x^n),x,3,(-1/6)/(n*x^(3*n))+1/8*b/(n*x^(2*n))-1/8*b^2/(n*x^n)-1/16*b^3*log(x)+1/16*b^3*log(2+b*x^n)/n],
[x^(-1+4*n)/(a+b*x^n)^2,x,3,-2*a*x^n/(b^3*n)+1/2*x^(2*n)/(b^2*n)+a^3/(b^4*n*(a+b*x^n))+3*a^2*log(a+b*x^n)/(b^4*n)],
[x^(-1+3*n)/(a+b*x^n)^2,x,3,x^n/(b^2*n)-a^2/(b^3*n*(a+b*x^n))-2*a*log(a+b*x^n)/(b^3*n)],
[x^(-1+2*n)/(a+b*x^n)^2,x,3,a/(b^2*n*(a+b*x^n))+log(a+b*x^n)/(b^2*n)],
[x^(-1+n)/(a+b*x^n)^2,x,1,(-1)/(b*n*(a+b*x^n))],
[1/(x*(a+b*x^n)^2),x,3,1/(a*n*(a+b*x^n))+log(x)/a^2-log(a+b*x^n)/(a^2*n)],
[x^(-1-n)/(a+b*x^n)^2,x,3,(-1)/(a^2*n*x^n)-b/(a^2*n*(a+b*x^n))-2*b*log(x)/a^3+2*b*log(a+b*x^n)/(a^3*n)],
[x^(-1-2*n)/(a+b*x^n)^2,x,3,(-1/2)/(a^2*n*x^(2*n))+2*b/(a^3*n*x^n)+b^2/(a^3*n*(a+b*x^n))+3*b^2*log(x)/a^4-3*b^2*log(a+b*x^n)/(a^4*n)],
[x^(-1-3*n)/(a+b*x^n)^2,x,3,(-1/3)/(a^2*n*x^(3*n))+b/(a^3*n*x^(2*n))-3*b^2/(a^4*n*x^n)-b^3/(a^4*n*(a+b*x^n))-4*b^3*log(x)/a^5+4*b^3*log(a+b*x^n)/(a^5*n)],
[x^(-1+4*n)/(a+b*x^n)^3,x,3,x^n/(b^3*n)+1/2*a^3/(b^4*n*(a+b*x^n)^2)-3*a^2/(b^4*n*(a+b*x^n))-3*a*log(a+b*x^n)/(b^4*n)],
[x^(-1+3*n)/(a+b*x^n)^3,x,3,-1/2*a^2/(b^3*n*(a+b*x^n)^2)+2*a/(b^3*n*(a+b*x^n))+log(a+b*x^n)/(b^3*n)],
[x^(-1+2*n)/(a+b*x^n)^3,x,1,1/2*x^(2*n)/(a*n*(a+b*x^n)^2)],
[x^(-1+n)/(a+b*x^n)^3,x,1,(-1/2)/(b*n*(a+b*x^n)^2)],
[1/(x*(a+b*x^n)^3),x,3,1/2/(a*n*(a+b*x^n)^2)+1/(a^2*n*(a+b*x^n))+log(x)/a^3-log(a+b*x^n)/(a^3*n)],
[x^(-1-n)/(a+b*x^n)^3,x,3,(-1)/(a^3*n*x^n)-1/2*b/(a^2*n*(a+b*x^n)^2)-2*b/(a^3*n*(a+b*x^n))-3*b*log(x)/a^4+3*b*log(a+b*x^n)/(a^4*n)],
[x^(-1-2*n)/(a+b*x^n)^3,x,3,(-1/2)/(a^3*n*x^(2*n))+3*b/(a^4*n*x^n)+1/2*b^2/(a^3*n*(a+b*x^n)^2)+3*b^2/(a^4*n*(a+b*x^n))+6*b^2*log(x)/a^5-6*b^2*log(a+b*x^n)/(a^5*n)],
[x^(-1-1/2*n)/(a+b*x^n),x,4,(-2)/(a*n*x^(1/2*n))+2*arctan(sqrt(a)/(x^(1/2*n)*sqrt(b)))*sqrt(b)/(a^(3/2)*n)],
[x^(-1-2/3*n)/(a+b*x^n),x,8,(-3/2)/(a*n*x^(2/3*n))-b^(2/3)*log(a^(1/3)+b^(1/3)*x^(1/3*n))/(a^(5/3)*n)+1/2*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3*n)+b^(2/3)*x^(2/3*n))/(a^(5/3)*n)+b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*x^(1/3*n))/(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(5/3)*n)],
[x^(-1-3/4*n)/(a+b*x^n),x,11,(-4/3)/(a*n*x^(3/4*n))+b^(3/4)*log(-a^(1/4)*b^(1/4)*x^(1/4*n)*sqrt(2)+sqrt(a)+x^(1/2*n)*sqrt(b))/(a^(7/4)*n*sqrt(2))-b^(3/4)*log(a^(1/4)*b^(1/4)*x^(1/4*n)*sqrt(2)+sqrt(a)+x^(1/2*n)*sqrt(b))/(a^(7/4)*n*sqrt(2))+b^(3/4)*arctan(1-b^(1/4)*x^(1/4*n)*sqrt(2)/a^(1/4))*sqrt(2)/(a^(7/4)*n)-b^(3/4)*arctan(1+b^(1/4)*x^(1/4*n)*sqrt(2)/a^(1/4))*sqrt(2)/(a^(7/4)*n)],
[x^(-1-n)/(a+b*x^n),x,3,(-1)/(a*n*x^n)-b*log(x)/a^2+b*log(a+b*x^n)/(a^2*n)],
[x^(-1-1/2*n)/(a+b*x^n),x,4,(-2)/(a*n*x^(1/2*n))+2*arctan(sqrt(a)/(x^(1/2*n)*sqrt(b)))*sqrt(b)/(a^(3/2)*n)],
[x^(-1-1/3*n)/(a+b*x^n),x,9,(-3)/(a*n*x^(1/3*n))+b^(1/3)*log(b^(1/3)+a^(1/3)/x^(1/3*n))/(a^(4/3)*n)-1/2*b^(1/3)*log(b^(2/3)+a^(2/3)/x^(2/3*n)-a^(1/3)*b^(1/3)/x^(1/3*n))/(a^(4/3)*n)-b^(1/3)*arctan((b^(1/3)-2*a^(1/3)/x^(1/3*n))/(b^(1/3)*sqrt(3)))*sqrt(3)/(a^(4/3)*n)],
[x^(-1-1/4*n)/(a+b*x^n),x,12,(-4)/(a*n*x^(1/4*n))-b^(1/4)*log(-a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(5/4)*n*sqrt(2))+b^(1/4)*log(a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(5/4)*n*sqrt(2))-b^(1/4)*arctan(1-a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(5/4)*n)+b^(1/4)*arctan(1+a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(5/4)*n)],
[x^(-1-3/2*n)/(a+b*x^n),x,5,(-2/3)/(a*n*x^(3/2*n))+2*b/(a^2*n*x^(1/2*n))-2*b^(3/2)*arctan(sqrt(a)/(x^(1/2*n)*sqrt(b)))/(a^(5/2)*n)],
[x^(-1-4/3*n)/(a+b*x^n),x,10,(-3/4)/(a*n*x^(4/3*n))+3*b/(a^2*n*x^(1/3*n))-b^(4/3)*log(b^(1/3)+a^(1/3)/x^(1/3*n))/(a^(7/3)*n)+1/2*b^(4/3)*log(b^(2/3)+a^(2/3)/x^(2/3*n)-a^(1/3)*b^(1/3)/x^(1/3*n))/(a^(7/3)*n)+b^(4/3)*arctan((1-2*a^(1/3)/(b^(1/3)*x^(1/3*n)))/sqrt(3))*sqrt(3)/(a^(7/3)*n)],
[x^(-1-5/4*n)/(a+b*x^n),x,13,(-4/5)/(a*n*x^(5/4*n))+4*b/(a^2*n*x^(1/4*n))+b^(5/4)*log(-a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(9/4)*n*sqrt(2))-b^(5/4)*log(a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(9/4)*n*sqrt(2))+b^(5/4)*arctan(1-a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(9/4)*n)-b^(5/4)*arctan(1+a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(9/4)*n)],

# Integrands of the form x^(k n-1) (a+b x^n)^(p/2)

# p>0
[x^(-1+4*n)*(a+b*x^n)^(1/2),x,3,-2/3*a^3*(a+b*x^n)^(3/2)/(b^4*n)+6/5*a^2*(a+b*x^n)^(5/2)/(b^4*n)-6/7*a*(a+b*x^n)^(7/2)/(b^4*n)+2/9*(a+b*x^n)^(9/2)/(b^4*n)],
[x^(-1+3*n)*(a+b*x^n)^(1/2),x,3,2/3*a^2*(a+b*x^n)^(3/2)/(b^3*n)-4/5*a*(a+b*x^n)^(5/2)/(b^3*n)+2/7*(a+b*x^n)^(7/2)/(b^3*n)],
[x^(-1+2*n)*(a+b*x^n)^(1/2),x,3,-2/3*a*(a+b*x^n)^(3/2)/(b^2*n)+2/5*(a+b*x^n)^(5/2)/(b^2*n)],
[x^(-1+n)*(a+b*x^n)^(1/2),x,1,2/3*(a+b*x^n)^(3/2)/(b*n)],
[(a+b*x^n)^(1/2)/x,x,4,-2*arctanh(sqrt(a+b*x^n)/sqrt(a))*sqrt(a)/n+2*sqrt(a+b*x^n)/n],
[x^(-1-n)*(a+b*x^n)^(1/2),x,4,-b*arctanh(sqrt(a+b*x^n)/sqrt(a))/(n*sqrt(a))-sqrt(a+b*x^n)/(n*x^n)],
[x^(-1-2*n)*(a+b*x^n)^(1/2),x,5,1/4*b^2*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(3/2)*n)-1/2*sqrt(a+b*x^n)/(n*x^(2*n))-1/4*b*sqrt(a+b*x^n)/(a*n*x^n)],
[x^(-1-3*n)*(a+b*x^n)^(1/2),x,6,-1/8*b^3*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(5/2)*n)-1/3*sqrt(a+b*x^n)/(n*x^(3*n))-1/12*b*sqrt(a+b*x^n)/(a*n*x^(2*n))+1/8*b^2*sqrt(a+b*x^n)/(a^2*n*x^n)],
[x^(-1-4*n)*(a+b*x^n)^(1/2),x,7,5/64*b^4*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(7/2)*n)-1/4*sqrt(a+b*x^n)/(n*x^(4*n))-1/24*b*sqrt(a+b*x^n)/(a*n*x^(3*n))+5/96*b^2*sqrt(a+b*x^n)/(a^2*n*x^(2*n))-5/64*b^3*sqrt(a+b*x^n)/(a^3*n*x^n)],

# p<0
[x^(-1+4*n)/(a+b*x^n)^(1/2),x,3,2*a^2*(a+b*x^n)^(3/2)/(b^4*n)-6/5*a*(a+b*x^n)^(5/2)/(b^4*n)+2/7*(a+b*x^n)^(7/2)/(b^4*n)-2*a^3*sqrt(a+b*x^n)/(b^4*n)],
[x^(-1+3*n)/(a+b*x^n)^(1/2),x,3,-4/3*a*(a+b*x^n)^(3/2)/(b^3*n)+2/5*(a+b*x^n)^(5/2)/(b^3*n)+2*a^2*sqrt(a+b*x^n)/(b^3*n)],
[x^(-1+2*n)/(a+b*x^n)^(1/2),x,3,2/3*(a+b*x^n)^(3/2)/(b^2*n)-2*a*sqrt(a+b*x^n)/(b^2*n)],
[x^(-1+n)/(a+b*x^n)^(1/2),x,1,2*sqrt(a+b*x^n)/(b*n)],
[1/(x*(a+b*x^n)^(1/2)),x,3,-2*arctanh(sqrt(a+b*x^n)/sqrt(a))/(n*sqrt(a))],
[x^(-1-n)/(a+b*x^n)^(1/2),x,4,b*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(3/2)*n)-sqrt(a+b*x^n)/(a*n*x^n)],
[x^(-1-2*n)/(a+b*x^n)^(1/2),x,5,-3/4*b^2*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(5/2)*n)-1/2*sqrt(a+b*x^n)/(a*n*x^(2*n))+3/4*b*sqrt(a+b*x^n)/(a^2*n*x^n)],
[x^(-1-3*n)/(a+b*x^n)^(1/2),x,6,5/8*b^3*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(7/2)*n)-1/3*sqrt(a+b*x^n)/(a*n*x^(3*n))+5/12*b*sqrt(a+b*x^n)/(a^2*n*x^(2*n))-5/8*b^2*sqrt(a+b*x^n)/(a^3*n*x^n)],
[x^(-1-4*n)/(a+b*x^n)^(1/2),x,7,-35/64*b^4*arctanh(sqrt(a+b*x^n)/sqrt(a))/(a^(9/2)*n)-1/4*sqrt(a+b*x^n)/(a*n*x^(4*n))+7/24*b*sqrt(a+b*x^n)/(a^2*n*x^(3*n))-35/96*b^2*sqrt(a+b*x^n)/(a^3*n*x^(2*n))+35/64*b^3*sqrt(a+b*x^n)/(a^4*n*x^n)],

