# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.3 General\1.2.3.2 (d x)^m (a+b x^n+c x^(2 n))^p.txt"

lst:=[

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

# Integrands of the form (d x)^m (a+b x^3+c x^6)^p with a=0
[(a*x^3+b*x^6)^(5/3),x,2,-3/88*a*(a*x^3+b*x^6)^(8/3)/(b^2*x^8)+1/11*(a*x^3+b*x^6)^(8/3)/(b*x^5)],
[(a*x^3+b*x^6)^(2/3),x,1,1/5*(a*x^3+b*x^6)^(5/3)/(b*x^5)],
[1/(a*x^3+b*x^6)^(2/3),x,1,-(a*x^3+b*x^6)^(1/3)/(a*x^2)],
[1/(a*x^3+b*x^6)^(5/3),x,3,1/2/(a*x^2*(a*x^3+b*x^6)^(2/3))-3/4*(a*x^3+b*x^6)^(1/3)/(a^2*x^5)+9/4*b*(a*x^3+b*x^6)^(1/3)/(a^3*x^2)],
[1/(-x^3+x^6),x,8,1/2/x^2+1/3*log(1-x)-1/6*log(1+x+x^2)-arctan((1+2*x)/sqrt(3))/sqrt(3)],

# Integrands of the form (d x)^m (a+b x^3+c x^6)^p with b^2-4 a c=0

# Integrands of the form x^m (a^2+2 a b x^3+b^2 x^6)^p

# Integrands of the form x^m (a^2+2 a b x^2+b^3 x^6)^(p/2)

# p>0
[x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6),x,3,1/6*a*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/9*b*x^9*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6),x,3,1/5*a*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/8*b*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6),x,3,1/4*a*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/7*b*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6),x,2,1/6*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b],
[x*sqrt(a^2+2*a*b*x^3+b^2*x^6),x,3,1/2*a*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/5*b*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[sqrt(a^2+2*a*b*x^3+b^2*x^6),x,2,a*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/4*b*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x,x,3,1/3*b*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+a*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^2,x,3,-a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+1/2*b*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^3,x,3,-1/2*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+b*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^4,x,3,-1/3*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+b*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^5,x,3,-1/4*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^6,x,3,-1/5*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-1/2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^7,x,3,-1/6*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-1/3*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^8,x,3,-1/7*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-1/4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^9,x,3,-1/8*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-1/5*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^10,x,3,-1/9*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^9*(a+b*x^3))-1/6*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))],
[sqrt(a^2+2*a*b*x^3+b^2*x^6)/x^11,x,3,-1/10*a*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-1/7*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))],
[x^9*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,1/10*a^3*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/13*a^2*b*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/16*a*b^2*x^16*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/19*b^3*x^19*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^8*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,-4,1/12*a^2*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^3-2/15*a*(a+b*x^3)^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^3+1/18*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^3],
[x^7*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,1/8*a^3*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/11*a^2*b*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/14*a*b^2*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/17*b^3*x^17*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^6*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,1/7*a^3*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/10*a^2*b*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/13*a*b^2*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/16*b^3*x^16*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^5*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,4,-1/12*a*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^2+1/15*(a+b*x^3)^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^2],
[x^4*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,1/5*a^3*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/8*a^2*b*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/11*a*b^2*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/14*b^3*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^3*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,1/4*a^3*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/7*a^2*b*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/10*a*b^2*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/13*b^3*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^2*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,2,1/12*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/b],
[x*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,1/2*a^3*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/5*a^2*b*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/8*a*b^2*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/11*b^3*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,a^3*x*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/(a+b*x^3)^3+3/4*a^2*b*x^4*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/(a+b*x^3)^3+3/7*a*b^2*x^7*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/(a+b*x^3)^3+1/10*b^3*x^10*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/(a+b*x^3)^3],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x,x,4,a^2*b*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/2*a*b^2*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/9*b^3*x^9*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+a^3*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^2,x,3,-a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+3/2*a^2*b*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/5*a*b^2*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/8*b^3*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^3,x,3,-1/2*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+3*a^2*b*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3/4*a*b^2*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/7*b^3*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^4,x,4,-1/3*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+a*b^2*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/6*b^3*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3*a^2*b*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^5,x,3,-1/4*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-3*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+3/2*a*b^2*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/5*b^3*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^6,x,3,-1/5*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-3/2*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+3*a*b^2*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/4*b^3*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^7,x,4,-1/6*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+1/3*b^3*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+3*a*b^2*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^8,x,3,-1/7*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-3/4*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-3*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+1/2*b^3*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^9,x,3,-1/8*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-3/5*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-3/2*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+b^3*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^10,x,4,-1/9*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^9*(a+b*x^3))-1/2*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+b^3*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^11,x,3,-1/10*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-3/7*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-3/4*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^12,x,3,-1/11*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-3/8*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-3/5*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-1/2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^13,x,2,-1/12*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a*x^12)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^14,x,3,-1/13*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^13*(a+b*x^3))-3/10*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-3/7*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-1/4*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^15,x,3,-1/14*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^14*(a+b*x^3))-3/11*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-3/8*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-1/5*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^16,x,4,-1/15*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a*x^15)+1/60*b*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a^2*x^12)],
[(a^2+2*a*b*x^3+b^2*x^6)^(3/2)/x^17,x,3,-1/16*a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^16*(a+b*x^3))-3/13*a^2*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^13*(a+b*x^3))-3/10*a*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-1/7*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))],
[x^13*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/14*a^5*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/17*a^4*b*x^17*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/2*a^3*b^2*x^20*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/23*a^2*b^3*x^23*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/26*a*b^4*x^26*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/29*b^5*x^29*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^12*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/13*a^5*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/16*a^4*b*x^16*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/19*a^3*b^2*x^19*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/11*a^2*b^3*x^22*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/5*a*b^4*x^25*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/28*b^5*x^28*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^11*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,4,-1/18*a^3*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^4+1/7*a^2*(a+b*x^3)^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^4-1/8*a*(a+b*x^3)^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^4+1/27*(a+b*x^3)^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^4],
[x^10*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/11*a^5*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/14*a^4*b*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/17*a^3*b^2*x^17*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/2*a^2*b^3*x^20*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/23*a*b^4*x^23*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/26*b^5*x^26*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^9*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/10*a^5*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/13*a^4*b*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/8*a^3*b^2*x^16*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/19*a^2*b^3*x^19*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/22*a*b^4*x^22*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/25*b^5*x^25*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^8*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,4,1/18*a^2*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^3-2/21*a*(a+b*x^3)^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^3+1/24*(a+b*x^3)^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^3],
[x^7*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/8*a^5*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/11*a^4*b*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/7*a^3*b^2*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/17*a^2*b^3*x^17*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/4*a*b^4*x^20*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/23*b^5*x^23*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^6*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/7*a^5*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/2*a^4*b*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/13*a^3*b^2*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/8*a^2*b^3*x^16*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/19*a*b^4*x^19*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/22*b^5*x^22*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^5*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,4,-1/18*a*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^2+1/21*(a+b*x^3)^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/b^2],
[x^4*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/5*a^5*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/8*a^4*b*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/11*a^3*b^2*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/7*a^2*b^3*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/17*a*b^4*x^17*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/20*b^5*x^20*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^3*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/4*a^5*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/7*a^4*b*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+a^3*b^2*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/13*a^2*b^3*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/16*a*b^4*x^16*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/19*b^5*x^19*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[x^2*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,2,1/18*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/b],
[x*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,1/2*a^5*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+a^4*b*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/4*a^3*b^2*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/11*a^2*b^3*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/14*a*b^4*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/17*b^5*x^17*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,a^5*x*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/(a+b*x^3)^5+5/4*a^4*b*x^4*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/(a+b*x^3)^5+10/7*a^3*b^2*x^7*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/(a+b*x^3)^5+a^2*b^3*x^10*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/(a+b*x^3)^5+5/13*a*b^4*x^13*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/(a+b*x^3)^5+1/16*b^5*x^16*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/(a+b*x^3)^5],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x,x,4,5/3*a^4*b*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/3*a^3*b^2*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/9*a^2*b^3*x^9*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/12*a*b^4*x^12*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/15*b^5*x^15*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+a^5*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^2,x,3,-a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+5/2*a^4*b*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+2*a^3*b^2*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/4*a^2*b^3*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/11*a*b^4*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/14*b^5*x^14*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^3,x,3,-1/2*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+5*a^4*b*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/2*a^3*b^2*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10/7*a^2*b^3*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/2*a*b^4*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/13*b^5*x^13*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^4,x,4,-1/3*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+10/3*a^3*b^2*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/3*a^2*b^3*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/9*a*b^4*x^9*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/12*b^5*x^12*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5*a^4*b*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^5,x,3,-1/4*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-5*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+5*a^3*b^2*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+2*a^2*b^3*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/8*a*b^4*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/11*b^5*x^11*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^6,x,3,-1/5*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-5/2*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+10*a^3*b^2*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/2*a^2*b^3*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/7*a*b^4*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/10*b^5*x^10*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^7,x,4,-1/6*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-5/3*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+10/3*a^2*b^3*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/6*a*b^4*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/9*b^5*x^9*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10*a^3*b^2*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^8,x,3,-1/7*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-5/4*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-10*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+5*a^2*b^3*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+a*b^4*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/8*b^5*x^8*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^9,x,3,-1/8*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-5*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+10*a^2*b^3*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5/4*a*b^4*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/7*b^5*x^7*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^10,x,4,-1/9*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^9*(a+b*x^3))-5/6*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-10/3*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+5/3*a*b^4*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/6*b^5*x^6*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+10*a^2*b^3*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^11,x,3,-1/10*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-5/7*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-5/2*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-10*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+5/2*a*b^4*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/5*b^5*x^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^12,x,3,-1/11*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-5/8*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-2*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-5*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+5*a*b^4*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+1/4*b^5*x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^13,x,4,-1/12*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^12*(a+b*x^3))-5/9*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^9*(a+b*x^3))-5/3*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-10/3*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+1/3*b^5*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)+5*a*b^4*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^14,x,3,-1/13*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^13*(a+b*x^3))-1/2*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-10/7*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-5/2*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-5*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))+1/2*b^5*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^15,x,3,-1/14*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^14*(a+b*x^3))-5/11*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-5/4*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-2*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-5/2*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))+b^5*x*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^16,x,4,-1/15*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^15*(a+b*x^3))-5/12*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^12*(a+b*x^3))-10/9*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^9*(a+b*x^3))-5/3*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^6*(a+b*x^3))-5/3*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^3*(a+b*x^3))+b^5*log(x)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a+b*x^3)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^17,x,3,-1/16*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^16*(a+b*x^3))-5/13*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^13*(a+b*x^3))-a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-10/7*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-5/4*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))-b^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^18,x,3,-1/17*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^17*(a+b*x^3))-5/14*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^14*(a+b*x^3))-10/11*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-5/4*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))-1/2*b^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^2*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^19,x,2,-1/18*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a*x^18)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^20,x,3,-1/19*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^19*(a+b*x^3))-5/16*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^16*(a+b*x^3))-10/13*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^13*(a+b*x^3))-a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-5/7*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))-1/4*b^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^4*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^21,x,3,-1/20*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^20*(a+b*x^3))-5/17*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^17*(a+b*x^3))-5/7*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^14*(a+b*x^3))-10/11*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-5/8*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))-1/5*b^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^5*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^22,x,4,-1/21*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a*x^21)+1/126*b*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a^2*x^18)],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^23,x,3,-1/22*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^22*(a+b*x^3))-5/19*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^19*(a+b*x^3))-5/8*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^16*(a+b*x^3))-10/13*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^13*(a+b*x^3))-1/2*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^10*(a+b*x^3))-1/7*b^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^7*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^24,x,3,-1/23*a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^23*(a+b*x^3))-1/4*a^4*b*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^20*(a+b*x^3))-10/17*a^3*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^17*(a+b*x^3))-5/7*a^2*b^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^14*(a+b*x^3))-5/11*a*b^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^11*(a+b*x^3))-1/8*b^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(x^8*(a+b*x^3))],
[(a^2+2*a*b*x^3+b^2*x^6)^(5/2)/x^25,x,5,-1/24*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a*x^24)+1/84*b*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a^2*x^21)-1/504*b^2*(a+b*x^3)^5*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(a^3*x^18)],

# p<0
[x^4/sqrt(a^2+2*a*b*x^3+b^2*x^6),x,8,1/2*x^2*(a+b*x^3)/(b*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/3*a^(2/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(b^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/6*a^(2/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+a^(2/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^3/sqrt(a^2+2*a*b*x^3+b^2*x^6),x,8,x*(a+b*x^3)/(b*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/3*a^(1/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(b^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6*a^(1/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+a^(1/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(4/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^2/sqrt(a^2+2*a*b*x^3+b^2*x^6),x,3,1/3*(a+b*x^3)*log(a+b*x^3)/(b*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x/sqrt(a^2+2*a*b*x^3+b^2*x^6),x,7,-1/3*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(2/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(2/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/sqrt(a^2+2*a*b*x^3+b^2*x^6),x,7,1/3*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(1/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/6*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x*sqrt(a^2+2*a*b*x^3+b^2*x^6)),x,5,(a+b*x^3)*log(x)/(a*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/3*(a+b*x^3)*log(a+b*x^3)/(a*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6)),x,8,(-a-b*x^3)/(a*x*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/3*b^(1/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/6*b^(1/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+b^(1/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6)),x,8,1/2*(-a-b*x^3)/(a*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/3*b^(2/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6*b^(2/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+b^(2/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^4*sqrt(a^2+2*a*b*x^3+b^2*x^6)),x,4,1/3*(-a-b*x^3)/(a*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-b*(a+b*x^3)*log(x)/(a^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/3*b*(a+b*x^3)*log(a+b*x^3)/(a^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^4/(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,9,1/9*x^2/(a*b*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/6*x^2/(b*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/27*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/54*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/9*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(5/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^3/(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,9,1/18*x/(a*b*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/6*x/(b*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/27*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/54*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/9*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(4/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^2/(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,2,(-1/6)/(b*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x/(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,9,2/9*x^2/(a^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6*x^2/(a*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-2/27*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(2/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/27*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(2/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-2/9*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(2/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,9,1/6*x*(a+b*x^3)/(a*(a^2+2*a*b*x^3+b^2*x^6)^(3/2))+5/18*x*(a+b*x^3)^2/(a^2*(a^2+2*a*b*x^3+b^2*x^6)^(3/2))+5/27*(a+b*x^3)^3*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(1/3)*(a^2+2*a*b*x^3+b^2*x^6)^(3/2))-5/54*(a+b*x^3)^3*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(1/3)*(a^2+2*a*b*x^3+b^2*x^6)^(3/2))-5/9*(a+b*x^3)^3*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(1/3)*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)*sqrt(3))],
[1/(x*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)),x,4,1/3/(a^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6/(a*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+(a+b*x^3)*log(x)/(a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/3*(a+b*x^3)*log(a+b*x^3)/(a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^2*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)),x,10,7/18/(a^2*x*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6/(a*x*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-14/9*(a+b*x^3)/(a^3*x*sqrt(a^2+2*a*b*x^3+b^2*x^6))+14/27*b^(1/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-7/27*b^(1/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+14/9*b^(1/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^3*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)),x,10,4/9/(a^2*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6/(a*x^2*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-10/9*(a+b*x^3)/(a^3*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))-20/27*b^(2/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+10/27*b^(2/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+20/9*b^(2/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^4*(a^2+2*a*b*x^3+b^2*x^6)^(3/2)),x,4,-2/3*b/(a^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/6*b/(a^2*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/3*(-a-b*x^3)/(a^3*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-3*b*(a+b*x^3)*log(x)/(a^4*sqrt(a^2+2*a*b*x^3+b^2*x^6))+b*(a+b*x^3)*log(a+b*x^3)/(a^4*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^6/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,11,5/486*x/(a^2*b^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/12*x^4/(b*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/27*x/(b^2*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/162*x/(a*b^2*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+5/729*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(7/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-5/1458*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(7/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-5/243*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(7/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^5/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,4,1/12*a/(b^2*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+(-1/9)/(b^2*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^4/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,11,7/243*x^2/(a^3*b*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/12*x^2/(b*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/54*x^2/(a*b*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+7/324*x^2/(a^2*b*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-7/729*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+7/1458*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(5/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-7/243*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(5/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^3/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,11,5/243*x/(a^3*b*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/12*x/(b*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/108*x/(a*b*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/81*x/(a^2*b*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+10/729*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-5/729*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(4/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-10/243*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(4/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[x^2/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,2,(-1/12)/(b*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^(3/2))],
[x/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,11,35/243*x^2/(a^4*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/12*x^2/(a*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+5/54*x^2/(a^2*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+35/324*x^2/(a^3*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-35/729*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(13/3)*b^(2/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+35/1458*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(13/3)*b^(2/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-35/243*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(13/3)*b^(2/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,11,1/12*x*(a+b*x^3)/(a*(a^2+2*a*b*x^3+b^2*x^6)^(5/2))+11/108*x*(a+b*x^3)^2/(a^2*(a^2+2*a*b*x^3+b^2*x^6)^(5/2))+11/81*x*(a+b*x^3)^3/(a^3*(a^2+2*a*b*x^3+b^2*x^6)^(5/2))+55/243*x*(a+b*x^3)^4/(a^4*(a^2+2*a*b*x^3+b^2*x^6)^(5/2))+110/729*(a+b*x^3)^5*log(a^(1/3)+b^(1/3)*x)/(a^(14/3)*b^(1/3)*(a^2+2*a*b*x^3+b^2*x^6)^(5/2))-55/729*(a+b*x^3)^5*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(14/3)*b^(1/3)*(a^2+2*a*b*x^3+b^2*x^6)^(5/2))-110/243*(a+b*x^3)^5*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(14/3)*b^(1/3)*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)*sqrt(3))],
[1/(x*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)),x,4,1/3/(a^4*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/12/(a*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/9/(a^2*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/6/(a^3*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+(a+b*x^3)*log(x)/(a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/3*(a+b*x^3)*log(a+b*x^3)/(a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^2*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)),x,12,455/972/(a^4*x*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/12/(a*x*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+13/108/(a^2*x*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+65/324/(a^3*x*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-455/243*(a+b*x^3)/(a^5*x*sqrt(a^2+2*a*b*x^3+b^2*x^6))+455/729*b^(1/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(16/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-455/1458*b^(1/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(16/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+455/243*b^(1/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(16/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^3*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)),x,12,154/243/(a^4*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/12/(a*x^2*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))+7/54/(a^2*x^2*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))+77/324/(a^3*x^2*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))-385/243*(a+b*x^3)/(a^5*x^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))-770/729*b^(2/3)*(a+b*x^3)*log(a^(1/3)+b^(1/3)*x)/(a^(17/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+385/729*b^(2/3)*(a+b*x^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(17/3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+770/243*b^(2/3)*(a+b*x^3)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(17/3)*sqrt(3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[1/(x^4*(a^2+2*a*b*x^3+b^2*x^6)^(5/2)),x,4,-4/3*b/(a^5*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/12*b/(a^2*(a+b*x^3)^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-2/9*b/(a^3*(a+b*x^3)^2*sqrt(a^2+2*a*b*x^3+b^2*x^6))-1/2*b/(a^4*(a+b*x^3)*sqrt(a^2+2*a*b*x^3+b^2*x^6))+1/3*(-a-b*x^3)/(a^5*x^3*sqrt(a^2+2*a*b*x^3+b^2*x^6))-5*b*(a+b*x^3)*log(x)/(a^6*sqrt(a^2+2*a*b*x^3+b^2*x^6))+5/3*b*(a+b*x^3)*log(a+b*x^3)/(a^6*sqrt(a^2+2*a*b*x^3+b^2*x^6))],

# Integrands of the form x^(m/2) (a^2+2 a b x^3+b^2 x^6)^p

# p>0

# p<0

# Integrands of the form x^(m/2) (a^2+2 a b x^3+b^2 x^6)^(p/2)

# p>0

# p<0

# Integrands of the form (d x)^m (a^2+2 a b x^2+b^3 x^6)^p with m symbolic
[(d*x)^m*(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,3,a^5*(d*x)^(1+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d*(1+m)*(a+b*x^3))+5*a^4*b*(d*x)^(4+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^4*(4+m)*(a+b*x^3))+10*a^3*b^2*(d*x)^(7+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^7*(7+m)*(a+b*x^3))+10*a^2*b^3*(d*x)^(10+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^10*(10+m)*(a+b*x^3))+5*a*b^4*(d*x)^(13+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^13*(13+m)*(a+b*x^3))+b^5*(d*x)^(16+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^16*(16+m)*(a+b*x^3))],
[(d*x)^m*(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,3,a^3*(d*x)^(1+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d*(1+m)*(a+b*x^3))+3*a^2*b*(d*x)^(4+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^4*(4+m)*(a+b*x^3))+3*a*b^2*(d*x)^(7+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^7*(7+m)*(a+b*x^3))+b^3*(d*x)^(10+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^10*(10+m)*(a+b*x^3))],
[(d*x)^m*(a^2+2*a*b*x^3+b^2*x^6)^(1/2),x,3,a*(d*x)^(1+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d*(1+m)*(a+b*x^3))+b*(d*x)^(4+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6)/(d^4*(4+m)*(a+b*x^3))],
[(d*x)^m/(a^2+2*a*b*x^3+b^2*x^6)^(1/2),x,2,(d*x)^(1+m)*(a+b*x^3)*hypergeom([1,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(a*d*(1+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[(d*x)^m/(a^2+2*a*b*x^3+b^2*x^6)^(3/2),x,2,(d*x)^(1+m)*(a+b*x^3)*hypergeom([3,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(a^3*d*(1+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],
[(d*x)^m/(a^2+2*a*b*x^3+b^2*x^6)^(5/2),x,2,(d*x)^(1+m)*(a+b*x^3)*hypergeom([5,1/3*(1+m)],[1/3*(4+m)],-b*x^3/a)/(a^5*d*(1+m)*sqrt(a^2+2*a*b*x^3+b^2*x^6))],

# Integrands of the form (d x)^m (a^2+2 a b x^2+b^3 x^6)^p with p symbolic
[(d*x)^m*(a^2+2*a*b*x^3+b^2*x^6)^p,x,2,(d*x)^(1+m)*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([1/3*(1+m),-2*p],[1/3*(4+m)],-b*x^3/a)/(d*(1+m)*(1+b*x^3/a)^(2*p))],
[x^11*(a^2+2*a*b*x^3+b^2*x^6)^p,x,4,-1/3*a^3*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^4*(1+2*p))+1/2*a^2*(a+b*x^3)^2*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^4*(1+p))-a*(a+b*x^3)^3*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^4*(3+2*p))+1/6*(a+b*x^3)^4*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^4*(2+p))],
[x^8*(a^2+2*a*b*x^3+b^2*x^6)^p,x,4,1/3*a^2*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^3*(1+2*p))-1/3*a*(a+b*x^3)^2*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^3*(1+p))+1/3*(a+b*x^3)^3*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^3*(3+2*p))],
[x^5*(a^2+2*a*b*x^3+b^2*x^6)^p,x,4,-1/3*a*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^2*(1+2*p))+1/6*(a+b*x^3)^2*(a^2+2*a*b*x^3+b^2*x^6)^p/(b^2*(1+p))],
[x^4*(a^2+2*a*b*x^3+b^2*x^6)^p,x,2,1/5*x^5*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([5/3,-2*p],[8/3],-b*x^3/a)/(1+b*x^3/a)^(2*p)],
[x^3*(a^2+2*a*b*x^3+b^2*x^6)^p,x,2,1/4*x^4*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([4/3,-2*p],[7/3],-b*x^3/a)/(1+b*x^3/a)^(2*p)],
[x^2*(a^2+2*a*b*x^3+b^2*x^6)^p,x,2,1/3*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p/(b*(1+2*p))],
[x*(a^2+2*a*b*x^3+b^2*x^6)^p,x,2,1/2*x^2*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([1,5/3+2*p],[5/3],-b*x^3/a)/a,1/2*x^2*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([2/3,-2*p],[5/3],-b*x^3/a)/(1+b*x^3/a)^(2*p)],
[(a^2+2*a*b*x^3+b^2*x^6)^p,x,3,x*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([1,4/3+2*p],[4/3],-b*x^3/a)/a,x*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([1/3,-2*p],[4/3],-b*x^3/a)/(1+b*x^3/a)^(2*p)],
[(a^2+2*a*b*x^3+b^2*x^6)^p/x,x,3,-1/3*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([1,1+2*p],[2*(1+p)],1+b*x^3/a)/(a*(1+2*p))],
[(a^2+2*a*b*x^3+b^2*x^6)^p/x^2,x,2,-(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([-1/3,-2*p],[2/3],-b*x^3/a)/(x*(1+b*x^3/a)^(2*p))],
[(a^2+2*a*b*x^3+b^2*x^6)^p/x^3,x,2,-1/2*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([-2/3,-2*p],[1/3],-b*x^3/a)/(x^2*(1+b*x^3/a)^(2*p))],
[(a^2+2*a*b*x^3+b^2*x^6)^p/x^4,x,3,1/3*b*(a+b*x^3)*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([2,1+2*p],[2*(1+p)],1+b*x^3/a)/(a^2*(1+2*p))],
[(a^2+2*a*b*x^3+b^2*x^6)^p/x^5,x,2,-1/4*(a^2+2*a*b*x^3+b^2*x^6)^p*hypergeom([-4/3,-2*p],[-1/3],-b*x^3/a)/(x^4*(1+b*x^3/a)^(2*p))],

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

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

# p>0

# p<0
[x^8/(a+b*x^3+c*x^6),x,6,1/3*x^3/c-1/6*b*log(a+b*x^3+c*x^6)/c^2-1/3*(b^2-2*a*c)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[x^5/(a+b*x^3+c*x^6),x,5,1/6*log(a+b*x^3+c*x^6)/c+1/3*b*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[x^2/(a+b*x^3+c*x^6),x,3,-2/3*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[1/(x*(a+b*x^3+c*x^6)),x,7,log(x)/a-1/6*log(a+b*x^3+c*x^6)/a+1/3*b*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[1/(x^4*(a+b*x^3+c*x^6)),x,8,(-1/3)/(a*x^3)-b*log(x)/a^2+1/6*b*log(a+b*x^3+c*x^6)/a^2-1/3*(b^2-2*a*c)*arctanh((b+2*c*x^3)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[x^7/(a+b*x^3+c*x^6),x,14,1/2*x^2/c+1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b-sqrt(b^2-4*a*c))^(1/3))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b-sqrt(b^2-4*a*c))^(1/3))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(1/3))+1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b+sqrt(b^2-4*a*c))^(1/3))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*(b+sqrt(b^2-4*a*c))^(1/3))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(2/3)*c^(5/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(1/3))],
[x^6/(a+b*x^3+c*x^6),x,14,x/c-1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b-sqrt(b^2-4*a*c))^(2/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b-sqrt(b^2-4*a*c))^(2/3))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3))-1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b+sqrt(b^2-4*a*c))^(2/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b+sqrt(b^2-4*a*c))^(2/3))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(2/3))],
[x^4/(a+b*x^3+c*x^6),x,13,1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(b-sqrt(b^2-4*a*c))^(2/3)/(2^(2/3)*c^(2/3)*sqrt(b^2-4*a*c))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(b-sqrt(b^2-4*a*c))^(2/3)/(2^(2/3)*c^(2/3)*sqrt(b^2-4*a*c))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b-sqrt(b^2-4*a*c))^(2/3)/(2^(2/3)*c^(2/3)*sqrt(3)*sqrt(b^2-4*a*c))-1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(b+sqrt(b^2-4*a*c))^(2/3)/(2^(2/3)*c^(2/3)*sqrt(b^2-4*a*c))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(b+sqrt(b^2-4*a*c))^(2/3)/(2^(2/3)*c^(2/3)*sqrt(b^2-4*a*c))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+sqrt(b^2-4*a*c))^(2/3)/(2^(2/3)*c^(2/3)*sqrt(3)*sqrt(b^2-4*a*c))],
[x^3/(a+b*x^3+c*x^6),x,13,-1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(b-sqrt(b^2-4*a*c))^(1/3)/(2^(1/3)*c^(1/3)*sqrt(b^2-4*a*c))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(b-sqrt(b^2-4*a*c))^(1/3)/(2^(1/3)*c^(1/3)*sqrt(b^2-4*a*c))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b-sqrt(b^2-4*a*c))^(1/3)/(2^(1/3)*c^(1/3)*sqrt(3)*sqrt(b^2-4*a*c))+1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(b+sqrt(b^2-4*a*c))^(1/3)/(2^(1/3)*c^(1/3)*sqrt(b^2-4*a*c))-1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(b+sqrt(b^2-4*a*c))^(1/3)/(2^(1/3)*c^(1/3)*sqrt(b^2-4*a*c))-arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+sqrt(b^2-4*a*c))^(1/3)/(2^(1/3)*c^(1/3)*sqrt(3)*sqrt(b^2-4*a*c))],
[x/(a+b*x^3+c*x^6),x,13,-1/3*2^(1/3)*c^(1/3)*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))/((b-sqrt(b^2-4*a*c))^(1/3)*sqrt(b^2-4*a*c))+1/3*c^(1/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))/(2^(2/3)*(b-sqrt(b^2-4*a*c))^(1/3)*sqrt(b^2-4*a*c))-2^(1/3)*c^(1/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))/(sqrt(3)*(b-sqrt(b^2-4*a*c))^(1/3)*sqrt(b^2-4*a*c))+1/3*2^(1/3)*c^(1/3)*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))/(sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(1/3))-1/3*c^(1/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))/(2^(2/3)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(1/3))+2^(1/3)*c^(1/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))/(sqrt(3)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(1/3))],
[1/(a+b*x^3+c*x^6),x,13,1/3*2^(2/3)*c^(2/3)*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))/((b-sqrt(b^2-4*a*c))^(2/3)*sqrt(b^2-4*a*c))-1/3*c^(2/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))/(2^(1/3)*(b-sqrt(b^2-4*a*c))^(2/3)*sqrt(b^2-4*a*c))-2^(2/3)*c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))/(sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3)*sqrt(b^2-4*a*c))-1/3*2^(2/3)*c^(2/3)*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))/(sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(2/3))+1/3*c^(2/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))/(2^(1/3)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(2/3))+2^(2/3)*c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))/(sqrt(3)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(2/3))],
[1/(x^2*(a+b*x^3+c*x^6)),x,14,(-1)/(a*x)+1/3*c^(1/3)*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(1+b/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b-sqrt(b^2-4*a*c))^(1/3))-1/6*c^(1/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(1+b/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b-sqrt(b^2-4*a*c))^(1/3))+c^(1/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(1+b/sqrt(b^2-4*a*c))/(2^(2/3)*a*sqrt(3)*(b-sqrt(b^2-4*a*c))^(1/3))+1/3*c^(1/3)*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(1-b/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b+sqrt(b^2-4*a*c))^(1/3))-1/6*c^(1/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(1-b/sqrt(b^2-4*a*c))/(2^(2/3)*a*(b+sqrt(b^2-4*a*c))^(1/3))+c^(1/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(1-b/sqrt(b^2-4*a*c))/(2^(2/3)*a*sqrt(3)*(b+sqrt(b^2-4*a*c))^(1/3))],
[1/(x^3*(a+b*x^3+c*x^6)),x,14,(-1/2)/(a*x^2)-1/3*c^(2/3)*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(1+b/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b-sqrt(b^2-4*a*c))^(2/3))+1/6*c^(2/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(1+b/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b-sqrt(b^2-4*a*c))^(2/3))+c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(1+b/sqrt(b^2-4*a*c))/(2^(1/3)*a*sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3))-1/3*c^(2/3)*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(1-b/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b+sqrt(b^2-4*a*c))^(2/3))+1/6*c^(2/3)*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(1-b/sqrt(b^2-4*a*c))/(2^(1/3)*a*(b+sqrt(b^2-4*a*c))^(2/3))+c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(1-b/sqrt(b^2-4*a*c))/(2^(1/3)*a*sqrt(3)*(b+sqrt(b^2-4*a*c))^(2/3))],
[x^11/(3+4*x^3+x^6),x,6,-4/3*x^3+1/6*x^6-1/6*log(1+x^3)+9/2*log(3+x^3)],
[x^8/(3+4*x^3+x^6),x,5,1/3*x^3+1/6*log(1+x^3)-3/2*log(3+x^3)],
[x^5/(3+4*x^3+x^6),x,4,-1/6*log(1+x^3)+1/2*log(3+x^3)],
[x^2/(3+4*x^3+x^6),x,4,-1/3*arctanh(2+x^3),1/6*log(1+x^3)-1/6*log(3+x^3)],
[1/(x*(3+4*x^3+x^6)),x,6,1/3*log(x)-1/6*log(1+x^3)+1/18*log(3+x^3)],
[1/(x^4*(3+4*x^3+x^6)),x,4,(-1/9)/x^3-4/9*log(x)+1/6*log(1+x^3)-1/54*log(3+x^3)],
[1/(x^7*(3+4*x^3+x^6)),x,4,(-1/18)/x^6+4/27/x^3+13/27*log(x)-1/6*log(1+x^3)+1/162*log(3+x^3)],
[x^10/(3+4*x^3+x^6),x,15,-2*x^2+1/5*x^5-9/2*3^(1/6)*arctan((3^(1/3)-2*x)/3^(5/6))+1/6*log(1+x)-3/2*3^(2/3)*log(3^(1/3)+x)-1/12*log(1-x+x^2)+3/4*3^(2/3)*log(3^(2/3)-3^(1/3)*x+x^2)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x^9/(3+4*x^3+x^6),x,15,-4*x+1/4*x^4-3/2*3^(5/6)*arctan((3^(1/3)-2*x)/3^(5/6))-1/6*log(1+x)+3/2*3^(1/3)*log(3^(1/3)+x)+1/12*log(1-x+x^2)-3/4*3^(1/3)*log(3^(2/3)-3^(1/3)*x+x^2)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x^7/(3+4*x^3+x^6),x,14,1/2*x^2+3/2*3^(1/6)*arctan((3^(1/3)-2*x)/3^(5/6))-1/6*log(1+x)+1/2*3^(2/3)*log(3^(1/3)+x)+1/12*log(1-x+x^2)-1/4*3^(2/3)*log(3^(2/3)-3^(1/3)*x+x^2)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x^6/(3+4*x^3+x^6),x,14,x+1/2*3^(5/6)*arctan((3^(1/3)-2*x)/3^(5/6))+1/6*log(1+x)-1/2*3^(1/3)*log(3^(1/3)+x)-1/12*log(1-x+x^2)+1/4*3^(1/3)*log(3^(2/3)-3^(1/3)*x+x^2)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x^4/(3+4*x^3+x^6),x,13,-1/2*3^(1/6)*arctan((3^(1/3)-2*x)/3^(5/6))+1/6*log(1+x)-1/2*log(3^(1/3)+x)/3^(1/3)-1/12*log(1-x+x^2)+1/4*log(3^(2/3)-3^(1/3)*x+x^2)/3^(1/3)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x^3/(3+4*x^3+x^6),x,13,-1/2*arctan((3^(1/3)-2*x)/3^(5/6))/3^(1/6)-1/6*log(1+x)+1/2*log(3^(1/3)+x)/3^(2/3)+1/12*log(1-x+x^2)-1/4*log(3^(2/3)-3^(1/3)*x+x^2)/3^(2/3)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x/(3+4*x^3+x^6),x,13,1/2*arctan((3^(1/3)-2*x)/3^(5/6))/3^(5/6)-1/6*log(1+x)+1/6*log(3^(1/3)+x)/3^(1/3)+1/12*log(1-x+x^2)-1/12*log(3^(2/3)-3^(1/3)*x+x^2)/3^(1/3)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[1/(3+4*x^3+x^6),x,13,1/6*arctan((3^(1/3)-2*x)/3^(5/6))/3^(1/6)+1/6*log(1+x)-1/6*log(3^(1/3)+x)/3^(2/3)-1/12*log(1-x+x^2)+1/12*log(3^(2/3)-3^(1/3)*x+x^2)/3^(2/3)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[1/(x^2*(3+4*x^3+x^6)),x,14,(-1/3)/x-1/6*arctan((3^(1/3)-2*x)/3^(5/6))/3^(5/6)+1/6*log(1+x)-1/18*log(3^(1/3)+x)/3^(1/3)-1/12*log(1-x+x^2)+1/36*log(3^(2/3)-3^(1/3)*x+x^2)/3^(1/3)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[1/(x^3*(3+4*x^3+x^6)),x,14,(-1/6)/x^2-1/18*arctan((3^(1/3)-2*x)/3^(5/6))/3^(1/6)-1/6*log(1+x)+1/18*log(3^(1/3)+x)/3^(2/3)+1/12*log(1-x+x^2)-1/36*log(3^(2/3)-3^(1/3)*x+x^2)/3^(2/3)+1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[1/(x^5*(3+4*x^3+x^6)),x,15,(-1/12)/x^4+4/9/x+1/18*arctan((3^(1/3)-2*x)/3^(5/6))/3^(5/6)-1/6*log(1+x)+1/54*log(3^(1/3)+x)/3^(1/3)+1/12*log(1-x+x^2)-1/108*log(3^(2/3)-3^(1/3)*x+x^2)/3^(1/3)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[1/(x^6*(3+4*x^3+x^6)),x,15,(-1/15)/x^5+2/9/x^2+1/54*arctan((3^(1/3)-2*x)/3^(5/6))/3^(1/6)+1/6*log(1+x)-1/54*log(3^(1/3)+x)/3^(2/3)-1/12*log(1-x+x^2)+1/108*log(3^(2/3)-3^(1/3)*x+x^2)/3^(2/3)-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[x^6/(1-x^3+x^6),x,14,x+1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))+1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))+1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))-1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))],
[x^5/(1-x^3+x^6),x,5,1/6*log(1-x^3+x^6)-1/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[x^4/(1-x^3+x^6),x,13,-1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))+1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))+1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))+1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))],
[x^3/(1-x^3+x^6),x,13,1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))],
[x^2/(1-x^3+x^6),x,3,-2/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[x/(1-x^3+x^6),x,13,1/3*I*2^(1/3)*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))/(1-I*sqrt(3))^(1/3)-1/3*I*2^(1/3)*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))/(1+I*sqrt(3))^(1/3)+1/3*I*2^(1/3)*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))/((1-I*sqrt(3))^(1/3)*sqrt(3))-1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))/(2^(2/3)*(1-I*sqrt(3))^(1/3)*sqrt(3))-1/3*I*2^(1/3)*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))/((1+I*sqrt(3))^(1/3)*sqrt(3))+1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))/(2^(2/3)*(1+I*sqrt(3))^(1/3)*sqrt(3))],
[1/(1-x^3+x^6),x,13,-1/3*(-1)^(13/18)*arctan((1+2*(-1)^(1/9)*x)/sqrt(3))+1/3*(-1)^(5/18)*arctan((1-2*(-1)^(8/9)*x)/sqrt(3))-1/9*(-1)^(5/18)*(log(2)+3*log((-1)^(1/9)-x))/sqrt(3)+1/3*(-1)^(13/18)*log(-2^(1/3)*((-1)^(8/9)+x))/sqrt(3)-1/6*(-1)^(13/18)*log(-2^(2/3)*((-1)^(7/9)+((-1)^(8/9)-x)*x))/sqrt(3)+1/6*(-1)^(5/18)*log(2^(2/3)*((-1)^(2/9)+x*((-1)^(1/9)+x)))/sqrt(3),-1/3*I*2^(2/3)*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))/(1-I*sqrt(3))^(2/3)+1/3*I*2^(2/3)*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))/(1+I*sqrt(3))^(2/3)+1/3*I*2^(2/3)*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))/((1-I*sqrt(3))^(2/3)*sqrt(3))-1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))/(2^(1/3)*(1-I*sqrt(3))^(2/3)*sqrt(3))-1/3*I*2^(2/3)*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))/((1+I*sqrt(3))^(2/3)*sqrt(3))+1/3*I*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))/(2^(1/3)*(1+I*sqrt(3))^(2/3)*sqrt(3))],
[1/(x*(1-x^3+x^6)),x,7,log(x)-1/6*log(1-x^3+x^6)-1/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[1/(x^2*(1-x^3+x^6)),x,14,(-1)/x+1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))],
[1/(x^3*(1-x^3+x^6)),x,14,(-1/2)/x^2-1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(1/3)*(1-I*sqrt(3))^(2/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))+1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(1/3)*(1+I*sqrt(3))^(2/3))],
[1/(x^4*(1-x^3+x^6)),x,8,(-1/3)/x^3+log(x)-1/6*log(1-x^3+x^6)+1/3*arctan((1-2*x^3)/sqrt(3))/sqrt(3)],
[1/(x^5*(1-x^3+x^6)),x,16,(-1/4)/x^4+(-1)/x+1/3*arctan((1+2*2^(1/3)*x/(1+I*sqrt(3))^(1/3))/sqrt(3))*(I-sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1+I*sqrt(3))^(1/3))*(3-I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1+I*sqrt(3))^(1/3)+(1+I*sqrt(3))^(2/3))*(3-I*sqrt(3))/(2^(2/3)*(1+I*sqrt(3))^(1/3))-1/9*log(-2^(1/3)*x+(1-I*sqrt(3))^(1/3))*(3+I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))+1/18*log(2^(2/3)*x^2+2^(1/3)*x*(1-I*sqrt(3))^(1/3)+(1-I*sqrt(3))^(2/3))*(3+I*sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))-1/3*arctan((1+2*2^(1/3)*x/(1-I*sqrt(3))^(1/3))/sqrt(3))*(I+sqrt(3))/(2^(2/3)*(1-I*sqrt(3))^(1/3))],
[1/(2+x^3+x^6),x,13,-1/3*I*2^(2/3)*log(2^(1/3)*x+(1-I*sqrt(7))^(1/3))/((1-I*sqrt(7))^(2/3)*sqrt(7))+1/3*I*log(2^(2/3)*x^2-2^(1/3)*x*(1-I*sqrt(7))^(1/3)+(1-I*sqrt(7))^(2/3))/(2^(1/3)*(1-I*sqrt(7))^(2/3)*sqrt(7))+1/3*I*2^(2/3)*log(2^(1/3)*x+(1+I*sqrt(7))^(1/3))/((1+I*sqrt(7))^(2/3)*sqrt(7))-1/3*I*log(2^(2/3)*x^2-2^(1/3)*x*(1+I*sqrt(7))^(1/3)+(1+I*sqrt(7))^(2/3))/(2^(1/3)*(1+I*sqrt(7))^(2/3)*sqrt(7))+I*2^(2/3)*arctan((1-2*2^(1/3)*x/(1-I*sqrt(7))^(1/3))/sqrt(3))/((1-I*sqrt(7))^(2/3)*sqrt(21))-I*2^(2/3)*arctan((1-2*2^(1/3)*x/(1+I*sqrt(7))^(1/3))/sqrt(3))/((1+I*sqrt(7))^(2/3)*sqrt(21))],
[x^2/(2+x^3+x^6),x,3,2/3*arctan((1+2*x^3)/sqrt(7))/sqrt(7)],
[x^3/(2+x^3+x^6),x,13,1/21*log(2^(1/3)*x+(1+I*sqrt(7))^(1/3))*(7-I*sqrt(7))/(2^(1/3)*(1+I*sqrt(7))^(2/3))-1/42*log(2^(2/3)*x^2-2^(1/3)*x*(1+I*sqrt(7))^(1/3)+(1+I*sqrt(7))^(2/3))*(7-I*sqrt(7))/(2^(1/3)*(1+I*sqrt(7))^(2/3))+1/21*log(2^(1/3)*x+(1-I*sqrt(7))^(1/3))*(7+I*sqrt(7))/(2^(1/3)*(1-I*sqrt(7))^(2/3))-1/42*log(2^(2/3)*x^2-2^(1/3)*x*(1-I*sqrt(7))^(1/3)+(1-I*sqrt(7))^(2/3))*(7+I*sqrt(7))/(2^(1/3)*(1-I*sqrt(7))^(2/3))-I*arctan((1-2*2^(1/3)*x/(1-I*sqrt(7))^(1/3))/sqrt(3))*(1-I*sqrt(7))^(1/3)/(2^(1/3)*sqrt(21))+I*arctan((1-2*2^(1/3)*x/(1+I*sqrt(7))^(1/3))/sqrt(3))*(1+I*sqrt(7))^(1/3)/(2^(1/3)*sqrt(21))],