# Integrands of the form x^m (a+b x^n)^p with m symbolic
[x^m*(a+b*x^n)^3,x,2,a^3*x^(1+m)/(1+m)+3*a^2*b*x^(1+m+n)/(1+m+n)+3*a*b^2*x^(1+m+2*n)/(1+m+2*n)+b^3*x^(1+m+3*n)/(1+m+3*n)],
[x^m*(a+b*x^n)^2,x,2,a^2*x^(1+m)/(1+m)+2*a*b*x^(1+m+n)/(1+m+n)+b^2*x^(1+m+2*n)/(1+m+2*n)],
[x^m*(a+b*x^n),x,2,a*x^(1+m)/(1+m)+b*x^(1+m+n)/(1+m+n)],
[x^m/(a+b*x^n),x,1,x^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a*(1+m))],
[x^m/(a+b*x^n)^2,x,1,x^(1+m)*hypergeom([2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a^2*(1+m))],
[x^m/(a+b*x^n)^3,x,1,x^(1+m)*hypergeom([3,(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a^3*(1+m))],
[x^m*(a+b*x^n)^(3/2),x,2,x^(1+m)*(a+b*x^n)^(5/2)*hypergeom([1,5/2+(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a*(1+m)),a*x^(1+m)*hypergeom([-3/2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/((1+m)*sqrt(1+b*x^n/a))],
[x^m*(a+b*x^n)^(1/2),x,2,x^(1+m)*(a+b*x^n)^(3/2)*hypergeom([1,3/2+(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a*(1+m)),x^(1+m)*hypergeom([-1/2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/((1+m)*sqrt(1+b*x^n/a))],
[x^m/(a+b*x^n)^(1/2),x,2,x^(1+m)*hypergeom([1,1/2+(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(1+m)),x^(1+m)*hypergeom([1/2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m)*sqrt(a+b*x^n))],
[x^m/(a+b*x^n)^(3/2),x,2,x^(1+m)*hypergeom([1,-1/2+(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a*(1+m)*sqrt(a+b*x^n)),x^(1+m)*hypergeom([3/2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a*(1+m)*sqrt(a+b*x^n))],
[x^m/(a+b*x^n)^(5/2),x,2,x^(1+m)*hypergeom([1,-3/2+(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a*(1+m)*(a+b*x^n)^(3/2)),x^(1+m)*hypergeom([5/2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(a^2*(1+m)*sqrt(a+b*x^n))],
[x^(3+2*n)/sqrt(a+b*x^n),x,2,1/2*x^(2*(2+n))*hypergeom([1,1/2*(5+8/n)],[3+4/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(2+n)),1/2*x^(2*(2+n))*hypergeom([1/2,2*(1+2/n)],[3+4/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((2+n)*sqrt(a+b*x^n))],
[x^(3+n)/sqrt(a+b*x^n),x,2,x^(4+n)*hypergeom([1,1/2*(3+8/n)],[2*(1+2/n)],-b*x^n/a)*sqrt(a+b*x^n)/(a*(4+n)),x^(4+n)*hypergeom([1/2,(4+n)/n],[2*(1+2/n)],-b*x^n/a)*sqrt(1+b*x^n/a)/((4+n)*sqrt(a+b*x^n))],
[x^(3-n)/sqrt(a+b*x^n),x,2,x^(4-n)*hypergeom([1,1/2*(-1+8/n)],[4/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(4-n)),x^(4-n)*hypergeom([1/2,-1+4/n],[4/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((4-n)*sqrt(a+b*x^n))],
[x^(3-2*n)/sqrt(a+b*x^n),x,2,1/2*x^(4-2*n)*hypergeom([1,1/2*(-3+8/n)],[-1+4/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(2-n)),1/2*x^(4-2*n)*hypergeom([1/2,-2*(1+(-2)/n)],[-1+4/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((2-n)*sqrt(a+b*x^n))],
[x^(m+2*n)/sqrt(a+b*x^n),x,2,x^(1+m+2*n)*hypergeom([1,1/2*(5+2*(1+m)/n)],[(1+m+3*n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(1+m+2*n)),x^(1+m+2*n)*hypergeom([1/2,(1+m+2*n)/n],[(1+m+3*n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m+2*n)*sqrt(a+b*x^n))],
[x^(m+n)/sqrt(a+b*x^n),x,2,x^(1+m+n)*hypergeom([1,1/2+(1+m+n)/n],[(1+m+2*n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(1+m+n)),x^(1+m+n)*hypergeom([1/2,(1+m+n)/n],[(1+m+2*n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m+n)*sqrt(a+b*x^n))],
[x^(m-n)/sqrt(a+b*x^n),x,2,x^(1+m-n)*hypergeom([1,1/2*(-1+2*(1+m)/n)],[(1+m)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(1+m-n)),x^(1+m-n)*hypergeom([1/2,(1+m-n)/n],[(1+m)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m-n)*sqrt(a+b*x^n))],
[x^(m-2*n)/sqrt(a+b*x^n),x,2,x^(1+m-2*n)*hypergeom([1,1/2*(-3+2*(1+m)/n)],[(1+m-n)/n],-b*x^n/a)*sqrt(a+b*x^n)/(a*(1+m-2*n)),x^(1+m-2*n)*hypergeom([1/2,(1+m-2*n)/n],[(1+m-n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m-2*n)*sqrt(a+b*x^n))],
[-1/2*b*n*x^(-1+m+n)/(a+b*x^n)^(3/2)+m*x^(-1+m)/sqrt(a+b*x^n),x,-5,x^m/sqrt(a+b*x^n),-b*x^(m+n)/(a*sqrt(a+b*x^n))+x^m*sqrt(a+b*x^n)/a],
[x^(-1+7/2*n)/sqrt(a+b*x^n),x,5,-5/8*a^3*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(b^(7/2)*n)+5/8*a^2*x^(1/2*n)*sqrt(a+b*x^n)/(b^3*n)-5/12*a*x^(3/2*n)*sqrt(a+b*x^n)/(b^2*n)+1/3*x^(5/2*n)*sqrt(a+b*x^n)/(b*n)],
[x^(-1+5/2*n)/sqrt(a+b*x^n),x,4,3/4*a^2*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(b^(5/2)*n)-3/4*a*x^(1/2*n)*sqrt(a+b*x^n)/(b^2*n)+1/2*x^(3/2*n)*sqrt(a+b*x^n)/(b*n)],
[x^(-1+3/2*n)/sqrt(a+b*x^n),x,3,-a*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(b^(3/2)*n)+x^(1/2*n)*sqrt(a+b*x^n)/(b*n)],
[x^(-1+1/2*n)/sqrt(a+b*x^n),x,3,2*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(n*sqrt(b))],
[x^(-1-1/2*n)/sqrt(a+b*x^n),x,1,-2*sqrt(a+b*x^n)/(a*n*x^(1/2*n))],
[x^(-1-3/2*n)/sqrt(a+b*x^n),x,2,-2/3*sqrt(a+b*x^n)/(a*n*x^(3/2*n))+4/3*b*sqrt(a+b*x^n)/(a^2*n*x^(1/2*n))],
[x^(-1-5/2*n)/sqrt(a+b*x^n),x,3,-2/5*sqrt(a+b*x^n)/(a*n*x^(5/2*n))+8/15*b*sqrt(a+b*x^n)/(a^2*n*x^(3/2*n))-16/15*b^2*sqrt(a+b*x^n)/(a^3*n*x^(1/2*n))],
[x^(-1-7/2*n)/sqrt(a+b*x^n),x,4,-2/7*sqrt(a+b*x^n)/(a*n*x^(7/2*n))+12/35*b*sqrt(a+b*x^n)/(a^2*n*x^(5/2*n))-16/35*b^2*sqrt(a+b*x^n)/(a^3*n*x^(3/2*n))+32/35*b^3*sqrt(a+b*x^n)/(a^4*n*x^(1/2*n))],
[x^m/sqrt(a+b*x^(-2+m)),x,2,x^(1+m)*hypergeom([1,-3/2*m/(2-m)],[(1-2*m)/(2-m)],-b*x^(-2+m)/a)*sqrt(a+b*x^(-2+m))/(a*(1+m)),x^(1+m)*hypergeom([1/2,(-1-m)/(2-m)],[(1-2*m)/(2-m)],-b*x^(-2+m)/a)*sqrt(1+b*x^(-2+m)/a)/((1+m)*sqrt(a+b*x^(-2+m)))],
[x^m/sqrt(a+b*x^(2-m)),x,2,x^(1+m)*hypergeom([1,1/2*(4+m)/(2-m)],[3/(2-m)],-b*x^(2-m)/a)*sqrt(a+b*x^(2-m))/(a*(1+m)),x^(1+m)*hypergeom([1/2,(1+m)/(2-m)],[3/(2-m)],-b*x^(2-m)/a)*sqrt(1+b*x^(2-m)/a)/((1+m)*sqrt(a+b*x^(2-m)))],
[6*a*x^2/(b*(4+m)*sqrt(a+b*x^(-2+m)))+x^m/sqrt(a+b*x^(-2+m)),x,-5,2*x^3*sqrt(a+b*x^(-2+m))/(b*(4+m)),2*x^(1+m)*(b+a*x^(2-m))/(b*(4+m)*sqrt(a+b*x^(-2+m)))],
[1/2*x^(-1+m)*(2*a*m+b*(2*m-n)*x^n)/(a+b*x^n)^(3/2),x,2,x^m/sqrt(a+b*x^n)],
[-1/2*b*n*x^(-1+m+n)/(a+b*x^n)^(3/2)+m*x^(-1+m)/sqrt(a+b*x^n),x,-5,x^m/sqrt(a+b*x^n),-b*x^(m+n)/(a*sqrt(a+b*x^n))+x^m*sqrt(a+b*x^n)/a],

#  The substitution m->-1-2*n/3 transforms the first integrand into the second. 
[x^m/(a+b*x^(3*(1+m)))^(1/3),x,2,-1/2*log(b^(1/3)*x^(1+m)-(a+b*x^(3*(1+m)))^(1/3))/(b^(1/3)*(1+m))+arctan((1+2*b^(1/3)*x^(1+m)/(a+b*x^(3*(1+m)))^(1/3))/sqrt(3))/(b^(1/3)*(1+m)*sqrt(3))],
[x^m*(a+b/x^(3/2*(1+m)))^(2/3),x,3,x^(1+m)*(a+b/x^(3/2*(1+m)))^(2/3)/(1+m)+b^(2/3)*log(b^(1/3)*x^(1/2*(-1-m))-(a+b/x^(3/2*(1+m)))^(1/3))/(1+m)-2*b^(2/3)*arctan((1+2*b^(1/3)*x^(1/2*(-1-m))/(a+b/x^(3/2*(1+m)))^(1/3))/sqrt(3))/((1+m)*sqrt(3)),x^(1+m)*(a+b/x^(3/2*(1+m)))^(2/3)/(1+m)+b^(2/3)*log(-x^(1/2*(-1-m))*(b^(1/3)-x^(1/2*(1+m))*(a+b/x^(3/2*(1+m)))^(1/3)))/(1+m)-2*b^(2/3)*arctan((1+2*b^(1/3)*x^(1/2*(-1-m))/(a+b/x^(3/2*(1+m)))^(1/3))/sqrt(3))/((1+m)*sqrt(3))],
[x^(-1+1/3*n)/(a+b*x^n)^(1/3),x,2,-3/2*log(b^(1/3)*x^(1/3*n)-(a+b*x^n)^(1/3))/(b^(1/3)*n)+arctan((1+2*b^(1/3)*x^(1/3*n)/(a+b*x^n)^(1/3))/sqrt(3))*sqrt(3)/(b^(1/3)*n)],
[x^(-1-2/3*n)*(a+b*x^n)^(2/3),x,3,-3/2*(a+b*x^n)^(2/3)/(n*x^(2/3*n))-3/2*b^(2/3)*log(b^(1/3)*x^(1/3*n)-(a+b*x^n)^(1/3))/n+b^(2/3)*arctan((1+2*b^(1/3)*x^(1/3*n)/(a+b*x^n)^(1/3))/sqrt(3))*sqrt(3)/n],

# Integrands of the form x^m (a+b x^n)^p with p symbolic
[x^m*(a+b*x^n)^p,x,2,x^(1+m)*(a+b*x^n)^(1+p)*hypergeom([1,1+(1+m)/n+p],[(1+m+n)/n],-b*x^n/a)/(a*(1+m)),x^(1+m)*(a+b*x^n)^p*hypergeom([(1+m)/n,-p],[(1+m+n)/n],-b*x^n/a)/((1+m)*(1+b*x^n/a)^p)],
[(a+b*x^n)^(-4+(-1)/n),x,4,x*(a+b*x^n)^(-3+(-1)/n)/(a*(1+3*n))+3*n*x*(a+b*x^n)^(-2+(-1)/n)/(a^2*(1+5*n+6*n^2))+6*n^2*x*(a+b*x^n)^(-1+(-1)/n)/(a^3*(1+6*n+11*n^2+6*n^3))+6*n^3*x/(a^4*(1+6*n+11*n^2+6*n^3)*(a+b*x^n)^(1/n))],
[(a+b*x^n)^(-3+(-1)/n),x,3,x*(a+b*x^n)^(-2+(-1)/n)/(a*(1+2*n))+2*n*x*(a+b*x^n)^(-1+(-1)/n)/(a^2*(1+3*n+2*n^2))+2*n^2*x/(a^3*(1+3*n+2*n^2)*(a+b*x^n)^(1/n))],
[(a+b*x^n)^(-2+(-1)/n),x,2,x*(a+b*x^n)^(-1+(-1)/n)/(a*(1+n))+n*x/(a^2*(1+n)*(a+b*x^n)^(1/n))],
[(a+b*x^n)^(-1+(-1)/n),x,1,x/(a*(a+b*x^n)^(1/n))],
[1/(a+b*x^n)^(1/n),x,2,x*(1+b*x^n/a)^(1/n)*hypergeom([1/n,1/n],[1+1/n],-b*x^n/a)/(a+b*x^n)^(1/n)],
[(a+b*x^n)^(1+(-1)/n),x,2,a*x*(1+b*x^n/a)^(1/n)*hypergeom([-1+1/n,1/n],[1+1/n],-b*x^n/a)/(a+b*x^n)^(1/n)],
[(a+b*x^n)^(2+(-1)/n),x,2,a^2*x*(1+b*x^n/a)^(1/n)*hypergeom([-2+1/n,1/n],[1+1/n],-b*x^n/a)/(a+b*x^n)^(1/n)],
[x^m*(b*x^n)^p,x,2,x^(1+m)*(b*x^n)^p/(1+m+n*p)],
[x^2*(b*x^n)^p,x,2,x^3*(b*x^n)^p/(3+n*p)],
[x*(b*x^n)^p,x,2,x^2*(b*x^n)^p/(2+n*p)],
[(b*x^n)^p,x,2,x*(b*x^n)^p/(1+n*p)],
[(b*x^n)^p/x,x,2,(b*x^n)^p/(n*p)],
[(b*x^n)^p/x^2,x,2,-(b*x^n)^p/((1-n*p)*x)],
[(b*x^n)^p/x^3,x,2,-(b*x^n)^p/((2-n*p)*x^2)],
[(b*x^n)^p/x^4,x,2,-(b*x^n)^p/((3-n*p)*x^3)],
[x^(-1+n)*(a+b*x^n)^p,x,1,(a+b*x^n)^(1+p)/(b*n*(1+p))],
[x^(-1+2*n)*(a+b*x^n)^p,x,3,-a*(a+b*x^n)^(1+p)/(b^2*n*(1+p))+(a+b*x^n)^(2+p)/(b^2*n*(2+p))],
[x^(-1+3*n)*(a+b*x^n)^p,x,3,a^2*(a+b*x^n)^(1+p)/(b^3*n*(1+p))-2*a*(a+b*x^n)^(2+p)/(b^3*n*(2+p))+(a+b*x^n)^(3+p)/(b^3*n*(3+p))],
[x^(-1+4*n)*(a+b*x^n)^p,x,3,-a^3*(a+b*x^n)^(1+p)/(b^4*n*(1+p))+3*a^2*(a+b*x^n)^(2+p)/(b^4*n*(2+p))-3*a*(a+b*x^n)^(3+p)/(b^4*n*(3+p))+(a+b*x^n)^(4+p)/(b^4*n*(4+p))],

#  Integrands of the form x^m*(a+b*x^n)^p where n*p+n+m+1=0 
[x^(-1-n-n*p)*(a+b*x^n)^p,x,1,-(a+b*x^n)^(1+p)/(a*n*(1+p)*x^(n*(1+p)))],
[x^(-1-9*n)*(a+b*x^n)^8,x,1,-1/9*(a+b*x^n)^9/(a*n*x^(9*n))],
[x^(-4-3*p)*(a+b*x^3)^p,x,1,-1/3*(a+b*x^3)^(1+p)/(a*(1+p)*x^(3*(1+p)))],
[(a+b*x^3)^8/x^28,x,1,-1/27*(a+b*x^3)^9/(a*x^27)],
[1/(x*(a+b*x^n)),x,4,log(x)/a-log(a+b*x^n)/(a*n)],
[1/(x*(a+b*x^3)),x,4,log(x)/a-1/3*log(a+b*x^3)/a],
[1/(a+b*x^n)^((1+4*n)/n),x,4,x*(a+b*x^n)^(-3+(-1)/n)/(a*(1+3*n))+3*n*x*(a+b*x^n)^(-2+(-1)/n)/(a^2*(1+5*n+6*n^2))+6*n^3*x/(a^4*(1+6*n+11*n^2+6*n^3)*(a+b*x^n)^(1/n))+6*n^2*x/(a^3*(1+6*n+11*n^2+6*n^3)*(a+b*x^n)^((1+n)/n))],
[1/(a+b*x^n)^((1+3*n)/n),x,3,x*(a+b*x^n)^(-2+(-1)/n)/(a*(1+2*n))+2*n^2*x/(a^3*(1+3*n+2*n^2)*(a+b*x^n)^(1/n))+2*n*x/(a^2*(1+3*n+2*n^2)*(a+b*x^n)^((1+n)/n))],
[1/(a+b*x^n)^((1+2*n)/n),x,2,n*x/(a^2*(1+n)*(a+b*x^n)^(1/n))+x/(a*(1+n)*(a+b*x^n)^((1+n)/n))],
[1/(a+b*x^n)^((1+n)/n),x,1,x/(a*(a+b*x^n)^(1/n))],
[1/(a+b*x^n)^(1/n),x,2,x*(1+b*x^n/a)^(1/n)*hypergeom([1/n,1/n],[1+1/n],-b*x^n/a)/(a+b*x^n)^(1/n)],
[1/(a+b*x^n)^((1-n)/n),x,2,x*(1+b*x^n/a)^(-1+1/n)*hypergeom([-1+1/n,1/n],[1+1/n],-b*x^n/a)/(a+b*x^n)^((1-n)/n)],
[1/(a+b*x^n)^((1-2*n)/n),x,2,x*(a+b*x^n)^(2+(-1)/n)*(1+b*x^n/a)^(-2+1/n)*hypergeom([-2+1/n,1/n],[1+1/n],-b*x^n/a)],
[1/(x*(a+b/x^n)),x,2,log(b+a*x^n)/(a*n)],

# Integrands of the form x^m (a+b x^(n (m + 1)))^p
[x^m/(a+b*x^(1+m)),x,1,log(a+b*x^(1+m))/(b*(1+m))],
[x^m*(a+b*x^(1+m))^n,x,1,(a+b*x^(1+m))^(1+n)/(b*(1+m)*(1+n))],
[x^m*(a+b*x^(2+2*m))^3,x,2,a^3*x^(1+m)/(1+m)+a^2*b*x^(3*(1+m))/(1+m)+3/5*a*b^2*x^(5*(1+m))/(1+m)+1/7*b^3*x^(7*(1+m))/(1+m)],
[x^m*(a+b*x^(2+2*m))^2,x,2,a^2*x^(1+m)/(1+m)+2/3*a*b*x^(3*(1+m))/(1+m)+1/5*b^2*x^(5*(1+m))/(1+m)],
[x^m*(a+b*x^(2+2*m)),x,2,a*x^(1+m)/(1+m)+1/3*b*x^(3*(1+m))/(1+m)],
[x^m/(a+b*x^(2+2*m)),x,2,arctan(x^(1+m)*sqrt(b)/sqrt(a))/((1+m)*sqrt(a)*sqrt(b))],
[x^m/(a+b*x^(2+2*m))^2,x,3,1/2*x^(1+m)/(a*(1+m)*(a+b*x^(2*(1+m))))+1/2*arctan(x^(1+m)*sqrt(b)/sqrt(a))/(a^(3/2)*(1+m)*sqrt(b))],
[x^m/(a+b*x^(2+2*m))^3,x,4,1/4*x^(1+m)/(a*(1+m)*(a+b*x^(2*(1+m)))^2)+3/8*x^(1+m)/(a^2*(1+m)*(a+b*x^(2*(1+m))))+3/8*arctan(x^(1+m)*sqrt(b)/sqrt(a))/(a^(5/2)*(1+m)*sqrt(b))],
[x^m*(a+b*x^(2+2*m))^(5/2),x,6,5/24*a*x^(1+m)*(a+b*x^(2*(1+m)))^(3/2)/(1+m)+1/6*x^(1+m)*(a+b*x^(2*(1+m)))^(5/2)/(1+m)+5/16*a^3*arctanh(x^(1+m)*sqrt(b)/sqrt(a+b*x^(2*(1+m))))/((1+m)*sqrt(b))+5/16*a^2*x^(1+m)*sqrt(a+b*x^(2*(1+m)))/(1+m)],
[x^m*(a+b*x^(2+2*m))^(3/2),x,5,1/4*x^(1+m)*(a+b*x^(2*(1+m)))^(3/2)/(1+m)+3/8*a^2*arctanh(x^(1+m)*sqrt(b)/sqrt(a+b*x^(2*(1+m))))/((1+m)*sqrt(b))+3/8*a*x^(1+m)*sqrt(a+b*x^(2*(1+m)))/(1+m)],
[x^m*(a+b*x^(2+2*m))^(1/2),x,4,1/2*a*arctanh(x^(1+m)*sqrt(b)/sqrt(a+b*x^(2*(1+m))))/((1+m)*sqrt(b))+1/2*x^(1+m)*sqrt(a+b*x^(2*(1+m)))/(1+m)],
[x^m/(a+b*x^(2+2*m))^(1/2),x,3,arctanh(x^(1+m)*sqrt(b)/sqrt(a+b*x^(2*(1+m))))/((1+m)*sqrt(b))],
[x^m/(a+b*x^(2+2*m))^(3/2),x,1,x^(1+m)/(a*(1+m)*sqrt(a+b*x^(2*(1+m))))],
[x^m/(a+b*x^(2+2*m))^(5/2),x,2,x^(1+m)/(a*(1+m)*(a+b*x^(2*(1+m)))^(3/2))+2/3*b*x^(3*(1+m))/(a^2*(1+m)*(a+b*x^(2*(1+m)))^(3/2))],
[x^m/(a+b*x^(2+2*m))^(7/2),x,3,x^(1+m)/(a*(1+m)*(a+b*x^(2*(1+m)))^(5/2))+4/3*b*x^(3*(1+m))/(a^2*(1+m)*(a+b*x^(2*(1+m)))^(5/2))+8/15*b^2*x^(5*(1+m))/(a^3*(1+m)*(a+b*x^(2*(1+m)))^(5/2))],
[x^n*sqrt(1+x^(1+n)),x,1,2/3*(1+x^(1+n))^(3/2)/(1+n)],
[x^n*sqrt(a^2+x^(1+n)),x,1,2/3*(a^2+x^(1+n))^(3/2)/(1+n)],

# Integrands of the form (c x)^m (a+b x^n)^p

# Integrands of the form (c x)^m (a+b x^n)^p

# p>0
[(c*x)^m*(a+b*x^n)^2,x,6,2*a*b*x^(1+n)*(c*x)^m/(1+m+n)+b^2*x^(1+2*n)*(c*x)^m/(1+m+2*n)+a^2*(c*x)^(1+m)/(c*(1+m))],
[(c*x)^m*(a+b*x^3)^2,x,2,a^2*(c*x)^(1+m)/(c*(1+m))+2*a*b*(c*x)^(4+m)/(c^4*(4+m))+b^2*(c*x)^(7+m)/(c^7*(7+m))],
[(c*x)^m*(a+b*x^2)^2,x,2,a^2*(c*x)^(1+m)/(c*(1+m))+2*a*b*(c*x)^(3+m)/(c^3*(3+m))+b^2*(c*x)^(5+m)/(c^5*(5+m))],
[(c*x)^m*(a+b*x)^2,x,2,a^2*(c*x)^(1+m)/(c*(1+m))+2*a*b*(c*x)^(2+m)/(c^2*(2+m))+b^2*(c*x)^(3+m)/(c^3*(3+m))],
[(a+b/x)^2*(c*x)^m,x,2,-b^2*c*(c*x)^(-1+m)/(1-m)+2*a*b*(c*x)^m/m+a^2*(c*x)^(1+m)/(c*(1+m))],
[(a+b/x^2)^2*(c*x)^m,x,2,-b^2*c^3*(c*x)^(-3+m)/(3-m)-2*a*b*c*(c*x)^(-1+m)/(1-m)+a^2*(c*x)^(1+m)/(c*(1+m))],
[(a+b/x^3)^2*(c*x)^m,x,2,-b^2*c^5*(c*x)^(-5+m)/(5-m)-2*a*b*c^2*(c*x)^(-2+m)/(2-m)+a^2*(c*x)^(1+m)/(c*(1+m))],

# p<0
[(c*x)^(-1-1/2*n)/(a+b*x^n),x,5,(-2)/(a*c*n*(c*x)^(1/2*n))+2*x^(1/2*n)*arctan(sqrt(a)/(x^(1/2*n)*sqrt(b)))*sqrt(b)/(a^(3/2)*c*n*(c*x)^(1/2*n))],
[(c*x)^(-1-2/3*n)/(a+b*x^n),x,9,(-3/2)/(a*c*n*(c*x)^(2/3*n))-b^(2/3)*x^(2/3*n)*log(a^(1/3)+b^(1/3)*x^(1/3*n))/(a^(5/3)*c*n*(c*x)^(2/3*n))+1/2*b^(2/3)*x^(2/3*n)*log(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/3*n)+b^(2/3)*x^(2/3*n))/(a^(5/3)*c*n*(c*x)^(2/3*n))+b^(2/3)*x^(2/3*n)*arctan((a^(1/3)-2*b^(1/3)*x^(1/3*n))/(a^(1/3)*sqrt(3)))*sqrt(3)/(a^(5/3)*c*n*(c*x)^(2/3*n))],
[(c*x)^(-1-3/4*n)/(a+b*x^n),x,12,(-4/3)/(a*c*n*(c*x)^(3/4*n))+b^(3/4)*x^(3/4*n)*log(-a^(1/4)*b^(1/4)*x^(1/4*n)*sqrt(2)+sqrt(a)+x^(1/2*n)*sqrt(b))/(a^(7/4)*c*n*(c*x)^(3/4*n)*sqrt(2))-b^(3/4)*x^(3/4*n)*log(a^(1/4)*b^(1/4)*x^(1/4*n)*sqrt(2)+sqrt(a)+x^(1/2*n)*sqrt(b))/(a^(7/4)*c*n*(c*x)^(3/4*n)*sqrt(2))+b^(3/4)*x^(3/4*n)*arctan(1-b^(1/4)*x^(1/4*n)*sqrt(2)/a^(1/4))*sqrt(2)/(a^(7/4)*c*n*(c*x)^(3/4*n))-b^(3/4)*x^(3/4*n)*arctan(1+b^(1/4)*x^(1/4*n)*sqrt(2)/a^(1/4))*sqrt(2)/(a^(7/4)*c*n*(c*x)^(3/4*n))],
[(c*x)^(-1-n)/(a+b*x^n),x,4,(-1)/(a*c*n*(c*x)^n)-b*x^n*log(x)/(a^2*c*(c*x)^n)+b*x^n*log(a+b*x^n)/(a^2*c*n*(c*x)^n)],
[(c*x)^(-1-1/2*n)/(a+b*x^n),x,5,(-2)/(a*c*n*(c*x)^(1/2*n))+2*x^(1/2*n)*arctan(sqrt(a)/(x^(1/2*n)*sqrt(b)))*sqrt(b)/(a^(3/2)*c*n*(c*x)^(1/2*n))],
[(c*x)^(-1-1/3*n)/(a+b*x^n),x,10,(-3)/(a*c*n*(c*x)^(1/3*n))+b^(1/3)*x^(1/3*n)*log(b^(1/3)+a^(1/3)/x^(1/3*n))/(a^(4/3)*c*n*(c*x)^(1/3*n))-1/2*b^(1/3)*x^(1/3*n)*log(b^(2/3)+a^(2/3)/x^(2/3*n)-a^(1/3)*b^(1/3)/x^(1/3*n))/(a^(4/3)*c*n*(c*x)^(1/3*n))-b^(1/3)*x^(1/3*n)*arctan((b^(1/3)-2*a^(1/3)/x^(1/3*n))/(b^(1/3)*sqrt(3)))*sqrt(3)/(a^(4/3)*c*n*(c*x)^(1/3*n))],
[(c*x)^(-1-1/4*n)/(a+b*x^n),x,13,(-4)/(a*c*n*(c*x)^(1/4*n))-b^(1/4)*x^(1/4*n)*log(-a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(5/4)*c*n*(c*x)^(1/4*n)*sqrt(2))+b^(1/4)*x^(1/4*n)*log(a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(5/4)*c*n*(c*x)^(1/4*n)*sqrt(2))-b^(1/4)*x^(1/4*n)*arctan(1-a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(5/4)*c*n*(c*x)^(1/4*n))+b^(1/4)*x^(1/4*n)*arctan(1+a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(5/4)*c*n*(c*x)^(1/4*n))],
[(c*x)^(-1-3/2*n)/(a+b*x^n),x,6,(-2/3)/(a*c*n*(c*x)^(3/2*n))+2*b*x^n/(a^2*c*n*(c*x)^(3/2*n))-2*b^(3/2)*x^(3/2*n)*arctan(sqrt(a)/(x^(1/2*n)*sqrt(b)))/(a^(5/2)*c*n*(c*x)^(3/2*n))],
[(c*x)^(-1-4/3*n)/(a+b*x^n),x,11,(-3/4)/(a*c*n*(c*x)^(4/3*n))+3*b*x^n/(a^2*c*n*(c*x)^(4/3*n))-b^(4/3)*x^(4/3*n)*log(b^(1/3)+a^(1/3)/x^(1/3*n))/(a^(7/3)*c*n*(c*x)^(4/3*n))+1/2*b^(4/3)*x^(4/3*n)*log(b^(2/3)+a^(2/3)/x^(2/3*n)-a^(1/3)*b^(1/3)/x^(1/3*n))/(a^(7/3)*c*n*(c*x)^(4/3*n))+b^(4/3)*x^(4/3*n)*arctan((1-2*a^(1/3)/(b^(1/3)*x^(1/3*n)))/sqrt(3))*sqrt(3)/(a^(7/3)*c*n*(c*x)^(4/3*n))],
[(c*x)^(-1-5/4*n)/(a+b*x^n),x,14,(-4/5)/(a*c*n*(c*x)^(5/4*n))+4*b*x^n/(a^2*c*n*(c*x)^(5/4*n))+b^(5/4)*x^(5/4*n)*log(-a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(9/4)*c*n*(c*x)^(5/4*n)*sqrt(2))-b^(5/4)*x^(5/4*n)*log(a^(1/4)*b^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(a)/x^(1/2*n)+sqrt(b))/(a^(9/4)*c*n*(c*x)^(5/4*n)*sqrt(2))+b^(5/4)*x^(5/4*n)*arctan(1-a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(9/4)*c*n*(c*x)^(5/4*n))-b^(5/4)*x^(5/4*n)*arctan(1+a^(1/4)*sqrt(2)/(b^(1/4)*x^(1/4*n)))*sqrt(2)/(a^(9/4)*c*n*(c*x)^(5/4*n))],
[(c*x)^(4+n)/(a+b*x^n),x,1,(c*x)^(5+n)*hypergeom([1,(5+n)/n],[2+5/n],-b*x^n/a)/(a*c*(5+n))],
[(c*x)^(3+n)/(a+b*x^n),x,1,(c*x)^(4+n)*hypergeom([1,(4+n)/n],[2*(1+2/n)],-b*x^n/a)/(a*c*(4+n))],
[(c*x)^(2+n)/(a+b*x^n),x,1,(c*x)^(3+n)*hypergeom([1,(3+n)/n],[2+3/n],-b*x^n/a)/(a*c*(3+n))],
[(c*x)^(1+n)/(a+b*x^n),x,1,(c*x)^(2+n)*hypergeom([1,(2+n)/n],[2*(1+1/n)],-b*x^n/a)/(a*c*(2+n))],
[(c*x)^n/(a+b*x^n),x,1,(c*x)^(1+n)*hypergeom([1,1+1/n],[2+1/n],-b*x^n/a)/(a*c*(1+n))],
[(c*x)^(-1+n)/(a+b*x^n),x,2,(c*x)^n*log(a+b*x^n)/(b*c*n*x^n)],
[(c*x)^(-2+n)/(a+b*x^n),x,1,-(c*x)^(-1+n)*hypergeom([1,(-1+n)/n],[2+(-1)/n],-b*x^n/a)/(a*c*(1-n))],
[(c*x)^(-3+n)/(a+b*x^n),x,1,-(c*x)^(-2+n)*hypergeom([1,(-2+n)/n],[2*(1+(-1)/n)],-b*x^n/a)/(a*c*(2-n))],
[(c*x)^(-1+n)/(a+b*x^n)^2,x,1,(c*x)^n/(a*c*n*(a+b*x^n))],

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

# p>0

# p<0
[(c*x)^(-1+7/2*n)/sqrt(a+b*x^n),x,6,-5/8*a^3*(c*x)^(7/2*n)*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(b^(7/2)*c*n*x^(7/2*n))+5/8*a^2*(c*x)^(7/2*n)*sqrt(a+b*x^n)/(b^3*c*n*x^(3*n))-5/12*a*(c*x)^(7/2*n)*sqrt(a+b*x^n)/(b^2*c*n*x^(2*n))+1/3*(c*x)^(7/2*n)*sqrt(a+b*x^n)/(b*c*n*x^n)],
[(c*x)^(-1+5/2*n)/sqrt(a+b*x^n),x,5,3/4*a^2*(c*x)^(5/2*n)*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(b^(5/2)*c*n*x^(5/2*n))-3/4*a*(c*x)^(5/2*n)*sqrt(a+b*x^n)/(b^2*c*n*x^(2*n))+1/2*(c*x)^(5/2*n)*sqrt(a+b*x^n)/(b*c*n*x^n)],
[(c*x)^(-1+3/2*n)/sqrt(a+b*x^n),x,4,-a*(c*x)^(3/2*n)*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(b^(3/2)*c*n*x^(3/2*n))+(c*x)^(3/2*n)*sqrt(a+b*x^n)/(b*c*n*x^n)],
[(c*x)^(-1+1/2*n)/sqrt(a+b*x^n),x,4,2*(c*x)^(1/2*n)*arctanh(x^(1/2*n)*sqrt(b)/sqrt(a+b*x^n))/(c*n*x^(1/2*n)*sqrt(b))],
[(c*x)^(-1-1/2*n)/sqrt(a+b*x^n),x,1,-2*sqrt(a+b*x^n)/(a*c*n*(c*x)^(1/2*n))],
[(c*x)^(-1-3/2*n)/sqrt(a+b*x^n),x,2,4/3*(a+b*x^n)^(3/2)/(a^2*c*n*(c*x)^(3/2*n))-2*sqrt(a+b*x^n)/(a*c*n*(c*x)^(3/2*n))],
[(c*x)^(-1-5/2*n)/sqrt(a+b*x^n),x,3,8/3*(a+b*x^n)^(3/2)/(a^2*c*n*(c*x)^(5/2*n))-16/15*(a+b*x^n)^(5/2)/(a^3*c*n*(c*x)^(5/2*n))-2*sqrt(a+b*x^n)/(a*c*n*(c*x)^(5/2*n))],
[(c*x)^(-1-7/2*n)/sqrt(a+b*x^n),x,4,4*(a+b*x^n)^(3/2)/(a^2*c*n*(c*x)^(7/2*n))-16/5*(a+b*x^n)^(5/2)/(a^3*c*n*(c*x)^(7/2*n))+32/35*(a+b*x^n)^(7/2)/(a^4*c*n*(c*x)^(7/2*n))-2*sqrt(a+b*x^n)/(a*c*n*(c*x)^(7/2*n))],

# Integrands of the form (c x)^m (a+b x^n)^p with p symbolic
[(c*x)^m*(a+b*x^n)^p,x,2,(c*x)^(1+m)*(a+b*x^n)^(1+p)*hypergeom([1,1+(1+m)/n+p],[(1+m+n)/n],-b*x^n/a)/(a*c*(1+m)),(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([(1+m)/n,-p],[(1+m+n)/n],-b*x^n/a)/(c*(1+m)*(1+b*x^n/a)^p)],
[(c*x)^(3*n)*(a+b*x^n)^p,x,2,(c*x)^(1+3*n)*(a+b*x^n)^(1+p)*hypergeom([1,4+1/n+p],[4+1/n],-b*x^n/a)/(a*c*(1+3*n)),(c*x)^(1+3*n)*(a+b*x^n)^p*hypergeom([3+1/n,-p],[4+1/n],-b*x^n/a)/(c*(1+3*n)*(1+b*x^n/a)^p)],
[(c*x)^(2*n)*(a+b*x^n)^p,x,2,(c*x)^(1+2*n)*(a+b*x^n)^(1+p)*hypergeom([1,3+1/n+p],[3+1/n],-b*x^n/a)/(a*c*(1+2*n)),(c*x)^(1+2*n)*(a+b*x^n)^p*hypergeom([2+1/n,-p],[3+1/n],-b*x^n/a)/(c*(1+2*n)*(1+b*x^n/a)^p)],
[(c*x)^n*(a+b*x^n)^p,x,2,(c*x)^(1+n)*(a+b*x^n)^(1+p)*hypergeom([1,2+1/n+p],[2+1/n],-b*x^n/a)/(a*c*(1+n)),(c*x)^(1+n)*(a+b*x^n)^p*hypergeom([1+1/n,-p],[2+1/n],-b*x^n/a)/(c*(1+n)*(1+b*x^n/a)^p)],
[(a+b*x^n)^p,x,2,x*(a+b*x^n)^(1+p)*hypergeom([1,1+1/n+p],[1+1/n],-b*x^n/a)/a,x*(a+b*x^n)^p*hypergeom([1/n,-p],[1+1/n],-b*x^n/a)/(1+b*x^n/a)^p],
[(a+b*x^n)^p/(c*x)^n,x,2,(c*x)^(1-n)*(a+b*x^n)^(1+p)*hypergeom([1,1/n+p],[1/n],-b*x^n/a)/(a*c*(1-n)),(c*x)^(1-n)*(a+b*x^n)^p*hypergeom([-1+1/n,-p],[1/n],-b*x^n/a)/(c*(1-n)*(1+b*x^n/a)^p)],
[(a+b*x^n)^p/(c*x)^(2*n),x,2,(c*x)^(1-2*n)*(a+b*x^n)^(1+p)*hypergeom([1,-1+1/n+p],[-1+1/n],-b*x^n/a)/(a*c*(1-2*n)),(c*x)^(1-2*n)*(a+b*x^n)^p*hypergeom([-2+1/n,-p],[-1+1/n],-b*x^n/a)/(c*(1-2*n)*(1+b*x^n/a)^p)],
[(a+b*x^n)^p/(c*x)^(3*n),x,2,(c*x)^(1-3*n)*(a+b*x^n)^(1+p)*hypergeom([1,-2+1/n+p],[-2+1/n],-b*x^n/a)/(a*c*(1-3*n)),(c*x)^(1-3*n)*(a+b*x^n)^p*hypergeom([-3+1/n,-p],[-2+1/n],-b*x^n/a)/(c*(1-3*n)*(1+b*x^n/a)^p)],
[(c*x)^(-1-n*p)*(a+b*x^n)^p,x,2,-(a+b*x^n)^p*hypergeom([-p,-p],[1-p],-b*x^n/a)/(c*n*p*(c*x)^(n*p)*(1+b*x^n/a)^p)],
[(c*x)^(-1-n-n*p)*(a+b*x^n)^p,x,1,-(a+b*x^n)^(1+p)/(a*c*n*(1+p)*(c*x)^(n*(1+p)))],
[(c*x)^(-1-2*n-n*p)*(a+b*x^n)^p,x,2,-(a+b*x^n)^(1+p)/(a*c*n*(1+p)*(c*x)^(n*(2+p)))+(a+b*x^n)^(2+p)/(a^2*c*n*(1+p)*(2+p)*(c*x)^(n*(2+p)))],
[(c*x)^(-1-3*n-n*p)*(a+b*x^n)^p,x,3,-(a+b*x^n)^(1+p)/(a*c*n*(1+p)*(c*x)^(n*(3+p)))+2*(a+b*x^n)^(2+p)/(a^2*c*n*(1+p)*(2+p)*(c*x)^(n*(3+p)))-2*(a+b*x^n)^(3+p)/(a^3*c*n*(1+p)*(2+p)*(3+p)*(c*x)^(n*(3+p)))],
[(c*x)^(-1-4*n-n*p)*(a+b*x^n)^p,x,4,-(a+b*x^n)^(1+p)/(a*c*n*(1+p)*(c*x)^(n*(4+p)))+3*(a+b*x^n)^(2+p)/(a^2*c*n*(1+p)*(2+p)*(c*x)^(n*(4+p)))-6*(a+b*x^n)^(3+p)/(a^3*c*n*(1+p)*(2+p)*(3+p)*(c*x)^(n*(4+p)))+6*(a+b*x^n)^(4+p)/(a^4*c*n*(2+3*p+p^2)*(12+7*p+p^2)*(c*x)^(n*(4+p)))],