# Integrands of the form x^m (a+b x^3+c x^6)^(p/2)

# p>0
[x^14*sqrt(a+b*x^3+c*x^6),x,7,-1/20*b*x^6*(a+b*x^3+c*x^6)^(3/2)/c^2+1/18*x^9*(a+b*x^3+c*x^6)^(3/2)/c-1/2880*(7*b*(15*b^2-28*a*c)-6*c*(21*b^2-20*a*c)*x^3)*(a+b*x^3+c*x^6)^(3/2)/c^4-1/3072*(b^2-4*a*c)*(21*b^4-56*a*b^2*c+16*a^2*c^2)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(11/2)+1/1536*(21*b^4-56*a*b^2*c+16*a^2*c^2)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^5],
[x^11*sqrt(a+b*x^3+c*x^6),x,6,1/15*x^6*(a+b*x^3+c*x^6)^(3/2)/c+1/720*(35*b^2-32*a*c-42*b*c*x^3)*(a+b*x^3+c*x^6)^(3/2)/c^3+1/768*b*(7*b^2-12*a*c)*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(9/2)-1/384*b*(7*b^2-12*a*c)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^4],
[x^8*sqrt(a+b*x^3+c*x^6),x,6,-5/72*b*(a+b*x^3+c*x^6)^(3/2)/c^2+1/12*x^3*(a+b*x^3+c*x^6)^(3/2)/c-1/384*(b^2-4*a*c)*(5*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(7/2)+1/192*(5*b^2-4*a*c)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^3],
[x^5*sqrt(a+b*x^3+c*x^6),x,5,1/9*(a+b*x^3+c*x^6)^(3/2)/c+1/48*b*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(5/2)-1/24*b*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^2],
[x^2*sqrt(a+b*x^3+c*x^6),x,4,-1/24*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(3/2)+1/12*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c],
[sqrt(a+b*x^3+c*x^6)/x,x,7,-1/3*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))*sqrt(a)+1/6*b*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/sqrt(c)+1/3*sqrt(a+b*x^3+c*x^6)],
[sqrt(a+b*x^3+c*x^6)/x^4,x,7,-1/6*b*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/sqrt(a)+1/3*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))*sqrt(c)-1/3*sqrt(a+b*x^3+c*x^6)/x^3],
[sqrt(a+b*x^3+c*x^6)/x^7,x,4,1/24*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(3/2)-1/12*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a*x^6)],
[sqrt(a+b*x^3+c*x^6)/x^10,x,5,-1/9*(a+b*x^3+c*x^6)^(3/2)/(a*x^9)-1/48*b*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(5/2)+1/24*b*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^2*x^6)],
[sqrt(a+b*x^3+c*x^6)/x^13,x,6,-1/12*(a+b*x^3+c*x^6)^(3/2)/(a*x^12)+5/72*b*(a+b*x^3+c*x^6)^(3/2)/(a^2*x^9)+1/384*(b^2-4*a*c)*(5*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(7/2)-1/192*(5*b^2-4*a*c)*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^3*x^6)],
[sqrt(a+b*x^3+c*x^6)/x^16,x,7,-1/15*(a+b*x^3+c*x^6)^(3/2)/(a*x^15)+7/120*b*(a+b*x^3+c*x^6)^(3/2)/(a^2*x^12)-1/720*(35*b^2-32*a*c)*(a+b*x^3+c*x^6)^(3/2)/(a^3*x^9)-1/768*b*(7*b^2-12*a*c)*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(9/2)+1/384*b*(7*b^2-12*a*c)*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^4*x^6)],
[x^3*sqrt(a+b*x^3+c*x^6),x,2,1/4*x^4*AppellF1(4/3,-1/2,-1/2,7/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[x*sqrt(a+b*x^3+c*x^6),x,2,1/2*x^2*AppellF1(2/3,-1/2,-1/2,5/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[sqrt(a+b*x^3+c*x^6),x,2,x*AppellF1(1/3,-1/2,-1/2,4/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[sqrt(a+b*x^3+c*x^6)/x^2,x,2,-AppellF1(-1/3,-1/2,-1/2,2/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(x*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[sqrt(a+b*x^3+c*x^6)/x^3,x,2,-1/2*AppellF1(-2/3,-1/2,-1/2,1/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(x^2*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[x^14*(a+b*x^3+c*x^6)^(3/2),x,8,1/6144*(33*b^4-72*a*b^2*c+16*a^2*c^2)*(b+2*c*x^3)*(a+b*x^3+c*x^6)^(3/2)/c^5-11/336*b*x^6*(a+b*x^3+c*x^6)^(5/2)/c^2+1/24*x^9*(a+b*x^3+c*x^6)^(5/2)/c-1/13440*(3*b*(77*b^2-124*a*c)-10*c*(33*b^2-28*a*c)*x^3)*(a+b*x^3+c*x^6)^(5/2)/c^4+1/32768*(b^2-4*a*c)^2*(33*b^4-72*a*b^2*c+16*a^2*c^2)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(13/2)-1/16384*(b^2-4*a*c)*(33*b^4-72*a*b^2*c+16*a^2*c^2)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^6],
[x^11*(a+b*x^3+c*x^6)^(3/2),x,7,-1/384*b*(3*b^2-4*a*c)*(b+2*c*x^3)*(a+b*x^3+c*x^6)^(3/2)/c^4+1/21*x^6*(a+b*x^3+c*x^6)^(5/2)/c+1/840*(21*b^2-16*a*c-30*b*c*x^3)*(a+b*x^3+c*x^6)^(5/2)/c^3-1/2048*b*(b^2-4*a*c)^2*(3*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(11/2)+1/1024*b*(b^2-4*a*c)*(3*b^2-4*a*c)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^5],
[x^8*(a+b*x^3+c*x^6)^(3/2),x,7,1/576*(7*b^2-4*a*c)*(b+2*c*x^3)*(a+b*x^3+c*x^6)^(3/2)/c^3-7/180*b*(a+b*x^3+c*x^6)^(5/2)/c^2+1/18*x^3*(a+b*x^3+c*x^6)^(5/2)/c+1/3072*(b^2-4*a*c)^2*(7*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(9/2)-1/1536*(b^2-4*a*c)*(7*b^2-4*a*c)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^4],
[x^5*(a+b*x^3+c*x^6)^(3/2),x,6,-1/48*b*(b+2*c*x^3)*(a+b*x^3+c*x^6)^(3/2)/c^2+1/15*(a+b*x^3+c*x^6)^(5/2)/c-1/256*b*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(7/2)+1/128*b*(b^2-4*a*c)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^3],
[x^2*(a+b*x^3+c*x^6)^(3/2),x,5,1/24*(b+2*c*x^3)*(a+b*x^3+c*x^6)^(3/2)/c+1/128*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(5/2)-1/64*(b^2-4*a*c)*(b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^2],
[(a+b*x^3+c*x^6)^(3/2)/x,x,8,1/9*(a+b*x^3+c*x^6)^(3/2)-1/3*a^(3/2)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))-1/48*b*(b^2-12*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(3/2)+1/24*(b^2+8*a*c+2*b*c*x^3)*sqrt(a+b*x^3+c*x^6)/c],
[(a+b*x^3+c*x^6)^(3/2)/x^4,x,8,-1/3*(a+b*x^3+c*x^6)^(3/2)/x^3-1/2*b*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))*sqrt(a)+1/8*(b^2+4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/sqrt(c)+1/4*(3*b+2*c*x^3)*sqrt(a+b*x^3+c*x^6)],
[(a+b*x^3+c*x^6)^(3/2)/x^7,x,8,-1/6*(a+b*x^3+c*x^6)^(3/2)/x^6-1/8*(b^2+4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/sqrt(a)+1/2*b*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))*sqrt(c)-1/4*(b-2*c*x^3)*sqrt(a+b*x^3+c*x^6)/x^3],
[(a+b*x^3+c*x^6)^(3/2)/x^10,x,8,-1/9*(a+b*x^3+c*x^6)^(3/2)/x^9+1/48*b*(b^2-12*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(3/2)+1/3*c^(3/2)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))-1/24*(2*a*b+(b^2+8*a*c)*x^3)*sqrt(a+b*x^3+c*x^6)/(a*x^6)],
[(a+b*x^3+c*x^6)^(3/2)/x^13,x,5,-1/24*(2*a+b*x^3)*(a+b*x^3+c*x^6)^(3/2)/(a*x^12)-1/128*(b^2-4*a*c)^2*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(5/2)+1/64*(b^2-4*a*c)*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^2*x^6)],
[(a+b*x^3+c*x^6)^(3/2)/x^16,x,6,1/48*b*(2*a+b*x^3)*(a+b*x^3+c*x^6)^(3/2)/(a^2*x^12)-1/15*(a+b*x^3+c*x^6)^(5/2)/(a*x^15)+1/256*b*(b^2-4*a*c)^2*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(7/2)-1/128*b*(b^2-4*a*c)*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^3*x^6)],
[(a+b*x^3+c*x^6)^(3/2)/x^19,x,7,-1/576*(7*b^2-4*a*c)*(2*a+b*x^3)*(a+b*x^3+c*x^6)^(3/2)/(a^3*x^12)-1/18*(a+b*x^3+c*x^6)^(5/2)/(a*x^18)+7/180*b*(a+b*x^3+c*x^6)^(5/2)/(a^2*x^15)-1/3072*(b^2-4*a*c)^2*(7*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(9/2)+1/1536*(b^2-4*a*c)*(7*b^2-4*a*c)*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^4*x^6)],
[(a+b*x^3+c*x^6)^(3/2)/x^22,x,8,1/384*b*(3*b^2-4*a*c)*(2*a+b*x^3)*(a+b*x^3+c*x^6)^(3/2)/(a^4*x^12)-1/21*(a+b*x^3+c*x^6)^(5/2)/(a*x^21)+1/28*b*(a+b*x^3+c*x^6)^(5/2)/(a^2*x^18)-1/840*(21*b^2-16*a*c)*(a+b*x^3+c*x^6)^(5/2)/(a^3*x^15)+1/2048*b*(b^2-4*a*c)^2*(3*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(11/2)-1/1024*b*(b^2-4*a*c)*(3*b^2-4*a*c)*(2*a+b*x^3)*sqrt(a+b*x^3+c*x^6)/(a^5*x^6)],
[x^3*(a+b*x^3+c*x^6)^(3/2),x,2,1/4*a*x^4*AppellF1(4/3,-3/2,-3/2,7/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[x*(a+b*x^3+c*x^6)^(3/2),x,2,1/2*a*x^2*AppellF1(2/3,-3/2,-3/2,5/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[(a+b*x^3+c*x^6)^(3/2),x,2,a*x*AppellF1(1/3,-3/2,-3/2,4/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[(a+b*x^3+c*x^6)^(3/2)/x^2,x,2,-a*AppellF1(-1/3,-3/2,-3/2,2/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(x*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[(a+b*x^3+c*x^6)^(3/2)/x^3,x,2,-1/2*a*AppellF1(-2/3,-3/2,-3/2,1/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(x^2*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],

# p<0
[x^14/sqrt(a+b*x^3+c*x^6),x,6,1/384*(35*b^4-120*a*b^2*c+48*a^2*c^2)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(9/2)-7/72*b*x^6*sqrt(a+b*x^3+c*x^6)/c^2+1/12*x^9*sqrt(a+b*x^3+c*x^6)/c-1/576*(5*b*(21*b^2-44*a*c)-2*c*(35*b^2-36*a*c)*x^3)*sqrt(a+b*x^3+c*x^6)/c^4],
[x^11/sqrt(a+b*x^3+c*x^6),x,5,-1/48*b*(5*b^2-12*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(7/2)+1/9*x^6*sqrt(a+b*x^3+c*x^6)/c+1/72*(15*b^2-16*a*c-10*b*c*x^3)*sqrt(a+b*x^3+c*x^6)/c^3],
[x^8/sqrt(a+b*x^3+c*x^6),x,5,1/24*(3*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(5/2)-1/4*b*sqrt(a+b*x^3+c*x^6)/c^2+1/6*x^3*sqrt(a+b*x^3+c*x^6)/c],
[x^5/sqrt(a+b*x^3+c*x^6),x,4,-1/6*b*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(3/2)+1/3*sqrt(a+b*x^3+c*x^6)/c],
[x^2/sqrt(a+b*x^3+c*x^6),x,3,1/3*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/sqrt(c)],
[1/(x*sqrt(a+b*x^3+c*x^6)),x,3,-1/3*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/sqrt(a)],
[1/(x^4*sqrt(a+b*x^3+c*x^6)),x,4,1/6*b*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(3/2)-1/3*sqrt(a+b*x^3+c*x^6)/(a*x^3)],
[1/(x^7*sqrt(a+b*x^3+c*x^6)),x,5,-1/24*(3*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(5/2)-1/6*sqrt(a+b*x^3+c*x^6)/(a*x^6)+1/4*b*sqrt(a+b*x^3+c*x^6)/(a^2*x^3)],
[1/(x^10*sqrt(a+b*x^3+c*x^6)),x,6,1/48*b*(5*b^2-12*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(7/2)-1/9*sqrt(a+b*x^3+c*x^6)/(a*x^9)+5/36*b*sqrt(a+b*x^3+c*x^6)/(a^2*x^6)-1/72*(15*b^2-16*a*c)*sqrt(a+b*x^3+c*x^6)/(a^3*x^3)],
[1/(x^13*sqrt(a+b*x^3+c*x^6)),x,7,-1/384*(35*b^4-120*a*b^2*c+48*a^2*c^2)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(9/2)-1/12*sqrt(a+b*x^3+c*x^6)/(a*x^12)+7/72*b*sqrt(a+b*x^3+c*x^6)/(a^2*x^9)-1/288*(35*b^2-36*a*c)*sqrt(a+b*x^3+c*x^6)/(a^3*x^6)+5/576*b*(21*b^2-44*a*c)*sqrt(a+b*x^3+c*x^6)/(a^4*x^3)],
[x^3/sqrt(a+b*x^3+c*x^6),x,2,1/4*x^4*AppellF1(4/3,1/2,1/2,7/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^3+c*x^6)],
[x/sqrt(a+b*x^3+c*x^6),x,2,1/2*x^2*AppellF1(2/3,1/2,1/2,5/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^3+c*x^6)],
[1/sqrt(a+b*x^3+c*x^6),x,2,x*AppellF1(1/3,1/2,1/2,4/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^3+c*x^6)],
[1/(x^2*sqrt(a+b*x^3+c*x^6)),x,2,-AppellF1(-1/3,1/2,1/2,2/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(x*sqrt(a+b*x^3+c*x^6))],
[1/(x^3*sqrt(a+b*x^3+c*x^6)),x,2,-1/2*AppellF1(-2/3,1/2,1/2,1/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(x^2*sqrt(a+b*x^3+c*x^6))],
[x^14/(a+b*x^3+c*x^6)^(3/2),x,6,1/8*(5*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(7/2)+2/3*x^9*(2*a+b*x^3)/((b^2-4*a*c)*sqrt(a+b*x^3+c*x^6))-2/3*b*x^6*sqrt(a+b*x^3+c*x^6)/(c*(b^2-4*a*c))-1/12*(b*(15*b^2-52*a*c)-2*c*(5*b^2-12*a*c)*x^3)*sqrt(a+b*x^3+c*x^6)/(c^3*(b^2-4*a*c))],
[x^11/(a+b*x^3+c*x^6)^(3/2),x,5,-1/2*b*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(5/2)+2/3*x^6*(2*a+b*x^3)/((b^2-4*a*c)*sqrt(a+b*x^3+c*x^6))+1/3*(3*b^2-8*a*c-2*b*c*x^3)*sqrt(a+b*x^3+c*x^6)/(c^2*(b^2-4*a*c))],
[x^8/(a+b*x^3+c*x^6)^(3/2),x,5,1/3*arctanh(1/2*(b+2*c*x^3)/(sqrt(c)*sqrt(a+b*x^3+c*x^6)))/c^(3/2)+2/3*x^3*(2*a+b*x^3)/((b^2-4*a*c)*sqrt(a+b*x^3+c*x^6))-2/3*b*sqrt(a+b*x^3+c*x^6)/(c*(b^2-4*a*c))],
[x^5/(a+b*x^3+c*x^6)^(3/2),x,2,2/3*(2*a+b*x^3)/((b^2-4*a*c)*sqrt(a+b*x^3+c*x^6))],
[x^2/(a+b*x^3+c*x^6)^(3/2),x,2,-2/3*(b+2*c*x^3)/((b^2-4*a*c)*sqrt(a+b*x^3+c*x^6))],
[1/(x*(a+b*x^3+c*x^6)^(3/2)),x,5,-1/3*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(3/2)+2/3*(b^2-2*a*c+b*c*x^3)/(a*(b^2-4*a*c)*sqrt(a+b*x^3+c*x^6))],
[1/(x^4*(a+b*x^3+c*x^6)^(3/2)),x,5,1/2*b*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(5/2)+2/3*(b^2-2*a*c+b*c*x^3)/(a*(b^2-4*a*c)*x^3*sqrt(a+b*x^3+c*x^6))-1/3*(3*b^2-8*a*c)*sqrt(a+b*x^3+c*x^6)/(a^2*(b^2-4*a*c)*x^3)],
[1/(x^7*(a+b*x^3+c*x^6)^(3/2)),x,6,-1/8*(5*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(7/2)+2/3*(b^2-2*a*c+b*c*x^3)/(a*(b^2-4*a*c)*x^6*sqrt(a+b*x^3+c*x^6))-1/6*(5*b^2-12*a*c)*sqrt(a+b*x^3+c*x^6)/(a^2*(b^2-4*a*c)*x^6)+1/12*b*(15*b^2-52*a*c)*sqrt(a+b*x^3+c*x^6)/(a^3*(b^2-4*a*c)*x^3)],
[1/(x^10*(a+b*x^3+c*x^6)^(3/2)),x,7,5/48*b*(7*b^2-12*a*c)*arctanh(1/2*(2*a+b*x^3)/(sqrt(a)*sqrt(a+b*x^3+c*x^6)))/a^(9/2)+2/3*(b^2-2*a*c+b*c*x^3)/(a*(b^2-4*a*c)*x^9*sqrt(a+b*x^3+c*x^6))-1/9*(7*b^2-16*a*c)*sqrt(a+b*x^3+c*x^6)/(a^2*(b^2-4*a*c)*x^9)+1/36*b*(35*b^2-116*a*c)*sqrt(a+b*x^3+c*x^6)/(a^3*(b^2-4*a*c)*x^6)-1/72*(105*b^4-460*a*b^2*c+256*a^2*c^2)*sqrt(a+b*x^3+c*x^6)/(a^4*(b^2-4*a*c)*x^3)],
[x^3/(a+b*x^3+c*x^6)^(3/2),x,2,1/4*x^4*AppellF1(4/3,3/2,3/2,7/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^3+c*x^6))],
[x/(a+b*x^3+c*x^6)^(3/2),x,2,1/2*x^2*AppellF1(2/3,3/2,3/2,5/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^3+c*x^6))],
[1/(a+b*x^3+c*x^6)^(3/2),x,2,x*AppellF1(1/3,3/2,3/2,4/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^3+c*x^6))],
[1/(x^2*(a+b*x^3+c*x^6)^(3/2)),x,2,-AppellF1(-1/3,3/2,3/2,2/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(a*x*sqrt(a+b*x^3+c*x^6))],
[1/(x^3*(a+b*x^3+c*x^6)^(3/2)),x,2,-1/2*AppellF1(-2/3,3/2,3/2,1/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(a*x^2*sqrt(a+b*x^3+c*x^6))],

# Integrands of the form (d x)^m (a+b x^3+c x^6)^p with m symbolic
[(d*x)^m*(a+b*x^3+c*x^6)^2,x,2,a^2*(d*x)^(1+m)/(d*(1+m))+2*a*b*(d*x)^(4+m)/(d^4*(4+m))+(b^2+2*a*c)*(d*x)^(7+m)/(d^7*(7+m))+2*b*c*(d*x)^(10+m)/(d^10*(10+m))+c^2*(d*x)^(13+m)/(d^13*(13+m))],
[(d*x)^m*(a+b*x^3+c*x^6),x,2,a*(d*x)^(1+m)/(d*(1+m))+b*(d*x)^(4+m)/(d^4*(4+m))+c*(d*x)^(7+m)/(d^7*(7+m))],
[(d*x)^m/(a+b*x^3+c*x^6),x,3,2*c*(d*x)^(1+m)*hypergeom([1,1/3*(1+m)],[1/3*(4+m)],-2*c*x^3/(b-sqrt(b^2-4*a*c)))/(d*(1+m)*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c))-2*c*(d*x)^(1+m)*hypergeom([1,1/3*(1+m)],[1/3*(4+m)],-2*c*x^3/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[(d*x)^m/(a+b*x^3+c*x^6)^2,x,4,1/3*(d*x)^(1+m)*(b^2-2*a*c+b*c*x^3)/(a*(b^2-4*a*c)*d*(a+b*x^3+c*x^6))-1/3*c*(d*x)^(1+m)*hypergeom([1,1/3*(1+m)],[1/3*(4+m)],-2*c*x^3/(b+sqrt(b^2-4*a*c)))*(b^2*(2-m)-4*a*c*(5-m)-b*(2-m)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^(3/2)*d*(1+m)*(b+sqrt(b^2-4*a*c)))+1/3*c*(d*x)^(1+m)*hypergeom([1,1/3*(1+m)],[1/3*(4+m)],-2*c*x^3/(b-sqrt(b^2-4*a*c)))*(b^2*(2-m)-4*a*c*(5-m)+b*(2-m)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^(3/2)*d*(1+m)*(b-sqrt(b^2-4*a*c)))],
[(d*x)^m*(a+b*x^3+c*x^6)^(3/2),x,2,a*(d*x)^(1+m)*AppellF1(1/3*(1+m),-3/2,-3/2,1/3*(4+m),-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(d*(1+m)*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[(d*x)^m*(a+b*x^3+c*x^6)^(1/2),x,2,(d*x)^(1+m)*AppellF1(1/3*(1+m),-1/2,-1/2,1/3*(4+m),-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^3+c*x^6)/(d*(1+m)*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c))))],
[(d*x)^m/(a+b*x^3+c*x^6)^(1/2),x,2,(d*x)^(1+m)*AppellF1(1/3*(1+m),1/2,1/2,1/3*(4+m),-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*sqrt(a+b*x^3+c*x^6))],
[(d*x)^m/(a+b*x^3+c*x^6)^(3/2),x,2,(d*x)^(1+m)*AppellF1(1/3*(1+m),3/2,3/2,1/3*(4+m),-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))/(a*d*(1+m)*sqrt(a+b*x^3+c*x^6))],