# Integrands of the form (c+d x)^m (b (c+d x)^n)^p

# Integrands of the form (c+d x)^m (b (c+d x)^2)^p
[(c*(a+b*x)^2)^(5/2),x,3,1/6*c^2*(a+b*x)^5*sqrt(c*(a+b*x)^2)/b],
[(c*(a+b*x)^2)^(3/2),x,3,1/4*c*(a+b*x)^3*sqrt(c*(a+b*x)^2)/b],
[(c*(a+b*x)^2)^(1/2),x,3,1/2*(a+b*x)*sqrt(c*(a+b*x)^2)/b],
[1/(c*(a+b*x)^2)^(1/2),x,3,(a+b*x)*log(a+b*x)/(b*sqrt(c*(a+b*x)^2))],
[1/(c*(a+b*x)^2)^(3/2),x,3,(-1/2)/(b*c*(a+b*x)*sqrt(c*(a+b*x)^2))],
[1/(c*(a+b*x)^2)^(5/2),x,3,(-1/4)/(b*c^2*(a+b*x)^3*sqrt(c*(a+b*x)^2))],
[sqrt((3+5*x)^2),x,3,1/10*(3+5*x)*sqrt((3+5*x)^2)],
[sqrt((6+10*x)^2),x,3,1/5*(3+5*x)*sqrt((3+5*x)^2)],
[1/sqrt((3+5*x)^2),x,3,1/5*(3+5*x)*log(3+5*x)/sqrt((3+5*x)^2)],
[1/sqrt((6+10*x)^2),x,3,1/10*(3+5*x)*log(3+5*x)/sqrt((3+5*x)^2),1/10*(3+5*x)*log(6+10*x)/sqrt((3+5*x)^2)],
[1/sqrt(-(2+3*x)^2),x,3,1/3*(2+3*x)*log(2+3*x)/sqrt(-(2+3*x)^2)],

# Integrands of the form (c+d x)^m (b (c+d x)^3)^p
[(c*(a+b*x)^3)^(5/2),x,3,2/17*c^2*(a+b*x)^7*sqrt(c*(a+b*x)^3)/b],
[(c*(a+b*x)^3)^(3/2),x,3,2/11*c*(a+b*x)^4*sqrt(c*(a+b*x)^3)/b],
[(c*(a+b*x)^3)^(1/2),x,3,2/5*(a+b*x)*sqrt(c*(a+b*x)^3)/b],
[1/(c*(a+b*x)^3)^(1/2),x,3,-2*(a+b*x)/(b*sqrt(c*(a+b*x)^3))],
[1/(c*(a+b*x)^3)^(3/2),x,3,(-2/7)/(b*c*(a+b*x)^2*sqrt(c*(a+b*x)^3))],
[1/(c*(a+b*x)^3)^(5/2),x,3,(-2/13)/(b*c^2*(a+b*x)^5*sqrt(c*(a+b*x)^3))],

# Integrands of the form (c+d x)^m (b (c+d x)^(-1))^p
[(c/(a+b*x))^(5/2),x,3,-2/3*c^2*sqrt(c/(a+b*x))/(b*(a+b*x))],
[(c/(a+b*x))^(3/2),x,3,-2*c*sqrt(c/(a+b*x))/b],
[(c/(a+b*x))^(1/2),x,3,2*(a+b*x)*sqrt(c/(a+b*x))/b],
[1/(c/(a+b*x))^(1/2),x,3,2/3*(a+b*x)/(b*sqrt(c/(a+b*x)))],
[1/(c/(a+b*x))^(3/2),x,3,2/5*(a+b*x)^2/(b*c*sqrt(c/(a+b*x)))],
[1/(c/(a+b*x))^(5/2),x,3,2/7*(a+b*x)^3/(b*c^2*sqrt(c/(a+b*x)))],

# Integrands of the form (c+d x)^m (b (c+d x)^(-2))^p
[(c/(a+b*x)^2)^(5/2),x,3,-1/4*c^2*sqrt(c/(a+b*x)^2)/(b*(a+b*x)^3)],
[(c/(a+b*x)^2)^(3/2),x,3,-1/2*c*sqrt(c/(a+b*x)^2)/(b*(a+b*x))],
[(c/(a+b*x)^2)^(1/2),x,3,(a+b*x)*log(a+b*x)*sqrt(c/(a+b*x)^2)/b],
[1/(c/(a+b*x)^2)^(1/2),x,3,1/2*(a+b*x)/(b*sqrt(c/(a+b*x)^2))],
[1/(c/(a+b*x)^2)^(3/2),x,3,1/4*(a+b*x)^3/(b*c*sqrt(c/(a+b*x)^2))],
[1/(c/(a+b*x)^2)^(5/2),x,3,1/6*(a+b*x)^5/(b*c^2*sqrt(c/(a+b*x)^2))],

# Integrands of the form (c+d x)^m (b (c+d x)^(-3))^p
[(c/(a+b*x)^3)^(5/2),x,3,-2/13*c^2*sqrt(c/(a+b*x)^3)/(b*(a+b*x)^5)],
[(c/(a+b*x)^3)^(3/2),x,3,-2/7*c*sqrt(c/(a+b*x)^3)/(b*(a+b*x)^2)],
[(c/(a+b*x)^3)^(1/2),x,3,-2*(a+b*x)*sqrt(c/(a+b*x)^3)/b],
[1/(c/(a+b*x)^3)^(1/2),x,3,2/5*(a+b*x)/(b*sqrt(c/(a+b*x)^3))],
[1/(c/(a+b*x)^3)^(3/2),x,3,2/11*(a+b*x)^4/(b*c*sqrt(c/(a+b*x)^3))],
[1/(c/(a+b*x)^3)^(5/2),x,3,2/17*(a+b*x)^7/(b*c^2*sqrt(c/(a+b*x)^3))],

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

# Integrands of the form (c+d x)^m (a+b (c+d x)^n)^p

# Integrands of the form (c+d x)^m (a+b (c+d x)^2)^p

# p>0
[(c+d*x)^3*(a+b*(c+d*x)^2),x,3,1/4*a*(c+d*x)^4/d+1/6*b*(c+d*x)^6/d],
[(c+d*x)^3*(a+b*(c+d*x)^2)^2,x,4,1/4*a^2*(c+d*x)^4/d+1/3*a*b*(c+d*x)^6/d+1/8*b^2*(c+d*x)^8/d],
[(c+d*x)^3*(a+b*(c+d*x)^2)^3,x,4,-1/8*a*(a+b*(c+d*x)^2)^4/(b^2*d)+1/10*(a+b*(c+d*x)^2)^5/(b^2*d)],

# p<0
[(2+x)/(1+(2+x)^2),x,2,1/2*log(1+(2+x)^2)],
[(2+x)/(1+(2+x)^2)^2,x,2,(-1/2)/(1+(2+x)^2)],
[(2+x)/(1+(2+x)^2)^3,x,2,(-1/4)/(1+(2+x)^2)^2],

# p symbolic
[(c+d*x)^5*(a+b*(c+d*x)^2)^p,x,4,1/2*a^2*(a+b*(c+d*x)^2)^(1+p)/(b^3*d*(1+p))-a*(a+b*(c+d*x)^2)^(2+p)/(b^3*d*(2+p))+1/2*(a+b*(c+d*x)^2)^(3+p)/(b^3*d*(3+p))],
[(c+d*x)^4*(a+b*(c+d*x)^2)^p,x,3,1/5*(c+d*x)^5*(a+b*(c+d*x)^2)^(1+p)*hypergeom([1,7/2+p],[7/2],-b*(c+d*x)^2/a)/(a*d),1/5*(c+d*x)^5*(a+b*(c+d*x)^2)^p*hypergeom([5/2,-p],[7/2],-b*(c+d*x)^2/a)/(d*(1+b*(c+d*x)^2/a)^p)],
[(c+d*x)^3*(a+b*(c+d*x)^2)^p,x,4,-1/2*a*(a+b*(c+d*x)^2)^(1+p)/(b^2*d*(1+p))+1/2*(a+b*(c+d*x)^2)^(2+p)/(b^2*d*(2+p))],
[(c+d*x)^2*(a+b*(c+d*x)^2)^p,x,3,1/3*(c+d*x)^3*(a+b*(c+d*x)^2)^(1+p)*hypergeom([1,5/2+p],[5/2],-b*(c+d*x)^2/a)/(a*d),1/3*(c+d*x)^3*(a+b*(c+d*x)^2)^p*hypergeom([3/2,-p],[5/2],-b*(c+d*x)^2/a)/(d*(1+b*(c+d*x)^2/a)^p)],
[(c+d*x)*(a+b*(c+d*x)^2)^p,x,2,1/2*(a+b*(c+d*x)^2)^(1+p)/(b*d*(1+p))],
[(a+b*(c+d*x)^2)^p/(c+d*x),x,3,-1/2*(a+b*(c+d*x)^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*(c+d*x)^2/a)/(a*d*(1+p))],

# Integrands of the form (c+d x)^m (a+b (c+d x)^3)^p

# p>0
[(c+d*x)^3*(a+b*(c+d*x)^3),x,3,1/4*a*(c+d*x)^4/d+1/7*b*(c+d*x)^7/d],
[(c+d*x)^3*(a+b*(c+d*x)^3)^2,x,3,1/4*a^2*(c+d*x)^4/d+2/7*a*b*(c+d*x)^7/d+1/10*b^2*(c+d*x)^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*a^2*b*(c+d*x)^7/d+3/10*a*b^2*(c+d*x)^10/d+1/13*b^3*(c+d*x)^13/d],
[(c*e+d*e*x)^3*(a+b*(c+d*x)^3),x,3,1/4*a*e^3*(c+d*x)^4/d+1/7*b*e^3*(c+d*x)^7/d],
[(c*e+d*e*x)^3*(a+b*(c+d*x)^3)^2,x,3,1/4*a^2*e^3*(c+d*x)^4/d+2/7*a*b*e^3*(c+d*x)^7/d+1/10*b^2*e^3*(c+d*x)^10/d],
[(c*e+d*e*x)^3*(a+b*(c+d*x)^3)^3,x,3,1/4*a^3*e^3*(c+d*x)^4/d+3/7*a^2*b*e^3*(c+d*x)^7/d+3/10*a*b^2*e^3*(c+d*x)^10/d+1/13*b^3*e^3*(c+d*x)^13/d],