# Integrands of the form (d x)^m (a+b x^3+c x^6)^p with p symbolic
[(d*x)^m*(a+b*x^3+c*x^6)^p,x,2,(d*x)^(1+m)*(a+b*x^3+c*x^6)^p*AppellF1(1/3*(1+m),-p,-p,1/3*(4+m),-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[x^8*(a+b*x^3+c*x^6)^p,x,4,-1/6*b*(2+p)*(a+b*x^3+c*x^6)^(1+p)/(c^2*(3+5*p+2*p^2))+1/3*x^3*(a+b*x^3+c*x^6)^(1+p)/(c*(3+2*p))+1/3*2^p*(2*a*c-b^2*(2+p))*(a+b*x^3+c*x^6)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x^3+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x^3+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/(c^2*(1+p)*(3+2*p)*sqrt(b^2-4*a*c))],
[x^5*(a+b*x^3+c*x^6)^p,x,3,1/6*(a+b*x^3+c*x^6)^(1+p)/(c*(1+p))+1/3*2^p*b*(a+b*x^3+c*x^6)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x^3+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x^3+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/(c*(1+p)*sqrt(b^2-4*a*c))],
[x^2*(a+b*x^3+c*x^6)^p,x,2,-1/3*2^(1+p)*(a+b*x^3+c*x^6)^(1+p)*hypergeom([-p,1+p],[2+p],1/2*(b+2*c*x^3+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))*((-b-2*c*x^3+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))^(-1-p)/((1+p)*sqrt(b^2-4*a*c))],
[x^4*(a+b*x^3+c*x^6)^p,x,2,1/5*x^5*(a+b*x^3+c*x^6)^p*AppellF1(5/3,-p,-p,8/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/((1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[x^3*(a+b*x^3+c*x^6)^p,x,2,1/4*x^4*(a+b*x^3+c*x^6)^p*AppellF1(4/3,-p,-p,7/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/((1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[x*(a+b*x^3+c*x^6)^p,x,2,1/2*x^2*(a+b*x^3+c*x^6)^p*AppellF1(2/3,-p,-p,5/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/((1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[(a+b*x^3+c*x^6)^p,x,2,x*(a+b*x^3+c*x^6)^p*AppellF1(1/3,-p,-p,4/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/((1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[(a+b*x^3+c*x^6)^p/x,x,3,1/3*2^(-1+2*p)*(a+b*x^3+c*x^6)^p*AppellF1(-2*p,-p,-p,1-2*p,1/2*(-b+sqrt(b^2-4*a*c))/(c*x^3),1/2*(-b-sqrt(b^2-4*a*c))/(c*x^3))/(p*((b+2*c*x^3-sqrt(b^2-4*a*c))/(c*x^3))^p*((b+2*c*x^3+sqrt(b^2-4*a*c))/(c*x^3))^p)],
[(a+b*x^3+c*x^6)^p/x^2,x,2,-(a+b*x^3+c*x^6)^p*AppellF1(-1/3,-p,-p,2/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/(x*(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[(a+b*x^3+c*x^6)^p/x^3,x,2,-1/2*(a+b*x^3+c*x^6)^p*AppellF1(-2/3,-p,-p,1/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/(x^2*(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[(a+b*x^3+c*x^6)^p/x^4,x,3,-1/3*4^p*(a+b*x^3+c*x^6)^p*AppellF1(1-2*p,-p,-p,2*(1-p),1/2*(-b+sqrt(b^2-4*a*c))/(c*x^3),1/2*(-b-sqrt(b^2-4*a*c))/(c*x^3))/((1-2*p)*x^3*((b+2*c*x^3-sqrt(b^2-4*a*c))/(c*x^3))^p*((b+2*c*x^3+sqrt(b^2-4*a*c))/(c*x^3))^p)],
[(a+b*x^3+c*x^6)^p/x^5,x,2,-1/4*(a+b*x^3+c*x^6)^p*AppellF1(-4/3,-p,-p,-1/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/(x^4*(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[(a+b*x^3+c*x^6)^p/x^6,x,2,-1/5*(a+b*x^3+c*x^6)^p*AppellF1(-5/3,-p,-p,-2/3,-2*c*x^3/(b-sqrt(b^2-4*a*c)),-2*c*x^3/(b+sqrt(b^2-4*a*c)))/(x^5*(1+2*c*x^3/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^3/(b+sqrt(b^2-4*a*c)))^p)],
[(a+b*x^3+c*x^6)^p/x^7,x,3,-1/3*2^(-1+2*p)*(a+b*x^3+c*x^6)^p*AppellF1(2*(1-p),-p,-p,3-2*p,1/2*(-b+sqrt(b^2-4*a*c))/(c*x^3),1/2*(-b-sqrt(b^2-4*a*c))/(c*x^3))/((1-p)*x^6*((b+2*c*x^3-sqrt(b^2-4*a*c))/(c*x^3))^p*((b+2*c*x^3+sqrt(b^2-4*a*c))/(c*x^3))^p)],

# Integrands of the form (d x)^m (a+b x^4+c x^8)^p

# Integrands of the form (d x)^m (a+b x^4+c x^8)^p with b^2-4 a c=0

# Integrands of the form x^m (a+b x^4+c x^8)^p

# p>0

# p<0
[x^m/(1+2*x^4+x^8),x,2,x^(1+m)*hypergeom([2,1/4*(1+m)],[1/4*(5+m)],-x^4)/(1+m)],
[x^9/(1+2*x^4+x^8),x,5,3/4*x^2-1/4*x^6/(1+x^4)-3/4*arctan(x^2)],
[x^7/(1+2*x^4+x^8),x,4,1/4/(1+x^4)+1/4*log(1+x^4)],
[x^5/(1+2*x^4+x^8),x,4,-1/4*x^2/(1+x^4)+1/4*arctan(x^2)],
[x^3/(1+2*x^4+x^8),x,2,(-1/4)/(1+x^4)],
[x/(1+2*x^4+x^8),x,4,1/4*x^2/(1+x^4)+1/4*arctan(x^2)],
[1/(x*(1+2*x^4+x^8)),x,4,1/4/(1+x^4)+log(x)-1/4*log(1+x^4)],
[1/(x^3*(1+2*x^4+x^8)),x,5,(-3/4)/x^2+1/4/(x^2*(1+x^4))-3/4*arctan(x^2)],
[1/(x^5*(1+2*x^4+x^8)),x,4,(-1/4)/x^4+(-1/4)/(1+x^4)-2*log(x)+1/2*log(1+x^4)],
[1/(x^7*(1+2*x^4+x^8)),x,6,(-5/12)/x^6+5/4/x^2+1/4/(x^6*(1+x^4))+5/4*arctan(x^2)],
[x^8/(1+2*x^4+x^8),x,12,5/4*x-1/4*x^5/(1+x^4)+5/8*arctan(1-x*sqrt(2))/sqrt(2)-5/8*arctan(1+x*sqrt(2))/sqrt(2)+5/16*log(1+x^2-x*sqrt(2))/sqrt(2)-5/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^6/(1+2*x^4+x^8),x,11,-1/4*x^3/(1+x^4)-3/8*arctan(1-x*sqrt(2))/sqrt(2)+3/8*arctan(1+x*sqrt(2))/sqrt(2)+3/16*log(1+x^2-x*sqrt(2))/sqrt(2)-3/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^4/(1+2*x^4+x^8),x,11,-1/4*x/(1+x^4)-1/8*arctan(1-x*sqrt(2))/sqrt(2)+1/8*arctan(1+x*sqrt(2))/sqrt(2)-1/16*log(1+x^2-x*sqrt(2))/sqrt(2)+1/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^2/(1+2*x^4+x^8),x,11,1/4*x^3/(1+x^4)-1/8*arctan(1-x*sqrt(2))/sqrt(2)+1/8*arctan(1+x*sqrt(2))/sqrt(2)+1/16*log(1+x^2-x*sqrt(2))/sqrt(2)-1/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(1+2*x^4+x^8),x,11,1/4*x/(1+x^4)-3/8*arctan(1-x*sqrt(2))/sqrt(2)+3/8*arctan(1+x*sqrt(2))/sqrt(2)-3/16*log(1+x^2-x*sqrt(2))/sqrt(2)+3/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^2*(1+2*x^4+x^8)),x,12,(-5/4)/x+1/4/(x*(1+x^4))+5/8*arctan(1-x*sqrt(2))/sqrt(2)-5/8*arctan(1+x*sqrt(2))/sqrt(2)-5/16*log(1+x^2-x*sqrt(2))/sqrt(2)+5/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^4*(1+2*x^4+x^8)),x,12,(-7/12)/x^3+1/4/(x^3*(1+x^4))+7/8*arctan(1-x*sqrt(2))/sqrt(2)-7/8*arctan(1+x*sqrt(2))/sqrt(2)+7/16*log(1+x^2-x*sqrt(2))/sqrt(2)-7/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^6*(1+2*x^4+x^8)),x,13,(-9/20)/x^5+9/4/x+1/4/(x^5*(1+x^4))-9/8*arctan(1-x*sqrt(2))/sqrt(2)+9/8*arctan(1+x*sqrt(2))/sqrt(2)+9/16*log(1+x^2-x*sqrt(2))/sqrt(2)-9/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[1/(x^8*(1+2*x^4+x^8)),x,13,(-11/28)/x^7+11/12/x^3+1/4/(x^7*(1+x^4))-11/8*arctan(1-x*sqrt(2))/sqrt(2)+11/8*arctan(1+x*sqrt(2))/sqrt(2)-11/16*log(1+x^2-x*sqrt(2))/sqrt(2)+11/16*log(1+x^2+x*sqrt(2))/sqrt(2)],
[x^m/(1-2*x^4+x^8),x,2,x^(1+m)*hypergeom([2,1/4*(1+m)],[1/4*(5+m)],x^4)/(1+m)],
[x^9/(1-2*x^4+x^8),x,5,3/4*x^2+1/4*x^6/(1-x^4)-3/4*arctanh(x^2)],
[x^7/(1-2*x^4+x^8),x,4,1/4/(1-x^4)+1/4*log(1-x^4)],
[x^5/(1-2*x^4+x^8),x,4,1/4*x^2/(1-x^4)-1/4*arctanh(x^2)],
[x^3/(1-2*x^4+x^8),x,2,1/4/(1-x^4)],
[x/(1-2*x^4+x^8),x,4,1/4*x^2/(1-x^4)+1/4*arctanh(x^2)],
[1/(x*(1-2*x^4+x^8)),x,4,1/4/(1-x^4)+log(x)-1/4*log(1-x^4)],
[1/(x^3*(1-2*x^4+x^8)),x,5,(-3/4)/x^2+1/4/(x^2*(1-x^4))+3/4*arctanh(x^2)],
[1/(x^5*(1-2*x^4+x^8)),x,4,(-1/4)/x^4+1/4/(1-x^4)+2*log(x)-1/2*log(1-x^4)],
[1/(x^7*(1-2*x^4+x^8)),x,6,(-5/12)/x^6+(-5/4)/x^2+1/4/(x^6*(1-x^4))+5/4*arctanh(x^2)],
[x^8/(1-2*x^4+x^8),x,6,5/4*x+1/4*x^5/(1-x^4)-5/8*arctan(x)-5/8*arctanh(x)],
[x^6/(1-2*x^4+x^8),x,5,1/4*x^3/(1-x^4)+3/8*arctan(x)-3/8*arctanh(x)],
[x^4/(1-2*x^4+x^8),x,5,1/4*x/(1-x^4)-1/8*arctan(x)-1/8*arctanh(x)],
[x^2/(1-2*x^4+x^8),x,5,1/4*x^3/(1-x^4)-1/8*arctan(x)+1/8*arctanh(x)],
[1/(1-2*x^4+x^8),x,5,1/4*x/(1-x^4)+3/8*arctan(x)+3/8*arctanh(x)],
[1/(x^2*(1-2*x^4+x^8)),x,6,(-5/4)/x+1/4/(x*(1-x^4))-5/8*arctan(x)+5/8*arctanh(x)],
[1/(x^4*(1-2*x^4+x^8)),x,6,(-7/12)/x^3+1/4/(x^3*(1-x^4))+7/8*arctan(x)+7/8*arctanh(x)],
[1/(x^6*(1-2*x^4+x^8)),x,7,(-9/20)/x^5+(-9/4)/x+1/4/(x^5*(1-x^4))-9/8*arctan(x)+9/8*arctanh(x)],
[1/(x^8*(1-2*x^4+x^8)),x,7,(-11/28)/x^7+(-11/12)/x^3+1/4/(x^7*(1-x^4))+11/8*arctan(x)+11/8*arctanh(x)],

# Integrands of the form x^m (a+b x^4+c x^8)^(p/2)

# p>0

# p<0

# Integrands of the form x^(m/2) (a+b x^4+c x^8)^p

# p>0

# p<0

# Integrands of the form x^(m/2) (a+b x^4+c x^8)^(p/2)

# p>0

# p<0

# Integrands of the form (d x)^m (a+b x^4+c x^8)^p

# Integrands of the form x^m (a+b x^4+c x^8)^p

# p>0

# p<0
[x^m/(a+b*x^4+c*x^8),x,3,2*c*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],-2*c*x^4/(b-sqrt(b^2-4*a*c)))/((1+m)*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c))-2*c*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],-2*c*x^4/(b+sqrt(b^2-4*a*c)))/((1+m)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[x^11/(a+b*x^4+c*x^8),x,6,1/4*x^4/c-1/8*b*log(a+b*x^4+c*x^8)/c^2-1/4*(b^2-2*a*c)*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[x^9/(a+b*x^4+c*x^8),x,5,1/2*x^2/c-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^7/(a+b*x^4+c*x^8),x,5,1/8*log(a+b*x^4+c*x^8)/c+1/4*b*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[x^5/(a+b*x^4+c*x^8),x,4,-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))+1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[x^3/(a+b*x^4+c*x^8),x,3,-1/2*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[x/(a+b*x^4+c*x^8),x,4,arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)/(sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)/(sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(x*(a+b*x^4+c*x^8)),x,7,log(x)/a-1/8*log(a+b*x^4+c*x^8)/a+1/4*b*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[1/(x^3*(a+b*x^4+c*x^8)),x,5,(-1/2)/(a*x^2)-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(1+b/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x^2*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(1-b/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(x^5*(a+b*x^4+c*x^8)),x,8,(-1/4)/(a*x^4)-b*log(x)/a^2+1/8*b*log(a+b*x^4+c*x^8)/a^2-1/4*(b^2-2*a*c)*arctanh((b+2*c*x^4)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[x^10/(a+b*x^4+c*x^8),x,8,1/3*x^3/c-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b-sqrt(b^2-4*a*c))^(1/4))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b+sqrt(b^2-4*a*c))^(1/4))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[x^8/(a+b*x^4+c*x^8),x,8,x/c+1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))],
[x^6/(a+b*x^4+c*x^8),x,7,-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))+1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))],
[x^4/(a+b*x^4+c*x^8),x,7,1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))-1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))-1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))],
[x^2/(a+b*x^4+c*x^8),x,7,-c^(1/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))/(2^(3/4)*(-b-sqrt(b^2-4*a*c))^(1/4)*sqrt(b^2-4*a*c))+c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))/(2^(3/4)*(-b-sqrt(b^2-4*a*c))^(1/4)*sqrt(b^2-4*a*c))+c^(1/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))/(2^(3/4)*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))-c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))/(2^(3/4)*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[1/(a+b*x^4+c*x^8),x,7,c^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))/(2^(1/4)*(-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c))+c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))/(2^(1/4)*(-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c))-c^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))/(2^(1/4)*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))-c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))/(2^(1/4)*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))],
[1/(x^2*(a+b*x^4+c*x^8)),x,8,(-1)/(a*x)-1/2*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b-sqrt(b^2-4*a*c))^(1/4))+1/2*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b-sqrt(b^2-4*a*c))^(1/4))-1/2*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b+sqrt(b^2-4*a*c))^(1/4))+1/2*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b+sqrt(b^2-4*a*c))^(1/4))],
[1/(x^4*(a+b*x^4+c*x^8)),x,8,(-1/3)/(a*x^3)+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b+sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b+sqrt(b^2-4*a*c))^(3/4))],
[x^m/(1+x^4+x^8),x,3,-2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],-2*x^4/(1+I*sqrt(3)))/((1+m)*(I-sqrt(3))*sqrt(3))+2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],-2*x^4/(1-I*sqrt(3)))/((1+m)*sqrt(3)*(I+sqrt(3)))],
[x^11/(1+x^4+x^8),x,6,1/4*x^4-1/8*log(1+x^4+x^8)-1/4*arctan((1+2*x^4)/sqrt(3))/sqrt(3)],
[x^9/(1+x^4+x^8),x,7,1/2*x^2+1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)-1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[x^7/(1+x^4+x^8),x,5,1/8*log(1+x^4+x^8)-1/4*arctan((1+2*x^4)/sqrt(3))/sqrt(3)],
[x^5/(1+x^4+x^8),x,10,1/8*log(1-x^2+x^4)-1/8*log(1+x^2+x^4)-1/4*arctan((1-2*x^2)/sqrt(3))/sqrt(3)+1/4*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[x^3/(1+x^4+x^8),x,3,1/2*arctan((1+2*x^4)/sqrt(3))/sqrt(3)],
[x/(1+x^4+x^8),x,10,-1/8*log(1-x^2+x^4)+1/8*log(1+x^2+x^4)-1/4*arctan((1-2*x^2)/sqrt(3))/sqrt(3)+1/4*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[1/(x*(1+x^4+x^8)),x,7,log(x)-1/8*log(1+x^4+x^8)-1/4*arctan((1+2*x^4)/sqrt(3))/sqrt(3)],
[1/(x^3*(1+x^4+x^8)),x,7,(-1/2)/x^2+1/2*arctan((1-2*x^2)/sqrt(3))/sqrt(3)-1/2*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[1/(x^5*(1+x^4+x^8)),x,8,(-1/4)/x^4-log(x)+1/8*log(1+x^4+x^8)-1/4*arctan((1+2*x^4)/sqrt(3))/sqrt(3)],
[1/(x^7*(1+x^4+x^8)),x,13,(-1/6)/x^6+1/2/x^2+1/8*log(1-x^2+x^4)-1/8*log(1+x^2+x^4)-1/4*arctan((1-2*x^2)/sqrt(3))/sqrt(3)+1/4*arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[x^8/(1+x^4+x^8),x,20,x+1/4*arctan(-2*x+sqrt(3))-1/4*arctan(2*x+sqrt(3))+1/8*log(1-x+x^2)-1/8*log(1+x+x^2)+1/4*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)+1/8*log(1+x^2-x*sqrt(3))/sqrt(3)-1/8*log(1+x^2+x*sqrt(3))/sqrt(3)],
[x^6/(1+x^4+x^8),x,9,-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*arctan((1+2*x)/sqrt(3))/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^4/(1+x^4+x^8),x,19,-1/4*arctan(-2*x+sqrt(3))+1/4*arctan(2*x+sqrt(3))-1/8*log(1-x+x^2)+1/8*log(1+x+x^2)+1/4*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)+1/8*log(1+x^2-x*sqrt(3))/sqrt(3)-1/8*log(1+x^2+x*sqrt(3))/sqrt(3)],
[x^2/(1+x^4+x^8),x,19,-1/4*arctan(-2*x+sqrt(3))+1/4*arctan(2*x+sqrt(3))+1/8*log(1-x+x^2)-1/8*log(1+x+x^2)+1/4*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)-1/8*log(1+x^2-x*sqrt(3))/sqrt(3)+1/8*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(1+x^4+x^8),x,9,-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*arctan((1+2*x)/sqrt(3))/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^2*(1+x^4+x^8)),x,20,(-1)/x+1/4*arctan(-2*x+sqrt(3))-1/4*arctan(2*x+sqrt(3))-1/8*log(1-x+x^2)+1/8*log(1+x+x^2)+1/4*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)-1/8*log(1+x^2-x*sqrt(3))/sqrt(3)+1/8*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(x^4*(1+x^4+x^8)),x,20,(-1/3)/x^3+1/4*arctan(-2*x+sqrt(3))-1/4*arctan(2*x+sqrt(3))+1/8*log(1-x+x^2)-1/8*log(1+x+x^2)+1/4*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)+1/8*log(1+x^2-x*sqrt(3))/sqrt(3)-1/8*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(x^6*(1+x^4+x^8)),x,12,(-1/5)/x^5+1/x-1/2*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*arctan((1+2*x)/sqrt(3))/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^8*(1+x^4+x^8)),x,22,(-1/7)/x^7+1/3/x^3-1/4*arctan(-2*x+sqrt(3))+1/4*arctan(2*x+sqrt(3))-1/8*log(1-x+x^2)+1/8*log(1+x+x^2)+1/4*arctan((1-2*x)/sqrt(3))/sqrt(3)-1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)+1/8*log(1+x^2-x*sqrt(3))/sqrt(3)-1/8*log(1+x^2+x*sqrt(3))/sqrt(3)],
[x^m/(1-x^4+x^8),x,3,-2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],2*x^4/(1+I*sqrt(3)))/((1+m)*(I-sqrt(3))*sqrt(3))+2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],2*x^4/(1-I*sqrt(3)))/((1+m)*sqrt(3)*(I+sqrt(3)))],
[x^11/(1-x^4+x^8),x,6,1/4*x^4+1/8*log(1-x^4+x^8)+1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[x^9/(1-x^4+x^8),x,5,1/2*x^2+1/4*log(1+x^4-x^2*sqrt(3))/sqrt(3)-1/4*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[x^7/(1-x^4+x^8),x,5,1/8*log(1-x^4+x^8)-1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[x^5/(1-x^4+x^8),x,10,-1/4*arctan(-2*x^2+sqrt(3))+1/4*arctan(2*x^2+sqrt(3))+1/8*log(1+x^4-x^2*sqrt(3))/sqrt(3)-1/8*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[x^3/(1-x^4+x^8),x,3,-1/2*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[x/(1-x^4+x^8),x,10,-1/4*arctan(-2*x^2+sqrt(3))+1/4*arctan(2*x^2+sqrt(3))-1/8*log(1+x^4-x^2*sqrt(3))/sqrt(3)+1/8*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[1/(x*(1-x^4+x^8)),x,7,log(x)-1/8*log(1-x^4+x^8)-1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[1/(x^3*(1-x^4+x^8)),x,5,(-1/2)/x^2-1/4*log(1+x^4-x^2*sqrt(3))/sqrt(3)+1/4*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[1/(x^5*(1-x^4+x^8)),x,8,(-1/4)/x^4+log(x)-1/8*log(1-x^4+x^8)+1/4*arctan((1-2*x^4)/sqrt(3))/sqrt(3)],
[1/(x^7*(1-x^4+x^8)),x,13,(-1/6)/x^6+(-1/2)/x^2+1/4*arctan(-2*x^2+sqrt(3))-1/4*arctan(2*x^2+sqrt(3))-1/8*log(1+x^4-x^2*sqrt(3))/sqrt(3)+1/8*log(1+x^4+x^2*sqrt(3))/sqrt(3)],
[x^8/(1-x^4+x^8),x,20,x-1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))+1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))+1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))],
[x^6/(1-x^4+x^8),x,19,-1/2*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)+1/2*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)-1/2*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)+1/2*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)+1/4*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(6)+1/4*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(6)-1/4*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(6)],
[x^4/(1-x^4+x^8),x,19,-1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2-sqrt(3)))+1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/8*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(3*(2-sqrt(3)))+1/8*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(3*(2-sqrt(3)))+1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2+sqrt(3)))-1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2+sqrt(3)))+1/8*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(3*(2+sqrt(3)))-1/8*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(3*(2+sqrt(3)))],
[x^2/(1-x^4+x^8),x,19,1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/8*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(3*(2+sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(3*(2+sqrt(3)))],
[1/(1-x^4+x^8),x,19,-1/2*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)+1/2*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)-1/2*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)+1/2*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(6)],
[1/(x^2*(1-x^4+x^8)),x,22,(-1)/x+1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(3*(2-sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))+1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(3*(2+sqrt(3)))],
[1/(x^4*(1-x^4+x^8)),x,20,(-1/3)/x^3+1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2+sqrt(3)))],
[1/(x^6*(1-x^4+x^8)),x,22,(-1/5)/x^5+(-1)/x+1/2*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)-1/2*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))/sqrt(6)+1/2*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)-1/2*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2-sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2-sqrt(3)))/sqrt(6)-1/4*log(1+x^2-x*sqrt(2+sqrt(3)))/sqrt(6)+1/4*log(1+x^2+x*sqrt(2+sqrt(3)))/sqrt(6)],
[1/(x^8*(1-x^4+x^8)),x,22,(-1/7)/x^7+(-1/3)/x^3-1/4*arctan((-2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((2*x+sqrt(2-sqrt(3)))/sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))-1/8*log(1+x^2-x*sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/8*log(1+x^2+x*sqrt(2+sqrt(3)))*sqrt(1/3*(2-sqrt(3)))+1/4*arctan((-2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/4*arctan((2*x+sqrt(2+sqrt(3)))/sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))+1/8*log(1+x^2-x*sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))-1/8*log(1+x^2+x*sqrt(2-sqrt(3)))*sqrt(1/3*(2+sqrt(3)))],
[x^m/(1+3*x^4+x^8),x,3,2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],-2*x^4/(3-sqrt(5)))/((1+m)*(3-sqrt(5))*sqrt(5))-2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],-2*x^4/(3+sqrt(5)))/((1+m)*sqrt(5)*(3+sqrt(5)))],
[x^11/(1+3*x^4+x^8),x,5,1/4*x^4-1/40*log(3+2*x^4-sqrt(5))*(15-7*sqrt(5))-1/40*log(3+2*x^4+sqrt(5))*(15+7*sqrt(5))],
[x^9/(1+3*x^4+x^8),x,5,1/2*x^2+1/2*arctan(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/5*(9-4*sqrt(5)))-1/2*arctan(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/5*(9+4*sqrt(5)))],
[x^7/(1+3*x^4+x^8),x,4,1/40*log(3+2*x^4-sqrt(5))*(5-3*sqrt(5))+1/40*log(3+2*x^4+sqrt(5))*(5+3*sqrt(5))],
[x^5/(1+3*x^4+x^8),x,4,-1/2*arctan(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/10*(3-sqrt(5)))+1/2*arctan(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/10*(3+sqrt(5)))],
[x^3/(1+3*x^4+x^8),x,3,-1/2*arctanh((3+2*x^4)/sqrt(5))/sqrt(5)],
[x/(1+3*x^4+x^8),x,4,1/2*arctan(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/10*(3+sqrt(5)))-arctan(x^2*sqrt(2/(3+sqrt(5))))/sqrt(10*(3+sqrt(5)))],
[1/(x*(1+3*x^4+x^8)),x,6,log(x)-1/40*log(3+2*x^4+sqrt(5))*(5-3*sqrt(5))-1/40*log(3+2*x^4-sqrt(5))*(5+3*sqrt(5))],
[1/(x^3*(1+3*x^4+x^8)),x,5,(-1/2)/x^2-1/4*arctan(x^2*sqrt(1/2*(3+sqrt(5))))*(3+sqrt(5))^(3/2)/sqrt(10)+1/2*arctan(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/5*(9-4*sqrt(5)))],
[1/(x^5*(1+3*x^4+x^8)),x,7,(-1/4)/x^4-3*log(x)+1/40*log(3+2*x^4+sqrt(5))*(15-7*sqrt(5))+1/40*log(3+2*x^4-sqrt(5))*(15+7*sqrt(5))],
[1/(x^7*(1+3*x^4+x^8)),x,6,(-1/6)/x^6+3/2/x^2-1/2*arctan(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/10*(123-55*sqrt(5)))+1/2*arctan(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/10*(123+55*sqrt(5)))],
[x^8/(1+3*x^4+x^8),x,20,x-1/2*arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))*(123-55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/2*arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))*(123-55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/4*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(123-55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/4*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(123-55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/2*arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))*(123+55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/2*arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))*(123+55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/4*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(123+55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/4*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(123+55*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))],
[x^6/(1+3*x^4+x^8),x,19,1/4*arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))*(3-sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))-1/4*arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))*(3-sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))-1/8*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(3-sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))+1/8*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(3-sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))-1/4*arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))*(3+sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))+1/4*arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))*(3+sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))+1/8*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(3+sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))-1/8*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(3+sqrt(5))^(3/4)/(2^(1/4)*sqrt(5))],
[x^4/(1+3*x^4+x^8),x,19,1/2*arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))*(3-sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/2*arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))*(3-sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/4*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(3-sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/4*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(3-sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/2*arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))*(3+sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/2*arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))*(3+sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/4*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(3+sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/4*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(3+sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))],
[x^2/(1+3*x^4+x^8),x,19,-1/2*arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))/(2^(1/4)*(3-sqrt(5))^(1/4)*sqrt(5))+1/2*arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))/(2^(1/4)*(3-sqrt(5))^(1/4)*sqrt(5))+1/4*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))/(2^(1/4)*(3-sqrt(5))^(1/4)*sqrt(5))-1/4*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))/(2^(1/4)*(3-sqrt(5))^(1/4)*sqrt(5))+1/2*arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))/(2^(1/4)*sqrt(5)*(3+sqrt(5))^(1/4))-1/2*arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))/(2^(1/4)*sqrt(5)*(3+sqrt(5))^(1/4))-1/4*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))/(2^(1/4)*sqrt(5)*(3+sqrt(5))^(1/4))+1/4*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))/(2^(1/4)*sqrt(5)*(3+sqrt(5))^(1/4))],
[1/(1+3*x^4+x^8),x,19,1/20*arctan(1-x*sqrt(-1+sqrt(5)))*sqrt(-20+10*sqrt(5))-1/20*arctan(1+x*sqrt(-1+sqrt(5)))*sqrt(-20+10*sqrt(5))+1/40*log(1+2*x^2+sqrt(5)-2*x*sqrt(1+sqrt(5)))*sqrt(-20+10*sqrt(5))-1/40*log(1+2*x^2+sqrt(5)+2*x*sqrt(1+sqrt(5)))*sqrt(-20+10*sqrt(5))-1/20*arctan(1-x*sqrt(1+sqrt(5)))*sqrt(20+10*sqrt(5))+1/20*arctan(1+x*sqrt(1+sqrt(5)))*sqrt(20+10*sqrt(5))-1/40*log(-1+2*x^2+sqrt(5)-2*x*sqrt(-1+sqrt(5)))*sqrt(20+10*sqrt(5))+1/40*log(-1+2*x^2+sqrt(5)+2*x*sqrt(-1+sqrt(5)))*sqrt(20+10*sqrt(5)),-arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))/(2^(3/4)*(3-sqrt(5))^(3/4)*sqrt(5))+arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))/(2^(3/4)*(3-sqrt(5))^(3/4)*sqrt(5))-1/2*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))/(2^(3/4)*(3-sqrt(5))^(3/4)*sqrt(5))+1/2*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))/(2^(3/4)*(3-sqrt(5))^(3/4)*sqrt(5))+arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))/(2^(3/4)*sqrt(5)*(3+sqrt(5))^(3/4))-arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))/(2^(3/4)*sqrt(5)*(3+sqrt(5))^(3/4))+1/2*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))/(2^(3/4)*sqrt(5)*(3+sqrt(5))^(3/4))-1/2*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))/(2^(3/4)*sqrt(5)*(3+sqrt(5))^(3/4))],
[1/(x^2*(1+3*x^4+x^8)),x,20,(-1)/x-1/20*arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))*(6150-2750*sqrt(5))^(1/4)+1/20*arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))*(6150-2750*sqrt(5))^(1/4)+1/40*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(6150-2750*sqrt(5))^(1/4)-1/40*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(6150-2750*sqrt(5))^(1/4)+1/4*arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))*(246+110*sqrt(5))^(1/4)/sqrt(5)-1/4*arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))*(246+110*sqrt(5))^(1/4)/sqrt(5)-1/8*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(246+110*sqrt(5))^(1/4)/sqrt(5)+1/8*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(246+110*sqrt(5))^(1/4)/sqrt(5)],
[1/(x^4*(1+3*x^4+x^8)),x,20,(-1/3)/x^3-1/2*arctan(1-2^(3/4)*x/(3+sqrt(5))^(1/4))*(843-377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/2*arctan(1+2^(3/4)*x/(3+sqrt(5))^(1/4))*(843-377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/4*log(2*x^2-2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(843-377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/4*log(2*x^2+2*2^(1/4)*x*(3+sqrt(5))^(1/4)+sqrt(2*(3+sqrt(5))))*(843-377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/2*arctan(1-2^(3/4)*x/(3-sqrt(5))^(1/4))*(843+377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/2*arctan(1+2^(3/4)*x/(3-sqrt(5))^(1/4))*(843+377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))+1/4*log(2*x^2-2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(843+377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))-1/4*log(2*x^2+2*2^(1/4)*x*(3-sqrt(5))^(1/4)+sqrt(2*(3-sqrt(5))))*(843+377*sqrt(5))^(1/4)/(2^(3/4)*sqrt(5))],
[x^m/(1-3*x^4+x^8),x,3,2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],2*x^4/(3-sqrt(5)))/((1+m)*(3-sqrt(5))*sqrt(5))-2*x^(1+m)*hypergeom([1,1/4*(1+m)],[1/4*(5+m)],2*x^4/(3+sqrt(5)))/((1+m)*sqrt(5)*(3+sqrt(5)))],
[x^11/(1-3*x^4+x^8),x,5,1/4*x^4+1/40*log(3-2*x^4-sqrt(5))*(15-7*sqrt(5))+1/40*log(3-2*x^4+sqrt(5))*(15+7*sqrt(5))],
[x^9/(1-3*x^4+x^8),x,5,1/2*x^2+1/2*arctanh(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/5*(9-4*sqrt(5)))-1/2*arctanh(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/5*(9+4*sqrt(5)))],
[x^7/(1-3*x^4+x^8),x,4,1/40*log(3-2*x^4-sqrt(5))*(5-3*sqrt(5))+1/40*log(3-2*x^4+sqrt(5))*(5+3*sqrt(5))],
[x^5/(1-3*x^4+x^8),x,4,1/2*arctanh(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/10*(3-sqrt(5)))-1/2*arctanh(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/10*(3+sqrt(5)))],
[x^3/(1-3*x^4+x^8),x,3,1/2*arctanh((3-2*x^4)/sqrt(5))/sqrt(5)],
[x/(1-3*x^4+x^8),x,4,1/2*arctanh(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/10*(3+sqrt(5)))-arctanh(x^2*sqrt(2/(3+sqrt(5))))/sqrt(10*(3+sqrt(5)))],
[1/(x*(1-3*x^4+x^8)),x,6,log(x)-1/40*log(3-2*x^4+sqrt(5))*(5-3*sqrt(5))-1/40*log(3-2*x^4-sqrt(5))*(5+3*sqrt(5))],
[1/(x^3*(1-3*x^4+x^8)),x,5,(-1/2)/x^2+1/4*arctanh(x^2*sqrt(1/2*(3+sqrt(5))))*(3+sqrt(5))^(3/2)/sqrt(10)-1/2*arctanh(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/5*(9-4*sqrt(5)))],
[1/(x^5*(1-3*x^4+x^8)),x,7,(-1/4)/x^4+3*log(x)-1/40*log(3-2*x^4+sqrt(5))*(15-7*sqrt(5))-1/40*log(3-2*x^4-sqrt(5))*(15+7*sqrt(5))],
[1/(x^7*(1-3*x^4+x^8)),x,6,(-1/6)/x^6+(-3/2)/x^2-1/2*arctanh(x^2*sqrt(2/(3+sqrt(5))))*sqrt(1/10*(123-55*sqrt(5)))+1/2*arctanh(x^2*sqrt(1/2*(3+sqrt(5))))*sqrt(1/10*(123+55*sqrt(5)))],
[x^8/(1-3*x^4+x^8),x,8,x+1/4*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(984-440*sqrt(5))^(1/4)/sqrt(5)+1/4*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(984-440*sqrt(5))^(1/4)/sqrt(5)-1/2*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(123+55*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))-1/2*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(123+55*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))],
[x^6/(1-3*x^4+x^8),x,7,-1/4*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(144-64*sqrt(5))^(1/4)/sqrt(5)+1/4*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(144-64*sqrt(5))^(1/4)/sqrt(5)+1/2*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(3+sqrt(5))^(3/4)/(2^(3/4)*sqrt(5))-1/2*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(3+sqrt(5))^(3/4)/(2^(3/4)*sqrt(5))],
[x^4/(1-3*x^4+x^8),x,7,1/2*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(3-sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))+1/2*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(3-sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))-1/2*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(3+sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))-1/2*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(3+sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))],
[x^2/(1-3*x^4+x^8),x,7,1/20*arctan(1/2*x*sqrt(-2+2*sqrt(5)))*sqrt(-10+10*sqrt(5))-1/20*arctanh(1/2*x*sqrt(-2+2*sqrt(5)))*sqrt(-10+10*sqrt(5))-1/20*arctan(1/2*x*sqrt(2+2*sqrt(5)))*sqrt(10+10*sqrt(5))+1/20*arctanh(1/2*x*sqrt(2+2*sqrt(5)))*sqrt(10+10*sqrt(5)),arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))/(2^(3/4)*sqrt(5)*(3+sqrt(5))^(1/4))-arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))/(2^(3/4)*sqrt(5)*(3+sqrt(5))^(1/4))-1/2*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(3+sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))+1/2*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(3+sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))],
[1/(1-3*x^4+x^8),x,7,-arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))/(2^(1/4)*sqrt(5)*(3+sqrt(5))^(3/4))-arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))/(2^(1/4)*sqrt(5)*(3+sqrt(5))^(3/4))+1/2*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(9+4*sqrt(5))^(1/4)/sqrt(5)+1/2*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(9+4*sqrt(5))^(1/4)/sqrt(5)],
[1/(x^2*(1-3*x^4+x^8)),x,8,(-1)/x+1/4*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(984-440*sqrt(5))^(1/4)/sqrt(5)-1/4*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(984-440*sqrt(5))^(1/4)/sqrt(5)-1/4*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(3+sqrt(5))^(5/4)/(2^(1/4)*sqrt(5))+1/4*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(3+sqrt(5))^(5/4)/(2^(1/4)*sqrt(5))],
[1/(x^4*(1-3*x^4+x^8)),x,8,(-1/3)/x^3-1/2*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(843-377*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))-1/2*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(843-377*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))+1/2*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(843+377*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))+1/2*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(843+377*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))],
[1/(x^6*(1-3*x^4+x^8)),x,9,(-1/5)/x^5+(-3)/x+1/2*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(2889-1292*sqrt(5))^(1/4)/sqrt(5)-1/2*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(2889-1292*sqrt(5))^(1/4)/sqrt(5)-1/2*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(2889+1292*sqrt(5))^(1/4)/sqrt(5)+1/2*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(2889+1292*sqrt(5))^(1/4)/sqrt(5)],
[1/(x^8*(1-3*x^4+x^8)),x,9,(-1/7)/x^7+(-1)/x^3-1/2*arctan(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(39603-17711*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))-1/2*arctanh(2^(1/4)*x*(1/(3+sqrt(5)))^(1/4))*(39603-17711*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))+1/2*arctan(x*(3+sqrt(5))^(1/4)/2^(1/4))*(39603+17711*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))+1/2*arctanh(x*(3+sqrt(5))^(1/4)/2^(1/4))*(39603+17711*sqrt(5))^(1/4)/(2^(1/4)*sqrt(5))],
[x^3/(2+3*x^4+x^8),x,4,1/4*log(1+x^4)-1/4*log(2+x^4)],
[x^11/(2+3*x^4+x^8),x,5,1/4*x^4+1/4*log(1+x^4)-log(2+x^4)],