# p<0
[(c+d*x)^4/(a+b*(c+d*x)^3),x,8,1/2*(c+d*x)^2/(b*d)+1/3*a^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(b^(5/3)*d)-1/6*a^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(b^(5/3)*d)+a^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(b^(5/3)*d*sqrt(3))],
[(c+d*x)^3/(a+b*(c+d*x)^3),x,8,x/b-1/3*a^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(b^(4/3)*d)+1/6*a^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(b^(4/3)*d)+a^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(b^(4/3)*d*sqrt(3))],
[(c+d*x)^2/(a+b*(c+d*x)^3),x,2,1/3*log(a+b*(c+d*x)^3)/(b*d)],
[(c+d*x)/(a+b*(c+d*x)^3),x,7,-1/3*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(1/3)*b^(2/3)*d)+1/6*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(1/3)*b^(2/3)*d)-arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*d*sqrt(3))],
[1/(a+b*(c+d*x)^3),x,7,1/3*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(1/3)*d)-1/6*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(1/3)*d)-arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*d*sqrt(3))],
[1/((c+d*x)*(a+b*(c+d*x)^3)),x,5,log(c+d*x)/(a*d)-1/3*log(a+b*(c+d*x)^3)/(a*d)],
[1/((c+d*x)^2*(a+b*(c+d*x)^3)),x,8,(-1)/(a*d*(c+d*x))+1/3*b^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(4/3)*d)-1/6*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(4/3)*d)+b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(4/3)*d*sqrt(3))],
[1/((c+d*x)^3*(a+b*(c+d*x)^3)),x,8,(-1/2)/(a*d*(c+d*x)^2)-1/3*b^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(5/3)*d)+1/6*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(5/3)*d)+b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*d*sqrt(3))],
[1/((c+d*x)^4*(a+b*(c+d*x)^3)),x,4,(-1/3)/(a*d*(c+d*x)^3)-b*log(c+d*x)/(a^2*d)+1/3*b*log(a+b*(c+d*x)^3)/(a^2*d)],
[(c+d*x)^4/(a+b*(c+d*x)^3)^2,x,8,-1/3*(c+d*x)^2/(b*d*(a+b*(c+d*x)^3))-2/9*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(1/3)*b^(5/3)*d)+1/9*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(1/3)*b^(5/3)*d)-2/3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(5/3)*d*sqrt(3))],
[(c+d*x)^3/(a+b*(c+d*x)^3)^2,x,8,1/3*(-c-d*x)/(b*d*(a+b*(c+d*x)^3))+1/9*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(4/3)*d)-1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(4/3)*d)-1/3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(4/3)*d*sqrt(3))],
[(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,8,1/3*(c+d*x)^2/(a*d*(a+b*(c+d*x)^3))-1/9*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(4/3)*b^(2/3)*d)+1/18*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(4/3)*b^(2/3)*d)-1/3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(2/3)*d*sqrt(3))],
[1/(a+b*(c+d*x)^3)^2,x,8,1/3*(c+d*x)/(a*d*(a+b*(c+d*x)^3))+2/9*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(5/3)*b^(1/3)*d)-1/9*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(5/3)*b^(1/3)*d)-2/3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(1/3)*d*sqrt(3))],
[1/((c+d*x)*(a+b*(c+d*x)^3)^2),x,4,1/3/(a*d*(a+b*(c+d*x)^3))+log(c+d*x)/(a^2*d)-1/3*log(a+b*(c+d*x)^3)/(a^2*d)],
[1/((c+d*x)^2*(a+b*(c+d*x)^3)^2),x,9,(-4/3)/(a^2*d*(c+d*x))+1/3/(a*d*(c+d*x)*(a+b*(c+d*x)^3))+4/9*b^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(7/3)*d)-2/9*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(7/3)*d)+4/3*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(7/3)*d*sqrt(3))],
[1/((c+d*x)^3*(a+b*(c+d*x)^3)^2),x,9,(-5/6)/(a^2*d*(c+d*x)^2)+1/3/(a*d*(c+d*x)^2*(a+b*(c+d*x)^3))-5/9*b^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(8/3)*d)+5/18*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(8/3)*d)+5/3*b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(8/3)*d*sqrt(3))],
[1/((c+d*x)^4*(a+b*(c+d*x)^3)^2),x,4,(-1/3)/(a^2*d*(c+d*x)^3)-1/3*b/(a^2*d*(a+b*(c+d*x)^3))-2*b*log(c+d*x)/(a^3*d)+2/3*b*log(a+b*(c+d*x)^3)/(a^3*d)],
[(c+d*x)^4/(a+b*(c+d*x)^3)^3,x,9,-1/6*(c+d*x)^2/(b*d*(a+b*(c+d*x)^3)^2)+1/9*(c+d*x)^2/(a*b*d*(a+b*(c+d*x)^3))-1/27*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(4/3)*b^(5/3)*d)+1/54*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(4/3)*b^(5/3)*d)-1/9*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(5/3)*d*sqrt(3))],
[(c+d*x)^3/(a+b*(c+d*x)^3)^3,x,9,1/6*(-c-d*x)/(b*d*(a+b*(c+d*x)^3)^2)+1/18*(c+d*x)/(a*b*d*(a+b*(c+d*x)^3))+1/27*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(5/3)*b^(4/3)*d)-1/54*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(5/3)*b^(4/3)*d)-1/9*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(4/3)*d*sqrt(3))],
[(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,9,1/6*(c+d*x)^2/(a*d*(a+b*(c+d*x)^3)^2)+2/9*(c+d*x)^2/(a^2*d*(a+b*(c+d*x)^3))-2/27*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(7/3)*b^(2/3)*d)+1/27*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(7/3)*b^(2/3)*d)-2/9*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(2/3)*d*sqrt(3))],
[1/(a+b*(c+d*x)^3)^3,x,9,1/6*(c+d*x)/(a*d*(a+b*(c+d*x)^3)^2)+5/18*(c+d*x)/(a^2*d*(a+b*(c+d*x)^3))+5/27*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(8/3)*b^(1/3)*d)-5/54*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(8/3)*b^(1/3)*d)-5/9*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(1/3)*d*sqrt(3))],
[1/((c+d*x)*(a+b*(c+d*x)^3)^3),x,4,1/6/(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)-1/3*log(a+b*(c+d*x)^3)/(a^3*d)],
[1/((c+d*x)^2*(a+b*(c+d*x)^3)^3),x,10,(-14/9)/(a^3*d*(c+d*x))+1/6/(a*d*(c+d*x)*(a+b*(c+d*x)^3)^2)+7/18/(a^2*d*(c+d*x)*(a+b*(c+d*x)^3))+14/27*b^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(10/3)*d)-7/27*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(10/3)*d)+14/9*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(10/3)*d*sqrt(3))],
[1/((c+d*x)^3*(a+b*(c+d*x)^3)^3),x,10,(-10/9)/(a^3*d*(c+d*x)^2)+1/6/(a*d*(c+d*x)^2*(a+b*(c+d*x)^3)^2)+4/9/(a^2*d*(c+d*x)^2*(a+b*(c+d*x)^3))-20/27*b^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(11/3)*d)+10/27*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(11/3)*d)+20/9*b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(11/3)*d*sqrt(3))],
[1/((c+d*x)^4*(a+b*(c+d*x)^3)^3),x,4,(-1/3)/(a^3*d*(c+d*x)^3)-1/6*b/(a^2*d*(a+b*(c+d*x)^3)^2)-2/3*b/(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)],
[(c*e+d*e*x)^4/(a+b*(c+d*x)^3),x,8,1/2*e^4*(c+d*x)^2/(b*d)+1/3*a^(2/3)*e^4*log(a^(1/3)+b^(1/3)*(c+d*x))/(b^(5/3)*d)-1/6*a^(2/3)*e^4*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(b^(5/3)*d)+a^(2/3)*e^4*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(b^(5/3)*d*sqrt(3))],
[(c*e+d*e*x)^3/(a+b*(c+d*x)^3),x,8,e^3*x/b-1/3*a^(1/3)*e^3*log(a^(1/3)+b^(1/3)*(c+d*x))/(b^(4/3)*d)+1/6*a^(1/3)*e^3*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(b^(4/3)*d)+a^(1/3)*e^3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(b^(4/3)*d*sqrt(3))],
[(c*e+d*e*x)^2/(a+b*(c+d*x)^3),x,2,1/3*e^2*log(a+b*(c+d*x)^3)/(b*d)],
[(c*e+d*e*x)/(a+b*(c+d*x)^3),x,7,-1/3*e*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(1/3)*b^(2/3)*d)+1/6*e*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(1/3)*b^(2/3)*d)-e*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*d*sqrt(3))],
[1/((c*e+d*e*x)*(a+b*(c+d*x)^3)),x,5,log(c+d*x)/(a*d*e)-1/3*log(a+b*(c+d*x)^3)/(a*d*e)],
[1/((c*e+d*e*x)^2*(a+b*(c+d*x)^3)),x,8,(-1)/(a*d*e^2*(c+d*x))+1/3*b^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(4/3)*d*e^2)-1/6*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(4/3)*d*e^2)+b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(4/3)*d*e^2*sqrt(3))],
[1/((c*e+d*e*x)^3*(a+b*(c+d*x)^3)),x,8,(-1/2)/(a*d*e^3*(c+d*x)^2)-1/3*b^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(5/3)*d*e^3)+1/6*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(5/3)*d*e^3)+b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*d*e^3*sqrt(3))],
[1/((c*e+d*e*x)^4*(a+b*(c+d*x)^3)),x,4,(-1/3)/(a*d*e^4*(c+d*x)^3)-b*log(c+d*x)/(a^2*d*e^4)+1/3*b*log(a+b*(c+d*x)^3)/(a^2*d*e^4)],
[(c*e+d*e*x)^4/(a+b*(c+d*x)^3)^2,x,8,-1/3*e^4*(c+d*x)^2/(b*d*(a+b*(c+d*x)^3))-2/9*e^4*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(1/3)*b^(5/3)*d)+1/9*e^4*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(1/3)*b^(5/3)*d)-2/3*e^4*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(5/3)*d*sqrt(3))],
[(c*e+d*e*x)^3/(a+b*(c+d*x)^3)^2,x,8,-1/3*e^3*(c+d*x)/(b*d*(a+b*(c+d*x)^3))+1/9*e^3*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(4/3)*d)-1/18*e^3*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(4/3)*d)-1/3*e^3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(4/3)*d*sqrt(3))],
[(c*e+d*e*x)^2/(a+b*(c+d*x)^3)^2,x,2,-1/3*e^2/(b*d*(a+b*(c+d*x)^3))],
[(c*e+d*e*x)/(a+b*(c+d*x)^3)^2,x,8,1/3*e*(c+d*x)^2/(a*d*(a+b*(c+d*x)^3))-1/9*e*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(4/3)*b^(2/3)*d)+1/18*e*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(4/3)*b^(2/3)*d)-1/3*e*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(2/3)*d*sqrt(3))],
[1/((c*e+d*e*x)*(a+b*(c+d*x)^3)^2),x,4,1/3/(a*d*e*(a+b*(c+d*x)^3))+log(c+d*x)/(a^2*d*e)-1/3*log(a+b*(c+d*x)^3)/(a^2*d*e)],
[1/((c*e+d*e*x)^2*(a+b*(c+d*x)^3)^2),x,9,(-4/3)/(a^2*d*e^2*(c+d*x))+1/3/(a*d*e^2*(c+d*x)*(a+b*(c+d*x)^3))+4/9*b^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(7/3)*d*e^2)-2/9*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(7/3)*d*e^2)+4/3*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(7/3)*d*e^2*sqrt(3))],
[1/((c*e+d*e*x)^3*(a+b*(c+d*x)^3)^2),x,9,(-5/6)/(a^2*d*e^3*(c+d*x)^2)+1/3/(a*d*e^3*(c+d*x)^2*(a+b*(c+d*x)^3))-5/9*b^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(8/3)*d*e^3)+5/18*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(8/3)*d*e^3)+5/3*b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(8/3)*d*e^3*sqrt(3))],
[1/((c*e+d*e*x)^4*(a+b*(c+d*x)^3)^2),x,4,(-1/3)/(a^2*d*e^4*(c+d*x)^3)-1/3*b/(a^2*d*e^4*(a+b*(c+d*x)^3))-2*b*log(c+d*x)/(a^3*d*e^4)+2/3*b*log(a+b*(c+d*x)^3)/(a^3*d*e^4)],
[(c*e+d*e*x)^4/(a+b*(c+d*x)^3)^3,x,9,-1/6*e^4*(c+d*x)^2/(b*d*(a+b*(c+d*x)^3)^2)+1/9*e^4*(c+d*x)^2/(a*b*d*(a+b*(c+d*x)^3))-1/27*e^4*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(4/3)*b^(5/3)*d)+1/54*e^4*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(4/3)*b^(5/3)*d)-1/9*e^4*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(5/3)*d*sqrt(3))],
[(c*e+d*e*x)^3/(a+b*(c+d*x)^3)^3,x,9,-1/6*e^3*(c+d*x)/(b*d*(a+b*(c+d*x)^3)^2)+1/18*e^3*(c+d*x)/(a*b*d*(a+b*(c+d*x)^3))+1/27*e^3*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(5/3)*b^(4/3)*d)-1/54*e^3*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(5/3)*b^(4/3)*d)-1/9*e^3*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(4/3)*d*sqrt(3))],
[(c*e+d*e*x)^2/(a+b*(c+d*x)^3)^3,x,2,-1/6*e^2/(b*d*(a+b*(c+d*x)^3)^2)],
[(c*e+d*e*x)/(a+b*(c+d*x)^3)^3,x,9,1/6*e*(c+d*x)^2/(a*d*(a+b*(c+d*x)^3)^2)+2/9*e*(c+d*x)^2/(a^2*d*(a+b*(c+d*x)^3))-2/27*e*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(7/3)*b^(2/3)*d)+1/27*e*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(7/3)*b^(2/3)*d)-2/9*e*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(2/3)*d*sqrt(3))],
[1/((c*e+d*e*x)*(a+b*(c+d*x)^3)^3),x,4,1/6/(a*d*e*(a+b*(c+d*x)^3)^2)+1/3/(a^2*d*e*(a+b*(c+d*x)^3))+log(c+d*x)/(a^3*d*e)-1/3*log(a+b*(c+d*x)^3)/(a^3*d*e)],
[1/((c*e+d*e*x)^2*(a+b*(c+d*x)^3)^3),x,10,(-14/9)/(a^3*d*e^2*(c+d*x))+1/6/(a*d*e^2*(c+d*x)*(a+b*(c+d*x)^3)^2)+7/18/(a^2*d*e^2*(c+d*x)*(a+b*(c+d*x)^3))+14/27*b^(1/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(10/3)*d*e^2)-7/27*b^(1/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(10/3)*d*e^2)+14/9*b^(1/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(10/3)*d*e^2*sqrt(3))],
[1/((c*e+d*e*x)^3*(a+b*(c+d*x)^3)^3),x,10,(-10/9)/(a^3*d*e^3*(c+d*x)^2)+1/6/(a*d*e^3*(c+d*x)^2*(a+b*(c+d*x)^3)^2)+4/9/(a^2*d*e^3*(c+d*x)^2*(a+b*(c+d*x)^3))-20/27*b^(2/3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(11/3)*d*e^3)+10/27*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(11/3)*d*e^3)+20/9*b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(11/3)*d*e^3*sqrt(3))],
[1/((c*e+d*e*x)^4*(a+b*(c+d*x)^3)^3),x,4,(-1/3)/(a^3*d*e^4*(c+d*x)^3)-1/6*b/(a^2*d*e^4*(a+b*(c+d*x)^3)^2)-2/3*b/(a^3*d*e^4*(a+b*(c+d*x)^3))-3*b*log(c+d*x)/(a^4*d*e^4)+b*log(a+b*(c+d*x)^3)/(a^4*d*e^4)],

# Integrands of the form (c+d x)^m (a+b (c+d x)^4)^p

# p>0
[(c+d*x)^3*(a+b*(c+d*x)^4)^p,x,2,1/4*(a+b*(c+d*x)^4)^(1+p)/(b*d*(1+p))],
[(c+d*x)^3*(a+b*(c+d*x)^4),x,3,1/8*(a+b*(c+d*x)^4)^2/(b*d),1/4*a*(c+d*x)^4/d+1/8*b*(c+d*x)^8/d],
[(c+d*x)^3*(a+b*(c+d*x)^4)^2,x,2,1/12*(a+b*(c+d*x)^4)^3/(b*d)],
[(c+d*x)^3*(a+b*(c+d*x)^4)^3,x,2,1/16*(a+b*(c+d*x)^4)^4/(b*d)],

# p<0
[(c+d*x)^3/(a+b*(c+d*x)^4),x,2,1/4*log(a+b*(c+d*x)^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)],
[1/sqrt(a+b*(c+d*x)^4),x,2,1/2*sqrt(cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*(c+d*x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+(c+d*x)^2*sqrt(b))*sqrt((a+b*(c+d*x)^4)/(sqrt(a)+(c+d*x)^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*d*sqrt(a+b*(c+d*x)^4))],
[x/sqrt(a+b*(c+d*x)^4),x,7,1/2*arctanh((c+d*x)^2*sqrt(b)/sqrt(a+b*(c+d*x)^4))/(d^2*sqrt(b))-1/2*c*sqrt(cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*(c+d*x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+(c+d*x)^2*sqrt(b))*sqrt((a+b*(c+d*x)^4)/(sqrt(a)+(c+d*x)^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*d^2*sqrt(a+b*(c+d*x)^4))],

# Integrands of the form (d x)^m (a+b (c x^q)^n)^p

# Integrands of the form (d x)^m (a+b (c x^1)^n)^p

# Integrands of the form (d x)^m (a+b (c x)^(n/2))^p with m symbolic

# p>0
[(d*x)^m*sqrt(a+b*(c*x)^(3/2)),x,5,x*(d*x)^m*hypergeom([-1/2,2/3*(1+m)],[1/3*(5+2*m)],-b*(c*x)^(3/2)/a)*sqrt(a+b*(c*x)^(3/2))/((1+m)*sqrt(1+b*(c*x)^(3/2)/a))],
[(d*x)^m*sqrt(a+b*(c*x)^(1/2)),x,5,-4/3*a*(d*x)^m*hypergeom([3/2,-1-2*m],[5/2],1+b*sqrt(c*x)/a)*(a+b*sqrt(c*x))^(3/2)/(b^2*c*(-b*sqrt(c*x)/a)^(2*m))],
[(d*x)^m*sqrt(a+b/(c*x)^(1/2)),x,6,4/3*b^2*(d*x)^m*hypergeom([3/2,3+2*m],[5/2],1+b/(a*sqrt(c*x)))*(a+b/sqrt(c*x))^(3/2)*(-b/(a*sqrt(c*x)))^(2*m)/(a^3*c)],
[(d*x)^m*sqrt(a+b/(c*x)^(3/2)),x,6,x*(d*x)^m*hypergeom([-1/2,-2/3*(1+m)],[1/3*(1-2*m)],-b/(a*(c*x)^(3/2)))*sqrt(a+b/(c*x)^(3/2))/((1+m)*sqrt(1+b/(a*(c*x)^(3/2))))],

# p<0

# Integrands of the form (d x)^m (a+b (c x)^n)^p with n and p symbolic
[(d*x)^m*(a+b*(c*x)^n)^p,x,3,(d*x)^(1+m)*(a+b*(c*x)^n)^p*hypergeom([(1+m)/n,-p],[(1+m+n)/n],-b*(c*x)^n/a)/(d*(1+m)*(1+b*(c*x)^n/a)^p)],
[x^2*(a+b*(c*x)^n)^p,x,4,1/3*x^3*(a+b*(c*x)^n)^p*hypergeom([3/n,-p],[(3+n)/n],-b*(c*x)^n/a)/(1+b*(c*x)^n/a)^p],
[x*(a+b*(c*x)^n)^p,x,4,1/2*x^2*(a+b*(c*x)^n)^p*hypergeom([2/n,-p],[(2+n)/n],-b*(c*x)^n/a)/(1+b*(c*x)^n/a)^p],
[(a+b*(c*x)^n)^p,x,3,x*(a+b*(c*x)^n)^p*hypergeom([1/n,-p],[1+1/n],-b*(c*x)^n/a)/(1+b*(c*x)^n/a)^p],
[(a+b*(c*x)^n)^p/x,x,4,-(a+b*(c*x)^n)^(1+p)*hypergeom([1,1+p],[2+p],1+b*(c*x)^n/a)/(a*n*(1+p))],
[(a+b*(c*x)^n)^p/x^2,x,4,-(a+b*(c*x)^n)^p*hypergeom([(-1)/n,-p],[(-1+n)/n],-b*(c*x)^n/a)/(x*(1+b*(c*x)^n/a)^p)],

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

# Integrands of the form x^m (a+b (c x^2)^(n/2))^(-1)
[1/(1+(x^2)^(3/2)),x,7,-1/6*x*log(1+x^2-sqrt(x^2))/sqrt(x^2)+1/3*x*log(1+sqrt(x^2))/sqrt(x^2)-x*arctan((1-2*sqrt(x^2))/sqrt(3))/(sqrt(3)*sqrt(x^2))],