# Integrands of the form x^m (a+b x^4+c x^8)^(p/2)

# Integrands of the form x^(m/2) (a+b x^4+c x^8)^p

# Integrands of the form x^(m/2) (a+b x^4+c x^8)^(p/2)

# Integrands of the form (d x)^m (a+b x^5+c x^10)^p

# Integrands of the form (d x)^m (a+b x^5+c x^10)^p with b^2-4 a c=0

# Integrands of the form (d x)^m (a+b x^5+c x^10)^p

# p>0

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

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with integer n<0

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

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

# p>0

# p<0
[x^3/(c+a/x^2+b/x),x,7,-b*(b^2-2*a*c)*x/c^4+1/2*(b^2-a*c)*x^2/c^3-1/3*b*x^3/c^2+1/4*x^4/c+1/2*(b^4-3*a*b^2*c+a^2*c^2)*log(a+b*x+c*x^2)/c^5+b*(b^4-5*a*b^2*c+5*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^5*sqrt(b^2-4*a*c))],
[x^2/(c+a/x^2+b/x),x,7,(b^2-a*c)*x/c^3-1/2*b*x^2/c^2+1/3*x^3/c-1/2*b*(b^2-2*a*c)*log(a+b*x+c*x^2)/c^4-(b^4-4*a*b^2*c+2*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*sqrt(b^2-4*a*c))],
[x/(c+a/x^2+b/x),x,7,-b*x/c^2+1/2*x^2/c+1/2*(b^2-a*c)*log(a+b*x+c*x^2)/c^3+b*(b^2-3*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[1/(c+a/x^2+b/x),x,6,x/c-1/2*b*log(a+b*x+c*x^2)/c^2-(b^2-2*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[1/((c+a/x^2+b/x)*x),x,5,1/2*log(a+b*x+c*x^2)/c+b*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[1/((c+a/x^2+b/x)*x^2),x,3,2*arctanh((b+2*a/x)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[1/((c+a/x^2+b/x)*x^3),x,7,log(x)/a-1/2*log(a+b*x+c*x^2)/a+b*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[1/((c+a/x^2+b/x)*x^4),x,8,(-1)/(a*x)-b*log(x)/a^2+1/2*b*log(a+b*x+c*x^2)/a^2-(b^2-2*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[1/((c+a/x^2+b/x)*x^5),x,8,(-1/2)/(a*x^2)+b/(a^2*x)+(b^2-a*c)*log(x)/a^3-1/2*(b^2-a*c)*log(a+b*x+c*x^2)/a^3+b*(b^2-3*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^3*sqrt(b^2-4*a*c))],
[1/((c+a/x^2+b/x)*x^6),x,8,(-1/3)/(a*x^3)+1/2*b/(a^2*x^2)+(-b^2+a*c)/(a^3*x)-b*(b^2-2*a*c)*log(x)/a^4+1/2*b*(b^2-2*a*c)*log(a+b*x+c*x^2)/a^4-(b^4-4*a*b^2*c+2*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^4*sqrt(b^2-4*a*c))],
[x/(c+a/x^2+b/x)^2,x,8,-b*(3*b^2-11*a*c)*x/(c^3*(b^2-4*a*c))+1/2*(3*b^2-8*a*c)*x^2/(c^2*(b^2-4*a*c))-b*x^3/(c*(b^2-4*a*c))+x^4*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+b*(3*b^4-20*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(3/2))+1/2*(3*b^2-2*a*c)*log(a+b*x+c*x^2)/c^4],
[1/(c+a/x^2+b/x)^2,x,8,2*(b^2-3*a*c)*x/(c^2*(b^2-4*a*c))-b*x^2/(c*(b^2-4*a*c))+x^3*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2))-2*(b^4-6*a*b^2*c+6*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(3/2))-b*log(a+b*x+c*x^2)/c^3],
[1/((c+a/x^2+b/x)^2*x),x,7,-b*x/(c*(b^2-4*a*c))+x^2*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+b*(b^2-6*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/2))+1/2*log(a+b*x+c*x^2)/c^2],
[1/((c+a/x^2+b/x)^2*x^2),x,4,(b+2*a/x)/((b^2-4*a*c)*(c+a/x^2+b/x))-4*a*arctanh((b+2*a/x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[1/((c+a/x^2+b/x)^2*x^3),x,4,(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2))-2*b*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[1/((c+a/x^2+b/x)^2*x^4),x,4,(-b-2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2))+4*c*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[1/((c+a/x^2+b/x)^2*x^5),x,8,(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*(a+b*x+c*x^2))+b*(b^2-6*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2))+log(x)/a^2-1/2*log(a+b*x+c*x^2)/a^2],
[1/((c+a/x^2+b/x)^2*x^6),x,8,-2*(b^2-3*a*c)/(a^2*(b^2-4*a*c)*x)+(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x*(a+b*x+c*x^2))-2*(b^4-6*a*b^2*c+6*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2))-2*b*log(x)/a^3+b*log(a+b*x+c*x^2)/a^3],
[1/((c+a/x^2+b/x)^2*x^7),x,8,1/2*(-3*b^2+8*a*c)/(a^2*(b^2-4*a*c)*x^2)+b*(3*b^2-11*a*c)/(a^3*(b^2-4*a*c)*x)+(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x^2*(a+b*x+c*x^2))+b*(3*b^4-20*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(3/2))+(3*b^2-2*a*c)*log(x)/a^4-1/2*(3*b^2-2*a*c)*log(a+b*x+c*x^2)/a^4],
[1/(c+a/x^2+b/x)^3,x,9,3*(b^4-7*a*b^2*c+10*a^2*c^2)*x/(c^3*(b^2-4*a*c)^2)-3/2*b*(b^2-6*a*c)*x^2/(c^2*(b^2-4*a*c)^2)+1/2*x^5*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+x^3*(a*(b^2-10*a*c)+b*(b^2-7*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))-3*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(5/2))-3/2*b*log(a+b*x+c*x^2)/c^4],
[1/((c+a/x^2+b/x)^3*x),x,8,-b*(b^2-7*a*c)*x/(c^2*(b^2-4*a*c)^2)+1/2*x^4*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+1/2*x^2*(a*(b^2-16*a*c)+b*(b^2-10*a*c)*x)/(c*(b^2-4*a*c)^2*(a+b*x+c*x^2))+b*(b^4-10*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(5/2))+1/2*log(a+b*x+c*x^2)/c^3],
[1/((c+a/x^2+b/x)^3*x^2),x,5,1/2*(b+2*a/x)/((b^2-4*a*c)*(c+a/x^2+b/x)^2)-3*a*(b+2*a/x)/((b^2-4*a*c)^2*(c+a/x^2+b/x))+12*a^2*arctanh((b+2*a/x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/((c+a/x^2+b/x)^3*x^3),x,5,-1/2*x^3*(b+2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3/2*b*x*(2*a+b*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))+6*a*b*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/((c+a/x^2+b/x)^3*x^4),x,5,1/2*x*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+(3*a*b+(b^2+2*a*c)*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-2*(b^2+2*a*c)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/((c+a/x^2+b/x)^3*x^5),x,5,1/2*(2*a+b*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)-3/2*b*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))+6*b*c*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/((c+a/x^2+b/x)^3*x^6),x,5,1/2*(-b-2*c*x)/((b^2-4*a*c)*(a+b*x+c*x^2)^2)+3*c*(b+2*c*x)/((b^2-4*a*c)^2*(a+b*x+c*x^2))-12*c^2*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/((c+a/x^2+b/x)^3*x^7),x,9,1/2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*(a+b*x+c*x^2)^2)+1/2*(2*b^4-15*a*b^2*c+16*a^2*c^2+2*b*c*(b^2-7*a*c)*x)/(a^2*(b^2-4*a*c)^2*(a+b*x+c*x^2))+b*(b^4-10*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2))+log(x)/a^3-1/2*log(a+b*x+c*x^2)/a^3],
[1/((c+a/x^2+b/x)^3*x^8),x,9,-3*(b^2-5*a*c)*(b^2-2*a*c)/(a^3*(b^2-4*a*c)^2*x)+1/2*(b^2-2*a*c+b*c*x)/(a*(b^2-4*a*c)*x*(a+b*x+c*x^2)^2)+1/2*(3*b^4-20*a*b^2*c+20*a^2*c^2+3*b*c*(b^2-6*a*c)*x)/(a^2*(b^2-4*a*c)^2*x*(a+b*x+c*x^2))-3*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(5/2))-3*b*log(x)/a^4+3/2*b*log(a+b*x+c*x^2)/a^4],
[x^2/(15+2/x^2+13/x),x,6,139/3375*x-13/450*x^2+1/45*x^3-16/567*log(2+3*x)+1/4375*log(1+5*x)],
[x/(15+2/x^2+13/x),x,6,-13/225*x+1/30*x^2+8/189*log(2+3*x)-1/875*log(1+5*x)],
[1/(15+2/x^2+13/x),x,5,1/15*x-4/63*log(2+3*x)+1/175*log(1+5*x)],
[1/((15+2/x^2+13/x)*x),x,4,2/21*log(2+3*x)-1/35*log(1+5*x)],
[1/((15+2/x^2+13/x)*x^2),x,4,1/7*log(5+1/x)-1/7*log(3+2/x)],
[1/((15+2/x^2+13/x)*x^3),x,6,1/2*log(x)+3/14*log(2+3*x)-5/7*log(1+5*x)],
[1/((15+2/x^2+13/x)*x^4),x,4,(-1/2)/x-13/4*log(x)-9/28*log(2+3*x)+25/7*log(1+5*x)],
[1/((15+2/x^2+13/x)*x^5),x,4,(-1/4)/x^2+13/4/x+139/8*log(x)+27/56*log(2+3*x)-125/7*log(1+5*x)],
[1/((15+2/x^2+13/x)*x^6),x,4,(-1/6)/x^3+13/8/x^2+(-139/8)/x-1417/16*log(x)-81/112*log(2+3*x)+625/7*log(1+5*x)],

# Integrands of the form (d x)^m (a+b/x+c/x^2)^(p/2)
[(a+c/x^2+b/x)^(5/2),x,9,-5/24*(a+c/x^2+b/x)^(3/2)*(7*b+6*c/x)+(a+c/x^2+b/x)^(5/2)*x+5/2*a^(3/2)*b*arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))+5/128*(b^4-24*a*b^2*c-48*a^2*c^2)*arctanh(1/2*(b+2*c/x)/(sqrt(c)*sqrt(a+c/x^2+b/x)))/c^(3/2)-5/64*(b*(b^2+44*a*c)+2*c*(b^2+12*a*c)/x)*sqrt(a+c/x^2+b/x)/c],
[(a+c/x^2+b/x)^(3/2),x,8,(a+c/x^2+b/x)^(3/2)*x+3/2*b*arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))*sqrt(a)-3/8*(b^2+4*a*c)*arctanh(1/2*(b+2*c/x)/(sqrt(c)*sqrt(a+c/x^2+b/x)))/sqrt(c)-3/4*(3*b+2*c/x)*sqrt(a+c/x^2+b/x)],
[(a+c/x^2+b/x)^(1/2),x,7,1/2*b*arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))/sqrt(a)-arctanh(1/2*(b+2*c/x)/(sqrt(c)*sqrt(a+c/x^2+b/x)))*sqrt(c)+x*sqrt(a+c/x^2+b/x)],
[1/(a+c/x^2+b/x)^(1/2),x,4,-1/2*b*arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))/a^(3/2)+x*sqrt(a+c/x^2+b/x)/a],
[1/(a+c/x^2+b/x)^(3/2),x,5,-3/2*b*arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))/a^(5/2)-2*(b^2-2*a*c+b*c/x)*x/(a*(b^2-4*a*c)*sqrt(a+c/x^2+b/x))+(3*b^2-8*a*c)*x*sqrt(a+c/x^2+b/x)/(a^2*(b^2-4*a*c))],
[1/(a+c/x^2+b/x)^(5/2),x,6,-2/3*(b^2-2*a*c+b*c/x)*x/(a*(b^2-4*a*c)*(a+c/x^2+b/x)^(3/2))-5/2*b*arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))/a^(7/2)-2/3*(5*b^4-32*a*b^2*c+32*a^2*c^2+b*c*(5*b^2-28*a*c)/x)*x/(a^2*(b^2-4*a*c)^2*sqrt(a+c/x^2+b/x))+1/3*(15*b^4-100*a*b^2*c+128*a^2*c^2)*x*sqrt(a+c/x^2+b/x)/(a^3*(b^2-4*a*c)^2)],

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

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

# p<0

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

# Integrands of the form (d x)^m (a+b/x^3+c/x^6)^p
[1/(c+a/x^6+b/x^3),x,15,x/c-1/3*log(2^(1/3)*c^(1/3)*x+(b-sqrt(b^2-4*a*c))^(1/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b-sqrt(b^2-4*a*c))^(2/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b-sqrt(b^2-4*a*c))^(2/3))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*sqrt(3)*(b-sqrt(b^2-4*a*c))^(2/3))-1/3*log(2^(1/3)*c^(1/3)*x+(b+sqrt(b^2-4*a*c))^(1/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b+sqrt(b^2-4*a*c))^(2/3))+1/6*log(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*(b+sqrt(b^2-4*a*c))^(2/3))+arctan((1-2*2^(1/3)*c^(1/3)*x/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*c^(4/3)*sqrt(3)*(b+sqrt(b^2-4*a*c))^(2/3))],

# Integrands of the form (d x)^m (a+b/x^4+c/x^8)^p
[1/(c+a/x^8+b/x^4),x,9,x/c+1/2*arctan(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*arctan(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/2*arctanh(2^(1/4)*c^(1/4)*x/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))],

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with fraction n>0

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

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

# Integrands of the form (d x)^m (a+b Sqrt[x]+c x)^p with b^2-4 a c=0
[(1/4*b^2/c+c*x+b*sqrt(x))^2,x,4,-1/160*b*(b+2*c*sqrt(x))^5/c^4+1/192*(b+2*c*sqrt(x))^6/c^4],

# Integrands of the form (d x)^m (a+b Sqrt[x]+c x)^(p/2) with b^2-4 a c=0
[1/(a^2+b^2*x+2*a*b*sqrt(x))^(1/2),x,4,-2*a*log(a+b*sqrt(x))*(a+b*sqrt(x))/(b^2*sqrt(a^2+b^2*x+2*a*b*sqrt(x)))+2*sqrt(a^2+b^2*x+2*a*b*sqrt(x))/b^2],

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

# Integrands of the form (d x)^m (a+b x^(1/3)+c x^(2/3))^(p/2) with b^2-4 a c=0
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(7/2),x,4,3/8*a^2*(a+b*x^(1/3))^7*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3-2/3*a*(a+b*x^(1/3))^8*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3+3/10*(a+b*x^(1/3))^9*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x,4,1/2*a^2*(a+b*x^(1/3))^5*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3-6/7*a*(a+b*x^(1/3))^6*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3+3/8*(a+b*x^(1/3))^7*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(3/2),x,3,3/4*a^2*(a+b*x^(1/3))^3*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3-6/5*a*(a+b*x^(1/3))^4*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3+1/2*(a+b*x^(1/3))^5*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/b^3],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x,4,a*x*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/(a+b*x^(1/3))+3/4*b*x^(4/3)*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))/(a+b*x^(1/3))],
[1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x,4,-3*a*(a+b*x^(1/3))*x^(1/3)/(b^2*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+3/2*(a+b*x^(1/3))*x^(2/3)/(b*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+3*a^2*(a+b*x^(1/3))*log(a+b*x^(1/3))/(b^3*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))],
[1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(3/2),x,4,6*a/(b^3*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))-3/2*a^2/(b^3*(a+b*x^(1/3))*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+3*(a+b*x^(1/3))*log(a+b*x^(1/3))/(b^3*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))],
[1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x,4,-3/4*a^2/(b^3*(a+b*x^(1/3))^3*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+2*a/(b^3*(a+b*x^(1/3))^2*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+(-3/2)/(b^3*(a+b*x^(1/3))*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))],
[1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(7/2),x,4,-1/2*a^2/(b^3*(a+b*x^(1/3))^5*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+6/5*a/(b^3*(a+b*x^(1/3))^4*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+(-3/4)/(b^3*(a+b*x^(1/3))^3*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))],
[1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(9/2),x,4,-3/8*a^2/(b^3*(a+b*x^(1/3))^7*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+6/7*a/(b^3*(a+b*x^(1/3))^6*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+(-1/2)/(b^3*(a+b*x^(1/3))^5*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))],
[1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(11/2),x,4,-3/10*a^2/(b^3*(a+b*x^(1/3))^9*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+2/3*a/(b^3*(a+b*x^(1/3))^8*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))+(-3/8)/(b^3*(a+b*x^(1/3))^7*sqrt(a^2+2*a*b*x^(1/3)+b^2*x^(2/3)))],

# Integrands of the form (d x)^m (a+b x^(1/3)+c x^(2/3))^p with b^2-4 a c=0 and p symbolic
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*(d*x)^m,x,4,(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*x*(d*x)^m*hypergeom([3*(1+m),-2*p],[4+3*m],-b*x^(1/3)/a)/((1+m)*(1+b*x^(1/3)/a)^(2*p))],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*x^2,x,4,3*a^9*(1+b*x^(1/3)/a)*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(1+2*p))-12*a^9*(1+b*x^(1/3)/a)^2*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(1+p))+84*a^9*(1+b*x^(1/3)/a)^3*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(3+2*p))-84*a^9*(1+b*x^(1/3)/a)^4*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(2+p))+210*a^9*(1+b*x^(1/3)/a)^5*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(5+2*p))-84*a^9*(1+b*x^(1/3)/a)^6*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(3+p))+84*a^9*(1+b*x^(1/3)/a)^7*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(7+2*p))-12*a^9*(1+b*x^(1/3)/a)^8*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(4+p))+3*a^9*(1+b*x^(1/3)/a)^9*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^9*(9+2*p))],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*x,x,4,-3*a^6*(1+b*x^(1/3)/a)*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^6*(1+2*p))+15/2*a^6*(1+b*x^(1/3)/a)^2*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^6*(1+p))-30*a^6*(1+b*x^(1/3)/a)^3*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^6*(3+2*p))+15*a^6*(1+b*x^(1/3)/a)^4*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^6*(2+p))-15*a^6*(1+b*x^(1/3)/a)^5*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^6*(5+2*p))+3/2*a^6*(1+b*x^(1/3)/a)^6*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^6*(3+p))],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p,x,4,3*a^2*(a+b*x^(1/3))*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^3*(1+2*p))-3*a*(a+b*x^(1/3))^2*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^3*(1+p))+3*(a+b*x^(1/3))^3*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(b^3*(3+2*p))],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/x,x,3,-3*(1+b*x^(1/3)/a)*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*hypergeom([1,1+2*p],[2*(1+p)],1+b*x^(1/3)/a)/(1+2*p)],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/x^2,x,3,3*b^3*(1+b*x^(1/3)/a)*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p*hypergeom([4,1+2*p],[2*(1+p)],1+b*x^(1/3)/a)/(a^3*(1+2*p))],
[(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/x^2-2/3*b^3*(1-2*p)*(1-p)*p*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(a^3*x),x,-7,-(a+b*x^(1/3))*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(a*x)+b*(1-p)*(a+b*x^(1/3))*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(a^2*x^(2/3))-b^2*(1-2*p)*(1-p)*(a+b*x^(1/3))*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^p/(a^3*x^(1/3))],

# Integrands of the form (d x)^m (a+b x^(1/4)+c x^(2/4))^p

# Integrands of the form (d x)^m (a+b x^(1/4)+c Sqrt[x])^(p/2) with b^2-4 a c=0
[1/(a^2+2*a*b*x^(1/4)+b^2*sqrt(x))^(3/2),x,4,-12*a^2/(b^4*sqrt(a^2+2*a*b*x^(1/4)+b^2*sqrt(x)))+2*a^3/(b^4*(a+b*x^(1/4))*sqrt(a^2+2*a*b*x^(1/4)+b^2*sqrt(x)))+4*(a+b*x^(1/4))*x^(1/4)/(b^3*sqrt(a^2+2*a*b*x^(1/4)+b^2*sqrt(x)))-12*a*(a+b*x^(1/4))*log(a+b*x^(1/4))/(b^4*sqrt(a^2+2*a*b*x^(1/4)+b^2*sqrt(x)))],

# Integrands of the form (d x)^m (a+b x^(1/6)+c x^(2/6))^p

# Integrands of the form (d x)^m (a+b x^(1/6)+c x^(1/3))^(p/2) with b^2-4 a c=0
[1/(a^2+2*a*b*x^(1/6)+b^2*x^(1/3))^(5/2),x,4,-60*a^2/(b^6*sqrt(a^2+2*a*b*x^(1/6)+b^2*x^(1/3)))+3/2*a^5/(b^6*(a+b*x^(1/6))^3*sqrt(a^2+2*a*b*x^(1/6)+b^2*x^(1/3)))-10*a^4/(b^6*(a+b*x^(1/6))^2*sqrt(a^2+2*a*b*x^(1/6)+b^2*x^(1/3)))+30*a^3/(b^6*(a+b*x^(1/6))*sqrt(a^2+2*a*b*x^(1/6)+b^2*x^(1/3)))+6*(a+b*x^(1/6))*x^(1/6)/(b^5*sqrt(a^2+2*a*b*x^(1/6)+b^2*x^(1/3)))-30*a*(a+b*x^(1/6))*log(a+b*x^(1/6))/(b^6*sqrt(a^2+2*a*b*x^(1/6)+b^2*x^(1/3)))],

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with fraction n<0

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

# Integrands of the form (d x)^m (a+b/x^(1/2)+c/x^(2/2)) with b^2-4 a c=0
[(a^2+b^2/x+2*a*b/x^(1/2))^(3/2),x,5,a^3*x*sqrt(a^2+b^2/x+2*a*b/sqrt(x))/(a+b/sqrt(x))+6*a*b^2*log(sqrt(x))*sqrt(a^2+b^2/x+2*a*b/sqrt(x))/(a+b/sqrt(x))-2*b^3*sqrt(a^2+b^2/x+2*a*b/sqrt(x))/((a+b/sqrt(x))*sqrt(x))+6*a^2*b*sqrt(x)*sqrt(a^2+b^2/x+2*a*b/sqrt(x))/(a+b/sqrt(x))],

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

# Integrands of the form (d x)^m (a+b/x^(1/3)+c/x^(2/3))^(p/2) with b^2-4 a c=0
[(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(7/2),x,5,-3/4*b^7*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/((a+b/x^(1/3))*x^(4/3))-7*a*b^6*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/((a+b/x^(1/3))*x)-63/2*a^2*b^5*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/((a+b/x^(1/3))*x^(2/3))-105*a^3*b^4*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/((a+b/x^(1/3))*x^(1/3))+63*a^5*b^2*x^(1/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+21/2*a^6*b*x^(2/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+a^7*x*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+105*a^4*b^3*log(x^(1/3))*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))],
[(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x,5,-3/2*b^5*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/((a+b/x^(1/3))*x^(2/3))-15*a*b^4*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/((a+b/x^(1/3))*x^(1/3))+30*a^3*b^2*x^(1/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+15/2*a^4*b*x^(2/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+a^5*x*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+30*a^2*b^3*log(x^(1/3))*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))],
[(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(3/2),x,5,9*a*b^2*x^(1/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+9/2*a^2*b*x^(2/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+a^3*x*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+3*b^3*log(x^(1/3))*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))],
[(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2),x,4,3/2*b*x^(2/3)*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))+a*x*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))/(a+b/x^(1/3))],
[1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2),x,5,3*b^2*(a+b/x^(1/3))*x^(1/3)/(a^3*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-3/2*b*(a+b/x^(1/3))*x^(2/3)/(a^2*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))+(a+b/x^(1/3))*x/(a*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-3*b^3*(a+b/x^(1/3))*log(b+a*x^(1/3))/(a^4*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))],
[1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(3/2),x,5,3/2*b^5*(a+b/x^(1/3))/(a^6*(b+a*x^(1/3))^2*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-15*b^4*(a+b/x^(1/3))/(a^6*(b+a*x^(1/3))*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))+18*b^2*(a+b/x^(1/3))*x^(1/3)/(a^5*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-9/2*b*(a+b/x^(1/3))*x^(2/3)/(a^4*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))+(a+b/x^(1/3))*x/(a^3*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-30*b^3*(a+b/x^(1/3))*log(b+a*x^(1/3))/(a^6*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))],
[1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x,5,3/4*b^7*(a+b/x^(1/3))/(a^8*(b+a*x^(1/3))^4*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-7*b^6*(a+b/x^(1/3))/(a^8*(b+a*x^(1/3))^3*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))+63/2*b^5*(a+b/x^(1/3))/(a^8*(b+a*x^(1/3))^2*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-105*b^4*(a+b/x^(1/3))/(a^8*(b+a*x^(1/3))*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))+45*b^2*(a+b/x^(1/3))*x^(1/3)/(a^7*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-15/2*b*(a+b/x^(1/3))*x^(2/3)/(a^6*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))+(a+b/x^(1/3))*x/(a^5*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))-105*b^3*(a+b/x^(1/3))*log(b+a*x^(1/3))/(a^8*sqrt(a^2+b^2/x^(2/3)+2*a*b/x^(1/3)))],

# Integrands of the form (d x)^m (a+b/x^(1/4)+c/x^(2/4))^p

# Integrands of the form (d x)^m (a+b/x^(1/4)+c/Sqrt[x])^(p/2) with b^2-4 a c=0
[(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))^(5/2),x,5,-4*b^5*sqrt(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))/((a+b/x^(1/4))*x^(1/4))+40*a^2*b^3*x^(1/4)*sqrt(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))/(a+b/x^(1/4))+20/3*a^4*b*x^(3/4)*sqrt(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))/(a+b/x^(1/4))+a^5*x*sqrt(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))/(a+b/x^(1/4))+20*a*b^4*log(x^(1/4))*sqrt(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))/(a+b/x^(1/4))+20*a^3*b^2*sqrt(x)*sqrt(a^2+2*a*b/x^(1/4)+b^2/sqrt(x))/(a+b/x^(1/4))],