# Integrands of the form x^m (a+b (c x^2)^(n/2))^(1/2)

# n>0
[x^5*sqrt(a+b*sqrt(c*x^2)),x,3,-2/3*a^5*(a+b*sqrt(c*x^2))^(3/2)/(b^6*c^3)+2*a^4*(a+b*sqrt(c*x^2))^(5/2)/(b^6*c^3)-20/7*a^3*(a+b*sqrt(c*x^2))^(7/2)/(b^6*c^3)+20/9*a^2*(a+b*sqrt(c*x^2))^(9/2)/(b^6*c^3)-10/11*a*(a+b*sqrt(c*x^2))^(11/2)/(b^6*c^3)+2/13*(a+b*sqrt(c*x^2))^(13/2)/(b^6*c^3)],
[x^3*sqrt(a+b*sqrt(c*x^2)),x,3,-2/3*a^3*(a+b*sqrt(c*x^2))^(3/2)/(b^4*c^2)+6/5*a^2*(a+b*sqrt(c*x^2))^(5/2)/(b^4*c^2)-6/7*a*(a+b*sqrt(c*x^2))^(7/2)/(b^4*c^2)+2/9*(a+b*sqrt(c*x^2))^(9/2)/(b^4*c^2)],
[x*sqrt(a+b*sqrt(c*x^2)),x,3,-2/3*a*(a+b*sqrt(c*x^2))^(3/2)/(b^2*c)+2/5*(a+b*sqrt(c*x^2))^(5/2)/(b^2*c)],
[sqrt(a+b*sqrt(c*x^2))/x,x,4,-2*arctanh(sqrt(a+b*sqrt(c*x^2))/sqrt(a))*sqrt(a)+2*sqrt(a+b*sqrt(c*x^2))],
[sqrt(a+b*sqrt(c*x^2))/x^3,x,5,1/4*b^2*c*arctanh(sqrt(a+b*sqrt(c*x^2))/sqrt(a))/a^(3/2)-1/2*sqrt(a+b*sqrt(c*x^2))/x^2-1/4*b*c*sqrt(a+b*sqrt(c*x^2))/(a*sqrt(c*x^2))],
[sqrt(a+b*sqrt(c*x^2))/x^5,x,7,5/64*b^4*c^2*arctanh(sqrt(a+b*sqrt(c*x^2))/sqrt(a))/a^(7/2)-1/4*sqrt(a+b*sqrt(c*x^2))/x^4+5/96*b^2*c*sqrt(a+b*sqrt(c*x^2))/(a^2*x^2)-1/24*b*c^2*sqrt(a+b*sqrt(c*x^2))/(a*(c*x^2)^(3/2))-5/64*b^3*c^2*sqrt(a+b*sqrt(c*x^2))/(a^3*sqrt(c*x^2))],
[x^4*sqrt(a+b*sqrt(c*x^2)),x,3,2/3*a^4*x^5*(a+b*sqrt(c*x^2))^(3/2)/(b^5*(c*x^2)^(5/2))-8/5*a^3*x^5*(a+b*sqrt(c*x^2))^(5/2)/(b^5*(c*x^2)^(5/2))+12/7*a^2*x^5*(a+b*sqrt(c*x^2))^(7/2)/(b^5*(c*x^2)^(5/2))-8/9*a*x^5*(a+b*sqrt(c*x^2))^(9/2)/(b^5*(c*x^2)^(5/2))+2/11*x^5*(a+b*sqrt(c*x^2))^(11/2)/(b^5*(c*x^2)^(5/2))],
[x^2*sqrt(a+b*sqrt(c*x^2)),x,3,2/3*a^2*x^3*(a+b*sqrt(c*x^2))^(3/2)/(b^3*(c*x^2)^(3/2))-4/5*a*x^3*(a+b*sqrt(c*x^2))^(5/2)/(b^3*(c*x^2)^(3/2))+2/7*x^3*(a+b*sqrt(c*x^2))^(7/2)/(b^3*(c*x^2)^(3/2))],
[sqrt(a+b*sqrt(c*x^2)),x,2,2/3*x*(a+b*sqrt(c*x^2))^(3/2)/(b*sqrt(c*x^2))],
[sqrt(a+b*sqrt(c*x^2))/x^2,x,4,-b*arctanh(sqrt(a+b*sqrt(c*x^2))/sqrt(a))*sqrt(c*x^2)/(x*sqrt(a))-sqrt(a+b*sqrt(c*x^2))/x],
[sqrt(a+b*sqrt(c*x^2))/x^4,x,6,-1/8*b^3*(c*x^2)^(3/2)*arctanh(sqrt(a+b*sqrt(c*x^2))/sqrt(a))/(a^(5/2)*x^3)-1/3*sqrt(a+b*sqrt(c*x^2))/x^3+1/8*b^2*c*sqrt(a+b*sqrt(c*x^2))/(a^2*x)-1/12*b*(c*x^2)^(3/2)*sqrt(a+b*sqrt(c*x^2))/(a*c*x^5)],
[sqrt(a+b*sqrt(c*x^2))/x^6,x,8,-7/128*b^5*(c*x^2)^(5/2)*arctanh(sqrt(a+b*sqrt(c*x^2))/sqrt(a))/(a^(9/2)*x^5)-1/5*sqrt(a+b*sqrt(c*x^2))/x^5+7/240*b^2*c*sqrt(a+b*sqrt(c*x^2))/(a^2*x^3)+7/128*b^4*c^2*sqrt(a+b*sqrt(c*x^2))/(a^4*x)-1/40*b*(c*x^2)^(5/2)*sqrt(a+b*sqrt(c*x^2))/(a*c^2*x^9)-7/192*b^3*(c*x^2)^(5/2)*sqrt(a+b*sqrt(c*x^2))/(a^3*c*x^7)],
[x^8*sqrt(a+b*(c*x^2)^(3/2)),x,4,2/9*a^2*x^9*(a+b*(c*x^2)^(3/2))^(3/2)/(b^3*(c*x^2)^(9/2))-4/15*a*x^9*(a+b*(c*x^2)^(3/2))^(5/2)/(b^3*(c*x^2)^(9/2))+2/21*x^9*(a+b*(c*x^2)^(3/2))^(7/2)/(b^3*(c*x^2)^(9/2))],
[x^5*sqrt(a+b*(c*x^2)^(3/2)),x,4,-2/9*a*(a+b*(c*x^2)^(3/2))^(3/2)/(b^2*c^3)+2/15*(a+b*(c*x^2)^(3/2))^(5/2)/(b^2*c^3)],
[x^2*sqrt(a+b*(c*x^2)^(3/2)),x,2,2/9*x^3*(a+b*(c*x^2)^(3/2))^(3/2)/(b*(c*x^2)^(3/2))],
[sqrt(a+b*(c*x^2)^(3/2))/x,x,5,-2/3*arctanh(sqrt(a+b*(c*x^2)^(3/2))/sqrt(a))*sqrt(a)+2/3*sqrt(a+b*(c*x^2)^(3/2))],
[sqrt(a+b*(c*x^2)^(3/2))/x^4,x,5,-1/3*b*(c*x^2)^(3/2)*arctanh(sqrt(a+b*(c*x^2)^(3/2))/sqrt(a))/(x^3*sqrt(a))-1/3*sqrt(a+b*(c*x^2)^(3/2))/x^3],
[x^3*sqrt(a+b*(c*x^2)^(3/2)),x,4,2/11*x^4*sqrt(a+b*(c*x^2)^(3/2))+6/55*a*sqrt(c*x^2)*sqrt(a+b*(c*x^2)^(3/2))/(b*c^2)-4/55*3^(3/4)*a^2*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(b^(4/3)*c^2*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[sqrt(a+b*(c*x^2)^(3/2)),x,3,2/5*x*sqrt(a+b*(c*x^2)^(3/2))+2/5*3^(3/4)*a*x*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(b^(1/3)*sqrt(c*x^2)*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[sqrt(a+b*(c*x^2)^(3/2))/x^3,x,3,-1/2*sqrt(a+b*(c*x^2)^(3/2))/x^2+1/2*3^(3/4)*b^(2/3)*c*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[sqrt(a+b*(c*x^2)^(3/2))/x^6,x,4,-1/5*sqrt(a+b*(c*x^2)^(3/2))/x^5-3/20*b*(c*x^2)^(5/2)*sqrt(a+b*(c*x^2)^(3/2))/(a*c*x^7)-1/20*3^(3/4)*b^(5/3)*(c*x^2)^(5/2)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(a*x^5*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[x^4*sqrt(a+b*(c*x^2)^(3/2)),x,6,2/13*x^5*sqrt(a+b*(c*x^2)^(3/2))+6/91*a*c*x^7*sqrt(a+b*(c*x^2)^(3/2))/(b*(c*x^2)^(5/2))-24/91*a^2*x^5*sqrt(a+b*(c*x^2)^(3/2))/(b^(5/3)*(c*x^2)^(5/2)*(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)))-8/91*3^(3/4)*a^(7/3)*x^5*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*sqrt(2)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(b^(5/3)*(c*x^2)^(5/2)*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))+12/91*3^(1/4)*a^(7/3)*x^5*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(b^(5/3)*(c*x^2)^(5/2)*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[x*sqrt(a+b*(c*x^2)^(3/2)),x,5,2/7*x^2*sqrt(a+b*(c*x^2)^(3/2))+6/7*a*sqrt(a+b*(c*x^2)^(3/2))/(b^(2/3)*c*(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)))+2/7*3^(3/4)*a^(4/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*sqrt(2)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(b^(2/3)*c*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))-3/7*3^(1/4)*a^(4/3)*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(b^(2/3)*c*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[sqrt(a+b*(c*x^2)^(3/2))/x^2,x,5,-sqrt(a+b*(c*x^2)^(3/2))/x+3*b^(1/3)*sqrt(c*x^2)*sqrt(a+b*(c*x^2)^(3/2))/(x*(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)))+3^(3/4)*a^(1/3)*b^(1/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(c*x^2)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(x*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))-3/2*3^(1/4)*a^(1/3)*b^(1/3)*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*sqrt(c*x^2)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(x*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],
[sqrt(a+b*(c*x^2)^(3/2))/x^5,x,6,-1/4*sqrt(a+b*(c*x^2)^(3/2))/x^4-3/8*b*c^2*sqrt(a+b*(c*x^2)^(3/2))/(a*sqrt(c*x^2))+3/8*b^(4/3)*c^2*sqrt(a+b*(c*x^2)^(3/2))/(a*(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)))+1/4*3^(3/4)*b^(4/3)*c^2*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(a^(2/3)*sqrt(2)*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))-3/16*3^(1/4)*b^(4/3)*c^2*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^2))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^2-a^(1/3)*b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2)/(a^(2/3)*sqrt(a+b*(c*x^2)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^2))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^2))^2))],

# n<0

# Integrands of the form (d x)^m (a+b (c x^2)^(n/2))^p with m symbolic

# p>0
[(d*x)^m*sqrt(a+b*(c*x^2)^(3/2)),x,3,(d*x)^(1+m)*hypergeom([-1/2,1/3*(1+m)],[1/3*(4+m)],-b*(c*x^2)^(3/2)/a)*sqrt(a+b*(c*x^2)^(3/2))/(d*(1+m)*sqrt(1+b*(c*x^2)^(3/2)/a))],
[(d*x)^m*sqrt(a+b*(c*x^2)^(1/2)),x,3,2/3*(d*x)^(1+m)*hypergeom([3/2,-m],[5/2],1+b*sqrt(c*x^2)/a)*(a+b*sqrt(c*x^2))^(3/2)/(b*d*sqrt(c*x^2)*(-b*sqrt(c*x^2)/a)^m)],
[(d*x)^m*sqrt(a+b/(c*x^2)^(1/2)),x,4,-2/3*b*(d*x)^(1+m)*hypergeom([3/2,2+m],[5/2],1+b/(a*sqrt(c*x^2)))*(a+b/sqrt(c*x^2))^(3/2)*(-b/(a*sqrt(c*x^2)))^m/(a^2*d*sqrt(c*x^2))],
[(d*x)^m*sqrt(a+b/(c*x^2)^(3/2)),x,4,(d*x)^(1+m)*hypergeom([-1/2,1/3*(-1-m)],[1/3*(2-m)],-b/(a*(c*x^2)^(3/2)))*sqrt(a+b/(c*x^2)^(3/2))/(d*(1+m)*sqrt(1+b/(a*(c*x^2)^(3/2))))],

# p<0

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

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

# Integrands of the form x^m (a+b (c x^3)^(n/2))^(1/2)

# n>0
[x^5*sqrt(a+b*sqrt(c*x^3)),x,4,-4/9*a^3*(a+b*sqrt(c*x^3))^(3/2)/(b^4*c^2)+4/5*a^2*(a+b*sqrt(c*x^3))^(5/2)/(b^4*c^2)-4/7*a*(a+b*sqrt(c*x^3))^(7/2)/(b^4*c^2)+4/27*(a+b*sqrt(c*x^3))^(9/2)/(b^4*c^2)],
[x^2*sqrt(a+b*sqrt(c*x^3)),x,4,-4/9*a*(a+b*sqrt(c*x^3))^(3/2)/(b^2*c)+4/15*(a+b*sqrt(c*x^3))^(5/2)/(b^2*c)],
[sqrt(a+b*sqrt(c*x^3))/x,x,5,-4/3*arctanh(sqrt(a+b*sqrt(c*x^3))/sqrt(a))*sqrt(a)+4/3*sqrt(a+b*sqrt(c*x^3))],
[sqrt(a+b*sqrt(c*x^3))/x^4,x,6,1/6*b^2*c*arctanh(sqrt(a+b*sqrt(c*x^3))/sqrt(a))/a^(3/2)-1/3*sqrt(a+b*sqrt(c*x^3))/x^3-1/6*b*c*sqrt(a+b*sqrt(c*x^3))/(a*sqrt(c*x^3))],
[x*sqrt(a+b*sqrt(c*x^3)),x,5,-8/55*3^(3/4)*a^2*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(b^(4/3)*c^(2/3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))+4/11*x^2*sqrt(a+b*sqrt(c*x^3))+12/55*a*x^2*sqrt(a+b*sqrt(c*x^3))/(b*sqrt(c*x^3))],
[sqrt(a+b*sqrt(c*x^3))/x^2,x,4,3^(3/4)*b^(2/3)*c^(1/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-sqrt(a+b*sqrt(c*x^3))/x],
[sqrt(a+b*sqrt(c*x^3))/x^5,x,6,7/160*3^(3/4)*b^(8/3)*c^(4/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(a^2*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-1/4*sqrt(a+b*sqrt(c*x^3))/x^4+21/160*b^2*c*sqrt(a+b*sqrt(c*x^3))/(a^2*x)-3/40*b*c^3*x^5*sqrt(a+b*sqrt(c*x^3))/(a*(c*x^3)^(5/2))],
[x^3*sqrt(a+b*sqrt(c*x^3)),x,8,160/1729*3^(3/4)*a^(10/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(b^(8/3)*c^(4/3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-240/1729*3^(1/4)*a^(10/3)*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(b^(8/3)*c^(4/3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-120/1729*a^2*x*sqrt(a+b*sqrt(c*x^3))/(b^2*c)+4/19*x^4*sqrt(a+b*sqrt(c*x^3))+480/1729*a^3*sqrt(a+b*sqrt(c*x^3))/(b^(8/3)*c^(4/3)*(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)))+12/247*a*x*sqrt(c*x^3)*sqrt(a+b*sqrt(c*x^3))/(b*c)],
[sqrt(a+b*sqrt(c*x^3)),x,6,4/7*3^(3/4)*a^(4/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(b^(2/3)*c^(1/3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-6/7*3^(1/4)*a^(4/3)*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(b^(2/3)*c^(1/3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))+4/7*x*sqrt(a+b*sqrt(c*x^3))+12/7*a*sqrt(a+b*sqrt(c*x^3))/(b^(2/3)*c^(1/3)*(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)))],
[sqrt(a+b*sqrt(c*x^3))/x^3,x,7,1/2*3^(3/4)*b^(4/3)*c^(2/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(a^(2/3)*sqrt(2)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-3/8*3^(1/4)*b^(4/3)*c^(2/3)*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)/(a^(2/3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3))^2)*sqrt(a+b*sqrt(c*x^3)))-1/2*sqrt(a+b*sqrt(c*x^3))/x^2+3/4*b^(4/3)*c^(2/3)*sqrt(a+b*sqrt(c*x^3))/(a*(a^(1/3)*(1+sqrt(3))+b^(1/3)*c^(2/3)*x^2/sqrt(c*x^3)))-3/4*b*c*x*sqrt(a+b*sqrt(c*x^3))/(a*sqrt(c*x^3))],
[x^17*sqrt(a+b*(c*x^3)^(3/2)),x,4,-4/27*a^3*(a+b*(c*x^3)^(3/2))^(3/2)/(b^4*c^6)+4/15*a^2*(a+b*(c*x^3)^(3/2))^(5/2)/(b^4*c^6)-4/21*a*(a+b*(c*x^3)^(3/2))^(7/2)/(b^4*c^6)+4/81*(a+b*(c*x^3)^(3/2))^(9/2)/(b^4*c^6)],
[x^8*sqrt(a+b*(c*x^3)^(3/2)),x,4,-4/27*a*(a+b*(c*x^3)^(3/2))^(3/2)/(b^2*c^3)+4/45*(a+b*(c*x^3)^(3/2))^(5/2)/(b^2*c^3)],
[sqrt(a+b*(c*x^3)^(3/2))/x,x,5,-4/9*arctanh(sqrt(a+b*(c*x^3)^(3/2))/sqrt(a))*sqrt(a)+4/9*sqrt(a+b*(c*x^3)^(3/2))],
[sqrt(a+b*(c*x^3)^(3/2))/x^10,x,6,1/18*b^2*c^3*arctanh(sqrt(a+b*(c*x^3)^(3/2))/sqrt(a))/a^(3/2)-1/9*sqrt(a+b*(c*x^3)^(3/2))/x^9-1/18*b*c^3*sqrt(a+b*(c*x^3)^(3/2))/(a*(c*x^3)^(3/2)),1/18*b^2*c^3*arctanh(sqrt(a+b*(c*x^3)^(3/2))/sqrt(a))/a^(3/2)-1/9*sqrt(a+b*(c*x^3)^(3/2))/x^9-1/18*b*c^6*x^9*sqrt(a+b*(c*x^3)^(3/2))/(a*(c*x^3)^(9/2))],
[x^2*sqrt(a+b*(c*x^3)^(3/2)),x,7,4/21*x^3*sqrt(a+b*(c*x^3)^(3/2))+4/7*a*sqrt(a+b*(c*x^3)^(3/2))/(b^(2/3)*c*(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3)))+4/7*a^(4/3)*EllipticF((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*(a^(1/3)+b^(1/3)*sqrt(c*x^3))*sqrt((a^(2/3)+b^(2/3)*c*x^3-a^(1/3)*b^(1/3)*sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3))^2)/(3^(1/4)*b^(2/3)*c*sqrt(a+b*(c*x^3)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3))^2))-2/7*3^(1/4)*a^(4/3)*EllipticE((a^(1/3)*(1-sqrt(3))+b^(1/3)*sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3)),sqrt(-7-4*sqrt(3)))*(a^(1/3)+b^(1/3)*sqrt(c*x^3))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+b^(2/3)*c*x^3-a^(1/3)*b^(1/3)*sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3))^2)/(b^(2/3)*c*sqrt(a+b*(c*x^3)^(3/2))*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*sqrt(c*x^3))/(a^(1/3)*(1+sqrt(3))+b^(1/3)*sqrt(c*x^3))^2))],
[x^9*sqrt(a+b*(c*x^3)^(3/2)),x,7,-792/19747*a^2*x*sqrt(a+b*(c*x^3)^(3/2))/(b^2*c^3)+4/49*x^10*sqrt(a+b*(c*x^3)^(3/2))+36/1519*a*x*(c*x^3)^(3/2)*sqrt(a+b*(c*x^3)^(3/2))/(b*c^3)+792/19747*a^3*x*hypergeom([2/9,1/2],[11/9],-b*(c*x^3)^(3/2)/a)*sqrt(1+b*(c*x^3)^(3/2)/a)/(b^2*c^3*sqrt(a+b*(c*x^3)^(3/2)))],
[sqrt(a+b*(c*x^3)^(3/2)),x,5,4/13*x*sqrt(a+b*(c*x^3)^(3/2))+9/13*a*x*hypergeom([2/9,1/2],[11/9],-b*(c*x^3)^(3/2)/a)*sqrt(1+b*(c*x^3)^(3/2)/a)/sqrt(a+b*(c*x^3)^(3/2))],
[sqrt(a+b*(c*x^3)^(3/2))/x^9,x,6,-1/8*sqrt(a+b*(c*x^3)^(3/2))/x^8-9/112*b*c^3*x*sqrt(a+b*(c*x^3)^(3/2))/(a*(c*x^3)^(3/2))-45/448*b^2*c^3*x*hypergeom([2/9,1/2],[11/9],-b*(c*x^3)^(3/2)/a)*sqrt(1+b*(c*x^3)^(3/2)/a)/(a*sqrt(a+b*(c*x^3)^(3/2))),-1/8*sqrt(a+b*(c*x^3)^(3/2))/x^8-9/112*b*c^5*x^7*sqrt(a+b*(c*x^3)^(3/2))/(a*(c*x^3)^(7/2))-45/448*b^2*c^3*x*hypergeom([2/9,1/2],[11/9],-b*(c*x^3)^(3/2)/a)*sqrt(1+b*(c*x^3)^(3/2)/a)/(a*sqrt(a+b*(c*x^3)^(3/2)))],