# Integrands of the form (d x)^m (a+b/x^(1/5)+c/x^(2/5))^p

# Integrands of the form (d x)^m (a+b/x^(1/5)+c/x^(2/5))^(p/2) with b^2-4 a c=0
[(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))^(5/2),x,5,25*a*b^4*x^(1/5)*sqrt(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))/(a+b/x^(1/5))+25*a^2*b^3*x^(2/5)*sqrt(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))/(a+b/x^(1/5))+50/3*a^3*b^2*x^(3/5)*sqrt(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))/(a+b/x^(1/5))+25/4*a^4*b*x^(4/5)*sqrt(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))/(a+b/x^(1/5))+a^5*x*sqrt(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))/(a+b/x^(1/5))+5*b^5*log(x^(1/5))*sqrt(a^2+b^2/x^(2/5)+2*a*b/x^(1/5))/(a+b/x^(1/5))],
[1/(a^2+2*a*b*x^(1/5)+b^2*x^(2/5))^(5/2),x,4,20*a/(b^5*sqrt(a^2+2*a*b*x^(1/5)+b^2*x^(2/5)))-5/4*a^4/(b^5*(a+b*x^(1/5))^3*sqrt(a^2+2*a*b*x^(1/5)+b^2*x^(2/5)))+20/3*a^3/(b^5*(a+b*x^(1/5))^2*sqrt(a^2+2*a*b*x^(1/5)+b^2*x^(2/5)))-15*a^2/(b^5*(a+b*x^(1/5))*sqrt(a^2+2*a*b*x^(1/5)+b^2*x^(2/5)))+5*(a+b*x^(1/5))*log(a+b*x^(1/5))/(b^5*sqrt(a^2+2*a*b*x^(1/5)+b^2*x^(2/5)))],

# Integrands of the form (d x)^m (a+b/x^(1/6)+c/x^(2/6))^p

# Integrands of the form (d x)^m (a+b/x^(1/6)+c/x^(1/3))^(p/2) with b^2-4 a c=0
[(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x,5,-6*b^7*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/((a+b/x^(1/6))*x^(1/6))+126*a^2*b^5*x^(1/6)*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/(a+b/x^(1/6))+105*a^3*b^4*x^(1/3)*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/(a+b/x^(1/6))+63/2*a^5*b^2*x^(2/3)*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/(a+b/x^(1/6))+42/5*a^6*b*x^(5/6)*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/(a+b/x^(1/6))+a^7*x*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/(a+b/x^(1/6))+42*a*b^6*log(x^(1/6))*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))/(a+b/x^(1/6))+70*a^4*b^3*sqrt(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))*sqrt(x)/(a+b/x^(1/6))],

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

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with a=0
[x^(-1+4*n)/(b*x^n+c*x^(2*n)),x,4,-b*x^n/(c^2*n)+1/2*x^(2*n)/(c*n)+b^2*log(b+c*x^n)/(c^3*n)],
[x^(-1+3*n)/(b*x^n+c*x^(2*n)),x,4,x^n/(c*n)-b*log(b+c*x^n)/(c^2*n)],
[x^(-1+2*n)/(b*x^n+c*x^(2*n)),x,2,log(b+c*x^n)/(c*n)],
[x^(-1+n)/(b*x^n+c*x^(2*n)),x,5,log(x)/b-log(b+c*x^n)/(b*n)],
[x^(-1-n)/(b*x^n+c*x^(2*n)),x,4,(-1/2)/(b*n*x^(2*n))+c/(b^2*n*x^n)+c^2*log(x)/b^3-c^2*log(b+c*x^n)/(b^3*n)],
[x^(-1-2*n)/(b*x^n+c*x^(2*n)),x,4,(-1/3)/(b*n*x^(3*n))+1/2*c/(b^2*n*x^(2*n))-c^2/(b^3*n*x^n)-c^3*log(x)/b^4+c^3*log(b+c*x^n)/(b^4*n)],
[x^(-1-3*n)/(b*x^n+c*x^(2*n)),x,4,(-1/4)/(b*n*x^(4*n))+1/3*c/(b^2*n*x^(3*n))-1/2*c^2/(b^3*n*x^(2*n))+c^3/(b^4*n*x^n)+c^4*log(x)/b^5-c^4*log(b+c*x^n)/(b^5*n)],
[x^(-1+1/4*n)/(b*x^n+c*x^(2*n)),x,12,(-4/3)/(b*n*x^(3/4*n))+c^(3/4)*log(-b^(1/4)*c^(1/4)*x^(1/4*n)*sqrt(2)+sqrt(b)+x^(1/2*n)*sqrt(c))/(b^(7/4)*n*sqrt(2))-c^(3/4)*log(b^(1/4)*c^(1/4)*x^(1/4*n)*sqrt(2)+sqrt(b)+x^(1/2*n)*sqrt(c))/(b^(7/4)*n*sqrt(2))+c^(3/4)*arctan(1-c^(1/4)*x^(1/4*n)*sqrt(2)/b^(1/4))*sqrt(2)/(b^(7/4)*n)-c^(3/4)*arctan(1+c^(1/4)*x^(1/4*n)*sqrt(2)/b^(1/4))*sqrt(2)/(b^(7/4)*n)],
[x^(-1+1/3*n)/(b*x^n+c*x^(2*n)),x,9,(-3/2)/(b*n*x^(2/3*n))-c^(2/3)*log(b^(1/3)+c^(1/3)*x^(1/3*n))/(b^(5/3)*n)+1/2*c^(2/3)*log(b^(2/3)-b^(1/3)*c^(1/3)*x^(1/3*n)+c^(2/3)*x^(2/3*n))/(b^(5/3)*n)+c^(2/3)*arctan((b^(1/3)-2*c^(1/3)*x^(1/3*n))/(b^(1/3)*sqrt(3)))*sqrt(3)/(b^(5/3)*n)],
[x^(-1+1/2*n)/(b*x^n+c*x^(2*n)),x,5,(-2)/(b*n*x^(1/2*n))+2*arctan(sqrt(b)/(x^(1/2*n)*sqrt(c)))*sqrt(c)/(b^(3/2)*n)],
[x^(-1-1/2*n)/(b*x^n+c*x^(2*n)),x,6,(-2/3)/(b*n*x^(3/2*n))+2*c/(b^2*n*x^(1/2*n))-2*c^(3/2)*arctan(sqrt(b)/(x^(1/2*n)*sqrt(c)))/(b^(5/2)*n)],
[x^(-1-1/3*n)/(b*x^n+c*x^(2*n)),x,11,(-3/4)/(b*n*x^(4/3*n))+3*c/(b^2*n*x^(1/3*n))-c^(4/3)*log(c^(1/3)+b^(1/3)/x^(1/3*n))/(b^(7/3)*n)+1/2*c^(4/3)*log(c^(2/3)+b^(2/3)/x^(2/3*n)-b^(1/3)*c^(1/3)/x^(1/3*n))/(b^(7/3)*n)+c^(4/3)*arctan((1-2*b^(1/3)/(c^(1/3)*x^(1/3*n)))/sqrt(3))*sqrt(3)/(b^(7/3)*n)],
[x^(-1-1/4*n)/(b*x^n+c*x^(2*n)),x,14,(-4/5)/(b*n*x^(5/4*n))+4*c/(b^2*n*x^(1/4*n))+c^(5/4)*log(-b^(1/4)*c^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(b)/x^(1/2*n)+sqrt(c))/(b^(9/4)*n*sqrt(2))-c^(5/4)*log(b^(1/4)*c^(1/4)*sqrt(2)/x^(1/4*n)+sqrt(b)/x^(1/2*n)+sqrt(c))/(b^(9/4)*n*sqrt(2))+c^(5/4)*arctan(1-b^(1/4)*sqrt(2)/(c^(1/4)*x^(1/4*n)))*sqrt(2)/(b^(9/4)*n)-c^(5/4)*arctan(1+b^(1/4)*sqrt(2)/(c^(1/4)*x^(1/4*n)))*sqrt(2)/(b^(9/4)*n)],
[x^(-1-n*(-1+p))*(b*x^n+c*x^(2*n))^p,x,1,(b*x^n+c*x^(2*n))^(1+p)/(c*n*(1+p)*x^(n*(1+p)))],
[x^(-1-n*(1+2*p))*(b*x^n+c*x^(2*n))^p,x,1,-(b*x^n+c*x^(2*n))^(1+p)/(b*n*(1+p)*x^(2*n*(1+p)))],

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with b^2-4 a c=0

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

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

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

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

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

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with m=k n-1
[x^(-1+4*n)/(a+b*x^n+c*x^(2*n)),x,7,-b*x^n/(c^2*n)+1/2*x^(2*n)/(c*n)+1/2*(b^2-a*c)*log(a+b*x^n+c*x^(2*n))/(c^3*n)+b*(b^2-3*a*c)*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(c^3*n*sqrt(b^2-4*a*c))],
[x^(-1+3*n)/(a+b*x^n+c*x^(2*n)),x,6,x^n/(c*n)-1/2*b*log(a+b*x^n+c*x^(2*n))/(c^2*n)-(b^2-2*a*c)*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(c^2*n*sqrt(b^2-4*a*c))],
[x^(-1+2*n)/(a+b*x^n+c*x^(2*n)),x,5,1/2*log(a+b*x^n+c*x^(2*n))/(c*n)+b*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(c*n*sqrt(b^2-4*a*c))],
[x^(-1+n)/(a+b*x^n+c*x^(2*n)),x,3,-2*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(n*sqrt(b^2-4*a*c))],
[x^(-1-n)/(a+b*x^n+c*x^(2*n)),x,8,(-1)/(a*n*x^n)-b*log(x)/a^2+1/2*b*log(a+b*x^n+c*x^(2*n))/(a^2*n)-(b^2-2*a*c)*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(a^2*n*sqrt(b^2-4*a*c))],
[x^(-1-2*n)/(a+b*x^n+c*x^(2*n)),x,8,(-1/2)/(a*n*x^(2*n))+b/(a^2*n*x^n)+(b^2-a*c)*log(x)/a^3-1/2*(b^2-a*c)*log(a+b*x^n+c*x^(2*n))/(a^3*n)+b*(b^2-3*a*c)*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(a^3*n*sqrt(b^2-4*a*c))],
[x^(-1-3*n)/(a+b*x^n+c*x^(2*n)),x,8,(-1/3)/(a*n*x^(3*n))+1/2*b/(a^2*n*x^(2*n))+(-b^2+a*c)/(a^3*n*x^n)-b*(b^2-2*a*c)*log(x)/a^4+1/2*b*(b^2-2*a*c)*log(a+b*x^n+c*x^(2*n))/(a^4*n)-(b^4-4*a*b^2*c+2*a^2*c^2)*arctanh((b+2*c*x^n)/sqrt(b^2-4*a*c))/(a^4*n*sqrt(b^2-4*a*c))],
[x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x,8,2*2^(3/4)*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x^(1/4*n)/(-b-sqrt(b^2-4*a*c))^(1/4))/(n*(-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c))+2*2^(3/4)*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x^(1/4*n)/(-b-sqrt(b^2-4*a*c))^(1/4))/(n*(-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c))-2*2^(3/4)*c^(3/4)*arctan(2^(1/4)*c^(1/4)*x^(1/4*n)/(-b+sqrt(b^2-4*a*c))^(1/4))/(n*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))-2*2^(3/4)*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*x^(1/4*n)/(-b+sqrt(b^2-4*a*c))^(1/4))/(n*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))],
[x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x,14,2^(2/3)*c^(2/3)*log(2^(1/3)*c^(1/3)*x^(1/3*n)+(b-sqrt(b^2-4*a*c))^(1/3))/(n*(b-sqrt(b^2-4*a*c))^(2/3)*sqrt(b^2-4*a*c))-c^(2/3)*log(2^(2/3)*c^(2/3)*x^(2/3*n)-2^(1/3)*c^(1/3)*x^(1/3*n)*(b-sqrt(b^2-4*a*c))^(1/3)+(b-sqrt(b^2-4*a*c))^(2/3))/(2^(1/3)*n*(b-sqrt(b^2-4*a*c))^(2/3)*sqrt(b^2-4*a*c))-2^(2/3)*c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x^(1/3*n)/(b-sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*sqrt(3)/(n*(b-sqrt(b^2-4*a*c))^(2/3)*sqrt(b^2-4*a*c))-2^(2/3)*c^(2/3)*log(2^(1/3)*c^(1/3)*x^(1/3*n)+(b+sqrt(b^2-4*a*c))^(1/3))/(n*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(2/3))+c^(2/3)*log(2^(2/3)*c^(2/3)*x^(2/3*n)-2^(1/3)*c^(1/3)*x^(1/3*n)*(b+sqrt(b^2-4*a*c))^(1/3)+(b+sqrt(b^2-4*a*c))^(2/3))/(2^(1/3)*n*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(2/3))+2^(2/3)*c^(2/3)*arctan((1-2*2^(1/3)*c^(1/3)*x^(1/3*n)/(b+sqrt(b^2-4*a*c))^(1/3))/sqrt(3))*sqrt(3)/(n*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(2/3))],
[x^(-1+1/2*n)/(a+b*x^n+c*x^(2*n)),x,4,2*arctan(x^(1/2*n)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(2)*sqrt(c)/(n*sqrt(b^2-4*a*c)*sqrt(b-sqrt(b^2-4*a*c)))-2*arctan(x^(1/2*n)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(2)*sqrt(c)/(n*sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^(-1-1/2*n)/(a+b*x^n+c*x^(2*n)),x,6,(-2)/(a*n*x^(1/2*n))+arctan(sqrt(2)*sqrt(a)/(x^(1/2*n)*sqrt(b-sqrt(b^2-4*a*c))))*sqrt(2)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^(3/2)*n*sqrt(b-sqrt(b^2-4*a*c)))+arctan(sqrt(2)*sqrt(a)/(x^(1/2*n)*sqrt(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^(3/2)*n*sqrt(b+sqrt(b^2-4*a*c)))],
[x^(-1-1/3*n)/(a+b*x^n+c*x^(2*n)),x,16,(-3)/(a*n*x^(1/3*n))+log(2^(1/3)*a^(1/3)/x^(1/3*n)+(b-sqrt(b^2-4*a*c))^(1/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*a^(4/3)*n*(b-sqrt(b^2-4*a*c))^(2/3))-1/2*log(2^(2/3)*a^(2/3)/x^(2/3*n)-2^(1/3)*a^(1/3)*(b-sqrt(b^2-4*a*c))^(1/3)/x^(1/3*n)+(b-sqrt(b^2-4*a*c))^(2/3))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*a^(4/3)*n*(b-sqrt(b^2-4*a*c))^(2/3))-arctan((1-2*2^(1/3)*a^(1/3)/(x^(1/3*n)*(b-sqrt(b^2-4*a*c))^(1/3)))/sqrt(3))*sqrt(3)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*a^(4/3)*n*(b-sqrt(b^2-4*a*c))^(2/3))+log(2^(1/3)*a^(1/3)/x^(1/3*n)+(b+sqrt(b^2-4*a*c))^(1/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*a^(4/3)*n*(b+sqrt(b^2-4*a*c))^(2/3))-1/2*log(2^(2/3)*a^(2/3)/x^(2/3*n)-2^(1/3)*a^(1/3)*(b+sqrt(b^2-4*a*c))^(1/3)/x^(1/3*n)+(b+sqrt(b^2-4*a*c))^(2/3))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*a^(4/3)*n*(b+sqrt(b^2-4*a*c))^(2/3))-arctan((1-2*2^(1/3)*a^(1/3)/(x^(1/3*n)*(b+sqrt(b^2-4*a*c))^(1/3)))/sqrt(3))*sqrt(3)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/3)*a^(4/3)*n*(b+sqrt(b^2-4*a*c))^(2/3))],
[x^(-1-1/4*n)/(a+b*x^n+c*x^(2*n)),x,10,(-4)/(a*n*x^(1/4*n))-2^(3/4)*arctan(2^(1/4)*a^(1/4)/(x^(1/4*n)*(-b-sqrt(b^2-4*a*c))^(1/4)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^(5/4)*n*(-b-sqrt(b^2-4*a*c))^(3/4))-2^(3/4)*arctanh(2^(1/4)*a^(1/4)/(x^(1/4*n)*(-b-sqrt(b^2-4*a*c))^(1/4)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^(5/4)*n*(-b-sqrt(b^2-4*a*c))^(3/4))-2^(3/4)*arctan(2^(1/4)*a^(1/4)/(x^(1/4*n)*(-b+sqrt(b^2-4*a*c))^(1/4)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^(5/4)*n*(-b+sqrt(b^2-4*a*c))^(3/4))-2^(3/4)*arctanh(2^(1/4)*a^(1/4)/(x^(1/4*n)*(-b+sqrt(b^2-4*a*c))^(1/4)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^(5/4)*n*(-b+sqrt(b^2-4*a*c))^(3/4))],

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

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

# p>0

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

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

# p>0
[x^3*sqrt(a+b*x^n+c*x^(2*n)),x,2,1/4*x^4*AppellF1(4/n,-1/2,-1/2,(4+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[x^2*sqrt(a+b*x^n+c*x^(2*n)),x,2,1/3*x^3*AppellF1(3/n,-1/2,-1/2,(3+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[x*sqrt(a+b*x^n+c*x^(2*n)),x,2,1/2*x^2*AppellF1(2/n,-1/2,-1/2,(2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[sqrt(a+b*x^n+c*x^(2*n)),x,2,x*AppellF1(1/n,-1/2,-1/2,1+1/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[sqrt(a+b*x^n+c*x^(2*n))/x,x,7,-arctanh(1/2*(2*a+b*x^n)/(sqrt(a)*sqrt(a+b*x^n+c*x^(2*n))))*sqrt(a)/n+1/2*b*arctanh(1/2*(b+2*c*x^n)/(sqrt(c)*sqrt(a+b*x^n+c*x^(2*n))))/(n*sqrt(c))+sqrt(a+b*x^n+c*x^(2*n))/n],
[sqrt(a+b*x^n+c*x^(2*n))/x^2,x,2,-AppellF1((-1)/n,-1/2,-1/2,(-1+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(x*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[sqrt(a+b*x^n+c*x^(2*n))/x^3,x,2,-1/2*AppellF1((-2)/n,-1/2,-1/2,(-2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(x^2*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[x^3*(a+b*x^n+c*x^(2*n))^(3/2),x,2,1/4*a*x^4*AppellF1(4/n,-3/2,-3/2,(4+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[x^2*(a+b*x^n+c*x^(2*n))^(3/2),x,2,1/3*a*x^3*AppellF1(3/n,-3/2,-3/2,(3+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[x*(a+b*x^n+c*x^(2*n))^(3/2),x,2,1/2*a*x^2*AppellF1(2/n,-3/2,-3/2,(2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[(a+b*x^n+c*x^(2*n))^(3/2),x,2,a*x*AppellF1(1/n,-3/2,-3/2,1+1/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[(a+b*x^n+c*x^(2*n))^(3/2)/x,x,8,1/3*(a+b*x^n+c*x^(2*n))^(3/2)/n-a^(3/2)*arctanh(1/2*(2*a+b*x^n)/(sqrt(a)*sqrt(a+b*x^n+c*x^(2*n))))/n-1/16*b*(b^2-12*a*c)*arctanh(1/2*(b+2*c*x^n)/(sqrt(c)*sqrt(a+b*x^n+c*x^(2*n))))/(c^(3/2)*n)+1/8*(b^2+8*a*c+2*b*c*x^n)*sqrt(a+b*x^n+c*x^(2*n))/(c*n)],
[(a+b*x^n+c*x^(2*n))^(3/2)/x^2,x,2,-a*AppellF1((-1)/n,-3/2,-3/2,(-1+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(x*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[(a+b*x^n+c*x^(2*n))^(3/2)/x^3,x,2,-1/2*a*AppellF1((-2)/n,-3/2,-3/2,(-2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(x^2*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],

# p<0
[x^3/sqrt(a+b*x^n+c*x^(2*n)),x,2,1/4*x^4*AppellF1(4/n,1/2,1/2,(4+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^n+c*x^(2*n))],
[x^2/sqrt(a+b*x^n+c*x^(2*n)),x,2,1/3*x^3*AppellF1(3/n,1/2,1/2,(3+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^n+c*x^(2*n))],
[x/sqrt(a+b*x^n+c*x^(2*n)),x,2,1/2*x^2*AppellF1(2/n,1/2,1/2,(2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^n+c*x^(2*n))],
[1/sqrt(a+b*x^n+c*x^(2*n)),x,2,x*AppellF1(1/n,1/2,1/2,1+1/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/sqrt(a+b*x^n+c*x^(2*n))],
[1/(x*sqrt(a+b*x^n+c*x^(2*n))),x,3,-arctanh(1/2*(2*a+b*x^n)/(sqrt(a)*sqrt(a+b*x^n+c*x^(2*n))))/(n*sqrt(a))],
[1/(x^2*sqrt(a+b*x^n+c*x^(2*n))),x,2,-AppellF1((-1)/n,1/2,1/2,(-1+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(x*sqrt(a+b*x^n+c*x^(2*n)))],
[1/(x^3*sqrt(a+b*x^n+c*x^(2*n))),x,2,-1/2*AppellF1((-2)/n,1/2,1/2,(-2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(x^2*sqrt(a+b*x^n+c*x^(2*n)))],
[x^3/(a+b*x^n+c*x^(2*n))^(3/2),x,2,1/4*x^4*AppellF1(4/n,3/2,3/2,(4+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^n+c*x^(2*n)))],
[x^2/(a+b*x^n+c*x^(2*n))^(3/2),x,2,1/3*x^3*AppellF1(3/n,3/2,3/2,(3+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^n+c*x^(2*n)))],
[x/(a+b*x^n+c*x^(2*n))^(3/2),x,2,1/2*x^2*AppellF1(2/n,3/2,3/2,(2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^n+c*x^(2*n)))],
[1/(a+b*x^n+c*x^(2*n))^(3/2),x,2,x*AppellF1(1/n,3/2,3/2,1+1/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*sqrt(a+b*x^n+c*x^(2*n)))],
[1/(x*(a+b*x^n+c*x^(2*n))^(3/2)),x,5,-arctanh(1/2*(2*a+b*x^n)/(sqrt(a)*sqrt(a+b*x^n+c*x^(2*n))))/(a^(3/2)*n)+2*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*n*sqrt(a+b*x^n+c*x^(2*n)))],
[1/(x^2*(a+b*x^n+c*x^(2*n))^(3/2)),x,2,-AppellF1((-1)/n,3/2,3/2,(-1+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*x*sqrt(a+b*x^n+c*x^(2*n)))],
[1/(x^3*(a+b*x^n+c*x^(2*n))^(3/2)),x,2,-1/2*AppellF1((-2)/n,3/2,3/2,(-2+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*x^2*sqrt(a+b*x^n+c*x^(2*n)))],

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with m symbolic
[(d*x)^m*(a+b*x^n+c*x^(2*n))^3,x,14,3*a^2*b*x^(1+n)*(d*x)^m/(1+m+n)+3*a*(b^2+a*c)*x^(1+2*n)*(d*x)^m/(1+m+2*n)+b*(b^2+6*a*c)*x^(1+3*n)*(d*x)^m/(1+m+3*n)+3*c*(b^2+a*c)*x^(1+4*n)*(d*x)^m/(1+m+4*n)+3*b*c^2*x^(1+5*n)*(d*x)^m/(1+m+5*n)+c^3*x^(1+6*n)*(d*x)^m/(1+m+6*n)+a^3*(d*x)^(1+m)/(d*(1+m))],
[(d*x)^m*(a+b*x^n+c*x^(2*n))^2,x,10,2*a*b*x^(1+n)*(d*x)^m/(1+m+n)+(b^2+2*a*c)*x^(1+2*n)*(d*x)^m/(1+m+2*n)+2*b*c*x^(1+3*n)*(d*x)^m/(1+m+3*n)+c^2*x^(1+4*n)*(d*x)^m/(1+m+4*n)+a^2*(d*x)^(1+m)/(d*(1+m))],
[(d*x)^m*(a+b*x^n+c*x^(2*n)),x,6,b*x^(1+n)*(d*x)^m/(1+m+n)+c*x^(1+2*n)*(d*x)^m/(1+m+2*n)+a*(d*x)^(1+m)/(d*(1+m))],
[(d*x)^m/(a+b*x^n+c*x^(2*n)),x,3,2*c*(d*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))/(d*(1+m)*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c))-2*c*(d*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[(d*x)^m/(a+b*x^n+c*x^(2*n))^2,x,5,(d*x)^(1+m)*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*d*n*(a+b*x^n+c*x^(2*n)))+c*(d*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(-b*(1+m-n)+(4*a*c*(1+m-2*n)-b^2*(1+m-n))/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*d*(1+m)*n*(b-sqrt(b^2-4*a*c)))-c*(d*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(4*a*c*(1+m-2*n)-b^2*(1+m-n)+b*(1+m-n)*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^(3/2)*d*(1+m)*n*(b+sqrt(b^2-4*a*c)))],
[(d*x)^m/(a+b*x^n+c*x^(2*n))^3,x,6,1/2*(d*x)^(1+m)*(b^2-2*a*c+b*c*x^n)/(a*(b^2-4*a*c)*d*n*(a+b*x^n+c*x^(2*n))^2)-1/2*(d*x)^(1+m)*(4*a^2*c^2*(1+m-4*n)-5*a*b^2*c*(1+m-3*n)+b^4*(1+m-2*n)-b*c*(2*a*c*(2+2*m-7*n)-b^2*(1+m-2*n))*x^n)/(a^2*(b^2-4*a*c)^2*d*n^2*(a+b*x^n+c*x^(2*n)))-1/2*c*(d*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-2*c*x^n/(b-sqrt(b^2-4*a*c)))*(-b^4*(1+m^2+m*(2-3*n)-3*n+2*n^2)+6*a*b^2*c*(1+m^2+m*(2-4*n)-4*n+3*n^2)-8*a^2*c^2*(1+m^2+m*(2-6*n)-6*n+8*n^2)+b*(2*a*c*(2+2*m-7*n)-b^2*(1+m-2*n))*(1+m-n)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(5/2)*d*(1+m)*n^2*(b-sqrt(b^2-4*a*c)))-1/2*c*(d*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-2*c*x^n/(b+sqrt(b^2-4*a*c)))*(b^4*(1+m^2+m*(2-3*n)-3*n+2*n^2)-6*a*b^2*c*(1+m^2+m*(2-4*n)-4*n+3*n^2)+8*a^2*c^2*(1+m^2+m*(2-6*n)-6*n+8*n^2)+b*(2*a*c*(2+2*m-7*n)-b^2*(1+m-2*n))*(1+m-n)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(5/2)*d*(1+m)*n^2*(b+sqrt(b^2-4*a*c)))],
[(d*x)^m*(a+b*x^n+c*x^(2*n))^(3/2),x,2,a*(d*x)^(1+m)*AppellF1((1+m)/n,-3/2,-3/2,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(d*(1+m)*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[(d*x)^m*(a+b*x^n+c*x^(2*n))^(1/2),x,2,(d*x)^(1+m)*AppellF1((1+m)/n,-1/2,-1/2,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^n+c*x^(2*n))/(d*(1+m)*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c))))],
[(d*x)^m/(a+b*x^n+c*x^(2*n))^(1/2),x,2,(d*x)^(1+m)*AppellF1((1+m)/n,1/2,1/2,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*sqrt(a+b*x^n+c*x^(2*n)))],
[(d*x)^m/(a+b*x^n+c*x^(2*n))^(3/2),x,2,(d*x)^(1+m)*AppellF1((1+m)/n,3/2,3/2,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))/(a*d*(1+m)*sqrt(a+b*x^n+c*x^(2*n)))],

# Integrands of the form (d x)^m (a+b x^n+c x^(2 n))^p with p symbolic
[(d*x)^m*(a+b*x^n+c*x^(2*n))^p,x,2,(d*x)^(1+m)*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+m)/n,-p,-p,(1+m+n)/n,-2*c*x^n/(b-sqrt(b^2-4*a*c)),-2*c*x^n/(b+sqrt(b^2-4*a*c)))/(d*(1+m)*(1+2*c*x^n/(b-sqrt(b^2-4*a*c)))^p*(1+2*c*x^n/(b+sqrt(b^2-4*a*c)))^p)],

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

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

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

# p>0
[(d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4),x,3,1/4*a*(d+e*x)^4/e+1/6*b*(d+e*x)^6/e+1/8*c*(d+e*x)^8/e],
[(d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,4,1/4*a^2*(d+e*x)^4/e+1/3*a*b*(d+e*x)^6/e+1/8*(b^2+2*a*c)*(d+e*x)^8/e+1/5*b*c*(d+e*x)^10/e+1/12*c^2*(d+e*x)^12/e],
[(d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,4,1/4*a^3*(d+e*x)^4/e+1/2*a^2*b*(d+e*x)^6/e+3/8*a*(b^2+a*c)*(d+e*x)^8/e+1/10*b*(b^2+6*a*c)*(d+e*x)^10/e+1/4*c*(b^2+a*c)*(d+e*x)^12/e+3/14*b*c^2*(d+e*x)^14/e+1/16*c^3*(d+e*x)^16/e],
[(d*f+e*f*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4),x,3,1/4*a*f^3*(d+e*x)^4/e+1/6*b*f^3*(d+e*x)^6/e+1/8*c*f^3*(d+e*x)^8/e],
[(d*f+e*f*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,4,1/4*a^2*f^3*(d+e*x)^4/e+1/3*a*b*f^3*(d+e*x)^6/e+1/8*(b^2+2*a*c)*f^3*(d+e*x)^8/e+1/5*b*c*f^3*(d+e*x)^10/e+1/12*c^2*f^3*(d+e*x)^12/e],
[(d*f+e*f*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,4,1/4*a^3*f^3*(d+e*x)^4/e+1/2*a^2*b*f^3*(d+e*x)^6/e+3/8*a*(b^2+a*c)*f^3*(d+e*x)^8/e+1/10*b*(b^2+6*a*c)*f^3*(d+e*x)^10/e+1/4*c*(b^2+a*c)*f^3*(d+e*x)^12/e+3/14*b*c^2*f^3*(d+e*x)^14/e+1/16*c^3*f^3*(d+e*x)^16/e],

# p<0
[(d+e*x)^4/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,5,x/c-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)^3/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,6,1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(c*e)+1/2*b*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(c*e*sqrt(b^2-4*a*c))],
[(d+e*x)^2/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,4,-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c))/(e*sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))+arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(e*sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[(d+e*x)/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,4,-arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(e*sqrt(b^2-4*a*c))],
[1/((d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,8,log(d+e*x)/(a*e)-1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a*e)+1/2*b*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a*e*sqrt(b^2-4*a*c))],
[1/((d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,5,(-1)/(a*e*(d+e*x))-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(1+b/sqrt(b^2-4*a*c))/(a*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(1-b/sqrt(b^2-4*a*c))/(a*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,9,(-1/2)/(a*e*(d+e*x)^2)-b*log(d+e*x)/(a^2*e)+1/4*b*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^2*e)-1/2*(b^2-2*a*c)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^2*e*sqrt(b^2-4*a*c))],
[1/((d+e*x)^4*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,6,(-1/3)/(a*e*(d+e*x)^3)+b/(a^2*e*(d+e*x))+arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)^4/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,1/2*(d+e*x)*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2-4*a*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*e*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2+4*a*c+b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)^3/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,1/2*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))-b*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e)],
[(d+e*x)^2/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,-1/2*(d+e*x)*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(2*b-sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(2*b+sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,1/2*(-b-2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+2*c*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e)],
[1/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,1/2*(d/e+x)*(b^2-2*a*c+b*c*e^2*(d/e+x)^2)/(a*(b^2-4*a*c)*(a+b*e^2*(d/e+x)^2+c*e^4*(d/e+x)^4))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^2-12*a*c+b*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b^2-12*a*c-b*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,9,1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*b*(b^2-6*a*c)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*e)+log(d+e*x)/(a^2*e)-1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^2*e)],
[1/((d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,6,1/2*(-3*b^2+10*a*c)/(a^2*(b^2-4*a*c)*e*(d+e*x))+1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4))-1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^3-16*a*b*c+(3*b^2-10*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^3-16*a*b*c-(3*b^2-10*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,9,(-b^2+3*a*c)/(a^2*(b^2-4*a*c)*e*(d+e*x)^2)+1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4))-(b^4-6*a*b^2*c+6*a^2*c^2)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*e)-2*b*log(d+e*x)/(a^3*e)+1/2*b*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^3*e)],
[1/((d+e*x)^4*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,7,1/6*(-5*b^2+14*a*c)/(a^2*(b^2-4*a*c)*e*(d+e*x)^3)+1/2*b*(5*b^2-19*a*c)/(a^3*(b^2-4*a*c)*e*(d+e*x))+1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4-29*a*b^2*c+28*a^2*c^2+b*(5*b^2-19*a*c)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4-29*a*b^2*c+28*a^2*c^2-b*(5*b^2-19*a*c)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)^4/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,1/4*(d+e*x)*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)-1/8*(d+e*x)*(7*b^2-4*a*c+12*b*c*(d+e*x)^2)/((b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+3/4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^2+4*a*c-2*b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^2+4*a*c+2*b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)^3/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,1/4*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)-3/4*b*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+3*b*c*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e)],
[(d+e*x)^2/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,-1/4*(d+e*x)*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/8*(d+e*x)*(b*(b^2+8*a*c)+c*(b^2+20*a*c)*(d+e*x)^2)/(a*(b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^2+20*a*c+b*(b^2-52*a*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^2*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b^2+20*a*c-b*(b^2-52*a*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^2*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x)/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,1/4*(-b-2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+3/2*c*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))-6*c^2*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e)],
[1/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,1/4*(d/e+x)*(b^2-2*a*c+b*c*e^2*(d/e+x)^2)/(a*(b^2-4*a*c)*(a+b*e^2*(d/e+x)^2+c*e^4*(d/e+x)^4)^2)+1/8*(d/e+x)*((b^2-7*a*c)*(3*b^2-4*a*c)+3*b*c*(b^2-8*a*c)*e^2*(d/e+x)^2)/(a^2*(b^2-4*a*c)^2*(a+b*e^2*(d/e+x)^2+c*e^4*(d/e+x)^4))+3/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^4-10*a*b^2*c+56*a^2*c^2+b*(b^2-8*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(5/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b^4-10*a*b^2*c+56*a^2*c^2-b*(b^2-8*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(5/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3),x,10,1/4*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/4*(2*b^4-15*a*b^2*c+16*a^2*c^2+2*b*c*(b^2-7*a*c)*(d+e*x)^2)/(a^2*(b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*b*(b^4-10*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2)*e)+log(d+e*x)/(a^3*e)-1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^3*e)],
[1/((d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3),x,7,-3/8*(5*b^2-12*a*c)*(b^2-5*a*c)/(a^3*(b^2-4*a*c)^2*e*(d+e*x))+1/4*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/8*(5*b^4-35*a*b^2*c+36*a^2*c^2+b*c*(5*b^2-32*a*c)*(d+e*x)^2)/(a^2*(b^2-4*a*c)^2*e*(d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4))-3/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*((5*b^2-12*a*c)*(b^2-5*a*c)+b*(5*b^4-47*a*b^2*c+124*a^2*c^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^2*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*((5*b^2-12*a*c)*(b^2-5*a*c)+(-5*b^5+47*a*b^3*c-124*a^2*b*c^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^2*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3),x,10,-3/2*(b^2-5*a*c)*(b^2-2*a*c)/(a^3*(b^2-4*a*c)^2*e*(d+e*x)^2)+1/4*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*(d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/4*(3*b^4-20*a*b^2*c+20*a^2*c^2+3*b*c*(b^2-6*a*c)*(d+e*x)^2)/(a^2*(b^2-4*a*c)^2*e*(d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4))-3/2*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(5/2)*e)-3*b*log(d+e*x)/(a^4*e)+3/4*b*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^4*e)],
[(d*f+e*f*x)^4/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,5,f^4*x/c-f^4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-f^4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)^3/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,6,1/4*f^3*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(c*e)+1/2*b*f^3*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(c*e*sqrt(b^2-4*a*c))],
[(d*f+e*f*x)^2/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,4,-f^2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c))/(e*sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))+f^2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(e*sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[(d*f+e*f*x)/(a+b*(d+e*x)^2+c*(d+e*x)^4),x,4,-f*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(e*sqrt(b^2-4*a*c))],
[1/((d*f+e*f*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,8,log(d+e*x)/(a*e*f)-1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a*e*f)+1/2*b*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a*e*f*sqrt(b^2-4*a*c))],
[1/((d*f+e*f*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,5,(-1)/(a*e*f^2*(d+e*x))-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(1+b/sqrt(b^2-4*a*c))/(a*e*f^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(1-b/sqrt(b^2-4*a*c))/(a*e*f^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d*f+e*f*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,9,(-1/2)/(a*e*f^3*(d+e*x)^2)-b*log(d+e*x)/(a^2*e*f^3)+1/4*b*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^2*e*f^3)-1/2*(b^2-2*a*c)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^2*e*f^3*sqrt(b^2-4*a*c))],
[1/((d*f+e*f*x)^4*(a+b*(d+e*x)^2+c*(d+e*x)^4)),x,6,(-1/3)/(a*e*f^4*(d+e*x)^3)+b/(a^2*e*f^4*(d+e*x))+arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*e*f^4*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*e*f^4*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)^4/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,1/2*f^4*(d+e*x)*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*f^4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2-4*a*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*e*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*f^4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2+4*a*c+b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)^3/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,1/2*f^3*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))-b*f^3*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e)],
[(d*f+e*f*x)^2/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,-1/2*f^2*(d+e*x)*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+f^2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(2*b-sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-f^2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(2*b+sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)/(a+b*(d+e*x)^2+c*(d+e*x)^4)^2,x,5,-1/2*f*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+2*c*f*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*e)],
[1/((d*f+e*f*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,9,1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*b*(b^2-6*a*c)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*e*f)+log(d+e*x)/(a^2*e*f)-1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^2*e*f)],
[1/((d*f+e*f*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,6,1/2*(-3*b^2+10*a*c)/(a^2*(b^2-4*a*c)*e*f^2*(d+e*x))+1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f^2*(d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4))-1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^3-16*a*b*c+(3*b^2-10*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*e*f^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^3-16*a*b*c-(3*b^2-10*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*e*f^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d*f+e*f*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,9,(-b^2+3*a*c)/(a^2*(b^2-4*a*c)*e*f^3*(d+e*x)^2)+1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f^3*(d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4))-(b^4-6*a*b^2*c+6*a^2*c^2)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*e*f^3)-2*b*log(d+e*x)/(a^3*e*f^3)+1/2*b*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^3*e*f^3)],
[1/((d*f+e*f*x)^4*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2),x,7,1/6*(-5*b^2+14*a*c)/(a^2*(b^2-4*a*c)*e*f^4*(d+e*x)^3)+1/2*b*(5*b^2-19*a*c)/(a^3*(b^2-4*a*c)*e*f^4*(d+e*x))+1/2*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f^4*(d+e*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4-29*a*b^2*c+28*a^2*c^2+b*(5*b^2-19*a*c)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*e*f^4*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4-29*a*b^2*c+28*a^2*c^2-b*(5*b^2-19*a*c)*sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2)*e*f^4*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)^4/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,1/4*f^4*(d+e*x)*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)-1/8*f^4*(d+e*x)*(7*b^2-4*a*c+12*b*c*(d+e*x)^2)/((b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+3/4*f^4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^2+4*a*c-2*b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/4*f^4*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^2+4*a*c+2*b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)^3/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,1/4*f^3*(2*a+b*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)-3/4*b*f^3*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+3*b*c*f^3*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e)],
[(d*f+e*f*x)^2/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,-1/4*f^2*(d+e*x)*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/8*f^2*(d+e*x)*(b*(b^2+8*a*c)+c*(b^2+20*a*c)*(d+e*x)^2)/(a*(b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/8*f^2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^2+20*a*c+b*(b^2-52*a*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^2*e*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*f^2*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b^2+20*a*c-b*(b^2-52*a*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^2*e*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d*f+e*f*x)/(a+b*(d+e*x)^2+c*(d+e*x)^4)^3,x,6,-1/4*f*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)*e*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+3/2*c*f*(b+2*c*(d+e*x)^2)/((b^2-4*a*c)^2*e*(a+b*(d+e*x)^2+c*(d+e*x)^4))-6*c^2*f*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*e)],
[1/((d*f+e*f*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3),x,10,1/4*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/4*(2*b^4-15*a*b^2*c+16*a^2*c^2+2*b*c*(b^2-7*a*c)*(d+e*x)^2)/(a^2*(b^2-4*a*c)^2*e*f*(a+b*(d+e*x)^2+c*(d+e*x)^4))+1/2*b*(b^4-10*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2)*e*f)+log(d+e*x)/(a^3*e*f)-1/4*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^3*e*f)],
[1/((d*f+e*f*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3),x,7,-3/8*(5*b^2-12*a*c)*(b^2-5*a*c)/(a^3*(b^2-4*a*c)^2*e*f^2*(d+e*x))+1/4*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f^2*(d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/8*(5*b^4-35*a*b^2*c+36*a^2*c^2+b*c*(5*b^2-32*a*c)*(d+e*x)^2)/(a^2*(b^2-4*a*c)^2*e*f^2*(d+e*x)*(a+b*(d+e*x)^2+c*(d+e*x)^4))-3/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*((5*b^2-12*a*c)*(b^2-5*a*c)+b*(5*b^4-47*a*b^2*c+124*a^2*c^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^2*e*f^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/8*arctan((d+e*x)*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*((5*b^2-12*a*c)*(b^2-5*a*c)+(-5*b^5+47*a*b^3*c-124*a^2*b*c^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^2*e*f^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d*f+e*f*x)^3*(a+b*(d+e*x)^2+c*(d+e*x)^4)^3),x,10,-3/2*(b^2-5*a*c)*(b^2-2*a*c)/(a^3*(b^2-4*a*c)^2*e*f^3*(d+e*x)^2)+1/4*(b^2-2*a*c+b*c*(d+e*x)^2)/(a*(b^2-4*a*c)*e*f^3*(d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4)^2)+1/4*(3*b^4-20*a*b^2*c+20*a^2*c^2+3*b*c*(b^2-6*a*c)*(d+e*x)^2)/(a^2*(b^2-4*a*c)^2*e*f^3*(d+e*x)^2*(a+b*(d+e*x)^2+c*(d+e*x)^4))-3/2*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*(d+e*x)^2)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(5/2)*e*f^3)-3*b*log(d+e*x)/(a^4*e*f^3)+3/4*b*log(a+b*(d+e*x)^2+c*(d+e*x)^4)/(a^4*e*f^3)],

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

# Integrands of the form (d+e x)^m (a+b (d+e x)^3+c (d+e x)^6)^p
[x/sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6),x,7,-d*(d+e*x)*AppellF1(1/3,1/2,1/2,4/3,-2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)),-2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))/(e^2*sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6))+1/2*(d+e*x)^2*AppellF1(2/3,1/2,1/2,5/3,-2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)),-2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))/(e^2*sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6))],
[x^2/sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6),x,10,1/3*arctanh(1/2*(b+2*c*(d+e*x)^3)/(sqrt(c)*sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6)))/(e^3*sqrt(c))+d^2*(d+e*x)*AppellF1(1/3,1/2,1/2,4/3,-2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)),-2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))/(e^3*sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6))-d*(d+e*x)^2*AppellF1(2/3,1/2,1/2,5/3,-2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)),-2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*(d+e*x)^3/(b+sqrt(b^2-4*a*c)))/(e^3*sqrt(a+b*(d+e*x)^3+c*(d+e*x)^6))],

# Integrands of the form (d+e x)^m (a+b (d+e x)^7+c (d+e x)^14)^p

# Integrands of the form (d+e x)^m (a+b (d+e x)^7+c (d+e x)^14)^p
[(2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14),x,3,1/21*(2+3*x)^7+1/42*(2+3*x)^14+1/63*(2+3*x)^21],
[(2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14)^2,x,4,1/21*(2+3*x)^7+1/21*(2+3*x)^14+1/21*(2+3*x)^21+1/42*(2+3*x)^28+1/105*(2+3*x)^35]]:

# Integrands of the form (d+e x)^m (a+b (d+e x)^7+c (d+e x)^14)^(p/2)