# n<0

# Integrands of the form (d x)^m (a+b (c x^3)^(n/2))^p with m symbolic

# p>0
[(d*x)^m*sqrt(a+b*(c*x^3)^(3/2)),x,5,x*(d*x)^m*hypergeom([-1/2,2/9*(1+m)],[1+2/9*(1+m)],-b*(c*x^3)^(3/2)/a)*sqrt(a+b*(c*x^3)^(3/2))/((1+m)*sqrt(1+b*(c*x^3)^(3/2)/a))],
[(d*x)^m*sqrt(a+b*(c*x^3)^(1/2)),x,5,x*(d*x)^m*hypergeom([-1/2,2/3*(1+m)],[1/3*(5+2*m)],-b*sqrt(c*x^3)/a)*sqrt(a+b*sqrt(c*x^3))/((1+m)*sqrt(1+b*sqrt(c*x^3)/a))],
[(d*x)^m*sqrt(a+b/(c*x^3)^(1/2)),x,6,x*(d*x)^m*hypergeom([-1/2,-2/3*(1+m)],[1/3*(1-2*m)],-b/(a*sqrt(c*x^3)))*sqrt(a+b/sqrt(c*x^3))/((1+m)*sqrt(1+b/(a*sqrt(c*x^3))))],
[(d*x)^m*sqrt(a+b/(c*x^3)^(3/2)),x,6,x*(d*x)^m*hypergeom([-1/2,-2/9*(1+m)],[1/9*(7-2*m)],-b*c^3*x^9/(a*(c*x^3)^(9/2)))*sqrt(a+b*c^3*x^9/(c*x^3)^(9/2))/((1+m)*sqrt(1+b*c^3*x^9/(a*(c*x^3)^(9/2))))],

# p<0

# Integrands of the form (d x)^m (a+b (c x^-1)^n)^p

# Integrands of the form x^m (a+b (c /x)^(n/2))^(1/2)

# n>0
[x*sqrt(a+b*sqrt(c/x)),x,8,-5/32*b^4*c^2*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/a^(7/2)+1/12*b*c^2*sqrt(a+b*sqrt(c/x))/(a*(c/x)^(3/2))-5/48*b^2*c*x*sqrt(a+b*sqrt(c/x))/a^2+1/2*x^2*sqrt(a+b*sqrt(c/x))+5/32*b^3*c^2*sqrt(a+b*sqrt(c/x))/(a^3*sqrt(c/x)),-5/32*b^4*c^2*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/a^(7/2)-5/48*b^2*c*x*sqrt(a+b*sqrt(c/x))/a^2+1/2*x^2*sqrt(a+b*sqrt(c/x))+1/12*b*(c/x)^(3/2)*x^3*sqrt(a+b*sqrt(c/x))/(a*c)+5/32*b^3*c^2*sqrt(a+b*sqrt(c/x))/(a^3*sqrt(c/x))],
[sqrt(a+b*sqrt(c/x)),x,6,-1/2*b^2*c*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/a^(3/2)+x*sqrt(a+b*sqrt(c/x))+1/2*b*c*sqrt(a+b*sqrt(c/x))/(a*sqrt(c/x))],
[sqrt(a+b*sqrt(c/x))/x,x,5,4*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))*sqrt(a)-4*sqrt(a+b*sqrt(c/x))],
[sqrt(a+b*sqrt(c/x))/x^2,x,4,4/3*a*(a+b*sqrt(c/x))^(3/2)/(b^2*c)-4/5*(a+b*sqrt(c/x))^(5/2)/(b^2*c)],
[sqrt(a+b*sqrt(c/x))/x^3,x,4,4/3*a^3*(a+b*sqrt(c/x))^(3/2)/(b^4*c^2)-12/5*a^2*(a+b*sqrt(c/x))^(5/2)/(b^4*c^2)+12/7*a*(a+b*sqrt(c/x))^(7/2)/(b^4*c^2)-4/9*(a+b*sqrt(c/x))^(9/2)/(b^4*c^2)],
[sqrt(a+b*sqrt(c/x))/x^4,x,4,4/3*a^5*(a+b*sqrt(c/x))^(3/2)/(b^6*c^3)-4*a^4*(a+b*sqrt(c/x))^(5/2)/(b^6*c^3)+40/7*a^3*(a+b*sqrt(c/x))^(7/2)/(b^6*c^3)-40/9*a^2*(a+b*sqrt(c/x))^(9/2)/(b^6*c^3)+20/11*a*(a+b*sqrt(c/x))^(11/2)/(b^6*c^3)-4/13*(a+b*sqrt(c/x))^(13/2)/(b^6*c^3)],

# n<0

# Integrands of the form x^m (a+b (c /x)^(n/2))^(-1/2)

# n>0
[x/sqrt(a+b*sqrt(c/x)),x,8,35/32*b^4*c^2*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/a^(9/2)-7/12*b*c^2*sqrt(a+b*sqrt(c/x))/(a^2*(c/x)^(3/2))+35/48*b^2*c*x*sqrt(a+b*sqrt(c/x))/a^3+1/2*x^2*sqrt(a+b*sqrt(c/x))/a-35/32*b^3*c^2*sqrt(a+b*sqrt(c/x))/(a^4*sqrt(c/x)),35/32*b^4*c^2*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/a^(9/2)+35/48*b^2*c*x*sqrt(a+b*sqrt(c/x))/a^3+1/2*x^2*sqrt(a+b*sqrt(c/x))/a-7/12*b*(c/x)^(3/2)*x^3*sqrt(a+b*sqrt(c/x))/(a^2*c)-35/32*b^3*c^2*sqrt(a+b*sqrt(c/x))/(a^4*sqrt(c/x))],
[1/sqrt(a+b*sqrt(c/x)),x,6,3/2*b^2*c*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/a^(5/2)+x*sqrt(a+b*sqrt(c/x))/a-3/2*b*c*sqrt(a+b*sqrt(c/x))/(a^2*sqrt(c/x))],
[1/(x*sqrt(a+b*sqrt(c/x))),x,4,4*arctanh(sqrt(a+b*sqrt(c/x))/sqrt(a))/sqrt(a)],
[1/(x^2*sqrt(a+b*sqrt(c/x))),x,4,-4/3*(a+b*sqrt(c/x))^(3/2)/(b^2*c)+4*a*sqrt(a+b*sqrt(c/x))/(b^2*c)],
[1/(x^3*sqrt(a+b*sqrt(c/x))),x,4,-4*a^2*(a+b*sqrt(c/x))^(3/2)/(b^4*c^2)+12/5*a*(a+b*sqrt(c/x))^(5/2)/(b^4*c^2)-4/7*(a+b*sqrt(c/x))^(7/2)/(b^4*c^2)+4*a^3*sqrt(a+b*sqrt(c/x))/(b^4*c^2)],
[1/(x^4*sqrt(a+b*sqrt(c/x))),x,4,-20/3*a^4*(a+b*sqrt(c/x))^(3/2)/(b^6*c^3)+8*a^3*(a+b*sqrt(c/x))^(5/2)/(b^6*c^3)-40/7*a^2*(a+b*sqrt(c/x))^(7/2)/(b^6*c^3)+20/9*a*(a+b*sqrt(c/x))^(9/2)/(b^6*c^3)-4/11*(a+b*sqrt(c/x))^(11/2)/(b^6*c^3)+4*a^5*sqrt(a+b*sqrt(c/x))/(b^6*c^3)],
[1/sqrt(1+sqrt(1/x)),x,6,3/2*arctanh(sqrt(1+sqrt(1/x)))+x*sqrt(1+sqrt(1/x))-3/2*sqrt(1+sqrt(1/x))/sqrt(1/x)],

# n<0

# Integrands of the form (d x)^m (a+b (c x^-1)^(n/2))^(p/2) with m symbolic

# p>0
[(d*x)^m*sqrt(a+b*(c/x)^(3/2)),x,6,x*(d*x)^m*hypergeom([-1/2,-2/3*(1+m)],[1/3*(1-2*m)],-b*c^3/(a*(c/x)^(3/2)*x^3))*sqrt(a+b*c^3/((c/x)^(3/2)*x^3))/((1+m)*sqrt(1+b*c^3/(a*(c/x)^(3/2)*x^3)))],
[(d*x)^m*sqrt(a+b*(c/x)^(1/2)),x,6,4/3*x^(1+m)*hypergeom([1,1/2*(-1-4*m)],[5/2],(a+b*sqrt(c/x))/a)*(a+b*sqrt(c/x))^(3/2)/a,4/3*b^2*c*(d*x)^m*hypergeom([3/2,3+2*m],[5/2],1+b*sqrt(c/x)/a)*(-b*sqrt(c/x)/a)^(2*m)*(a+b*sqrt(c/x))^(3/2)/a^3],
[(d*x)^m*sqrt(a+b/(c/x)^(1/2)),x,5,-4/3*a*c*(d*x)^m*hypergeom([3/2,-1-2*m],[5/2],1+b/(a*sqrt(c/x)))*(a+b/sqrt(c/x))^(3/2)/(b^2*(-b/(a*sqrt(c/x)))^(2*m))],
[(d*x)^m*sqrt(a+b/(c/x)^(3/2)),x,5,x*(d*x)^m*hypergeom([-1/2,2/3*(1+m)],[1/3*(5+2*m)],-b*(c/x)^(3/2)*x^3/(a*c^3))*sqrt(a+b*(c/x)^(3/2)*x^3/c^3)/((1+m)*sqrt(1+b*(c/x)^(3/2)*x^3/(a*c^3)))],

# p<0
[(d*x)^m/sqrt(a+b*(c/x)^(3/2)),x,6,x*(d*x)^m*hypergeom([1/2,-2/3*(1+m)],[1/3*(1-2*m)],-b*c^3/(a*(c/x)^(3/2)*x^3))*sqrt(1+b*c^3/(a*(c/x)^(3/2)*x^3))/((1+m)*sqrt(a+b*c^3/((c/x)^(3/2)*x^3)))],
[(d*x)^m/sqrt(a+b*(c/x)^(1/2)),x,6,4*x^(1+m)*hypergeom([1,1/2*(-3-4*m)],[3/2],(a+b*sqrt(c/x))/a)*sqrt(a+b*sqrt(c/x))/a,4*b^2*c*(d*x)^m*hypergeom([1/2,3+2*m],[3/2],1+b*sqrt(c/x)/a)*(-b*sqrt(c/x)/a)^(2*m)*sqrt(a+b*sqrt(c/x))/a^3],
[(d*x)^m/sqrt(a+b/(c/x)^(1/2)),x,5,-4*a*c*(d*x)^m*hypergeom([1/2,-1-2*m],[3/2],1+b/(a*sqrt(c/x)))*sqrt(a+b/sqrt(c/x))/(b^2*(-b/(a*sqrt(c/x)))^(2*m))],
[(d*x)^m/sqrt(a+b/(c/x)^(3/2)),x,5,x*(d*x)^m*hypergeom([1/2,2/3*(1+m)],[1/3*(5+2*m)],-b*(c/x)^(3/2)*x^3/(a*c^3))*sqrt(1+b*(c/x)^(3/2)*x^3/(a*c^3))/((1+m)*sqrt(a+b*(c/x)^(3/2)*x^3/c^3))],

# Integrands of the form (d x)^m (a+b (c x^q)^n)^p

# Integrands of the form x^m (a+b (c x^n)^(1/n))^p

# p>0
[a+b*(c*x^n)^(1/n),x,3,a*x+1/2*b*x*(c*x^n)^(1/n)],
[(a+b*(c*x^n)^(1/n))^2,x,2,1/3*x*(a+b*(c*x^n)^(1/n))^3/(b*(c*x^n)^(1/n))],
[(a+b*(c*x^n)^(1/n))^3,x,2,1/4*x*(a+b*(c*x^n)^(1/n))^4/(b*(c*x^n)^(1/n))],

# p<0
[x^3/(a+b*(c*x^n)^(1/n)),x,3,a^2*x^4/(b^3*(c*x^n)^(3/n))-1/2*a*x^4/(b^2*(c*x^n)^(2/n))+1/3*x^4/(b*(c*x^n)^(1/n))-a^3*x^4*log(a+b*(c*x^n)^(1/n))/(b^4*(c*x^n)^(4/n))],
[x^2/(a+b*(c*x^n)^(1/n)),x,3,-a*x^3/(b^2*(c*x^n)^(2/n))+1/2*x^3/(b*(c*x^n)^(1/n))+a^2*x^3*log(a+b*(c*x^n)^(1/n))/(b^3*(c*x^n)^(3/n))],
[x/(a+b*(c*x^n)^(1/n)),x,3,x^2/(b*(c*x^n)^(1/n))-a*x^2*log(a+b*(c*x^n)^(1/n))/(b^2*(c*x^n)^(2/n))],
[1/(a+b*(c*x^n)^(1/n)),x,2,x*log(a+b*(c*x^n)^(1/n))/(b*(c*x^n)^(1/n))],
[1/(x*(a+b*(c*x^n)^(1/n))),x,4,log(x)/a-log(a+b*(c*x^n)^(1/n))/a],
[1/(x^2*(a+b*(c*x^n)^(1/n))),x,3,(-1)/(a*x)-b*(c*x^n)^(1/n)*log(x)/(a^2*x)+b*(c*x^n)^(1/n)*log(a+b*(c*x^n)^(1/n))/(a^2*x)],
[1/(x^3*(a+b*(c*x^n)^(1/n))),x,3,(-1/2)/(a*x^2)+b*(c*x^n)^(1/n)/(a^2*x^2)+b^2*(c*x^n)^(2/n)*log(x)/(a^3*x^2)-b^2*(c*x^n)^(2/n)*log(a+b*(c*x^n)^(1/n))/(a^3*x^2)],
[x^3/(a+b*(c*x^n)^(1/n))^2,x,3,-2*a*x^4/(b^3*(c*x^n)^(3/n))+1/2*x^4/(b^2*(c*x^n)^(2/n))+a^3*x^4/(b^4*(c*x^n)^(4/n)*(a+b*(c*x^n)^(1/n)))+3*a^2*x^4*log(a+b*(c*x^n)^(1/n))/(b^4*(c*x^n)^(4/n))],
[x^2/(a+b*(c*x^n)^(1/n))^2,x,3,x^3/(b^2*(c*x^n)^(2/n))-a^2*x^3/(b^3*(c*x^n)^(3/n)*(a+b*(c*x^n)^(1/n)))-2*a*x^3*log(a+b*(c*x^n)^(1/n))/(b^3*(c*x^n)^(3/n))],
[x/(a+b*(c*x^n)^(1/n))^2,x,3,a*x^2/(b^2*(c*x^n)^(2/n)*(a+b*(c*x^n)^(1/n)))+x^2*log(a+b*(c*x^n)^(1/n))/(b^2*(c*x^n)^(2/n))],
[1/(a+b*(c*x^n)^(1/n))^2,x,2,x/(a^2+a*b*(c*x^n)^(1/n)),-x/(b*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n)))],
[1/(x*(a+b*(c*x^n)^(1/n))^2),x,3,1/(a*(a+b*(c*x^n)^(1/n)))+log(x)/a^2-log(a+b*(c*x^n)^(1/n))/a^2],
[1/(x^2*(a+b*(c*x^n)^(1/n))^2),x,3,(-1)/(a^2*x)-b*(c*x^n)^(1/n)/(a^2*x*(a+b*(c*x^n)^(1/n)))-2*b*(c*x^n)^(1/n)*log(x)/(a^3*x)+2*b*(c*x^n)^(1/n)*log(a+b*(c*x^n)^(1/n))/(a^3*x)],
[1/(x^3*(a+b*(c*x^n)^(1/n))^2),x,3,(-1/2)/(a^2*x^2)+2*b*(c*x^n)^(1/n)/(a^3*x^2)+b^2*(c*x^n)^(2/n)/(a^3*x^2*(a+b*(c*x^n)^(1/n)))+3*b^2*(c*x^n)^(2/n)*log(x)/(a^4*x^2)-3*b^2*(c*x^n)^(2/n)*log(a+b*(c*x^n)^(1/n))/(a^4*x^2)],
[1/(a+b*(c*x^n)^(1/n))^3,x,2,-1/2*x/(b*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^2)],
[x/(1+(x^n)^(1/n))^2,x,3,x^2/((x^n)^(2/n)*(1+(x^n)^(1/n)))+x^2*log(1+(x^n)^(1/n))/(x^n)^(2/n)],

# p symbolic
[x^3*(a+b*(c*x^n)^(1/n))^p,x,3,-a^3*x^4*(a+b*(c*x^n)^(1/n))^(1+p)/(b^4*(1+p)*(c*x^n)^(4/n))+3*a^2*x^4*(a+b*(c*x^n)^(1/n))^(2+p)/(b^4*(2+p)*(c*x^n)^(4/n))-3*a*x^4*(a+b*(c*x^n)^(1/n))^(3+p)/(b^4*(3+p)*(c*x^n)^(4/n))+x^4*(a+b*(c*x^n)^(1/n))^(4+p)/(b^4*(4+p)*(c*x^n)^(4/n))],
[x^2*(a+b*(c*x^n)^(1/n))^p,x,3,a^2*x^3*(a+b*(c*x^n)^(1/n))^(1+p)/(b^3*(1+p)*(c*x^n)^(3/n))-2*a*x^3*(a+b*(c*x^n)^(1/n))^(2+p)/(b^3*(2+p)*(c*x^n)^(3/n))+x^3*(a+b*(c*x^n)^(1/n))^(3+p)/(b^3*(3+p)*(c*x^n)^(3/n))],
[x*(a+b*(c*x^n)^(1/n))^p,x,3,-a*x^2*(a+b*(c*x^n)^(1/n))^(1+p)/(b^2*(1+p)*(c*x^n)^(2/n))+x^2*(a+b*(c*x^n)^(1/n))^(2+p)/(b^2*(2+p)*(c*x^n)^(2/n))],
[(a+b*(c*x^n)^(1/n))^p,x,2,x*(a+b*(c*x^n)^(1/n))^(1+p)/(b*(1+p)*(c*x^n)^(1/n))],
[(a+b*(c*x^n)^(1/n))^p/x,x,2,-(a+b*(c*x^n)^(1/n))^(1+p)*hypergeom([1,1+p],[2+p],1+b*(c*x^n)^(1/n)/a)/(a*(1+p))],
[(a+b*(c*x^n)^(1/n))^p/x^2,x,2,b*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^(1+p)*hypergeom([2,1+p],[2+p],1+b*(c*x^n)^(1/n)/a)/(a^2*(1+p)*x)],

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

# Integrands of the form x^m (a+b (c x^n)^(3/n))^p
[(a+b*(c*x^n)^(3/n))^3,x,3,a^3*x+3/4*a^2*b*x*(c*x^n)^(3/n)+3/7*a*b^2*x*(c*x^n)^(6/n)+1/10*b^3*x*(c*x^n)^(9/n)],
[(a+b*(c*x^n)^(3/n))^2,x,3,a^2*x+1/2*a*b*x*(c*x^n)^(3/n)+1/7*b^2*x*(c*x^n)^(6/n)],
[a+b*(c*x^n)^(3/n),x,3,a*x+1/4*b*x*(c*x^n)^(3/n)],
[1/(a+b*(c*x^n)^(3/n)),x,7,1/3*x*log(a^(1/3)+b^(1/3)*(c*x^n)^(1/n))/(a^(2/3)*b^(1/3)*(c*x^n)^(1/n))-1/6*x*log(a^(2/3)-a^(1/3)*b^(1/3)*(c*x^n)^(1/n)+b^(2/3)*(c*x^n)^(2/n))/(a^(2/3)*b^(1/3)*(c*x^n)^(1/n))-x*arctan((a^(1/3)-2*b^(1/3)*(c*x^n)^(1/n))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*(c*x^n)^(1/n)*sqrt(3))],
[1/(a+b*(c*x^n)^(3/n))^2,x,8,1/3*x/(a*(a+b*(c*x^n)^(3/n)))+2/9*x*log(a^(1/3)+b^(1/3)*(c*x^n)^(1/n))/(a^(5/3)*b^(1/3)*(c*x^n)^(1/n))-1/9*x*log(a^(2/3)-a^(1/3)*b^(1/3)*(c*x^n)^(1/n)+b^(2/3)*(c*x^n)^(2/n))/(a^(5/3)*b^(1/3)*(c*x^n)^(1/n))-2/3*x*arctan((a^(1/3)-2*b^(1/3)*(c*x^n)^(1/n))/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(1/3)*(c*x^n)^(1/n)*sqrt(3))],
[1/(a+b*(c*x^n)^(3/n))^3,x,9,1/6*x/(a*(a+b*(c*x^n)^(3/n))^2)+5/18*x/(a^2*(a+b*(c*x^n)^(3/n)))+5/27*x*log(a^(1/3)+b^(1/3)*(c*x^n)^(1/n))/(a^(8/3)*b^(1/3)*(c*x^n)^(1/n))-5/54*x*log(a^(2/3)-a^(1/3)*b^(1/3)*(c*x^n)^(1/n)+b^(2/3)*(c*x^n)^(2/n))/(a^(8/3)*b^(1/3)*(c*x^n)^(1/n))-5/9*x*arctan((a^(1/3)-2*b^(1/3)*(c*x^n)^(1/n))/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(1/3)*(c*x^n)^(1/n)*sqrt(3))],

# Integrands of the form (d x)^m (a+b (c x^q)^n)^p with n, p and q symbolic
[(d*x)^m*(a+b*(c*x^q)^n)^p,x,3,(d*x)^(1+m)*(a+b*(c*x^q)^n)^p*hypergeom([-p,(1+m)/(n*q)],[1+(1+m)/(n*q)],-b*(c*x^q)^n/a)/(d*(1+m)*(1+b*(c*x^q)^n/a)^p)],
[x^2*(a+b*(c*x^q)^n)^p,x,3,1/3*x^3*(a+b*(c*x^q)^n)^p*hypergeom([-p,3/(n*q)],[1+3/(n*q)],-b*(c*x^q)^n/a)/(1+b*(c*x^q)^n/a)^p],
[x*(a+b*(c*x^q)^n)^p,x,3,1/2*x^2*(a+b*(c*x^q)^n)^p*hypergeom([-p,2/(n*q)],[1+2/(n*q)],-b*(c*x^q)^n/a)/(1+b*(c*x^q)^n/a)^p],
[(a+b*(c*x^q)^n)^p,x,3,x*(a+b*(c*x^q)^n)^p*hypergeom([-p,1/(n*q)],[1+1/(n*q)],-b*(c*x^q)^n/a)/(1+b*(c*x^q)^n/a)^p],
[(a+b*(c*x^q)^n)^p/x,x,3,-(a+b*(c*x^q)^n)^(1+p)*hypergeom([1,1+p],[2+p],1+b*(c*x^q)^n/a)/(a*n*(1+p)*q)],
[(a+b*(c*x^q)^n)^p/x^2,x,3,-(a+b*(c*x^q)^n)^p*hypergeom([-p,(-1)/(n*q)],[1+(-1)/(n*q)],-b*(c*x^q)^n/a)/(x*(1+b*(c*x^q)^n/a)^p)],

# Integrands of the form x^m (a+b (d/x)^n+c (d/x)^(2n))^p
[x^m*sqrt(a+c/x+b*sqrt(d/x)),x,4,x^(1+m)*AppellF1(-2*(1+m),-1/2,-1/2,-1-2*m,-2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)-sqrt(-4*a*c+b^2*d))),-2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)+sqrt(-4*a*c+b^2*d))))*sqrt(a+c/x+b*sqrt(d/x))/((1+m)*sqrt(1+2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)-sqrt(-4*a*c+b^2*d))))*sqrt(1+2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)+sqrt(-4*a*c+b^2*d)))))],
[x^2*sqrt(a+c/x+b*sqrt(d/x)),x,9,1/512*(4*a*c-b^2*d)*(16*a^2*c^2-56*a*b^2*c*d+21*b^4*d^2)*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/a^(11/2)-3/10*b*d^3*(a+c/x+b*sqrt(d/x))^(3/2)/(a^2*(d/x)^(5/2))+7/480*b*d^2*(28*a*c-15*b^2*d)*(a+c/x+b*sqrt(d/x))^(3/2)/(a^4*(d/x)^(3/2))-1/80*(20*a*c-21*b^2*d)*x^2*(a+c/x+b*sqrt(d/x))^(3/2)/a^3+1/3*x^3*(a+c/x+b*sqrt(d/x))^(3/2)/a+1/256*(16*a^2*c^2-56*a*b^2*c*d+21*b^4*d^2)*x*(2*a+b*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/a^5],
[x*sqrt(a+c/x+b*sqrt(d/x)),x,7,-1/64*(4*a*c-5*b^2*d)*(4*a*c-b^2*d)*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/a^(7/2)-5/12*b*d^2*(a+c/x+b*sqrt(d/x))^(3/2)/(a^2*(d/x)^(3/2))+1/2*x^2*(a+c/x+b*sqrt(d/x))^(3/2)/a-1/32*(4*a*c-5*b^2*d)*x*(2*a+b*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/a^3],
[sqrt(a+c/x+b*sqrt(d/x)),x,5,1/4*(4*a*c-b^2*d)*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/a^(3/2)+1/2*x*(2*a+b*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/a],
[sqrt(a+c/x+b*sqrt(d/x))/x,x,8,2*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(a)-b*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/sqrt(c)-2*sqrt(a+c/x+b*sqrt(d/x))],
[sqrt(a+c/x+b*sqrt(d/x))/x^2,x,6,1/8*b*(4*a*c-b^2*d)*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/c^(5/2)-2/3*(a+c/x+b*sqrt(d/x))^(3/2)/c+1/4*b*(b*d+2*c*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/c^2],
[sqrt(a+c/x+b*sqrt(d/x))/x^3,x,7,-1/128*b*(12*a*c-7*b^2*d)*(4*a*c-b^2*d)*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/c^(9/2)-2/5*(a+c/x+b*sqrt(d/x))^(3/2)/(c*x)+1/120*(a+c/x+b*sqrt(d/x))^(3/2)*(32*a*c-35*b^2*d+42*b*c*sqrt(d/x))/c^3-1/64*b*(12*a*c-7*b^2*d)*(b*d+2*c*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/c^4],
[sqrt(a+c/x+b*sqrt(d/x))/x^4,x,9,1/1024*b*(4*a*c-b^2*d)*(80*a^2*c^2-120*a*b^2*c*d+33*b^4*d^2)*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/c^(13/2)+11/42*b*(d/x)^(3/2)*(a+c/x+b*sqrt(d/x))^(3/2)/(c^2*d)-2/7*(a+c/x+b*sqrt(d/x))^(3/2)/(c*x^2)+1/140*(32*a*c-33*b^2*d)*(a+c/x+b*sqrt(d/x))^(3/2)/(c^3*x)-1/6720*(a+c/x+b*sqrt(d/x))^(3/2)*(1024*a^2*c^2-3276*a*b^2*c*d+1155*b^4*d^2+18*b*c*(148*a*c-77*b^2*d)*sqrt(d/x))/c^5+1/512*b*(80*a^2*c^2-120*a*b^2*c*d+33*b^4*d^2)*(b*d+2*c*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/c^6],
[x^m/sqrt(a+c/x+b*sqrt(d/x)),x,4,x^(1+m)*AppellF1(-2*(1+m),1/2,1/2,-1-2*m,-2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)-sqrt(-4*a*c+b^2*d))),-2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)+sqrt(-4*a*c+b^2*d))))*sqrt(1+2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)-sqrt(-4*a*c+b^2*d))))*sqrt(1+2*c*sqrt(d/x)/(sqrt(d)*(b*sqrt(d)+sqrt(-4*a*c+b^2*d))))/((1+m)*sqrt(a+c/x+b*sqrt(d/x)))],
[x^2/sqrt(a+c/x+b*sqrt(d/x)),x,10,-1/512*(320*a^3*c^3-1680*a^2*b^2*c^2*d+1260*a*b^4*c*d^2-231*b^6*d^3)*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/a^(13/2)-11/30*b*d^3*sqrt(a+c/x+b*sqrt(d/x))/(a^2*(d/x)^(5/2))+1/160*b*d^2*(156*a*c-77*b^2*d)*sqrt(a+c/x+b*sqrt(d/x))/(a^4*(d/x)^(3/2))+1/640*(400*a^2*c^2-1176*a*b^2*c*d+385*b^4*d^2)*x*sqrt(a+c/x+b*sqrt(d/x))/a^5-1/240*(100*a*c-99*b^2*d)*x^2*sqrt(a+c/x+b*sqrt(d/x))/a^3+1/3*x^3*sqrt(a+c/x+b*sqrt(d/x))/a-7/1280*b*d*(528*a^2*c^2-680*a*b^2*c*d+165*b^4*d^2)*sqrt(a+c/x+b*sqrt(d/x))/(a^6*sqrt(d/x))],
[x/sqrt(a+c/x+b*sqrt(d/x)),x,8,1/64*(48*a^2*c^2-120*a*b^2*c*d+35*b^4*d^2)*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/a^(9/2)-7/12*b*d^2*sqrt(a+c/x+b*sqrt(d/x))/(a^2*(d/x)^(3/2))-1/48*(36*a*c-35*b^2*d)*x*sqrt(a+c/x+b*sqrt(d/x))/a^3+1/2*x^2*sqrt(a+c/x+b*sqrt(d/x))/a+5/96*b*d*(44*a*c-21*b^2*d)*sqrt(a+c/x+b*sqrt(d/x))/(a^4*sqrt(d/x))],
[1/sqrt(a+c/x+b*sqrt(d/x)),x,6,-1/4*(4*a*c-3*b^2*d)*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/a^(5/2)+x*sqrt(a+c/x+b*sqrt(d/x))/a-3/2*b*d*sqrt(a+c/x+b*sqrt(d/x))/(a^2*sqrt(d/x))],
[1/(x*sqrt(a+c/x+b*sqrt(d/x))),x,4,2*arctanh(1/2*(2*a+b*sqrt(d/x))/(sqrt(a)*sqrt(a+c/x+b*sqrt(d/x))))/sqrt(a)],
[1/(x^2*sqrt(a+c/x+b*sqrt(d/x))),x,5,b*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/c^(3/2)-2*sqrt(a+c/x+b*sqrt(d/x))/c],
[1/(x^3*sqrt(a+c/x+b*sqrt(d/x))),x,6,-1/8*b*(12*a*c-5*b^2*d)*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/c^(7/2)-2/3*sqrt(a+c/x+b*sqrt(d/x))/(c*x)+1/12*(16*a*c-15*b^2*d+10*b*c*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/c^3],
[1/(x^4*sqrt(a+c/x+b*sqrt(d/x))),x,8,1/128*b*(240*a^2*c^2-280*a*b^2*c*d+63*b^4*d^2)*arctanh(1/2*(b*d+2*c*sqrt(d/x))/(sqrt(c)*sqrt(d)*sqrt(a+c/x+b*sqrt(d/x))))*sqrt(d)/c^(11/2)+9/20*b*(d/x)^(3/2)*sqrt(a+c/x+b*sqrt(d/x))/(c^2*d)-2/5*sqrt(a+c/x+b*sqrt(d/x))/(c*x^2)+1/120*(64*a*c-63*b^2*d)*sqrt(a+c/x+b*sqrt(d/x))/(c^3*x)-1/960*(1024*a^2*c^2-2940*a*b^2*c*d+945*b^4*d^2+14*b*c*(92*a*c-45*b^2*d)*sqrt(d/x))*sqrt(a+c/x+b*sqrt(d/x))/c^5],
[sqrt(1/x+sqrt(1/x)),x,2,4/3*(1/x+sqrt(1/x))^(3/2)/(1/x)^(3/2)],
[sqrt(2+1/x+sqrt(1/x)),x,5,7/8*arctanh(1/2*(4+sqrt(1/x))/(sqrt(2)*sqrt(2+1/x+sqrt(1/x))))/sqrt(2)+1/4*x*(4+sqrt(1/x))*sqrt(2+1/x+sqrt(1/x))],

# Integrands of the form (c x^n)^(1/n) (a+b (c x^n)^(m/n))^p
[(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x,4,-a*x*(a+b*(c*x^n)^(1/n))^(1+p)/(b^2*(1+p)*(c*x^n)^(1/n))+x*(a+b*(c*x^n)^(1/n))^(2+p)/(b^2*(2+p)*(c*x^n)^(1/n))],
[(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x,4,-1/4*a*x*(a+b*(c*x^n)^(1/n))^4/(b^2*(c*x^n)^(1/n))+1/5*x*(a+b*(c*x^n)^(1/n))^5/(b^2*(c*x^n)^(1/n))],
[(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^2,x,4,1/2*a^2*x*(c*x^n)^(1/n)+2/3*a*b*x*(c*x^n)^(2/n)+1/4*b^2*x*(c*x^n)^(3/n)],
[(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n)),x,5,1/2*a*x*(c*x^n)^(1/n)+1/3*b*x*(c*x^n)^(2/n)],
[(c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n)),x,4,x/b-a*x*log(a+b*(c*x^n)^(1/n))/(b^2*(c*x^n)^(1/n))],
[(c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^2,x,4,a*x/(b^2*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n)))+x*log(a+b*(c*x^n)^(1/n))/(b^2*(c*x^n)^(1/n))],
[(c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^3,x,3,1/2*x*(c*x^n)^(1/n)/(a*(a+b*(c*x^n)^(1/n))^2)],
[(c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^4,x,4,1/3*a*x/(b^2*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3)-1/2*x/(b^2*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^2)],
[(c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^5,x,4,1/4*a*x/(b^2*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^4)-1/3*x/(b^2*(c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3)]]:
