# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.3 General\1.1.3.8 P(x) (c x)^m (a+b x^n)^p.txt"

lst:=[

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

# Integrands of the form Pq(x) (a+b x^1)^p

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

# p>0

# p<0
[(c+d*x+e*x^2)/sqrt(a+b*x),x,2,2/3*(b*d-2*a*e)*(a+b*x)^(3/2)/b^3+2/5*e*(a+b*x)^(5/2)/b^3+2*(b^2*c-a*b*d+a^2*e)*sqrt(a+b*x)/b^3],
[(c+d*x+e*x^2)^2/sqrt(a+b*x),x,2,4/3*(b*d-2*a*e)*(b^2*c-a*b*d+a^2*e)*(a+b*x)^(3/2)/b^5-2/5*(6*a*b*d*e-6*a^2*e^2-b^2*(d^2+2*c*e))*(a+b*x)^(5/2)/b^5+4/7*e*(b*d-2*a*e)*(a+b*x)^(7/2)/b^5+2/9*e^2*(a+b*x)^(9/2)/b^5+2*(b^2*c-a*b*d+a^2*e)^2*sqrt(a+b*x)/b^5],
[(c+d*x+e*x^2)^3/sqrt(a+b*x),x,2,2*(b*d-2*a*e)*(b^2*c-a*b*d+a^2*e)^2*(a+b*x)^(3/2)/b^7-6/5*(b^2*c-a*b*d+a^2*e)*(5*a*b*d*e-5*a^2*e^2-b^2*(d^2+c*e))*(a+b*x)^(5/2)/b^7-2/7*(b*d-2*a*e)*(10*a*b*d*e-10*a^2*e^2-b^2*(d^2+6*c*e))*(a+b*x)^(7/2)/b^7-2/3*e*(5*a*b*d*e-5*a^2*e^2-b^2*(d^2+c*e))*(a+b*x)^(9/2)/b^7+6/11*e^2*(b*d-2*a*e)*(a+b*x)^(11/2)/b^7+2/13*e^3*(a+b*x)^(13/2)/b^7+2*(b^2*c-a*b*d+a^2*e)^3*sqrt(a+b*x)/b^7],
[(c+d*x+e*x^2+f*x^3)/sqrt(a+b*x),x,2,2/3*(b^2*d-2*a*b*e+3*a^2*f)*(a+b*x)^(3/2)/b^4+2/5*(b*e-3*a*f)*(a+b*x)^(5/2)/b^4+2/7*f*(a+b*x)^(7/2)/b^4+2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*sqrt(a+b*x)/b^4],
[(c+d*x+e*x^2+f*x^3)^2/sqrt(a+b*x),x,2,4/3*(b^2*d-2*a*b*e+3*a^2*f)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*(a+b*x)^(3/2)/b^7+2/5*(b^4*(d^2+2*c*e)-20*a^3*b*e*f+15*a^4*f^2-6*a*b^3*(d*e+c*f)+6*a^2*b^2*(e^2+2*d*f))*(a+b*x)^(5/2)/b^7+4/7*(10*a^2*b*e*f-10*a^3*f^2+b^3*(d*e+c*f)-2*a*b^2*(e^2+2*d*f))*(a+b*x)^(7/2)/b^7-2/9*(10*a*b*e*f-15*a^2*f^2-b^2*(e^2+2*d*f))*(a+b*x)^(9/2)/b^7+4/11*f*(b*e-3*a*f)*(a+b*x)^(11/2)/b^7+2/13*f^2*(a+b*x)^(13/2)/b^7+2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)^2*sqrt(a+b*x)/b^7],
[(c+d*x+e*x^2+f*x^3)^3/sqrt(a+b*x),x,2,2*(b^2*d-2*a*b*e+3*a^2*f)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)^2*(a+b*x)^(3/2)/b^10+6/5*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*(b^4*(d^2+c*e)-16*a^3*b*e*f+12*a^4*f^2-a*b^3*(5*d*e+3*c*f)+a^2*b^2*(5*e^2+9*d*f))*(a+b*x)^(5/2)/b^10-2/7*(168*a^5*b*e*f^2-84*a^6*f^3-b^6*(d^3+6*c*d*e+3*c^2*f)-105*a^4*b^2*f*(e^2+d*f)+12*a*b^5*(d^2*e+c*e^2+2*c*d*f)-30*a^2*b^4*(d*e^2+d^2*f+2*c*e*f)+20*a^3*b^3*(e^3+6*d*e*f+3*c*f^2))*(a+b*x)^(7/2)/b^10+2/3*(70*a^4*b*e*f^2-42*a^5*f^3-35*a^3*b^2*f*(e^2+d*f)+b^5*(d^2*e+c*e^2+2*c*d*f)-5*a*b^4*(d*e^2+d^2*f+2*c*e*f)+5*a^2*b^3*(e^3+6*d*e*f+3*c*f^2))*(a+b*x)^(9/2)/b^10-6/11*(56*a^3*b*e*f^2-42*a^4*f^3-21*a^2*b^2*f*(e^2+d*f)-b^4*(d*e^2+d^2*f+2*c*e*f)+2*a*b^3*(e^3+6*d*e*f+3*c*f^2))*(a+b*x)^(11/2)/b^10+2/13*(84*a^2*b*e*f^2-84*a^3*f^3-21*a*b^2*f*(e^2+d*f)+b^3*(e^3+6*d*e*f+3*c*f^2))*(a+b*x)^(13/2)/b^10-2/5*f*(8*a*b*e*f-12*a^2*f^2-b^2*(e^2+d*f))*(a+b*x)^(15/2)/b^10+6/17*f^2*(b*e-3*a*f)*(a+b*x)^(17/2)/b^10+2/19*f^3*(a+b*x)^(19/2)/b^10+2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)^3*sqrt(a+b*x)/b^10],

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

# Integrands of the form P1(x) (a+b x^3)^p
[(c+d*x)/(a+b*x^3),x,6,1/3*(b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(2/3))-1/6*(c-a^(1/3)*d/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))-(b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*sqrt(3))],
[(c+d*x)/(a+b*x^3)^2,x,7,1/3*x*(c+d*x)/(a*(a+b*x^3))+1/9*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(2/3))-1/18*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(2/3))-1/3*(2*b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(2/3)*sqrt(3))],
[(c+d*x)/(a+b*x^3)^3,x,8,1/6*x*(c+d*x)/(a*(a+b*x^3)^2)+1/18*x*(5*c+4*d*x)/(a^2*(a+b*x^3))+1/27*(5*b^(1/3)*c-2*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(2/3))-1/54*(5*b^(1/3)*c-2*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(2/3))-1/9*(5*b^(1/3)*c+2*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(2/3)*sqrt(3))],
[(c+d*x)/(a+b*x^3)^4,x,9,1/9*x*(c+d*x)/(a*(a+b*x^3)^3)+1/54*x*(8*c+7*d*x)/(a^2*(a+b*x^3)^2)+2/81*x*(10*c+7*d*x)/(a^3*(a+b*x^3))+2/243*(20*b^(1/3)*c-7*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(2/3))-1/243*(20*b^(1/3)*c-7*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(2/3))-2/81*(20*b^(1/3)*c+7*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(2/3)*sqrt(3))],
[(a+b*x)/(d+e*x^3),x,6,-1/3*(b*d^(1/3)-a*e^(1/3))*log(d^(1/3)+e^(1/3)*x)/(d^(2/3)*e^(2/3))-1/6*(a-b*d^(1/3)/e^(1/3))*log(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(2/3)*e^(1/3))-(b*d^(1/3)+a*e^(1/3))*arctan((d^(1/3)-2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(2/3)*e^(2/3)*sqrt(3))],
[(a+b*x)/(d-e*x^3),x,6,-1/3*(b*d^(1/3)+a*e^(1/3))*log(d^(1/3)-e^(1/3)*x)/(d^(2/3)*e^(2/3))+1/6*(b*d^(1/3)+a*e^(1/3))*log(d^(2/3)+d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(2/3)*e^(2/3))-(b*d^(1/3)-a*e^(1/3))*arctan((d^(1/3)+2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(2/3)*e^(2/3)*sqrt(3))],
[(1+x)/(1+x^3),x,3,-2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[(1-x)/(1-x^3),x,3,2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(1+x)/(1-x^3),x,3,-2/3*log(1-x)+1/3*log(1+x+x^2)],
[(1-x)/(1+x^3),x,3,2/3*log(1+x)-1/3*log(1-x+x^2)],
[(3-x)/(1-x^3),x,6,-2/3*log(1-x)+1/3*log(1+x+x^2)+4*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(c+d*x)/(c^3+d^3*x^3),x,3,-2*arctan((c-2*d*x)/(c*sqrt(3)))/(c*d*sqrt(3))],
[(c-d*x)/(c^3-d^3*x^3),x,3,2*arctan((c+2*d*x)/(c*sqrt(3)))/(c*d*sqrt(3))],
[(a^(1/3)*b^(1/3)*B+b^(2/3)*B*x)/(a+b*x^3),x,3,-2*B*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*sqrt(3))],
[(a^(1/3)*(-b)^(1/3)*B-(-b)^(2/3)*B*x)/(a+b*x^3),x,3,2*B*arctan((a^(1/3)+2*(-b)^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*sqrt(3))],

# Integrands of the form P2(x) (a+b x^3)^p
[-C*x^2/(a+b*x^3)+(B*x+C*x^2)/(a+b*x^3),x,12,-1/3*B*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(2/3))+1/6*B*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(2/3))-B*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*sqrt(3))],
[-C*x^2/(a+b*x^3)+(A+C*x^2)/(a+b*x^3),x,11,1/3*A*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(1/3))-1/6*A*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))-A*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*sqrt(3))],
[-C*x^2/(a+b*x^3)+(A+B*x+C*x^2)/(a+b*x^3),x,10,1/3*(A*b^(1/3)-a^(1/3)*B)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(2/3))-1/6*(A-a^(1/3)*B/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))-(A*b^(1/3)+a^(1/3)*B)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*sqrt(3))],
[(b*x+c*x^2)/(d+e*x^3),x,10,-1/3*b*log(d^(1/3)+e^(1/3)*x)/(d^(1/3)*e^(2/3))+1/6*b*log(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(1/3)*e^(2/3))+1/3*c*log(d+e*x^3)/e-b*arctan((d^(1/3)-2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(1/3)*e^(2/3)*sqrt(3))],
[(a+c*x^2)/(d-e*x^3),x,9,-1/3*a*log(d^(1/3)-e^(1/3)*x)/(d^(2/3)*e^(1/3))+1/6*a*log(d^(2/3)+d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/(d^(2/3)*e^(1/3))-1/3*c*log(d-e*x^3)/e+a*arctan((d^(1/3)+2*e^(1/3)*x)/(d^(1/3)*sqrt(3)))/(d^(2/3)*e^(1/3)*sqrt(3))],
[(2*a^2+b^2*x^2)/(a^3+b^3*x^3),x,4,log(a+b*x)/b-2*arctan((a-2*b*x)/(a*sqrt(3)))/(b*sqrt(3))],
[(2*a^2+b^2*x^2)/(a^3-b^3*x^3),x,4,-log(a-b*x)/b+2*arctan((a+2*b*x)/(a*sqrt(3)))/(b*sqrt(3))],
[(8*C+b^(2/3)*C*x^2)/(8+b*x^3),x,4,C*log(2+b^(1/3)*x)/b^(1/3)-2*C*arctan((1-b^(1/3)*x)/sqrt(3))/(b^(1/3)*sqrt(3))],
[(a^(2/3)*C+2*C*x^2)/(a+8*x^3),x,4,1/4*C*log(a^(1/3)+2*x)-1/2*C*arctan((a^(1/3)-4*x)/(a^(1/3)*sqrt(3)))/sqrt(3)],
[(8*C+(-b)^(2/3)*C*x^2)/(-8+b*x^3),x,4,-C*log(2+(-b)^(1/3)*x)/(-b)^(1/3)+2*C*arctan((1-(-b)^(1/3)*x)/sqrt(3))/((-b)^(1/3)*sqrt(3))],
[((-a)^(2/3)*C+2*C*x^2)/(a-8*x^3),x,4,-1/4*C*log((-a)^(1/3)+2*x)+1/2*C*arctan((1-4*x/(-a)^(1/3))/sqrt(3))/sqrt(3)],
[(2*(a/b)^(2/3)*C+C*x^2)/(a+b*x^3),x,4,C*log((a/b)^(1/3)+x)/b-2*C*arctan((1-2*x/(a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[(2*(-a/b)^(2/3)*C+C*x^2)/(a-b*x^3),x,4,-C*log((-a/b)^(1/3)+x)/b+2*C*arctan((1-2*x/(-a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[(2*(-a/b)^(2/3)*C+C*x^2)/(a+b*x^3),x,4,C*log((-a/b)^(1/3)-x)/b-2*C*arctan((1+2*x/(-a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[(2*(a/b)^(2/3)*C+C*x^2)/(a-b*x^3),x,4,-C*log((a/b)^(1/3)-x)/b+2*C*arctan((1+2*x/(a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[(2*a^(2/3)*C+b^(2/3)*C*x^2)/(a+b*x^3),x,4,C*log(a^(1/3)+b^(1/3)*x)/b^(1/3)-2*C*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(1/3)*sqrt(3))],
[(-2*a^(2/3)*C-(-b)^(2/3)*C*x^2)/(a+b*x^3),x,4,C*log(a^(1/3)-(-b)^(1/3)*x)/(-b)^(1/3)-2*C*arctan((a^(1/3)+2*(-b)^(1/3)*x)/(a^(1/3)*sqrt(3)))/((-b)^(1/3)*sqrt(3))],
[(-3+x^2)/(-1+x^3),x,6,-2/3*log(1-x)+5/6*log(1+x+x^2)+arctan((1+2*x)/sqrt(3))*sqrt(3)],
[(a^(1/3)*b^(1/3)*B+2*a^(2/3)*C+b^(2/3)*B*x+b^(2/3)*C*x^2)/(a+b*x^3),x,4,C*log(a^(1/3)+b^(1/3)*x)/b^(1/3)-2*(B/a^(1/3)+C/b^(1/3))*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/sqrt(3)],
[(a^(1/3)*(-b)^(1/3)*B-2*a^(2/3)*C-(-b)^(2/3)*B*x-(-b)^(2/3)*C*x^2)/(a+b*x^3),x,4,C*log(a^(1/3)-(-b)^(1/3)*x)/(-b)^(1/3)+2*(b*B+a^(1/3)*(-b)^(2/3)*C)*arctan((a^(1/3)+2*(-b)^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b*sqrt(3))],
[(B^2+B*C*x+C^2*x^2)/(-B^3+C^3*x^3),x,2,log(B-C*x)/C],
[(a^(2/3)*C-a^(1/3)*b^(1/3)*C*x+b^(2/3)*C*x^2)/(a+b*x^3),x,2,C*log(a^(1/3)+b^(1/3)*x)/b^(1/3)],
[((a/b)^(1/3)*B+2*(a/b)^(2/3)*C+B*x+C*x^2)/(a+b*x^3),x,4,C*log((a/b)^(1/3)+x)/b-2*(a/b)^(2/3)*(B+(a/b)^(1/3)*C)*arctan((1-2*x/(a/b)^(1/3))/sqrt(3))/(a*sqrt(3))],
[((-a/b)^(1/3)*B+2*(-a/b)^(2/3)*C+B*x+C*x^2)/(a-b*x^3),x,4,-C*log((-a/b)^(1/3)+x)/b+2*(B+(-a/b)^(1/3)*C)*arctan((1-2*x/(-a/b)^(1/3))/sqrt(3))/((-a/b)^(1/3)*b*sqrt(3))],
[(-(-a/b)^(1/3)*B+2*(-a/b)^(2/3)*C+B*x+C*x^2)/(a+b*x^3),x,4,C*log((-a/b)^(1/3)-x)/b+2*(B-(-a/b)^(1/3)*C)*arctan((1+2*x/(-a/b)^(1/3))/sqrt(3))/((-a/b)^(1/3)*b*sqrt(3))],
[(-(a/b)^(1/3)*B+2*(a/b)^(2/3)*C+B*x+C*x^2)/(a-b*x^3),x,4,-C*log((a/b)^(1/3)-x)/b-2*(a/b)^(2/3)*(B-(a/b)^(1/3)*C)*arctan((1+2*x/(a/b)^(1/3))/sqrt(3))/(a*sqrt(3))],
[(a+a*x+c*x^2)/(1-x^3),x,3,-1/3*(2*a+c)*log(1-x)+1/3*(a-c)*log(1+x+x^2)],
[(a+b*x+c*x^2)/(1-x^3),x,6,-1/3*(a+b+c)*log(1-x)+1/6*(a+b-2*c)*log(1+x+x^2)+(a-b)*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(1+x+x^2)/(1-x^3),x,2,-log(1-x)],
[(1-x+3*x^2)/(1-x^3),x,5,-log(1-x^3)+2*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(1+x+4*x^2)/(1-x^3),x,3,-2*log(1-x)-log(1+x+x^2)],

# Integrands of the form P4(x) (a+b x^3)^p
[(a+b*x^3)^3*(a*c+a*d*x+b*c*x^3+b*d*x^4),x,2,a^4*c*x+1/2*a^4*d*x^2+a^3*b*c*x^4+4/5*a^3*b*d*x^5+6/7*a^2*b^2*c*x^7+3/4*a^2*b^2*d*x^8+2/5*a*b^3*c*x^10+4/11*a*b^3*d*x^11+1/13*b^4*c*x^13+1/14*b^4*d*x^14],
[(a+b*x^3)^2*(a*c+a*d*x+b*c*x^3+b*d*x^4),x,2,a^3*c*x+1/2*a^3*d*x^2+3/4*a^2*b*c*x^4+3/5*a^2*b*d*x^5+3/7*a*b^2*c*x^7+3/8*a*b^2*d*x^8+1/10*b^3*c*x^10+1/11*b^3*d*x^11],
[(a+b*x^3)*(a*c+a*d*x+b*c*x^3+b*d*x^4),x,2,a^2*c*x+1/2*a^2*d*x^2+1/2*a*b*c*x^4+2/5*a*b*d*x^5+1/7*b^2*c*x^7+1/8*b^2*d*x^8],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3),x,2,c*x+1/2*d*x^2],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^2,x,7,1/3*(b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(2/3))-1/6*(c-a^(1/3)*d/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))-(b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*sqrt(3))],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^3,x,8,1/3*x*(c+d*x)/(a*(a+b*x^3))+1/9*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(2/3))-1/18*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(2/3))-1/3*(2*b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(2/3)*sqrt(3))],

# Integrands of the form P4(x) (a+b x^3)^(p/2)
[(a+b*x^3)^(3/2)*(a*c+a*d*x+b*c*x^3+b*d*x^4),x,7,30/46189*a*(247*c*x+187*d*x^2)*(a+b*x^3)^(3/2)+2/323*(19*c*x+17*d*x^2)*(a+b*x^3)^(5/2)+54/323323*a^2*(1729*c*x+935*d*x^2)*sqrt(a+b*x^3)+810/1729*a^3*d*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-405/1729*3^(1/4)*a^(10/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+54/323323*3^(3/4)*a^3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(1729*b^(1/3)*c-935*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(1/2)*(a*c+a*d*x+b*c*x^3+b*d*x^4),x,6,2/143*(13*c*x+11*d*x^2)*(a+b*x^3)^(3/2)+18/5005*a*(91*c*x+55*d*x^2)*sqrt(a+b*x^3)+54/91*a^2*d*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/91*3^(1/4)*a^(7/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+18/5005*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(91*b^(1/3)*c-55*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^(1/2),x,5,2/35*(7*c*x+5*d*x^2)*sqrt(a+b*x^3)+6/7*a*d*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/7*3^(1/4)*a^(4/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/35*3^(3/4)*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*c-5*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^(3/2),x,4,2*d*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*c-a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^(5/2),x,5,2/3*x*(c+d*x)/(a*sqrt(a+b*x^3))-2/3*d*sqrt(a+b*x^3)/(a*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*c+a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^(7/2),x,6,2/9*x*(c+d*x)/(a*(a+b*x^3)^(3/2))+2/27*x*(7*c+5*d*x)/(a^2*sqrt(a+b*x^3))-10/27*d*sqrt(a+b*x^3)/(a^2*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+5/9*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(5/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/27*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*c+5*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^2*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a*c+a*d*x+b*c*x^3+b*d*x^4)/(a+b*x^3)^(9/2),x,7,2/15*x*(c+d*x)/(a*(a+b*x^3)^(5/2))+2/135*x*(13*c+11*d*x)/(a^2*(a+b*x^3)^(3/2))+2/405*x*(91*c+55*d*x)/(a^3*sqrt(a+b*x^3))-22/81*d*sqrt(a+b*x^3)/(a^3*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+11/27*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(8/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/405*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(91*b^(1/3)*c+55*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^3*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^3)^(1/2),x,7,2/3*e*sqrt(a+b*x^3)/b+2/5*f*x*sqrt(a+b*x^3)/b+2/7*g*x^2*sqrt(a+b*x^3)/b+2/7*(7*b*d-4*a*g)*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-1/7*3^(1/4)*a^(1/3)*(7*b*d-4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/35*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*(5*b*c-2*a*f)-5*a^(1/3)*(7*b*d-4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^3)^(3/2),x,6,2/3*x*(b*c-a*f+(b*d-a*g)*x+b*e*x^2)/(a*b*sqrt(a+b*x^3))-2/3*e*sqrt(a+b*x^3)/(a*b)-2/3*(b*d-4*a*g)*sqrt(a+b*x^3)/(a*b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+(b*d-4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*(b*c+2*a*f)+a^(1/3)*(b*d-4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^3)^(5/2),x,5,2/9*x*(b*c-a*f+(b*d-a*g)*x+b*e*x^2)/(a*b*(a+b*x^3)^(3/2))-2/27*(3*a*e-x*(7*b*c+2*a*f+(5*b*d+4*a*g)*x))/(a^2*b*sqrt(a+b*x^3))-2/27*(5*b*d+4*a*g)*sqrt(a+b*x^3)/(a^2*b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+1/9*(5*b*d+4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(5/3)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/27*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*(7*b*c+2*a*f)+a^(1/3)*(5*b*d+4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^2*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^3)^(7/2),x,6,2/15*x*(b*c-a*f+(b*d-a*g)*x+b*e*x^2)/(a*b*(a+b*x^3)^(5/2))-2/135*(9*a*e-x*(13*b*c+2*a*f+(11*b*d+4*a*g)*x))/(a^2*b*(a+b*x^3)^(3/2))+2/405*x*(7*(13*b*c+2*a*f)+5*(11*b*d+4*a*g)*x)/(a^3*b*sqrt(a+b*x^3))-2/81*(11*b*d+4*a*g)*sqrt(a+b*x^3)/(a^3*b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+1/27*(11*b*d+4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(8/3)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/405*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*(13*b*c+2*a*f)+5*a^(1/3)*(11*b*d+4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^3*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],

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

# p>0

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

# Integrands of the form (c+d x) / (a+b x^3)^(p/2) with b c^3 - 2 (5+3 Sqrt[3]) a d^3=0

# p>0

# p<0
[(1+x+sqrt(3))/sqrt(1+x^3),x,3,2*sqrt(1+x^3)/(1+x+sqrt(3))-3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+4*3^(1/4)*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1-x+sqrt(3))/sqrt(1-x^3),x,3,-2*sqrt(1-x^3)/(1-x+sqrt(3))+3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-4*3^(1/4)*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(1-x+sqrt(3))/sqrt(-1+x^3),x,1,2*sqrt(-1+x^3)/(1-x-sqrt(3))-3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(1+x+sqrt(3))/sqrt(-1-x^3),x,1,-2*sqrt(-1-x^3)/(1+x-sqrt(3))+3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/sqrt(a+b*x^3),x,3,2*sqrt(a+b*x^3)/(b^(1/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+4*3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/sqrt(a-b*x^3),x,3,-2*sqrt(a-b*x^3)/(b^(1/3)*(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+3^(1/4)*a^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticE((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-4*3^(1/4)*a^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/sqrt(-a+b*x^3),x,1,2*sqrt(-a+b*x^3)/(b^(1/3)*(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))))-3^(1/4)*a^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticE((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b^(1/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/sqrt(-a-b*x^3),x,1,-2*sqrt(-a-b*x^3)/(b^(1/3)*(b^(1/3)*x+a^(1/3)*(1-sqrt(3))))+3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b^(1/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(1+(b/a)^(1/3)*x+sqrt(3))/sqrt(a+b*x^3),x,3,2*(b/a)^(1/3)*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*(b/a)^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(-a^(1/3)*(b/a)^(1/3)*(1-sqrt(3))+b^(1/3)*(1+sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(1-(b/a)^(1/3)*x+sqrt(3))/sqrt(a-b*x^3),x,3,-2*(b/a)^(1/3)*sqrt(a-b*x^3)/(b^(2/3)*(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+3^(1/4)*a^(1/3)*(b/a)^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticE((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-2*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(-a^(1/3)*(b/a)^(1/3)*(1-sqrt(3))+b^(1/3)*(1+sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(1-(b/a)^(1/3)*x+sqrt(3))/sqrt(-a+b*x^3),x,1,2*(b/a)^(2/3)*sqrt(-a+b*x^3)/(b*(1-(b/a)^(1/3)*x-sqrt(3)))-3^(1/4)*(1-(b/a)^(1/3)*x)*EllipticE((1-(b/a)^(1/3)*x+sqrt(3))/(1-(b/a)^(1/3)*x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(b/a)^(1/3)*x+(b/a)^(2/3)*x^2)/(1-(b/a)^(1/3)*x-sqrt(3))^2)*sqrt(2+sqrt(3))/((b/a)^(1/3)*sqrt(-a+b*x^3)*sqrt((-1+(b/a)^(1/3)*x)/(1-(b/a)^(1/3)*x-sqrt(3))^2))],
[(1+(b/a)^(1/3)*x+sqrt(3))/sqrt(-a-b*x^3),x,1,-2*(b/a)^(2/3)*sqrt(-a-b*x^3)/(b*(1+(b/a)^(1/3)*x-sqrt(3)))+3^(1/4)*(1+(b/a)^(1/3)*x)*EllipticE((1+(b/a)^(1/3)*x+sqrt(3))/(1+(b/a)^(1/3)*x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-(b/a)^(1/3)*x+(b/a)^(2/3)*x^2)/(1+(b/a)^(1/3)*x-sqrt(3))^2)*sqrt(2+sqrt(3))/((b/a)^(1/3)*sqrt(-a-b*x^3)*sqrt((-1-(b/a)^(1/3)*x)/(1+(b/a)^(1/3)*x-sqrt(3))^2))],

# Integrands of the form (c+d x) / (a+b x^3)^(p/2) with b c^3 - 2 (5-3 Sqrt[3]) a d^3=0

# p>0

# p<0
[(1+x-sqrt(3))/sqrt(1+x^3),x,1,2*sqrt(1+x^3)/(1+x+sqrt(3))-3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1-x-sqrt(3))/sqrt(1-x^3),x,1,-2*sqrt(1-x^3)/(1-x+sqrt(3))+3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(1-x-sqrt(3))/sqrt(-1+x^3),x,3,2*sqrt(-1+x^3)/(1-x-sqrt(3))+4*3^(1/4)*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(1+x-sqrt(3))/sqrt(-1-x^3),x,3,-2*sqrt(-1-x^3)/(1+x-sqrt(3))-4*3^(1/4)*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(-1-x+sqrt(3))/sqrt(1+x^3),x,1,-2*sqrt(1+x^3)/(1+x+sqrt(3))+3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(-1+x+sqrt(3))/sqrt(1-x^3),x,1,2*sqrt(1-x^3)/(1-x+sqrt(3))-3^(1/4)*(1-x)*EllipticE((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(-1+x+sqrt(3))/sqrt(-1+x^3),x,3,-2*sqrt(-1+x^3)/(1-x-sqrt(3))-4*3^(1/4)*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))+3^(1/4)*(1-x)*EllipticE((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(-1-x+sqrt(3))/sqrt(-1-x^3),x,3,2*sqrt(-1-x^3)/(1+x-sqrt(3))+4*3^(1/4)*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))-3^(1/4)*(1+x)*EllipticE((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/sqrt(a+b*x^3),x,1,2*sqrt(a+b*x^3)/(b^(1/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/sqrt(a-b*x^3),x,1,-2*sqrt(a-b*x^3)/(b^(1/3)*(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+3^(1/4)*a^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticE((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/sqrt(-a+b*x^3),x,3,2*sqrt(-a+b*x^3)/(b^(1/3)*(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))))+4*3^(1/4)*a^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(b^(1/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))-3^(1/4)*a^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticE((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b^(1/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/sqrt(-a-b*x^3),x,3,-2*sqrt(-a-b*x^3)/(b^(1/3)*(b^(1/3)*x+a^(1/3)*(1-sqrt(3))))-4*3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(b^(1/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))+3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b^(1/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(1+(b/a)^(1/3)*x-sqrt(3))/sqrt(a+b*x^3),x,1,2*(b/a)^(2/3)*sqrt(a+b*x^3)/(b*(1+(b/a)^(1/3)*x+sqrt(3)))-3^(1/4)*(1+(b/a)^(1/3)*x)*EllipticE((1+(b/a)^(1/3)*x-sqrt(3))/(1+(b/a)^(1/3)*x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-(b/a)^(1/3)*x+(b/a)^(2/3)*x^2)/(1+(b/a)^(1/3)*x+sqrt(3))^2)/((b/a)^(1/3)*sqrt(a+b*x^3)*sqrt((1+(b/a)^(1/3)*x)/(1+(b/a)^(1/3)*x+sqrt(3))^2))],
[(1-(b/a)^(1/3)*x-sqrt(3))/sqrt(a-b*x^3),x,1,-2*(b/a)^(2/3)*sqrt(a-b*x^3)/(b*(1-(b/a)^(1/3)*x+sqrt(3)))+3^(1/4)*(1-(b/a)^(1/3)*x)*EllipticE((1-(b/a)^(1/3)*x-sqrt(3))/(1-(b/a)^(1/3)*x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+(b/a)^(1/3)*x+(b/a)^(2/3)*x^2)/(1-(b/a)^(1/3)*x+sqrt(3))^2)/((b/a)^(1/3)*sqrt(a-b*x^3)*sqrt((1-(b/a)^(1/3)*x)/(1-(b/a)^(1/3)*x+sqrt(3))^2))],
[(1-(b/a)^(1/3)*x-sqrt(3))/sqrt(-a+b*x^3),x,3,2*(b/a)^(1/3)*sqrt(-a+b*x^3)/(b^(2/3)*(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))))-2*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*(b^(1/3)*(1-sqrt(3))-a^(1/3)*(b/a)^(1/3)*(1+sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))-3^(1/4)*a^(1/3)*(b/a)^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticE((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(1+(b/a)^(1/3)*x-sqrt(3))/sqrt(-a-b*x^3),x,3,-2*(b/a)^(1/3)*sqrt(-a-b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1-sqrt(3))))+2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*(b^(1/3)*(1-sqrt(3))-a^(1/3)*(b/a)^(1/3)*(1+sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))+3^(1/4)*a^(1/3)*(b/a)^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],

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

# p>0

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

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

# Integrands of the form P1(x) (a+b x^4)^p

# p>0

# p<0
[(c+d*x)/(a-b*x^4),x,7,1/2*c*arctan(b^(1/4)*x/a^(1/4))/(a^(3/4)*b^(1/4))+1/2*c*arctanh(b^(1/4)*x/a^(1/4))/(a^(3/4)*b^(1/4))+1/2*d*arctanh(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],
[(c+d*x)/(a+b*x^4),x,13,-1/2*c*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*sqrt(2))+1/2*c*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*sqrt(2))-1/4*c*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(3/4)*b^(1/4)*sqrt(2))+1/4*c*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(3/4)*b^(1/4)*sqrt(2))+1/2*d*arctan(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],
[(c+d*x)/(a-b*x^4)^2,x,8,1/4*x*(c+d*x)/(a*(a-b*x^4))+3/8*c*arctan(b^(1/4)*x/a^(1/4))/(a^(7/4)*b^(1/4))+3/8*c*arctanh(b^(1/4)*x/a^(1/4))/(a^(7/4)*b^(1/4))+1/4*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))],
[(c+d*x)/(a+b*x^4)^2,x,14,1/4*x*(c+d*x)/(a*(a+b*x^4))-3/8*c*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*b^(1/4)*sqrt(2))+3/8*c*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*b^(1/4)*sqrt(2))-3/16*c*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(7/4)*b^(1/4)*sqrt(2))+3/16*c*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(7/4)*b^(1/4)*sqrt(2))+1/4*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))],
[(c+d*x)/(a-b*x^4)^3,x,9,1/8*x*(c+d*x)/(a*(a-b*x^4)^2)+1/32*x*(7*c+6*d*x)/(a^2*(a-b*x^4))+21/64*c*arctan(b^(1/4)*x/a^(1/4))/(a^(11/4)*b^(1/4))+21/64*c*arctanh(b^(1/4)*x/a^(1/4))/(a^(11/4)*b^(1/4))+3/16*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))],
[(c+d*x)/(a+b*x^4)^3,x,15,1/8*x*(c+d*x)/(a*(a+b*x^4)^2)+1/32*x*(7*c+6*d*x)/(a^2*(a+b*x^4))-21/64*c*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*b^(1/4)*sqrt(2))+21/64*c*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*b^(1/4)*sqrt(2))-21/128*c*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(11/4)*b^(1/4)*sqrt(2))+21/128*c*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(11/4)*b^(1/4)*sqrt(2))+3/16*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))],
[(c+d*x)/(a-b*x^4)^4,x,10,1/12*x*(c+d*x)/(a*(a-b*x^4)^3)+1/96*x*(11*c+10*d*x)/(a^2*(a-b*x^4)^2)+1/384*x*(77*c+60*d*x)/(a^3*(a-b*x^4))+77/256*c*arctan(b^(1/4)*x/a^(1/4))/(a^(15/4)*b^(1/4))+77/256*c*arctanh(b^(1/4)*x/a^(1/4))/(a^(15/4)*b^(1/4))+5/32*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))],
[(c+d*x)/(a+b*x^4)^4,x,16,1/12*x*(c+d*x)/(a*(a+b*x^4)^3)+1/96*x*(11*c+10*d*x)/(a^2*(a+b*x^4)^2)+1/384*x*(77*c+60*d*x)/(a^3*(a+b*x^4))-77/256*c*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(15/4)*b^(1/4)*sqrt(2))+77/256*c*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))/(a^(15/4)*b^(1/4)*sqrt(2))-77/512*c*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(15/4)*b^(1/4)*sqrt(2))+77/512*c*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))/(a^(15/4)*b^(1/4)*sqrt(2))+5/32*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))],
[(c+d*x)/(1-x^4),x,7,1/2*c*arctan(x)+1/2*c*arctanh(x)+1/2*d*arctanh(x^2)],
[(c+d*x)/(1+x^4),x,13,1/2*d*arctan(x^2)-1/2*c*arctan(1-x*sqrt(2))/sqrt(2)+1/2*c*arctan(1+x*sqrt(2))/sqrt(2)-1/4*c*log(1+x^2-x*sqrt(2))/sqrt(2)+1/4*c*log(1+x^2+x*sqrt(2))/sqrt(2)],

# Integrands of the form P2(x) (a+b x^4)^p

# p>0

# p<0
[(c+d*x+e*x^2)/(a-b*x^4),x,7,1/2*d*arctanh(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))+1/2*arctan(b^(1/4)*x/a^(1/4))*(-e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4))],
[(c+d*x+e*x^2)/(a+b*x^4),x,13,1/2*d*arctan(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))-1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))],
[(c+d*x+e*x^2)/(a-b*x^4)^2,x,8,1/4*x*(c+d*x+e*x^2)/(a*(a-b*x^4))+1/4*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))+1/8*arctan(b^(1/4)*x/a^(1/4))*(-e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4))+1/8*arctanh(b^(1/4)*x/a^(1/4))*(e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4))],
[(c+d*x+e*x^2)/(a+b*x^4)^2,x,14,1/4*x*(c+d*x+e*x^2)/(a*(a+b*x^4))+1/4*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))],
[(c+d*x+e*x^2)/(a-b*x^4)^3,x,9,1/8*x*(c+d*x+e*x^2)/(a*(a-b*x^4)^2)+1/32*x*(7*c+6*d*x+5*e*x^2)/(a^2*(a-b*x^4))+3/16*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))+1/64*arctan(b^(1/4)*x/a^(1/4))*(-5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4))+1/64*arctanh(b^(1/4)*x/a^(1/4))*(5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4))],
[(c+d*x+e*x^2)/(a+b*x^4)^3,x,15,1/8*x*(c+d*x+e*x^2)/(a*(a+b*x^4)^2)+1/32*x*(7*c+6*d*x+5*e*x^2)/(a^2*(a+b*x^4))+3/16*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))-1/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))+1/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))-1/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))+1/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))],
[(c+d*x+e*x^2)/(a-b*x^4)^4,x,10,1/12*x*(c+d*x+e*x^2)/(a*(a-b*x^4)^3)+1/96*x*(11*c+10*d*x+9*e*x^2)/(a^2*(a-b*x^4)^2)+1/384*x*(77*c+60*d*x+45*e*x^2)/(a^3*(a-b*x^4))+5/32*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))+1/256*arctan(b^(1/4)*x/a^(1/4))*(-15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4))+1/256*arctanh(b^(1/4)*x/a^(1/4))*(15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4))],
[(c+d*x+e*x^2)/(a+b*x^4)^4,x,16,1/12*x*(c+d*x+e*x^2)/(a*(a+b*x^4)^3)+1/96*x*(11*c+10*d*x+9*e*x^2)/(a^2*(a+b*x^4)^2)+1/384*x*(77*c+60*d*x+45*e*x^2)/(a^3*(a+b*x^4))+5/32*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))],

# Integrands of the form P3(x) (a+b x^4)^p

# p>0
[a*(e+f*x^4)^2,x,3,a*e^2*x+2/5*a*e*f*x^5+1/9*a*f^2*x^9],
[b*x*(e+f*x^4)^2,x,3,1/2*b*e^2*x^2+1/3*b*e*f*x^6+1/10*b*f^2*x^10],
[(a+b*x)*(e+f*x^4)^2,x,2,a*e^2*x+1/2*b*e^2*x^2+2/5*a*e*f*x^5+1/3*b*e*f*x^6+1/9*a*f^2*x^9+1/10*b*f^2*x^10],
[c*x^2*(e+f*x^4)^2,x,3,1/3*c*e^2*x^3+2/7*c*e*f*x^7+1/11*c*f^2*x^11],
[(a+c*x^2)*(e+f*x^4)^2,x,2,a*e^2*x+1/3*c*e^2*x^3+2/5*a*e*f*x^5+2/7*c*e*f*x^7+1/9*a*f^2*x^9+1/11*c*f^2*x^11],
[(b*x+c*x^2)*(e+f*x^4)^2,x,3,1/2*b*e^2*x^2+1/3*c*e^2*x^3+1/3*b*e*f*x^6+2/7*c*e*f*x^7+1/10*b*f^2*x^10+1/11*c*f^2*x^11],
[(a+b*x+c*x^2)*(e+f*x^4)^2,x,2,a*e^2*x+1/2*b*e^2*x^2+1/3*c*e^2*x^3+2/5*a*e*f*x^5+1/3*b*e*f*x^6+2/7*c*e*f*x^7+1/9*a*f^2*x^9+1/10*b*f^2*x^10+1/11*c*f^2*x^11],
[d*x^3*(e+f*x^4)^2,x,2,1/12*d*(e+f*x^4)^3/f],
[(a+d*x^3)*(e+f*x^4)^2,x,4,a*e^2*x+2/5*a*e*f*x^5+1/9*a*f^2*x^9+1/12*d*(e+f*x^4)^3/f],
[(b*x+d*x^3)*(e+f*x^4)^2,x,4,1/2*b*e^2*x^2+1/3*b*e*f*x^6+1/10*b*f^2*x^10+1/12*d*(e+f*x^4)^3/f],
[(a+b*x+d*x^3)*(e+f*x^4)^2,x,3,a*e^2*x+1/2*b*e^2*x^2+2/5*a*e*f*x^5+1/3*b*e*f*x^6+1/9*a*f^2*x^9+1/10*b*f^2*x^10+1/12*d*(e+f*x^4)^3/f],
[(c*x^2+d*x^3)*(e+f*x^4)^2,x,4,1/3*c*e^2*x^3+2/7*c*e*f*x^7+1/11*c*f^2*x^11+1/12*d*(e+f*x^4)^3/f],
[(a+c*x^2+d*x^3)*(e+f*x^4)^2,x,3,a*e^2*x+1/3*c*e^2*x^3+2/5*a*e*f*x^5+2/7*c*e*f*x^7+1/9*a*f^2*x^9+1/11*c*f^2*x^11+1/12*d*(e+f*x^4)^3/f],
[(b*x+c*x^2+d*x^3)*(e+f*x^4)^2,x,4,1/2*b*e^2*x^2+1/3*c*e^2*x^3+1/3*b*e*f*x^6+2/7*c*e*f*x^7+1/10*b*f^2*x^10+1/11*c*f^2*x^11+1/12*d*(e+f*x^4)^3/f],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^2,x,3,a^2*c*x+1/2*a^2*d*x^2+1/3*a^2*e*x^3+2/5*a*b*c*x^5+1/3*a*b*d*x^6+2/7*a*b*e*x^7+1/9*b^2*c*x^9+1/10*b^2*d*x^10+1/11*b^2*e*x^11+1/12*f*(a+b*x^4)^3/b],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^3,x,3,a^3*c*x+1/2*a^3*d*x^2+1/3*a^3*e*x^3+3/5*a^2*b*c*x^5+1/2*a^2*b*d*x^6+3/7*a^2*b*e*x^7+1/3*a*b^2*c*x^9+3/10*a*b^2*d*x^10+3/11*a*b^2*e*x^11+1/13*b^3*c*x^13+1/14*b^3*d*x^14+1/15*b^3*e*x^15+1/16*f*(a+b*x^4)^4/b],

# p<0
[(c+d*x+e*x^2+f*x^3)/(a-b*x^4)^2,x,8,1/4*(a*f+b*x*(c+d*x+e*x^2))/(a*b*(a-b*x^4))+1/4*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))+1/8*arctan(b^(1/4)*x/a^(1/4))*(-e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4))+1/8*arctanh(b^(1/4)*x/a^(1/4))*(e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4))],
[(c+d*x+e*x^2+f*x^3)/(a-b*x^4)^3,x,9,1/32*x*(7*c+6*d*x+5*e*x^2)/(a^2*(a-b*x^4))+1/8*(a*f+b*x*(c+d*x+e*x^2))/(a*b*(a-b*x^4)^2)+3/16*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))+1/64*arctan(b^(1/4)*x/a^(1/4))*(-5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4))+1/64*arctanh(b^(1/4)*x/a^(1/4))*(5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4))],
[(c+d*x+e*x^2+f*x^3)/(a-b*x^4)^4,x,10,1/96*x*(11*c+10*d*x+9*e*x^2)/(a^2*(a-b*x^4)^2)+1/384*x*(77*c+60*d*x+45*e*x^2)/(a^3*(a-b*x^4))+1/12*(a*f+b*x*(c+d*x+e*x^2))/(a*b*(a-b*x^4)^3)+5/32*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))+1/256*arctan(b^(1/4)*x/a^(1/4))*(-15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4))+1/256*arctanh(b^(1/4)*x/a^(1/4))*(15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4))],
[a/(2+3*x^4),x,10,-1/4*a*arctan(1-6^(1/4)*x)/6^(1/4)+1/4*a*arctan(1+6^(1/4)*x)/6^(1/4)-1/8*a*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/8*a*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)],
[b*x/(2+3*x^4),x,3,1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)],
[(a+b*x)/(2+3*x^4),x,13,-1/4*a*arctan(1-6^(1/4)*x)/6^(1/4)+1/4*a*arctan(1+6^(1/4)*x)/6^(1/4)-1/8*a*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/8*a*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)],
[c*x^2/(2+3*x^4),x,10,-1/2*c*arctan(1-6^(1/4)*x)/6^(3/4)+1/2*c*arctan(1+6^(1/4)*x)/6^(3/4)+1/4*c*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)-1/4*c*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)],
[(a+c*x^2)/(2+3*x^4),x,9,-1/8*log(-6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)+1/8*log(6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)-1/4*arctan(1-6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)+1/4*arctan(1+6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)],
[(b*x+c*x^2)/(2+3*x^4),x,14,-1/2*c*arctan(1-6^(1/4)*x)/6^(3/4)+1/2*c*arctan(1+6^(1/4)*x)/6^(3/4)+1/4*c*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)-1/4*c*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)+1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)],
[(a+b*x+c*x^2)/(2+3*x^4),x,13,1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)-1/8*log(-6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)+1/8*log(6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)-1/4*arctan(1-6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)+1/4*arctan(1+6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)],
[d*x^3/(2+3*x^4),x,2,1/12*d*log(2+3*x^4)],
[(a+d*x^3)/(2+3*x^4),x,12,-1/4*a*arctan(1-6^(1/4)*x)/6^(1/4)+1/4*a*arctan(1+6^(1/4)*x)/6^(1/4)+1/12*d*log(2+3*x^4)-1/8*a*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/8*a*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)],
[(b*x+d*x^3)/(2+3*x^4),x,5,1/12*d*log(2+3*x^4)+1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)],
[(a+b*x+d*x^3)/(2+3*x^4),x,15,-1/4*a*arctan(1-6^(1/4)*x)/6^(1/4)+1/4*a*arctan(1+6^(1/4)*x)/6^(1/4)+1/12*d*log(2+3*x^4)-1/8*a*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/8*a*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(1/4)+1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)],
[(c*x^2+d*x^3)/(2+3*x^4),x,13,-1/2*c*arctan(1-6^(1/4)*x)/6^(3/4)+1/2*c*arctan(1+6^(1/4)*x)/6^(3/4)+1/12*d*log(2+3*x^4)+1/4*c*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)-1/4*c*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)],
[(a+c*x^2+d*x^3)/(2+3*x^4),x,12,1/12*d*log(2+3*x^4)-1/8*log(-6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)+1/8*log(6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)-1/4*arctan(1-6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)+1/4*arctan(1+6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)],
[(b*x+c*x^2+d*x^3)/(2+3*x^4),x,16,-1/2*c*arctan(1-6^(1/4)*x)/6^(3/4)+1/2*c*arctan(1+6^(1/4)*x)/6^(3/4)+1/12*d*log(2+3*x^4)+1/4*c*log(-6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)-1/4*c*log(6^(3/4)*x+3*x^2+sqrt(6))/6^(3/4)+1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)],
[(a+b*x+c*x^2+d*x^3)/(2+3*x^4),x,15,1/12*d*log(2+3*x^4)+1/2*b*arctan(x^2*sqrt(3/2))/sqrt(6)-1/8*log(-6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)+1/8*log(6^(3/4)*x+3*x^2+sqrt(6))*(-2*c+a*sqrt(6))/6^(3/4)-1/4*arctan(1-6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)+1/4*arctan(1+6^(1/4)*x)*(2*c+a*sqrt(6))/6^(3/4)],
[(1+x+x^2+x^3)/(1-x^4),x,2,-log(1-x)],
[(1+x+x^2+x^3)/(1+x^4),x,11,1/2*arctan(x^2)+1/4*log(1+x^4)-arctan(1-x*sqrt(2))/sqrt(2)+arctan(1+x*sqrt(2))/sqrt(2)],
[(1+x+x^2+x^3)/(a-b*x^4),x,9,-1/4*log(a-b*x^4)/b-1/2*arctan(b^(1/4)*x/a^(1/4))*(sqrt(a)-sqrt(b))/(a^(3/4)*b^(3/4))+1/2*arctanh(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(sqrt(a)+sqrt(b))/(a^(3/4)*b^(3/4))],
[(1+x+x^2+x^3)/(a+b*x^4),x,15,1/4*log(a+b*x^4)/b+1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(sqrt(a)-sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))-1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(sqrt(a)-sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))+1/2*arctan(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(sqrt(a)+sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(sqrt(a)+sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))],

# Integrands of the form P4(x) (a+b x^4)^p

# p>0

# p<0
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a-b*x^4),x,11,-g*x/b-1/4*f*log(a-b*x^4)/b+1/2*d*arctanh(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))+1/2*arctan(b^(1/4)*x/a^(1/4))*(b*c+a*g-e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(b*c+a*g+e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a-b*x^4)^2,x,8,1/4*x*(b*c+a*g+b*d*x+b*e*x^2+b*f*x^3)/(a*b*(a-b*x^4))+1/4*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))+1/8*arctan(b^(1/4)*x/a^(1/4))*(3*b*c-a*g-e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4))+1/8*arctanh(b^(1/4)*x/a^(1/4))*(3*b*c-a*g+e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a-b*x^4)^3,x,9,1/8*x*(b*c+a*g+b*d*x+b*e*x^2+b*f*x^3)/(a*b*(a-b*x^4)^2)+1/32*(4*a*f+x*(7*b*c-a*g+6*b*d*x+5*b*e*x^2))/(a^2*b*(a-b*x^4))+3/16*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))+1/64*arctan(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g-5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4))+1/64*arctanh(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g+5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a-b*x^4)^4,x,10,1/12*x*(b*c+a*g+b*d*x+b*e*x^2+b*f*x^3)/(a*b*(a-b*x^4)^3)+1/384*x*(7*(11*b*c-a*g)+60*b*d*x+45*b*e*x^2)/(a^3*b*(a-b*x^4))+1/96*(8*a*f+x*(11*b*c-a*g+10*b*d*x+9*b*e*x^2))/(a^2*b*(a-b*x^4)^2)+5/32*d*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))+1/256*arctan(b^(1/4)*x/a^(1/4))*(77*b*c-7*a*g-15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4))+1/256*arctanh(b^(1/4)*x/a^(1/4))*(77*b*c-7*a*g+15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^4),x,17,g*x/b+1/4*f*log(a+b*x^4)/b+1/2*d*arctan(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))-1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(b*c-a*g-e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(b*c-a*g-e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(b*c-a*g+e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(b*c-a*g+e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^4)^2,x,14,1/4*x*(b*c-a*g+b*d*x+b*e*x^2+b*f*x^3)/(a*b*(a+b*x^4))+1/4*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(3*b*c+a*g-e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(3*b*c+a*g-e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(3*b*c+a*g+e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(3*b*c+a*g+e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^4)^3,x,15,1/8*x*(b*c-a*g+b*d*x+b*e*x^2+b*f*x^3)/(a*b*(a+b*x^4)^2)+1/32*(-4*a*f+x*(7*b*c+a*g+6*b*d*x+5*b*e*x^2))/(a^2*b*(a+b*x^4))+3/16*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))-1/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(21*b*c+3*a*g-5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))+1/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(21*b*c+3*a*g-5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))-1/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(21*b*c+3*a*g+5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))+1/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(21*b*c+3*a*g+5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)/(a+b*x^4)^4,x,16,1/12*x*(b*c-a*g+b*d*x+b*e*x^2+b*f*x^3)/(a*b*(a+b*x^4)^3)+1/384*x*(7*(11*b*c+a*g)+60*b*d*x+45*b*e*x^2)/(a^3*b*(a+b*x^4))+1/96*(-8*a*f+x*(11*b*c+a*g+10*b*d*x+9*b*e*x^2))/(a^2*b*(a+b*x^4)^2)+5/32*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(77*b*c+7*a*g-15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(77*b*c+7*a*g-15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(77*b*c+7*a*g+15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(77*b*c+7*a*g+15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))],

# Integrands of the form Pq(x) (a+b x^4)^p with q>4

# p>0

# p<0
[(1-x^4)^3/(1+x+x^2+x^3)^3,x,2,-1/4*(1-x)^4],
[(1-x^4)^2/(1+x+x^2+x^3)^2,x,2,-1/3*(1-x)^3],
[(1-x^4)/(1+x+x^2+x^3),x,2,x-1/2*x^2],
[(1+x+x^2+x^3)/(1-x^4),x,2,-log(1-x)],
[(1+x+x^2+x^3)^2/(1-x^4)^2,x,2,1/(1-x)],
[(1+x+x^2+x^3)^3/(1-x^4)^3,x,2,1/2/(1-x)^2],
[(1+x+x^2+x^3)^4/(1-x^4)^4,x,2,1/3/(1-x)^3],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a-b*x^4),x,13,-g*x/b-1/2*h*x^2/b-1/4*f*log(a-b*x^4)/b+1/2*(b*d+a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))+1/2*arctan(b^(1/4)*x/a^(1/4))*(b*c+a*g-e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(b*c+a*g+e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a-b*x^4),x,13,-g*x/b-1/2*h*x^2/b-1/3*i*x^3/b-1/4*f*log(a-b*x^4)/b+1/2*(b*d+a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))-1/2*arctan(b^(1/4)*x/a^(1/4))*(b*e+a*i-(b*c+a*g)*sqrt(b)/sqrt(a))/(a^(1/4)*b^(7/4))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(b*e+a*i+(b*c+a*g)*sqrt(b)/sqrt(a))/(a^(1/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a-b*x^4),x,13,-g*x/b-1/2*h*x^2/b-1/3*i*x^3/b-1/4*j*x^4/b-1/4*(b*f+a*j)*log(a-b*x^4)/b^2+1/2*(b*d+a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))-1/2*arctan(b^(1/4)*x/a^(1/4))*(b*e+a*i-(b*c+a*g)*sqrt(b)/sqrt(a))/(a^(1/4)*b^(7/4))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(b*e+a*i+(b*c+a*g)*sqrt(b)/sqrt(a))/(a^(1/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^4),x,19,g*x/b+1/2*h*x^2/b+1/4*f*log(a+b*x^4)/b+1/2*(b*d-a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))-1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(b*c-a*g-e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(b*c-a*g-e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(b*c-a*g+e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(b*c-a*g+e*sqrt(a)*sqrt(b))/(a^(3/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a+b*x^4),x,19,g*x/b+1/2*h*x^2/b+1/3*i*x^3/b+1/4*f*log(a+b*x^4)/b+1/2*(b*d-a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))-1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a+b*x^4),x,19,g*x/b+1/2*h*x^2/b+1/3*i*x^3/b+1/4*j*x^4/b+1/4*(b*f-a*j)*log(a+b*x^4)/b^2+1/2*(b*d-a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))-1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e-a*i)*sqrt(a)+(b*c-a*g)*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a-b*x^4)^2,x,8,1/4*x*(b*c+a*g+(b*d+a*h)*x+b*e*x^2+b*f*x^3)/(a*b*(a-b*x^4))+1/4*(b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))+1/8*arctan(b^(1/4)*x/a^(1/4))*(3*b*c-a*g-e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4))+1/8*arctanh(b^(1/4)*x/a^(1/4))*(3*b*c-a*g+e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a-b*x^4)^2,x,8,1/4*x*(b*c+a*g+(b*d+a*h)*x+(b*e+a*i)*x^2+b*f*x^3)/(a*b*(a-b*x^4))+1/4*(b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))-1/8*arctan(b^(1/4)*x/a^(1/4))*(b*e-3*a*i-(3*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(5/4)*b^(7/4))+1/8*arctanh(b^(1/4)*x/a^(1/4))*(b*e-3*a*i+(3*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(5/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a-b*x^4)^2,x,10,1/4*x*(b*c+a*g+(b*d+a*h)*x+(b*e+a*i)*x^2+(b*f+a*j)*x^3)/(a*b*(a-b*x^4))+1/4*(b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))+1/4*j*log(a-b*x^4)/b^2-1/8*arctan(b^(1/4)*x/a^(1/4))*(b*e-3*a*i-(3*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(5/4)*b^(7/4))+1/8*arctanh(b^(1/4)*x/a^(1/4))*(b*e-3*a*i+(3*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(5/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^4)^2,x,14,1/4*x*(b*c-a*g+(b*d-a*h)*x+b*e*x^2+b*f*x^3)/(a*b*(a+b*x^4))+1/4*(b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(3*b*c+a*g-e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(3*b*c+a*g-e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(3*b*c+a*g+e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(3*b*c+a*g+e*sqrt(a)*sqrt(b))/(a^(7/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a+b*x^4)^2,x,14,1/4*x*(b*c-a*g+(b*d-a*h)*x+(b*e-a*i)*x^2+b*f*x^3)/(a*b*(a+b*x^4))+1/4*(b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a+b*x^4)^2,x,16,1/4*x*(b*c-a*g+(b*d-a*h)*x+(b*e-a*i)*x^2+(b*f-a*j)*x^3)/(a*b*(a+b*x^4))+1/4*(b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))+1/4*j*log(a+b*x^4)/b^2-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*((b*e+3*a*i)*sqrt(a)+(3*b*c+a*g)*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a-b*x^4)^3,x,9,1/8*x*(b*c+a*g+(b*d+a*h)*x+b*e*x^2+b*f*x^3)/(a*b*(a-b*x^4)^2)+1/32*(4*a*f+x*(7*b*c-a*g+2*(3*b*d-a*h)*x+5*b*e*x^2))/(a^2*b*(a-b*x^4))+1/16*(3*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))+1/64*arctan(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g-5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4))+1/64*arctanh(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g+5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a-b*x^4)^3,x,9,1/8*x*(b*c+a*g+(b*d+a*h)*x+(b*e+a*i)*x^2+b*f*x^3)/(a*b*(a-b*x^4)^2)+1/32*(4*a*f+x*(7*b*c-a*g+2*(3*b*d-a*h)*x+(5*b*e-3*a*i)*x^2))/(a^2*b*(a-b*x^4))+1/16*(3*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))-1/64*arctan(b^(1/4)*x/a^(1/4))*(5*b*e-3*a*i-3*(7*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(9/4)*b^(7/4))+1/64*arctanh(b^(1/4)*x/a^(1/4))*(5*b*e-3*a*i+3*(7*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(9/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a-b*x^4)^3,x,9,1/8*x*(b*c+a*g+(b*d+a*h)*x+(b*e+a*i)*x^2+(b*f+a*j)*x^3)/(a*b*(a-b*x^4)^2)+1/32*(4*a*(b*f-a*j)+x*(b*(7*b*c-a*g)+2*b*(3*b*d-a*h)*x+b*(5*b*e-3*a*i)*x^2))/(a^2*b^2*(a-b*x^4))+1/16*(3*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))-1/64*arctan(b^(1/4)*x/a^(1/4))*(5*b*e-3*a*i-3*(7*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(9/4)*b^(7/4))+1/64*arctanh(b^(1/4)*x/a^(1/4))*(5*b*e-3*a*i+3*(7*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(9/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^4)^3,x,15,1/8*x*(b*c-a*g+(b*d-a*h)*x+b*e*x^2+b*f*x^3)/(a*b*(a+b*x^4)^2)+1/32*(-4*a*f+x*(7*b*c+a*g+2*(3*b*d+a*h)*x+5*b*e*x^2))/(a^2*b*(a+b*x^4))+1/16*(3*b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))-1/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(21*b*c+3*a*g-5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))+1/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(21*b*c+3*a*g-5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))-1/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(21*b*c+3*a*g+5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))+1/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(21*b*c+3*a*g+5*e*sqrt(a)*sqrt(b))/(a^(11/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a+b*x^4)^3,x,15,1/8*x*(b*c-a*g+(b*d-a*h)*x+(b*e-a*i)*x^2+b*f*x^3)/(a*b*(a+b*x^4)^2)+1/32*(-4*a*f+x*(7*b*c+a*g+2*(3*b*d+a*h)*x+(5*b*e+3*a*i)*x^2))/(a^2*b*(a+b*x^4))+1/16*(3*b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))-1/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))+1/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))-1/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*((5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))+1/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*((5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a+b*x^4)^3,x,15,1/8*x*(b*c-a*g+(b*d-a*h)*x+(b*e-a*i)*x^2+(b*f-a*j)*x^3)/(a*b*(a+b*x^4)^2)+1/32*(-4*a*(b*f+a*j)+x*(b*(7*b*c+a*g)+2*b*(3*b*d+a*h)*x+b*(5*b*e+3*a*i)*x^2))/(a^2*b^2*(a+b*x^4))+1/16*(3*b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))-1/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))+1/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-(5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))-1/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*((5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))+1/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*((5*b*e+3*a*i)*sqrt(a)+3*(7*b*c+a*g)*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a-b*x^4)^4,x,10,1/12*x*(b*c+a*g+(b*d+a*h)*x+b*e*x^2+b*f*x^3)/(a*b*(a-b*x^4)^3)+1/384*x*(7*(11*b*c-a*g)+12*(5*b*d-a*h)*x+45*b*e*x^2)/(a^3*b*(a-b*x^4))+1/96*(8*a*f+x*(11*b*c-a*g+2*(5*b*d-a*h)*x+9*b*e*x^2))/(a^2*b*(a-b*x^4)^2)+1/32*(5*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))+1/256*arctan(b^(1/4)*x/a^(1/4))*(77*b*c-7*a*g-15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4))+1/256*arctanh(b^(1/4)*x/a^(1/4))*(77*b*c-7*a*g+15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a-b*x^4)^4,x,10,1/12*x*(b*c+a*g+(b*d+a*h)*x+(b*e+a*i)*x^2+b*f*x^3)/(a*b*(a-b*x^4)^3)+1/384*x*(7*(11*b*c-a*g)+12*(5*b*d-a*h)*x+15*(3*b*e-a*i)*x^2)/(a^3*b*(a-b*x^4))+1/96*(8*a*f+x*(11*b*c-a*g+2*(5*b*d-a*h)*x+3*(3*b*e-a*i)*x^2))/(a^2*b*(a-b*x^4)^2)+1/32*(5*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))+1/256*arctanh(b^(1/4)*x/a^(1/4))*(15*b*e-5*a*i+7*(11*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(13/4)*b^(7/4))+1/256*arctan(b^(1/4)*x/a^(1/4))*(-5*(3*b*e-a*i)+7*(11*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(13/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a-b*x^4)^4,x,10,1/12*x*(b*c+a*g+(b*d+a*h)*x+(b*e+a*i)*x^2+(b*f+a*j)*x^3)/(a*b*(a-b*x^4)^3)+1/384*x*(7*(11*b*c-a*g)+12*(5*b*d-a*h)*x+15*(3*b*e-a*i)*x^2)/(a^3*b*(a-b*x^4))+1/96*(4*a*(2*b*f-a*j)+x*(b*(11*b*c-a*g)+2*b*(5*b*d-a*h)*x+3*b*(3*b*e-a*i)*x^2))/(a^2*b^2*(a-b*x^4)^2)+1/32*(5*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))+1/256*arctanh(b^(1/4)*x/a^(1/4))*(15*b*e-5*a*i+7*(11*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(13/4)*b^(7/4))+1/256*arctan(b^(1/4)*x/a^(1/4))*(-5*(3*b*e-a*i)+7*(11*b*c-a*g)*sqrt(b)/sqrt(a))/(a^(13/4)*b^(7/4))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^4)^4,x,16,1/12*x*(b*c-a*g+(b*d-a*h)*x+b*e*x^2+b*f*x^3)/(a*b*(a+b*x^4)^3)+1/384*x*(7*(11*b*c+a*g)+12*(5*b*d+a*h)*x+45*b*e*x^2)/(a^3*b*(a+b*x^4))+1/96*(-8*a*f+x*(11*b*c+a*g+2*(5*b*d+a*h)*x+9*b*e*x^2))/(a^2*b*(a+b*x^4)^2)+1/32*(5*b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(77*b*c+7*a*g-15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(77*b*c+7*a*g-15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(77*b*c+7*a*g+15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(77*b*c+7*a*g+15*e*sqrt(a)*sqrt(b))/(a^(15/4)*b^(5/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/(a+b*x^4)^4,x,16,1/12*x*(b*c-a*g+(b*d-a*h)*x+(b*e-a*i)*x^2+b*f*x^3)/(a*b*(a+b*x^4)^3)+1/384*x*(7*(11*b*c+a*g)+12*(5*b*d+a*h)*x+15*(3*b*e+a*i)*x^2)/(a^3*b*(a+b*x^4))+1/96*(-8*a*f+x*(11*b*c+a*g+2*(5*b*d+a*h)*x+3*(3*b*e+a*i)*x^2))/(a^2*b*(a+b*x^4)^2)+1/32*(5*b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6+j*x^7)/(a+b*x^4)^4,x,16,1/12*x*(b*c-a*g+(b*d-a*h)*x+(b*e-a*i)*x^2+(b*f-a*j)*x^3)/(a*b*(a+b*x^4)^3)+1/384*x*(7*(11*b*c+a*g)+12*(5*b*d+a*h)*x+15*(3*b*e+a*i)*x^2)/(a^3*b*(a+b*x^4))+1/96*(-4*a*(2*b*f+a*j)+x*(b*(11*b*c+a*g)+2*b*(5*b*d+a*h)*x+3*b*(3*b*e+a*i)*x^2))/(a^2*b^2*(a+b*x^4)^2)+1/32*(5*b*d+a*h)*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(5*(3*b*e+a*i)*sqrt(a)+7*(11*b*c+a*g)*sqrt(b))/(a^(15/4)*b^(7/4)*sqrt(2))],

# Integrands of the form P1(x) (a+b x^4)^(p/2)

# p>0

# p<0
[(c+d*x)/sqrt(a+b*x^4),x,6,1/2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+1/2*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x)/sqrt(a-b*x^4),x,7,1/2*d*arctan(x^2*sqrt(b)/sqrt(a-b*x^4))/sqrt(b)+a^(1/4)*c*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(1/4)*sqrt(a-b*x^4))],
[(c+d*x)/sqrt(-a+b*x^4),x,7,1/2*d*arctanh(x^2*sqrt(b)/sqrt(-a+b*x^4))/sqrt(b)+a^(1/4)*c*EllipticF(b^(1/4)*x/a^(1/4),I)*sqrt(1-b*x^4/a)/(b^(1/4)*sqrt(-a+b*x^4))],
[(c+d*x)/sqrt(-a-b*x^4),x,6,1/2*d*arctan(x^2*sqrt(b)/sqrt(-a-b*x^4))/sqrt(b)+1/2*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*sqrt(-a-b*x^4))],

# Integrands of the form P2(x) (a+b x^4)^(p/2)

# p>0

# p<0
[(c+d*x+e*x^2)/sqrt(a+b*x^4),x,8,1/2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+e*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-a^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/2*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(e+c*sqrt(b)/sqrt(a))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],

# Integrands of the form P4(x) (a+b x^4)^(p/2)

# p>0

# p<0
[(a*g-b*g*x^4)/(a+b*x^4)^(3/2),x,1,g*x/sqrt(a+b*x^4)],
[(a*g+e*x-b*g*x^4)/(a+b*x^4)^(3/2),x,1,1/2*(2*a*g*x+e*x^2)/(a*sqrt(a+b*x^4))],
[(a*g+f*x^3-b*g*x^4)/(a+b*x^4)^(3/2),x,1,1/2*(-f+2*b*g*x)/(b*sqrt(a+b*x^4))],
[(a*g+e*x+f*x^3-b*g*x^4)/(a+b*x^4)^(3/2),x,1,1/2*(-a*f+2*a*b*g*x+b*e*x^2)/(a*b*sqrt(a+b*x^4))],
[(-1+x^4)/(1+x^4)^(3/2),x,1,-x/sqrt(1+x^4)],

# Integrands of the form P4(x) (a+b x^4)^(p/2) with q>4
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5+i*x^6)/sqrt(a+b*x^4),x,12,1/4*(2*b*d-a*h)*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/2*f*sqrt(a+b*x^4)/b+1/3*g*x*sqrt(a+b*x^4)/b+1/4*h*x^2*sqrt(a+b*x^4)/b+1/5*i*x^3*sqrt(a+b*x^4)/b+1/5*(5*b*e-3*a*i)*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))-1/5*a^(1/4)*(5*b*e-3*a*i)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))+1/30*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(15*b*e-9*a*i+5*(3*b*c-a*g)*sqrt(b)/sqrt(a))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],

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

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

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

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

# p>0

# p<0
[x^11*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,3,1/3*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^3/b^6-1/6*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^6/b^5+1/9*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^9/b^4+1/12*(b^2*d-a*b*e+a^2*f)*x^12/b^3+1/15*(b*e-a*f)*x^15/b^2+1/18*f*x^18/b-1/3*a^3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/b^7],
[x^8*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,3,-1/3*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^3/b^5+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^6/b^4+1/9*(b^2*d-a*b*e+a^2*f)*x^9/b^3+1/12*(b*e-a*f)*x^12/b^2+1/15*f*x^15/b+1/3*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/b^6],
[x^5*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,3,1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^3/b^4+1/6*(b^2*d-a*b*e+a^2*f)*x^6/b^3+1/9*(b*e-a*f)*x^9/b^2+1/12*f*x^12/b-1/3*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/b^5],
[x^2*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,3,1/3*(b^2*d-a*b*e+a^2*f)*x^3/b^3+1/6*(b*e-a*f)*x^6/b^2+1/9*f*x^9/b+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/b^4],
[(c+d*x^3+e*x^6+f*x^9)/(x*(a+b*x^3)),x,3,1/3*(b*e-a*f)*x^3/b^2+1/6*f*x^6/b+c*log(x)/a-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/(a*b^3)],
[(c+d*x^3+e*x^6+f*x^9)/(x^4*(a+b*x^3)),x,3,-1/3*c/(a*x^3)+1/3*f*x^3/b-(b*c-a*d)*log(x)/a^2+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/(a^2*b^2)],
[(c+d*x^3+e*x^6+f*x^9)/(x^7*(a+b*x^3)),x,3,-1/6*c/(a*x^6)+1/3*(b*c-a*d)/(a^2*x^3)+(b^2*c-a*b*d+a^2*e)*log(x)/a^3-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/(a^3*b)],
[(c+d*x^3+e*x^6+f*x^9)/(x^10*(a+b*x^3)),x,3,-1/9*c/(a*x^9)+1/6*(b*c-a*d)/(a^2*x^6)+1/3*(-b^2*c+a*b*d-a^2*e)/(a^3*x^3)-(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(x)/a^4+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/a^4],
[(c+d*x^3+e*x^6+f*x^9)/(x^13*(a+b*x^3)),x,3,-1/12*c/(a*x^12)+1/9*(b*c-a*d)/(a^2*x^9)+1/6*(-b^2*c+a*b*d-a^2*e)/(a^3*x^6)+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x^3)+b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(x)/a^5-1/3*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/a^5],
[(c+d*x^3+e*x^6+f*x^9)/(x^16*(a+b*x^3)),x,3,-1/15*c/(a*x^15)+1/12*(b*c-a*d)/(a^2*x^12)+1/9*(-b^2*c+a*b*d-a^2*e)/(a^3*x^9)+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x^6)-1/3*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^5*x^3)-b^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(x)/a^6+1/3*b^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a+b*x^3)/a^6],
[x^9*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,9,a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/b^6-1/4*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^4/b^5+1/7*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^7/b^4+1/10*(b^2*d-a*b*e+a^2*f)*x^10/b^3+1/13*(b*e-a*f)*x^13/b^2+1/16*f*x^16/b-1/3*a^(7/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(19/3)+1/6*a^(7/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(19/3)+a^(7/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(19/3)*sqrt(3))],
[x^7*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,9,-1/2*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/b^5+1/5*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^5/b^4+1/8*(b^2*d-a*b*e+a^2*f)*x^8/b^3+1/11*(b*e-a*f)*x^11/b^2+1/14*f*x^14/b-1/3*a^(5/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(17/3)+1/6*a^(5/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(17/3)-a^(5/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(17/3)*sqrt(3))],
[x^6*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,9,-a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/b^5+1/4*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^4/b^4+1/7*(b^2*d-a*b*e+a^2*f)*x^7/b^3+1/10*(b*e-a*f)*x^10/b^2+1/13*f*x^13/b+1/3*a^(4/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(16/3)-1/6*a^(4/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(16/3)-a^(4/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(16/3)*sqrt(3))],
[x^4*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,9,1/2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/b^4+1/5*(b^2*d-a*b*e+a^2*f)*x^5/b^3+1/8*(b*e-a*f)*x^8/b^2+1/11*f*x^11/b+1/3*a^(2/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(14/3)-1/6*a^(2/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(14/3)+a^(2/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(14/3)*sqrt(3))],
[x^3*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,9,(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/b^4+1/4*(b^2*d-a*b*e+a^2*f)*x^4/b^3+1/7*(b*e-a*f)*x^7/b^2+1/10*f*x^10/b-1/3*a^(1/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(13/3)+1/6*a^(1/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(13/3)+a^(1/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(13/3)*sqrt(3))],
[x*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,9,1/2*(b^2*d-a*b*e+a^2*f)*x^2/b^3+1/5*(b*e-a*f)*x^5/b^2+1/8*f*x^8/b-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(11/3))+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(11/3))-(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(11/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3),x,8,(b^2*d-a*b*e+a^2*f)*x/b^3+1/4*(b*e-a*f)*x^4/b^2+1/7*f*x^7/b+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(10/3))-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(10/3))-(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(10/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^2*(a+b*x^3)),x,8,-c/(a*x)+1/2*(b*e-a*f)*x^2/b^2+1/5*f*x^5/b+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(8/3))-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(8/3))+(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(8/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^3*(a+b*x^3)),x,8,-1/2*c/(a*x^2)+(b*e-a*f)*x/b^2+1/4*f*x^4/b-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(7/3))+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(7/3))+(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(7/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^5*(a+b*x^3)),x,8,-1/4*c/(a*x^4)+(b*c-a*d)/(a^2*x)+1/2*f*x^2/b-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(5/3))+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(5/3))-(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(5/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^6*(a+b*x^3)),x,8,-1/5*c/(a*x^5)+1/2*(b*c-a*d)/(a^2*x^2)+f*x/b+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(4/3))-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(4/3))-(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(4/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^8*(a+b*x^3)),x,8,-1/7*c/(a*x^7)+1/4*(b*c-a*d)/(a^2*x^4)+(-b^2*c+a*b*d-a^2*e)/(a^3*x)+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(2/3))-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(2/3))+(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(2/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^9*(a+b*x^3)),x,8,-1/8*c/(a*x^8)+1/5*(b*c-a*d)/(a^2*x^5)+1/2*(-b^2*c+a*b*d-a^2*e)/(a^3*x^2)-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(1/3))+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(1/3))+(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(1/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^11*(a+b*x^3)),x,8,-1/10*c/(a*x^10)+1/7*(b*c-a*d)/(a^2*x^7)+1/4*(-b^2*c+a*b*d-a^2*e)/(a^3*x^4)+(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x)-1/3*b^(1/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(13/3)+1/6*b^(1/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)-b^(1/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(13/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^12*(a+b*x^3)),x,8,-1/11*c/(a*x^11)+1/8*(b*c-a*d)/(a^2*x^8)+1/5*(-b^2*c+a*b*d-a^2*e)/(a^3*x^5)+1/2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x^2)+1/3*b^(2/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(14/3)-1/6*b^(2/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3)-b^(2/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(14/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^14*(a+b*x^3)),x,8,-1/13*c/(a*x^13)+1/10*(b*c-a*d)/(a^2*x^10)+1/7*(-b^2*c+a*b*d-a^2*e)/(a^3*x^7)+1/4*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x^4)-b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^5*x)+1/3*b^(4/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(16/3)-1/6*b^(4/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(16/3)+b^(4/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(16/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^15*(a+b*x^3)),x,8,-1/14*c/(a*x^14)+1/11*(b*c-a*d)/(a^2*x^11)+1/8*(-b^2*c+a*b*d-a^2*e)/(a^3*x^8)+1/5*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x^5)-1/2*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^5*x^2)-1/3*b^(5/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(17/3)+1/6*b^(5/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)+b^(5/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(17/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^17*(a+b*x^3)),x,8,-1/16*c/(a*x^16)+1/13*(b*c-a*d)/(a^2*x^13)+1/10*(-b^2*c+a*b*d-a^2*e)/(a^3*x^10)+1/7*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^4*x^7)-1/4*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^5*x^4)+b^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^6*x)-1/3*b^(7/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(19/3)+1/6*b^(7/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(19/3)-b^(7/3)*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(19/3)*sqrt(3))],
[x^11*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,3,-1/3*a*(2*b^3*c-3*a*b^2*d+4*a^2*b*e-5*a^3*f)*x^3/b^6+1/6*(b^3*c-2*a*b^2*d+3*a^2*b*e-4*a^3*f)*x^6/b^5+1/9*(b^2*d-2*a*b*e+3*a^2*f)*x^9/b^4+1/12*(b*e-2*a*f)*x^12/b^3+1/15*f*x^15/b^2+1/3*a^3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^7*(a+b*x^3))+1/3*a^2*(3*b^3*c-4*a*b^2*d+5*a^2*b*e-6*a^3*f)*log(a+b*x^3)/b^7],
[x^8*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,3,1/3*(b^3*c-2*a*b^2*d+3*a^2*b*e-4*a^3*f)*x^3/b^5+1/6*(b^2*d-2*a*b*e+3*a^2*f)*x^6/b^4+1/9*(b*e-2*a*f)*x^9/b^3+1/12*f*x^12/b^2-1/3*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^6*(a+b*x^3))-1/3*a*(2*b^3*c-3*a*b^2*d+4*a^2*b*e-5*a^3*f)*log(a+b*x^3)/b^6],
[x^5*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,3,1/3*(b^2*d-2*a*b*e+3*a^2*f)*x^3/b^4+1/6*(b*e-2*a*f)*x^6/b^3+1/9*f*x^9/b^2+1/3*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^5*(a+b*x^3))+1/3*(b^3*c-2*a*b^2*d+3*a^2*b*e-4*a^3*f)*log(a+b*x^3)/b^5],
[x^2*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,3,1/3*(b*e-2*a*f)*x^3/b^3+1/6*f*x^6/b^2+1/3*(-b^3*c+a*b^2*d-a^2*b*e+a^3*f)/(b^4*(a+b*x^3))+1/3*(b^2*d-2*a*b*e+3*a^2*f)*log(a+b*x^3)/b^4],
[(c+d*x^3+e*x^6+f*x^9)/(x*(a+b*x^3)^2),x,3,1/3*f*x^3/b^2+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a*b^3*(a+b*x^3))+c*log(x)/a^2-1/3*(b^3*c-a^2*b*e+2*a^3*f)*log(a+b*x^3)/(a^2*b^3)],
[(c+d*x^3+e*x^6+f*x^9)/(x^4*(a+b*x^3)^2),x,3,-1/3*c/(a^2*x^3)+1/3*(-b^3*c+a*b^2*d-a^2*b*e+a^3*f)/(a^2*b^2*(a+b*x^3))-(2*b*c-a*d)*log(x)/a^3+1/3*(2*b^3*c-a*b^2*d+a^3*f)*log(a+b*x^3)/(a^3*b^2)],
[(c+d*x^3+e*x^6+f*x^9)/(x^7*(a+b*x^3)^2),x,3,-1/6*c/(a^2*x^6)+1/3*(2*b*c-a*d)/(a^3*x^3)+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^3*b*(a+b*x^3))+(3*b^2*c-2*a*b*d+a^2*e)*log(x)/a^4-1/3*(3*b^2*c-2*a*b*d+a^2*e)*log(a+b*x^3)/a^4],
[(c+d*x^3+e*x^6+f*x^9)/(x^10*(a+b*x^3)^2),x,3,-1/9*c/(a^2*x^9)+1/6*(2*b*c-a*d)/(a^3*x^6)+1/3*(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x^3)+1/3*(-b^3*c+a*b^2*d-a^2*b*e+a^3*f)/(a^4*(a+b*x^3))-(4*b^3*c-3*a*b^2*d+2*a^2*b*e-a^3*f)*log(x)/a^5+1/3*(4*b^3*c-3*a*b^2*d+2*a^2*b*e-a^3*f)*log(a+b*x^3)/a^5],
[(c+d*x^3+e*x^6+f*x^9)/(x^13*(a+b*x^3)^2),x,3,-1/12*c/(a^2*x^12)+1/9*(2*b*c-a*d)/(a^3*x^9)+1/6*(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x^6)+1/3*(4*b^3*c-3*a*b^2*d+2*a^2*b*e-a^3*f)/(a^5*x^3)+1/3*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^5*(a+b*x^3))+b*(5*b^3*c-4*a*b^2*d+3*a^2*b*e-2*a^3*f)*log(x)/a^6-1/3*b*(5*b^3*c-4*a*b^2*d+3*a^2*b*e-2*a^3*f)*log(a+b*x^3)/a^6],
[x^9*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,9,-a*(2*b^3*c-3*a*b^2*d+4*a^2*b*e-5*a^3*f)*x/b^6+1/4*(b^3*c-2*a*b^2*d+3*a^2*b*e-4*a^3*f)*x^4/b^5+1/7*(b^2*d-2*a*b*e+3*a^2*f)*x^7/b^4+1/10*(b*e-2*a*f)*x^10/b^3+1/13*f*x^13/b^2-1/3*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^6*(a+b*x^3))+1/9*a^(4/3)*(7*b^3*c-10*a*b^2*d+13*a^2*b*e-16*a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(19/3)-1/18*a^(4/3)*(7*b^3*c-10*a*b^2*d+13*a^2*b*e-16*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(19/3)-1/3*a^(4/3)*(7*b^3*c-10*a*b^2*d+13*a^2*b*e-16*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(19/3)*sqrt(3))],
[x^7*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,12,1/2*(b^3*c-2*a*b^2*d+3*a^2*b*e-4*a^3*f)*x^2/b^5+1/5*(b^2*d-2*a*b*e+3*a^2*f)*x^5/b^4+1/8*(b*e-2*a*f)*x^8/b^3+1/11*f*x^11/b^2+1/3*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(b^5*(a+b*x^3))+1/9*a^(2/3)*(5*b^3*c-8*a*b^2*d+11*a^2*b*e-14*a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(17/3)-1/18*a^(2/3)*(5*b^3*c-8*a*b^2*d+11*a^2*b*e-14*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(17/3)+1/3*a^(2/3)*(5*b^3*c-8*a*b^2*d+11*a^2*b*e-14*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(17/3)*sqrt(3))],
[x^6*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,9,(b^3*c-2*a*b^2*d+3*a^2*b*e-4*a^3*f)*x/b^5+1/4*(b^2*d-2*a*b*e+3*a^2*f)*x^4/b^4+1/7*(b*e-2*a*f)*x^7/b^3+1/10*f*x^10/b^2+1/3*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^5*(a+b*x^3))-1/9*a^(1/3)*(4*b^3*c-7*a*b^2*d+10*a^2*b*e-13*a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(16/3)+1/18*a^(1/3)*(4*b^3*c-7*a*b^2*d+10*a^2*b*e-13*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(16/3)+1/3*a^(1/3)*(4*b^3*c-7*a*b^2*d+10*a^2*b*e-13*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(16/3)*sqrt(3))],
[x^4*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,11,1/2*(b^2*d-2*a*b*e+3*a^2*f)*x^2/b^4+1/5*(b*e-2*a*f)*x^5/b^3+1/8*f*x^8/b^2-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(b^4*(a+b*x^3))-1/9*(2*b^3*c-5*a*b^2*d+8*a^2*b*e-11*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(14/3))+1/18*(2*b^3*c-5*a*b^2*d+8*a^2*b*e-11*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(14/3))-1/3*(2*b^3*c-5*a*b^2*d+8*a^2*b*e-11*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(14/3)*sqrt(3))],
[x^3*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,9,(b^2*d-2*a*b*e+3*a^2*f)*x/b^4+1/4*(b*e-2*a*f)*x^4/b^3+1/7*f*x^7/b^2-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^4*(a+b*x^3))+1/9*(b^3*c-4*a*b^2*d+7*a^2*b*e-10*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(13/3))-1/18*(b^3*c-4*a*b^2*d+7*a^2*b*e-10*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(13/3))-1/3*(b^3*c-4*a*b^2*d+7*a^2*b*e-10*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(13/3)*sqrt(3))],
[x*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,10,1/2*(b*e-2*a*f)*x^2/b^3+1/5*f*x^5/b^2+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a*b^3*(a+b*x^3))-1/9*(b^3*c+2*a*b^2*d-5*a^2*b*e+8*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(11/3))+1/18*(b^3*c+2*a*b^2*d-5*a^2*b*e+8*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(11/3))-1/3*(b^3*c+2*a*b^2*d-5*a^2*b*e+8*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(11/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^2,x,9,(b*e-2*a*f)*x/b^3+1/4*f*x^4/b^2+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a*b^3*(a+b*x^3))+1/9*(2*b^3*c+a*b^2*d-4*a^2*b*e+7*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(10/3))-1/18*(2*b^3*c+a*b^2*d-4*a^2*b*e+7*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(10/3))-1/3*(2*b^3*c+a*b^2*d-4*a^2*b*e+7*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(10/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^2*(a+b*x^3)^2),x,9,-c/(a^2*x)+1/2*f*x^2/b^2-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^2*b^2*(a+b*x^3))+1/9*(4*b^3*c-a*b^2*d-2*a^2*b*e+5*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(8/3))-1/18*(4*b^3*c-a*b^2*d-2*a^2*b*e+5*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(8/3))+1/3*(4*b^3*c-a*b^2*d-2*a^2*b*e+5*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(8/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^3*(a+b*x^3)^2),x,9,-1/2*c/(a^2*x^2)+f*x/b^2-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^2*b^2*(a+b*x^3))-1/9*(5*b^3*c-2*a*b^2*d-a^2*b*e+4*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(7/3))+1/18*(5*b^3*c-2*a*b^2*d-a^2*b*e+4*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(7/3))+1/3*(5*b^3*c-2*a*b^2*d-a^2*b*e+4*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(7/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^5*(a+b*x^3)^2),x,9,-1/4*c/(a^2*x^4)+(2*b*c-a*d)/(a^3*x)+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^3*b*(a+b*x^3))-1/9*(7*b^3*c-4*a*b^2*d+a^2*b*e+2*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(5/3))+1/18*(7*b^3*c-4*a*b^2*d+a^2*b*e+2*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(5/3))-1/3*(7*b^3*c-4*a*b^2*d+a^2*b*e+2*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(5/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^6*(a+b*x^3)^2),x,9,-1/5*c/(a^2*x^5)+1/2*(2*b*c-a*d)/(a^3*x^2)+1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^3*b*(a+b*x^3))+1/9*(8*b^3*c-5*a*b^2*d+2*a^2*b*e+a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(4/3))-1/18*(8*b^3*c-5*a*b^2*d+2*a^2*b*e+a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(4/3))-1/3*(8*b^3*c-5*a*b^2*d+2*a^2*b*e+a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(4/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^8*(a+b*x^3)^2),x,9,-1/7*c/(a^2*x^7)+1/4*(2*b*c-a*d)/(a^3*x^4)+(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x)-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^4*(a+b*x^3))+1/9*(10*b^3*c-7*a*b^2*d+4*a^2*b*e-a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(13/3)*b^(2/3))-1/18*(10*b^3*c-7*a*b^2*d+4*a^2*b*e-a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(13/3)*b^(2/3))+1/3*(10*b^3*c-7*a*b^2*d+4*a^2*b*e-a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(13/3)*b^(2/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^9*(a+b*x^3)^2),x,9,-1/8*c/(a^2*x^8)+1/5*(2*b*c-a*d)/(a^3*x^5)+1/2*(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x^2)-1/3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^4*(a+b*x^3))-1/9*(11*b^3*c-8*a*b^2*d+5*a^2*b*e-2*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(14/3)*b^(1/3))+1/18*(11*b^3*c-8*a*b^2*d+5*a^2*b*e-2*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(14/3)*b^(1/3))+1/3*(11*b^3*c-8*a*b^2*d+5*a^2*b*e-2*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(14/3)*b^(1/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^11*(a+b*x^3)^2),x,9,-1/10*c/(a^2*x^10)+1/7*(2*b*c-a*d)/(a^3*x^7)+1/4*(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x^4)+(4*b^3*c-3*a*b^2*d+2*a^2*b*e-a^3*f)/(a^5*x)+1/3*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^5*(a+b*x^3))-1/9*b^(1/3)*(13*b^3*c-10*a*b^2*d+7*a^2*b*e-4*a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(16/3)+1/18*b^(1/3)*(13*b^3*c-10*a*b^2*d+7*a^2*b*e-4*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(16/3)-1/3*b^(1/3)*(13*b^3*c-10*a*b^2*d+7*a^2*b*e-4*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(16/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^12*(a+b*x^3)^2),x,9,-1/11*c/(a^2*x^11)+1/8*(2*b*c-a*d)/(a^3*x^8)+1/5*(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x^5)+1/2*(4*b^3*c-3*a*b^2*d+2*a^2*b*e-a^3*f)/(a^5*x^2)+1/3*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^5*(a+b*x^3))+1/9*b^(2/3)*(14*b^3*c-11*a*b^2*d+8*a^2*b*e-5*a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(17/3)-1/18*b^(2/3)*(14*b^3*c-11*a*b^2*d+8*a^2*b*e-5*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)-1/3*b^(2/3)*(14*b^3*c-11*a*b^2*d+8*a^2*b*e-5*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(17/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^14*(a+b*x^3)^2),x,9,-1/13*c/(a^2*x^13)+1/10*(2*b*c-a*d)/(a^3*x^10)+1/7*(-3*b^2*c+2*a*b*d-a^2*e)/(a^4*x^7)+1/4*(4*b^3*c-3*a*b^2*d+2*a^2*b*e-a^3*f)/(a^5*x^4)-b*(5*b^3*c-4*a*b^2*d+3*a^2*b*e-2*a^3*f)/(a^6*x)-1/3*b^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^6*(a+b*x^3))+1/9*b^(4/3)*(16*b^3*c-13*a*b^2*d+10*a^2*b*e-7*a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(19/3)-1/18*b^(4/3)*(16*b^3*c-13*a*b^2*d+10*a^2*b*e-7*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(19/3)+1/3*b^(4/3)*(16*b^3*c-13*a*b^2*d+10*a^2*b*e-7*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(19/3)*sqrt(3))],
[x^14*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,3,-1/3*a*(3*b^3*c-6*a*b^2*d+10*a^2*b*e-15*a^3*f)*x^3/b^7+1/6*(b^3*c-3*a*b^2*d+6*a^2*b*e-10*a^3*f)*x^6/b^6+1/9*(b^2*d-3*a*b*e+6*a^2*f)*x^9/b^5+1/12*(b*e-3*a*f)*x^12/b^4+1/15*f*x^15/b^3-1/6*a^4*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^8*(a+b*x^3)^2)+1/3*a^3*(4*b^3*c-5*a*b^2*d+6*a^2*b*e-7*a^3*f)/(b^8*(a+b*x^3))+1/3*a^2*(6*b^3*c-10*a*b^2*d+15*a^2*b*e-21*a^3*f)*log(a+b*x^3)/b^8],
[x^11*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,3,1/3*(b^3*c-3*a*b^2*d+6*a^2*b*e-10*a^3*f)*x^3/b^6+1/6*(b^2*d-3*a*b*e+6*a^2*f)*x^6/b^5+1/9*(b*e-3*a*f)*x^9/b^4+1/12*f*x^12/b^3+1/6*a^3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^7*(a+b*x^3)^2)-1/3*a^2*(3*b^3*c-4*a*b^2*d+5*a^2*b*e-6*a^3*f)/(b^7*(a+b*x^3))-1/3*a*(3*b^3*c-6*a*b^2*d+10*a^2*b*e-15*a^3*f)*log(a+b*x^3)/b^7],
[x^8*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,3,1/3*(b^2*d-3*a*b*e+6*a^2*f)*x^3/b^5+1/6*(b*e-3*a*f)*x^6/b^4+1/9*f*x^9/b^3-1/6*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^6*(a+b*x^3)^2)+1/3*a*(2*b^3*c-3*a*b^2*d+4*a^2*b*e-5*a^3*f)/(b^6*(a+b*x^3))+1/3*(b^3*c-3*a*b^2*d+6*a^2*b*e-10*a^3*f)*log(a+b*x^3)/b^6],
[x^5*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,3,1/3*(b*e-3*a*f)*x^3/b^4+1/6*f*x^6/b^3+1/6*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(b^5*(a+b*x^3)^2)+1/3*(-b^3*c+2*a*b^2*d-3*a^2*b*e+4*a^3*f)/(b^5*(a+b*x^3))+1/3*(b^2*d-3*a*b*e+6*a^2*f)*log(a+b*x^3)/b^5],
[x^2*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,3,1/3*f*x^3/b^3+1/6*(-b^3*c+a*b^2*d-a^2*b*e+a^3*f)/(b^4*(a+b*x^3)^2)+1/3*(-b^2*d+2*a*b*e-3*a^2*f)/(b^4*(a+b*x^3))+1/3*(b*e-3*a*f)*log(a+b*x^3)/b^4],
[(c+d*x^3+e*x^6+f*x^9)/(x*(a+b*x^3)^3),x,3,1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a*b^3*(a+b*x^3)^2)+1/3*(b^3*c-a^2*b*e+2*a^3*f)/(a^2*b^3*(a+b*x^3))+c*log(x)/a^3-1/3*(c/a^3-f/b^3)*log(a+b*x^3)],
[(c+d*x^3+e*x^6+f*x^9)/(x^4*(a+b*x^3)^3),x,3,-1/3*c/(a^3*x^3)+1/6*(-b^3*c+a*b^2*d-a^2*b*e+a^3*f)/(a^2*b^2*(a+b*x^3)^2)+1/3*(-2*b^3*c+a*b^2*d-a^3*f)/(a^3*b^2*(a+b*x^3))-(3*b*c-a*d)*log(x)/a^4+1/3*(3*b*c-a*d)*log(a+b*x^3)/a^4],
[(c+d*x^3+e*x^6+f*x^9)/(x^7*(a+b*x^3)^3),x,3,-1/6*c/(a^3*x^6)+1/3*(3*b*c-a*d)/(a^4*x^3)+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^3*b*(a+b*x^3)^2)+1/3*(3*b^2*c-2*a*b*d+a^2*e)/(a^4*(a+b*x^3))+(6*b^2*c-3*a*b*d+a^2*e)*log(x)/a^5-1/3*(6*b^2*c-3*a*b*d+a^2*e)*log(a+b*x^3)/a^5],
[(c+d*x^3+e*x^6+f*x^9)/(x^10*(a+b*x^3)^3),x,3,-1/9*c/(a^3*x^9)+1/6*(3*b*c-a*d)/(a^4*x^6)+1/3*(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x^3)+1/6*(-b^3*c+a*b^2*d-a^2*b*e+a^3*f)/(a^4*(a+b*x^3)^2)+1/3*(-4*b^3*c+3*a*b^2*d-2*a^2*b*e+a^3*f)/(a^5*(a+b*x^3))-(10*b^3*c-6*a*b^2*d+3*a^2*b*e-a^3*f)*log(x)/a^6+1/3*(10*b^3*c-6*a*b^2*d+3*a^2*b*e-a^3*f)*log(a+b*x^3)/a^6],
[(c+d*x^3+e*x^6+f*x^9)/(x^13*(a+b*x^3)^3),x,3,-1/12*c/(a^3*x^12)+1/9*(3*b*c-a*d)/(a^4*x^9)+1/6*(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x^6)+1/3*(10*b^3*c-6*a*b^2*d+3*a^2*b*e-a^3*f)/(a^6*x^3)+1/6*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)/(a^5*(a+b*x^3)^2)+1/3*b*(5*b^3*c-4*a*b^2*d+3*a^2*b*e-2*a^3*f)/(a^6*(a+b*x^3))+b*(15*b^3*c-10*a*b^2*d+6*a^2*b*e-3*a^3*f)*log(x)/a^7-1/3*b*(15*b^3*c-10*a*b^2*d+6*a^2*b*e-3*a^3*f)*log(a+b*x^3)/a^7],
[x^12*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,10,-a*(3*b^3*c-6*a*b^2*d+10*a^2*b*e-15*a^3*f)*x/b^7+1/4*(b^3*c-3*a*b^2*d+6*a^2*b*e-10*a^3*f)*x^4/b^6+1/7*(b^2*d-3*a*b*e+6*a^2*f)*x^7/b^5+1/10*(b*e-3*a*f)*x^10/b^4+1/13*f*x^13/b^3+1/6*a^3*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^7*(a+b*x^3)^2)-1/18*a^2*(19*b^3*c-25*a*b^2*d+31*a^2*b*e-37*a^3*f)*x/(b^7*(a+b*x^3))+1/27*a^(4/3)*(35*b^3*c-65*a*b^2*d+104*a^2*b*e-152*a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(22/3)-1/54*a^(4/3)*(35*b^3*c-65*a*b^2*d+104*a^2*b*e-152*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(22/3)-1/9*a^(4/3)*(35*b^3*c-65*a*b^2*d+104*a^2*b*e-152*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(22/3)*sqrt(3))],
[x^10*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,14,1/2*(b^3*c-3*a*b^2*d+6*a^2*b*e-10*a^3*f)*x^2/b^6+1/5*(b^2*d-3*a*b*e+6*a^2*f)*x^5/b^5+1/8*(b*e-3*a*f)*x^8/b^4+1/11*f*x^11/b^3-1/6*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(b^6*(a+b*x^3)^2)+1/9*a*(7*b^3*c-10*a*b^2*d+13*a^2*b*e-16*a^3*f)*x^2/(b^6*(a+b*x^3))+1/27*a^(2/3)*(20*b^3*c-44*a*b^2*d+77*a^2*b*e-119*a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(20/3)-1/54*a^(2/3)*(20*b^3*c-44*a*b^2*d+77*a^2*b*e-119*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(20/3)+1/9*a^(2/3)*(20*b^3*c-44*a*b^2*d+77*a^2*b*e-119*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(20/3)*sqrt(3))],
[x^9*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,10,(b^3*c-3*a*b^2*d+6*a^2*b*e-10*a^3*f)*x/b^6+1/4*(b^2*d-3*a*b*e+6*a^2*f)*x^4/b^5+1/7*(b*e-3*a*f)*x^7/b^4+1/10*f*x^10/b^3-1/6*a^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^6*(a+b*x^3)^2)+1/18*a*(13*b^3*c-19*a*b^2*d+25*a^2*b*e-31*a^3*f)*x/(b^6*(a+b*x^3))-1/27*a^(1/3)*(14*b^3*c-35*a*b^2*d+65*a^2*b*e-104*a^3*f)*log(a^(1/3)+b^(1/3)*x)/b^(19/3)+1/54*a^(1/3)*(14*b^3*c-35*a*b^2*d+65*a^2*b*e-104*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(19/3)+1/9*a^(1/3)*(14*b^3*c-35*a*b^2*d+65*a^2*b*e-104*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(19/3)*sqrt(3))],
[x^7*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,13,1/2*(b^2*d-3*a*b*e+6*a^2*f)*x^2/b^5+1/5*(b*e-3*a*f)*x^5/b^4+1/8*f*x^8/b^3+1/6*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(b^5*(a+b*x^3)^2)-1/9*(4*b^3*c-7*a*b^2*d+10*a^2*b*e-13*a^3*f)*x^2/(b^5*(a+b*x^3))-1/27*(5*b^3*c-20*a*b^2*d+44*a^2*b*e-77*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(17/3))+1/54*(5*b^3*c-20*a*b^2*d+44*a^2*b*e-77*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(17/3))-1/9*(5*b^3*c-20*a*b^2*d+44*a^2*b*e-77*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(17/3)*sqrt(3))],
[x^6*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,10,(b^2*d-3*a*b*e+6*a^2*f)*x/b^5+1/4*(b*e-3*a*f)*x^4/b^4+1/7*f*x^7/b^3+1/6*a*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^5*(a+b*x^3)^2)-1/18*(7*b^3*c-13*a*b^2*d+19*a^2*b*e-25*a^3*f)*x/(b^5*(a+b*x^3))+1/27*(2*b^3*c-14*a*b^2*d+35*a^2*b*e-65*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(16/3))-1/54*(2*b^3*c-14*a*b^2*d+35*a^2*b*e-65*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(16/3))-1/9*(2*b^3*c-14*a*b^2*d+35*a^2*b*e-65*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(16/3)*sqrt(3))],
[x^4*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,12,1/2*(b*e-3*a*f)*x^2/b^4+1/5*f*x^5/b^3-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(b^4*(a+b*x^3)^2)+1/9*(b^3*c-4*a*b^2*d+7*a^2*b*e-10*a^3*f)*x^2/(a*b^4*(a+b*x^3))-1/27*(b^3*c+5*a*b^2*d-20*a^2*b*e+44*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(14/3))+1/54*(b^3*c+5*a*b^2*d-20*a^2*b*e+44*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(14/3))-1/9*(b^3*c+5*a*b^2*d-20*a^2*b*e+44*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(14/3)*sqrt(3))],
[x^3*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,10,(b*e-3*a*f)*x/b^4+1/4*f*x^4/b^3-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(b^4*(a+b*x^3)^2)+1/18*(b^3*c-7*a*b^2*d+13*a^2*b*e-19*a^3*f)*x/(a*b^4*(a+b*x^3))+1/27*(b^3*c+2*a*b^2*d-14*a^2*b*e+35*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(13/3))-1/54*(b^3*c+2*a*b^2*d-14*a^2*b*e+35*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(13/3))-1/9*(b^3*c+2*a*b^2*d-14*a^2*b*e+35*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(13/3)*sqrt(3))],
[x*(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,10,1/2*f*x^2/b^3+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a*b^3*(a+b*x^3)^2)+1/9*(2*b^3*c+a*b^2*d-4*a^2*b*e+7*a^3*f)*x^2/(a^2*b^3*(a+b*x^3))-1/27*(2*b^3*c+a*b^2*d+5*a^2*b*e-20*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(11/3))+1/54*(2*b^3*c+a*b^2*d+5*a^2*b*e-20*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(11/3))-1/9*(2*b^3*c+a*b^2*d+5*a^2*b*e-20*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(11/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(a+b*x^3)^3,x,9,f*x/b^3+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a*b^3*(a+b*x^3)^2)+1/18*(5*b^3*c+a*b^2*d-7*a^2*b*e+13*a^3*f)*x/(a^2*b^3*(a+b*x^3))+1/27*(5*b^3*c+a*b^2*d+2*a^2*b*e-14*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(10/3))-1/54*(5*b^3*c+a*b^2*d+2*a^2*b*e-14*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(10/3))-1/9*(5*b^3*c+a*b^2*d+2*a^2*b*e-14*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(10/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^2*(a+b*x^3)^3),x,9,-c/(a^3*x)-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^2*b^2*(a+b*x^3)^2)-1/9*(5*b^3*c-2*a*b^2*d-a^2*b*e+4*a^3*f)*x^2/(a^3*b^2*(a+b*x^3))+1/27*(14*b^3*c-2*a*b^2*d-a^2*b*e-5*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(8/3))-1/54*(14*b^3*c-2*a*b^2*d-a^2*b*e-5*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(8/3))+1/9*(14*b^3*c-2*a*b^2*d-a^2*b*e-5*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(8/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^3*(a+b*x^3)^3),x,9,-1/2*c/(a^3*x^2)-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^2*b^2*(a+b*x^3)^2)-1/18*(11*b^3*c-5*a*b^2*d-a^2*b*e+7*a^3*f)*x/(a^3*b^2*(a+b*x^3))-1/27*(20*b^3*c-5*a*b^2*d-a^2*b*e-2*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(7/3))+1/54*(20*b^3*c-5*a*b^2*d-a^2*b*e-2*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(7/3))+1/9*(20*b^3*c-5*a*b^2*d-a^2*b*e-2*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(7/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^5*(a+b*x^3)^3),x,10,-1/4*c/(a^3*x^4)+(3*b*c-a*d)/(a^4*x)+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^3*b*(a+b*x^3)^2)+1/9*(8*b^3*c-5*a*b^2*d+2*a^2*b*e+a^3*f)*x^2/(a^4*b*(a+b*x^3))-1/27*(35*b^3*c-14*a*b^2*d+2*a^2*b*e+a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(13/3)*b^(5/3))+1/54*(35*b^3*c-14*a*b^2*d+2*a^2*b*e+a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(13/3)*b^(5/3))-1/9*(35*b^3*c-14*a*b^2*d+2*a^2*b*e+a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(13/3)*b^(5/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^6*(a+b*x^3)^3),x,10,-1/5*c/(a^3*x^5)+1/2*(3*b*c-a*d)/(a^4*x^2)+1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^3*b*(a+b*x^3)^2)+1/18*(17*b^3*c-11*a*b^2*d+5*a^2*b*e+a^3*f)*x/(a^4*b*(a+b*x^3))+1/27*(44*b^3*c-20*a*b^2*d+5*a^2*b*e+a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(14/3)*b^(4/3))-1/54*(44*b^3*c-20*a*b^2*d+5*a^2*b*e+a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(14/3)*b^(4/3))-1/9*(44*b^3*c-20*a*b^2*d+5*a^2*b*e+a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(14/3)*b^(4/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^8*(a+b*x^3)^3),x,10,-1/7*c/(a^3*x^7)+1/4*(3*b*c-a*d)/(a^4*x^4)+(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x)-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^4*(a+b*x^3)^2)-1/9*(11*b^3*c-8*a*b^2*d+5*a^2*b*e-2*a^3*f)*x^2/(a^5*(a+b*x^3))+1/27*(65*b^3*c-35*a*b^2*d+14*a^2*b*e-2*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(16/3)*b^(2/3))-1/54*(65*b^3*c-35*a*b^2*d+14*a^2*b*e-2*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(16/3)*b^(2/3))+1/9*(65*b^3*c-35*a*b^2*d+14*a^2*b*e-2*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(16/3)*b^(2/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^9*(a+b*x^3)^3),x,10,-1/8*c/(a^3*x^8)+1/5*(3*b*c-a*d)/(a^4*x^5)+1/2*(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x^2)-1/6*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^4*(a+b*x^3)^2)-1/18*(23*b^3*c-17*a*b^2*d+11*a^2*b*e-5*a^3*f)*x/(a^5*(a+b*x^3))-1/27*(77*b^3*c-44*a*b^2*d+20*a^2*b*e-5*a^3*f)*log(a^(1/3)+b^(1/3)*x)/(a^(17/3)*b^(1/3))+1/54*(77*b^3*c-44*a*b^2*d+20*a^2*b*e-5*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(17/3)*b^(1/3))+1/9*(77*b^3*c-44*a*b^2*d+20*a^2*b*e-5*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(17/3)*b^(1/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^11*(a+b*x^3)^3),x,10,-1/10*c/(a^3*x^10)+1/7*(3*b*c-a*d)/(a^4*x^7)+1/4*(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x^4)+(10*b^3*c-6*a*b^2*d+3*a^2*b*e-a^3*f)/(a^6*x)+1/6*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^5*(a+b*x^3)^2)+1/9*b*(14*b^3*c-11*a*b^2*d+8*a^2*b*e-5*a^3*f)*x^2/(a^6*(a+b*x^3))-1/27*b^(1/3)*(104*b^3*c-65*a*b^2*d+35*a^2*b*e-14*a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(19/3)+1/54*b^(1/3)*(104*b^3*c-65*a*b^2*d+35*a^2*b*e-14*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(19/3)-1/9*b^(1/3)*(104*b^3*c-65*a*b^2*d+35*a^2*b*e-14*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(19/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^12*(a+b*x^3)^3),x,10,-1/11*c/(a^3*x^11)+1/8*(3*b*c-a*d)/(a^4*x^8)+1/5*(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x^5)+1/2*(10*b^3*c-6*a*b^2*d+3*a^2*b*e-a^3*f)/(a^6*x^2)+1/6*b*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x/(a^5*(a+b*x^3)^2)+1/18*b*(29*b^3*c-23*a*b^2*d+17*a^2*b*e-11*a^3*f)*x/(a^6*(a+b*x^3))+1/27*b^(2/3)*(119*b^3*c-77*a*b^2*d+44*a^2*b*e-20*a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(20/3)-1/54*b^(2/3)*(119*b^3*c-77*a*b^2*d+44*a^2*b*e-20*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(20/3)-1/9*b^(2/3)*(119*b^3*c-77*a*b^2*d+44*a^2*b*e-20*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(20/3)*sqrt(3))],
[(c+d*x^3+e*x^6+f*x^9)/(x^14*(a+b*x^3)^3),x,10,-1/13*c/(a^3*x^13)+1/10*(3*b*c-a*d)/(a^4*x^10)+1/7*(-6*b^2*c+3*a*b*d-a^2*e)/(a^5*x^7)+1/4*(10*b^3*c-6*a*b^2*d+3*a^2*b*e-a^3*f)/(a^6*x^4)-b*(15*b^3*c-10*a*b^2*d+6*a^2*b*e-3*a^3*f)/(a^7*x)-1/6*b^2*(b^3*c-a*b^2*d+a^2*b*e-a^3*f)*x^2/(a^6*(a+b*x^3)^2)-1/9*b^2*(17*b^3*c-14*a*b^2*d+11*a^2*b*e-8*a^3*f)*x^2/(a^7*(a+b*x^3))+1/27*b^(4/3)*(152*b^3*c-104*a*b^2*d+65*a^2*b*e-35*a^3*f)*log(a^(1/3)+b^(1/3)*x)/a^(22/3)-1/54*b^(4/3)*(152*b^3*c-104*a*b^2*d+65*a^2*b*e-35*a^3*f)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(22/3)+1/9*b^(4/3)*(152*b^3*c-104*a*b^2*d+65*a^2*b*e-35*a^3*f)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(22/3)*sqrt(3))],

# Integrands of the form x^m P1(x) (a+b x^3)^p

# p>0

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

# Integrands of the form x^m P2(x) (a+b x^3)^p

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

# p<0
[x^3*(c+d*x+e*x^2)/(a+b*x^3),x,10,c*x/b+1/2*d*x^2/b+1/3*e*x^3/b-1/3*a^(1/3)*(b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/b^(5/3)+1/6*a^(1/3)*(c-a^(1/3)*d/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(4/3)-1/3*a*e*log(a+b*x^3)/b^2+a^(1/3)*(b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2)/(a+b*x^3),x,10,d*x/b+1/2*e*x^2/b-1/3*a^(1/3)*(b^(1/3)*d-a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/b^(5/3)+1/6*a^(1/3)*(d-a^(1/3)*e/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(4/3)+1/3*c*log(a+b*x^3)/b+a^(1/3)*(b^(1/3)*d+a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3))],
[x*(c+d*x+e*x^2)/(a+b*x^3),x,10,e*x/b-1/3*(b^(2/3)*c+a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(4/3))+1/6*(b^(2/3)*c+a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(4/3))+1/3*d*log(a+b*x^3)/b-(b^(2/3)*c-a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2)/(a+b*x^3),x,8,1/3*(b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(2/3))-1/6*(c-a^(1/3)*d/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))+1/3*e*log(a+b*x^3)/b-(b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x*(a+b*x^3)),x,10,c*log(x)/a+1/3*(b^(1/3)*d-a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(2/3))-1/6*(d-a^(1/3)*e/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(1/3))-1/3*c*log(a+b*x^3)/a-(b^(1/3)*d+a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^2*(a+b*x^3)),x,10,-c/(a*x)+d*log(x)/a+1/3*(b^(2/3)*c+a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(1/3))-1/6*(b^(2/3)*c+a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(1/3))-1/3*d*log(a+b*x^3)/a+(b^(2/3)*c-a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(1/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^3*(a+b*x^3)),x,10,-1/2*c/(a*x^2)-d/(a*x)+e*log(x)/a-1/3*b^(1/3)*(b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/a^(5/3)+1/6*b^(2/3)*(c-a^(1/3)*d/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)-1/3*e*log(a+b*x^3)/a+b^(1/3)*(b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2)/(a+b*x^3)^2,x,7,1/3*(-c-d*x-e*x^2)/(b*(a+b*x^3))+1/9*(b^(1/3)*d-2*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(5/3))-1/18*(d-2*a^(1/3)*e/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(4/3))-1/3*(b^(1/3)*d+2*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(5/3)*sqrt(3))],
[x*(c+d*x+e*x^2)/(a+b*x^3)^2,x,7,-1/3*x*(a*e-b*c*x-b*d*x^2)/(a*b*(a+b*x^3))-1/9*(b^(2/3)*c-a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(4/3))+1/18*(b^(2/3)*c-a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(4/3))-1/3*(b^(2/3)*c+a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2)/(a+b*x^3)^2,x,7,1/3*(-a*e+b*x*(c+d*x))/(a*b*(a+b*x^3))+1/9*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(2/3))-1/18*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(2/3))-1/3*(2*b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x*(a+b*x^3)^2),x,11,1/3*x*(a*d+a*e*x-b*c*x^2)/(a^2*(a+b*x^3))+c*log(x)/a^2+1/9*(2*b^(1/3)*d-a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(2/3))-1/18*(2*b^(1/3)*d-a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(2/3))-1/3*c*log(a+b*x^3)/a^2-1/3*(2*b^(1/3)*d+a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^2*(a+b*x^3)^2),x,11,-c/(a^2*x)+1/3*x*(a*e-b*c*x-b*d*x^2)/(a^2*(a+b*x^3))+d*log(x)/a^2+2/9*(2*b^(2/3)*c+a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(1/3))-1/9*(2*b^(2/3)*c+a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(1/3))-1/3*d*log(a+b*x^3)/a^2+2/3*(2*b^(2/3)*c-a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(1/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^3*(a+b*x^3)^2),x,11,-1/2*c/(a^2*x^2)-d/(a^2*x)-1/3*x*(b*c+b*d*x+b*e*x^2)/(a^2*(a+b*x^3))+e*log(x)/a^2-1/9*b^(1/3)*(5*b^(1/3)*c-4*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/a^(8/3)+1/18*b^(1/3)*(5*b^(1/3)*c-4*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)-1/3*e*log(a+b*x^3)/a^2+1/3*b^(1/3)*(5*b^(1/3)*c+4*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^4*(a+b*x^3)^2),x,11,-1/3*c/(a^2*x^3)-1/2*d/(a^2*x^2)-e/(a^2*x)-1/3*x*(b*d+b*e*x-b^2*c*x^2/a)/(a^2*(a+b*x^3))-2*b*c*log(x)/a^3-1/9*b^(1/3)*(5*b^(1/3)*d-4*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/a^(8/3)+1/18*b^(1/3)*(5*b^(1/3)*d-4*a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(8/3)+2/3*b*c*log(a+b*x^3)/a^3+1/3*b^(1/3)*(5*b^(1/3)*d+4*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2)/(a+b*x^3)^3,x,8,1/6*(-c-d*x-e*x^2)/(b*(a+b*x^3)^2)+1/18*x*(d+2*e*x)/(a*b*(a+b*x^3))+1/27*(b^(1/3)*d-a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(5/3))-1/54*(d-a^(1/3)*e/b^(1/3))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(4/3))-1/9*(b^(1/3)*d+a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(5/3)*sqrt(3))],
[x*(c+d*x+e*x^2)/(a+b*x^3)^3,x,8,-1/6*x*(a*e-b*c*x-b*d*x^2)/(a*b*(a+b*x^3)^2)+1/18*(-3*a*d+x*(a*e+4*b*c*x))/(a^2*b*(a+b*x^3))-1/27*(2*b^(2/3)*c-a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(4/3))+1/54*(2*b^(2/3)*c-a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(4/3))-1/9*(2*b^(2/3)*c+a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2)/(a+b*x^3)^3,x,8,1/18*x*(5*c+4*d*x)/(a^2*(a+b*x^3))+1/6*(-a*e+b*x*(c+d*x))/(a*b*(a+b*x^3)^2)+1/27*(5*b^(1/3)*c-2*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(2/3))-1/54*(5*b^(1/3)*c-2*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(2/3))-1/9*(5*b^(1/3)*c+2*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x*(a+b*x^3)^3),x,12,1/6*x*(a*d+a*e*x-b*c*x^2)/(a^2*(a+b*x^3)^2)+1/18*x*(5*a*d+4*a*e*x-9*b*c*x^2)/(a^3*(a+b*x^3))+c*log(x)/a^3+1/27*(5*b^(1/3)*d-2*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(2/3))-1/54*(5*b^(1/3)*d-2*a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(2/3))-1/3*c*log(a+b*x^3)/a^3-1/9*(5*b^(1/3)*d+2*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^2*(a+b*x^3)^3),x,12,-c/(a^3*x)+1/6*x*(a*e-b*c*x-b*d*x^2)/(a^2*(a+b*x^3)^2)+1/18*x*(5*a*e-10*b*c*x-9*b*d*x^2)/(a^3*(a+b*x^3))+d*log(x)/a^3+1/27*(14*b^(2/3)*c+5*a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(1/3))-1/54*(14*b^(2/3)*c+5*a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(1/3))-1/3*d*log(a+b*x^3)/a^3+1/9*(14*b^(2/3)*c-5*a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(1/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^3*(a+b*x^3)^3),x,12,-1/2*c/(a^3*x^2)-d/(a^3*x)-1/6*x*(b*c+b*d*x+b*e*x^2)/(a^2*(a+b*x^3)^2)-1/18*x*(11*b*c+10*b*d*x+9*b*e*x^2)/(a^3*(a+b*x^3))+e*log(x)/a^3-2/27*b^(1/3)*(10*b^(1/3)*c-7*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/a^(11/3)+1/27*b^(1/3)*(10*b^(1/3)*c-7*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/3)-1/3*e*log(a+b*x^3)/a^3+2/9*b^(1/3)*(10*b^(1/3)*c+7*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^4*(a+b*x^3)^3),x,12,-1/3*c/(a^3*x^3)-1/2*d/(a^3*x^2)-e/(a^3*x)-1/6*x*(b*d+b*e*x-b^2*c*x^2/a)/(a^2*(a+b*x^3)^2)-1/18*x*(11*b*d+10*b*e*x-15*b^2*c*x^2/a)/(a^3*(a+b*x^3))-3*b*c*log(x)/a^4-2/27*b^(1/3)*(10*b^(1/3)*d-7*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/a^(11/3)+1/27*b^(1/3)*(10*b^(1/3)*d-7*a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/3)+b*c*log(a+b*x^3)/a^4+2/9*b^(1/3)*(10*b^(1/3)*d+7*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2)/(a+b*x^3)^4,x,9,1/9*(-c-d*x-e*x^2)/(b*(a+b*x^3)^3)+1/54*x*(d+2*e*x)/(a*b*(a+b*x^3)^2)+1/162*x*(5*d+8*e*x)/(a^2*b*(a+b*x^3))+1/243*(5*b^(1/3)*d-4*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(5/3))-1/486*(5*b^(1/3)*d-4*a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(5/3))-1/81*(5*b^(1/3)*d+4*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(5/3)*sqrt(3))],
[x*(c+d*x+e*x^2)/(a+b*x^3)^4,x,9,-1/9*x*(a*e-b*c*x-b*d*x^2)/(a*b*(a+b*x^3)^3)+1/162*x*(5*a*e+28*b*c*x)/(a^3*b*(a+b*x^3))+1/54*(-6*a*d+x*(a*e+7*b*c*x))/(a^2*b*(a+b*x^3)^2)-1/243*(14*b^(2/3)*c-5*a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(4/3))+1/486*(14*b^(2/3)*c-5*a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(4/3))-1/81*(14*b^(2/3)*c+5*a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2)/(a+b*x^3)^4,x,9,1/54*x*(8*c+7*d*x)/(a^2*(a+b*x^3)^2)+2/81*x*(10*c+7*d*x)/(a^3*(a+b*x^3))+1/9*(-a*e+b*x*(c+d*x))/(a*b*(a+b*x^3)^3)+2/243*(20*b^(1/3)*c-7*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(2/3))-1/243*(20*b^(1/3)*c-7*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(2/3))-2/81*(20*b^(1/3)*c+7*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x*(a+b*x^3)^4),x,13,1/9*x*(a*d+a*e*x-b*c*x^2)/(a^2*(a+b*x^3)^3)+1/54*x*(8*a*d+7*a*e*x-15*b*c*x^2)/(a^3*(a+b*x^3)^2)+1/162*x*(40*a*d+28*a*e*x-99*b*c*x^2)/(a^4*(a+b*x^3))+c*log(x)/a^4+2/243*(20*b^(1/3)*d-7*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(2/3))-1/243*(20*b^(1/3)*d-7*a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(2/3))-1/3*c*log(a+b*x^3)/a^4-2/81*(20*b^(1/3)*d+7*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^2*(a+b*x^3)^4),x,13,-c/(a^4*x)+1/9*x*(a*e-b*c*x-b*d*x^2)/(a^2*(a+b*x^3)^3)+1/54*x*(8*a*e-16*b*c*x-15*b*d*x^2)/(a^3*(a+b*x^3)^2)+1/162*x*(40*a*e-118*b*c*x-99*b*d*x^2)/(a^4*(a+b*x^3))+d*log(x)/a^4+20/243*(7*b^(2/3)*c+2*a^(2/3)*e)*log(a^(1/3)+b^(1/3)*x)/(a^(13/3)*b^(1/3))-10/243*(7*b^(2/3)*c+2*a^(2/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(13/3)*b^(1/3))-1/3*d*log(a+b*x^3)/a^4+20/81*(7*b^(2/3)*c-2*a^(2/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(13/3)*b^(1/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^3*(a+b*x^3)^4),x,13,-1/2*c/(a^4*x^2)-d/(a^4*x)-1/9*x*(b*c+b*d*x+b*e*x^2)/(a^2*(a+b*x^3)^3)-1/54*x*(17*b*c+16*b*d*x+15*b*e*x^2)/(a^3*(a+b*x^3)^2)-1/162*x*(139*b*c+118*b*d*x+99*b*e*x^2)/(a^4*(a+b*x^3))+e*log(x)/a^4-20/243*b^(1/3)*(11*b^(1/3)*c-7*a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/a^(14/3)+10/243*b^(1/3)*(11*b^(1/3)*c-7*a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3)-1/3*e*log(a+b*x^3)/a^4+20/81*b^(1/3)*(11*b^(1/3)*c+7*a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(14/3)*sqrt(3))],
[(c+d*x+e*x^2)/(x^4*(a+b*x^3)^4),x,13,-1/3*c/(a^4*x^3)-1/2*d/(a^4*x^2)-e/(a^4*x)-1/9*x*(b*d+b*e*x-b^2*c*x^2/a)/(a^2*(a+b*x^3)^3)-1/54*x*(17*b*d+16*b*e*x-24*b^2*c*x^2/a)/(a^3*(a+b*x^3)^2)-1/162*x*(139*b*d+118*b*e*x-234*b^2*c*x^2/a)/(a^4*(a+b*x^3))-4*b*c*log(x)/a^5-20/243*b^(1/3)*(11*b^(1/3)*d-7*a^(1/3)*e)*log(a^(1/3)+b^(1/3)*x)/a^(14/3)+10/243*b^(1/3)*(11*b^(1/3)*d-7*a^(1/3)*e)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3)+4/3*b*c*log(a+b*x^3)/a^5+20/81*b^(1/3)*(11*b^(1/3)*d+7*a^(1/3)*e)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(14/3)*sqrt(3))],
[(2*a*x-x^2)/(a^3+x^3),x,5,-log(a+x)-2*arctan((a-2*x)/(a*sqrt(3)))/sqrt(3)],
[(2*a-x)*x/(a^3+x^3),x,4,-log(a+x)-2*arctan((a-2*x)/(a*sqrt(3)))/sqrt(3)],
[(2*a*x+x^2)/(a^3-x^3),x,5,-log(a-x)-2*arctan((a+2*x)/(a*sqrt(3)))/sqrt(3)],
[x*(2*a+x)/(a^3-x^3),x,4,-log(a-x)-2*arctan((a+2*x)/(a*sqrt(3)))/sqrt(3)],
[x*(-2*(a/b)^(1/3)*C+C*x)/(a+b*x^3),x,4,C*log((a/b)^(1/3)+x)/b+2*C*arctan((1-2*x/(a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[x*(-2*(-a/b)^(1/3)*C+C*x)/(a-b*x^3),x,4,-C*log((-a/b)^(1/3)+x)/b-2*C*arctan((1-2*x/(-a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[x*(2*(-a/b)^(1/3)*C+C*x)/(a+b*x^3),x,4,C*log((-a/b)^(1/3)-x)/b+2*C*arctan((1+2*x/(-a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],
[x*(2*(a/b)^(1/3)*C+C*x)/(a-b*x^3),x,4,-C*log((a/b)^(1/3)-x)/b-2*C*arctan((1+2*x/(a/b)^(1/3))/sqrt(3))/(b*sqrt(3))],

# Integrands of the form x^m P5(x) (a+b x^3)^p

# p>0
[x^4*(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/5*a*c*x^5+1/6*a*d*x^6+1/7*a*e*x^7+1/8*(b*c+a*f)*x^8+1/9*(b*d+a*g)*x^9+1/10*(b*e+a*h)*x^10+1/11*b*f*x^11+1/12*b*g*x^12+1/13*b*h*x^13],
[x^3*(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/4*a*c*x^4+1/5*a*d*x^5+1/6*a*e*x^6+1/7*(b*c+a*f)*x^7+1/8*(b*d+a*g)*x^8+1/9*(b*e+a*h)*x^9+1/10*b*f*x^10+1/11*b*g*x^11+1/12*b*h*x^12],
[x^2*(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/3*a*c*x^3+1/4*a*d*x^4+1/5*a*e*x^5+1/6*(b*c+a*f)*x^6+1/7*(b*d+a*g)*x^7+1/8*(b*e+a*h)*x^8+1/9*b*f*x^9+1/10*b*g*x^10+1/11*b*h*x^11],
[x*(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/2*a*c*x^2+1/3*a*d*x^3+1/4*a*e*x^4+1/5*(b*c+a*f)*x^5+1/6*(b*d+a*g)*x^6+1/7*(b*e+a*h)*x^7+1/8*b*f*x^8+1/9*b*g*x^9+1/10*b*h*x^10],
[(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,a*c*x+1/2*a*d*x^2+1/3*a*e*x^3+1/4*(b*c+a*f)*x^4+1/5*(b*d+a*g)*x^5+1/6*(b*e+a*h)*x^6+1/7*b*f*x^7+1/8*b*g*x^8+1/9*b*h*x^9],
[(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x,x,2,a*d*x+1/2*a*e*x^2+1/3*(b*c+a*f)*x^3+1/4*(b*d+a*g)*x^4+1/5*(b*e+a*h)*x^5+1/6*b*f*x^6+1/7*b*g*x^7+1/8*b*h*x^8+a*c*log(x)],
[(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^2,x,2,-a*c/x+a*e*x+1/2*(b*c+a*f)*x^2+1/3*(b*d+a*g)*x^3+1/4*(b*e+a*h)*x^4+1/5*b*f*x^5+1/6*b*g*x^6+1/7*b*h*x^7+a*d*log(x)],
[(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^3,x,2,-1/2*a*c/x^2-a*d/x+(b*c+a*f)*x+1/2*(b*d+a*g)*x^2+1/3*(b*e+a*h)*x^3+1/4*b*f*x^4+1/5*b*g*x^5+1/6*b*h*x^6+a*e*log(x)],
[(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^4,x,2,-1/3*a*c/x^3-1/2*a*d/x^2-a*e/x+(b*d+a*g)*x+1/2*(b*e+a*h)*x^2+1/3*b*f*x^3+1/4*b*g*x^4+1/5*b*h*x^5+(b*c+a*f)*log(x)],
[(a+b*x^3)*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^5,x,2,-1/4*a*c/x^4-1/3*a*d/x^3-1/2*a*e/x^2+(-b*c-a*f)/x+(b*e+a*h)*x+1/2*b*f*x^2+1/3*b*g*x^3+1/4*b*h*x^4+(b*d+a*g)*log(x)],
[x^4*(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/5*a^2*c*x^5+1/6*a^2*d*x^6+1/7*a^2*e*x^7+1/8*a*(2*b*c+a*f)*x^8+1/9*a*(2*b*d+a*g)*x^9+1/10*a*(2*b*e+a*h)*x^10+1/11*b*(b*c+2*a*f)*x^11+1/12*b*(b*d+2*a*g)*x^12+1/13*b*(b*e+2*a*h)*x^13+1/14*b^2*f*x^14+1/15*b^2*g*x^15+1/16*b^2*h*x^16],
[x^3*(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/4*a^2*c*x^4+1/5*a^2*d*x^5+1/6*a^2*e*x^6+1/7*a*(2*b*c+a*f)*x^7+1/8*a*(2*b*d+a*g)*x^8+1/9*a*(2*b*e+a*h)*x^9+1/10*b*(b*c+2*a*f)*x^10+1/11*b*(b*d+2*a*g)*x^11+1/12*b*(b*e+2*a*h)*x^12+1/13*b^2*f*x^13+1/14*b^2*g*x^14+1/15*b^2*h*x^15],
[x^2*(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,3,1/4*a^2*d*x^4+1/5*a^2*e*x^5+1/6*a^2*f*x^6+1/7*a*(2*b*d+a*g)*x^7+1/8*a*(2*b*e+a*h)*x^8+2/9*a*b*f*x^9+1/10*b*(b*d+2*a*g)*x^10+1/11*b*(b*e+2*a*h)*x^11+1/12*b^2*f*x^12+1/13*b^2*g*x^13+1/14*b^2*h*x^14+1/9*c*(a+b*x^3)^3/b],
[x*(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,3,1/2*a^2*c*x^2+1/4*a^2*e*x^4+1/5*a*(2*b*c+a*f)*x^5+1/6*a^2*g*x^6+1/7*a*(2*b*e+a*h)*x^7+1/8*b*(b*c+2*a*f)*x^8+2/9*a*b*g*x^9+1/10*b*(b*e+2*a*h)*x^10+1/11*b^2*f*x^11+1/12*b^2*g*x^12+1/13*b^2*h*x^13+1/9*d*(a+b*x^3)^3/b],
[(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,3,a^2*c*x+1/2*a^2*d*x^2+1/4*a*(2*b*c+a*f)*x^4+1/5*a*(2*b*d+a*g)*x^5+1/6*a^2*h*x^6+1/7*b*(b*c+2*a*f)*x^7+1/8*b*(b*d+2*a*g)*x^8+2/9*a*b*h*x^9+1/10*b^2*f*x^10+1/11*b^2*g*x^11+1/12*b^2*h*x^12+1/9*e*(a+b*x^3)^3/b],
[(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x,x,3,a^2*d*x+1/2*a^2*e*x^2+2/3*a*b*c*x^3+1/4*a*(2*b*d+a*g)*x^4+1/5*a*(2*b*e+a*h)*x^5+1/6*b^2*c*x^6+1/7*b*(b*d+2*a*g)*x^7+1/8*b*(b*e+2*a*h)*x^8+1/10*b^2*g*x^10+1/11*b^2*h*x^11+1/9*f*(a+b*x^3)^3/b+a^2*c*log(x)],
[(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^2,x,3,-a^2*c/x+a^2*e*x+1/2*a*(2*b*c+a*f)*x^2+2/3*a*b*d*x^3+1/4*a*(2*b*e+a*h)*x^4+1/5*b*(b*c+2*a*f)*x^5+1/6*b^2*d*x^6+1/7*b*(b*e+2*a*h)*x^7+1/8*b^2*f*x^8+1/10*b^2*h*x^10+1/9*g*(a+b*x^3)^3/b+a^2*d*log(x)],
[(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^3,x,3,-1/2*a^2*c/x^2-a^2*d/x+a*(2*b*c+a*f)*x+1/2*a*(2*b*d+a*g)*x^2+2/3*a*b*e*x^3+1/4*b*(b*c+2*a*f)*x^4+1/5*b*(b*d+2*a*g)*x^5+1/6*b^2*e*x^6+1/7*b^2*f*x^7+1/8*b^2*g*x^8+1/9*h*(a+b*x^3)^3/b+a^2*e*log(x)],
[(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^4,x,2,-1/3*a^2*c/x^3-1/2*a^2*d/x^2-a^2*e/x+a*(2*b*d+a*g)*x+1/2*a*(2*b*e+a*h)*x^2+1/3*b*(b*c+2*a*f)*x^3+1/4*b*(b*d+2*a*g)*x^4+1/5*b*(b*e+2*a*h)*x^5+1/6*b^2*f*x^6+1/7*b^2*g*x^7+1/8*b^2*h*x^8+a*(2*b*c+a*f)*log(x)],
[(a+b*x^3)^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^5,x,2,-1/4*a^2*c/x^4-1/3*a^2*d/x^3-1/2*a^2*e/x^2-a*(2*b*c+a*f)/x+a*(2*b*e+a*h)*x+1/2*b*(b*c+2*a*f)*x^2+1/3*b*(b*d+2*a*g)*x^3+1/4*b*(b*e+2*a*h)*x^4+1/5*b^2*f*x^5+1/6*b^2*g*x^6+1/7*b^2*h*x^7+a*(2*b*d+a*g)*log(x)],
[x^4*(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/5*a^3*c*x^5+1/6*a^3*d*x^6+1/7*a^3*e*x^7+1/8*a^2*(3*b*c+a*f)*x^8+1/9*a^2*(3*b*d+a*g)*x^9+1/10*a^2*(3*b*e+a*h)*x^10+3/11*a*b*(b*c+a*f)*x^11+1/4*a*b*(b*d+a*g)*x^12+3/13*a*b*(b*e+a*h)*x^13+1/14*b^2*(b*c+3*a*f)*x^14+1/15*b^2*(b*d+3*a*g)*x^15+1/16*b^2*(b*e+3*a*h)*x^16+1/17*b^3*f*x^17+1/18*b^3*g*x^18+1/19*b^3*h*x^19],
[x^3*(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,2,1/4*a^3*c*x^4+1/5*a^3*d*x^5+1/6*a^3*e*x^6+1/7*a^2*(3*b*c+a*f)*x^7+1/8*a^2*(3*b*d+a*g)*x^8+1/9*a^2*(3*b*e+a*h)*x^9+3/10*a*b*(b*c+a*f)*x^10+3/11*a*b*(b*d+a*g)*x^11+1/4*a*b*(b*e+a*h)*x^12+1/13*b^2*(b*c+3*a*f)*x^13+1/14*b^2*(b*d+3*a*g)*x^14+1/15*b^2*(b*e+3*a*h)*x^15+1/16*b^3*f*x^16+1/17*b^3*g*x^17+1/18*b^3*h*x^18],
[x^2*(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,3,1/4*a^3*d*x^4+1/5*a^3*e*x^5+1/6*a^3*f*x^6+1/7*a^2*(3*b*d+a*g)*x^7+1/8*a^2*(3*b*e+a*h)*x^8+1/3*a^2*b*f*x^9+3/10*a*b*(b*d+a*g)*x^10+3/11*a*b*(b*e+a*h)*x^11+1/4*a*b^2*f*x^12+1/13*b^2*(b*d+3*a*g)*x^13+1/14*b^2*(b*e+3*a*h)*x^14+1/15*b^3*f*x^15+1/16*b^3*g*x^16+1/17*b^3*h*x^17+1/12*c*(a+b*x^3)^4/b],
[x*(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,3,1/2*a^3*c*x^2+1/4*a^3*e*x^4+1/5*a^2*(3*b*c+a*f)*x^5+1/6*a^3*g*x^6+1/7*a^2*(3*b*e+a*h)*x^7+3/8*a*b*(b*c+a*f)*x^8+1/3*a^2*b*g*x^9+3/10*a*b*(b*e+a*h)*x^10+1/11*b^2*(b*c+3*a*f)*x^11+1/4*a*b^2*g*x^12+1/13*b^2*(b*e+3*a*h)*x^13+1/14*b^3*f*x^14+1/15*b^3*g*x^15+1/16*b^3*h*x^16+1/12*d*(a+b*x^3)^4/b],
[(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5),x,3,a^3*c*x+1/2*a^3*d*x^2+1/4*a^2*(3*b*c+a*f)*x^4+1/5*a^2*(3*b*d+a*g)*x^5+1/6*a^3*h*x^6+3/7*a*b*(b*c+a*f)*x^7+3/8*a*b*(b*d+a*g)*x^8+1/3*a^2*b*h*x^9+1/10*b^2*(b*c+3*a*f)*x^10+1/11*b^2*(b*d+3*a*g)*x^11+1/4*a*b^2*h*x^12+1/13*b^3*f*x^13+1/14*b^3*g*x^14+1/15*b^3*h*x^15+1/12*e*(a+b*x^3)^4/b],
[(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x,x,3,a^3*d*x+1/2*a^3*e*x^2+a^2*b*c*x^3+1/4*a^2*(3*b*d+a*g)*x^4+1/5*a^2*(3*b*e+a*h)*x^5+1/2*a*b^2*c*x^6+3/7*a*b*(b*d+a*g)*x^7+3/8*a*b*(b*e+a*h)*x^8+1/9*b^3*c*x^9+1/10*b^2*(b*d+3*a*g)*x^10+1/11*b^2*(b*e+3*a*h)*x^11+1/13*b^3*g*x^13+1/14*b^3*h*x^14+1/12*f*(a+b*x^3)^4/b+a^3*c*log(x)],
[(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^2,x,3,-a^3*c/x+a^3*e*x+1/2*a^2*(3*b*c+a*f)*x^2+a^2*b*d*x^3+1/4*a^2*(3*b*e+a*h)*x^4+3/5*a*b*(b*c+a*f)*x^5+1/2*a*b^2*d*x^6+3/7*a*b*(b*e+a*h)*x^7+1/8*b^2*(b*c+3*a*f)*x^8+1/9*b^3*d*x^9+1/10*b^2*(b*e+3*a*h)*x^10+1/11*b^3*f*x^11+1/13*b^3*h*x^13+1/12*g*(a+b*x^3)^4/b+a^3*d*log(x)],
[(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^3,x,3,-1/2*a^3*c/x^2-a^3*d/x+a^2*(3*b*c+a*f)*x+1/2*a^2*(3*b*d+a*g)*x^2+a^2*b*e*x^3+3/4*a*b*(b*c+a*f)*x^4+3/5*a*b*(b*d+a*g)*x^5+1/2*a*b^2*e*x^6+1/7*b^2*(b*c+3*a*f)*x^7+1/8*b^2*(b*d+3*a*g)*x^8+1/9*b^3*e*x^9+1/10*b^3*f*x^10+1/11*b^3*g*x^11+1/12*h*(a+b*x^3)^4/b+a^3*e*log(x)],
[(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^4,x,2,-1/3*a^3*c/x^3-1/2*a^3*d/x^2-a^3*e/x+a^2*(3*b*d+a*g)*x+1/2*a^2*(3*b*e+a*h)*x^2+a*b*(b*c+a*f)*x^3+3/4*a*b*(b*d+a*g)*x^4+3/5*a*b*(b*e+a*h)*x^5+1/6*b^2*(b*c+3*a*f)*x^6+1/7*b^2*(b*d+3*a*g)*x^7+1/8*b^2*(b*e+3*a*h)*x^8+1/9*b^3*f*x^9+1/10*b^3*g*x^10+1/11*b^3*h*x^11+a^2*(3*b*c+a*f)*log(x)],
[(a+b*x^3)^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/x^5,x,2,-1/4*a^3*c/x^4-1/3*a^3*d/x^3-1/2*a^3*e/x^2-a^2*(3*b*c+a*f)/x+a^2*(3*b*e+a*h)*x+3/2*a*b*(b*c+a*f)*x^2+a*b*(b*d+a*g)*x^3+3/4*a*b*(b*e+a*h)*x^4+1/5*b^2*(b*c+3*a*f)*x^5+1/6*b^2*(b*d+3*a*g)*x^6+1/7*b^2*(b*e+3*a*h)*x^7+1/8*b^3*f*x^8+1/9*b^3*g*x^9+1/10*b^3*h*x^10+a^2*(3*b*d+a*g)*log(x)],

# p<0
[x^4*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3),x,13,-a*(b*e-a*h)*x/b^3+1/2*(b*c-a*f)*x^2/b^2+1/3*(b*d-a*g)*x^3/b^2+1/4*(b*e-a*h)*x^4/b^2+1/5*f*x^5/b+1/6*g*x^6/b+1/7*h*x^7/b+1/3*a^(2/3)*(b^(2/3)*(b*c-a*f)+a^(2/3)*(b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/b^(10/3)-1/6*a^(2/3)*(b^(2/3)*(b*c-a*f)+a^(2/3)*(b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(10/3)-1/3*a*(b*d-a*g)*log(a+b*x^3)/b^3+a^(2/3)*(b^(5/3)*c-a^(2/3)*b*e-a*b^(2/3)*f+a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(10/3)*sqrt(3))],
[x^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3),x,13,(b*c-a*f)*x/b^2+1/2*(b*d-a*g)*x^2/b^2+1/3*(b*e-a*h)*x^3/b^2+1/4*f*x^4/b+1/5*g*x^5/b+1/6*h*x^6/b-1/3*a^(1/3)*(b^(1/3)*(b*c-a*f)-a^(1/3)*(b*d-a*g))*log(a^(1/3)+b^(1/3)*x)/b^(8/3)+1/6*a^(1/3)*(b^(1/3)*(b*c-a*f)-a^(1/3)*(b*d-a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(8/3)-1/3*a*(b*e-a*h)*log(a+b*x^3)/b^3+a^(1/3)*(b^(4/3)*c+a^(1/3)*b*d-a*b^(1/3)*f-a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(8/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3),x,13,(b*d-a*g)*x/b^2+1/2*(b*e-a*h)*x^2/b^2+1/3*f*x^3/b+1/4*g*x^4/b+1/5*h*x^5/b-1/3*a^(1/3)*(b^(1/3)*(b*d-a*g)-a^(1/3)*(b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/b^(8/3)+1/6*a^(1/3)*(b^(1/3)*(b*d-a*g)-a^(1/3)*(b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(8/3)+1/3*(b*c-a*f)*log(a+b*x^3)/b^2+a^(1/3)*(b^(4/3)*d+a^(1/3)*b*e-a*b^(1/3)*g-a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(8/3)*sqrt(3))],
[x*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3),x,13,(b*e-a*h)*x/b^2+1/2*f*x^2/b+1/3*g*x^3/b+1/4*h*x^4/b-1/3*(b^(2/3)*(b*c-a*f)+a^(2/3)*(b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(7/3))+1/6*(b^(2/3)*(b*c-a*f)+a^(2/3)*(b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(7/3))+1/3*(b*d-a*g)*log(a+b*x^3)/b^2-(b^(5/3)*c-a^(2/3)*b*e-a*b^(2/3)*f+a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(7/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3),x,10,f*x/b+1/2*g*x^2/b+1/3*h*x^3/b+1/3*(b^(1/3)*(b*c-a*f)-a^(1/3)*(b*d-a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(5/3))-1/6*(b^(1/3)*(b*c-a*f)-a^(1/3)*(b*d-a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(5/3))+1/3*(b*e-a*h)*log(a+b*x^3)/b^2-(b^(4/3)*c+a^(1/3)*b*d-a*b^(1/3)*f-a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(5/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x*(a+b*x^3)),x,10,g*x/b+1/2*h*x^2/b+c*log(x)/a+1/3*(b^(1/3)*(b*d-a*g)-a^(1/3)*(b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(5/3))-1/6*(b^(1/3)*(b*d-a*g)-a^(1/3)*(b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(5/3))-1/3*(b*c-a*f)*log(a+b*x^3)/(a*b)-(b^(4/3)*d+a^(1/3)*b*e-a*b^(1/3)*g-a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(5/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^2*(a+b*x^3)),x,10,-c/(a*x)+h*x/b+d*log(x)/a+1/3*(b^(2/3)*(b*c-a*f)+a^(2/3)*(b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(4/3))-1/6*(b^(2/3)*(b*c-a*f)+a^(2/3)*(b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(4/3))-1/3*(b*d-a*g)*log(a+b*x^3)/(a*b)+(b^(5/3)*c-a^(2/3)*b*e-a*b^(2/3)*f+a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^3*(a+b*x^3)),x,10,-1/2*c/(a*x^2)-d/(a*x)+e*log(x)/a-1/3*(b^(1/3)*(b*c-a*f)-a^(1/3)*(b*d-a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(2/3))+1/6*(b^(1/3)*(b*c-a*f)-a^(1/3)*(b*d-a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(2/3))-1/3*(b*e-a*h)*log(a+b*x^3)/(a*b)+(b^(4/3)*c+a^(1/3)*b*d-a*b^(1/3)*f-a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^4*(a+b*x^3)),x,10,-1/3*c/(a*x^3)-1/2*d/(a*x^2)-e/(a*x)-(b*c-a*f)*log(x)/a^2-1/3*(b^(1/3)*(b*d-a*g)-a^(1/3)*(b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(2/3))+1/6*(b^(1/3)*(b*d-a*g)-a^(1/3)*(b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(2/3))+1/3*(b*c-a*f)*log(a+b*x^3)/a^2+(b^(4/3)*d+a^(1/3)*b*e-a*b^(1/3)*g-a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(2/3)*sqrt(3))],
[x^4*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^2,x,11,(b*e-2*a*h)*x/b^3+1/2*f*x^2/b^2+1/3*g*x^3/b^2+1/4*h*x^4/b^2+1/3*x*(a*(b*e-a*h)-b*(b*c-a*f)*x-b*(b*d-a*g)*x^2)/(b^3*(a+b*x^3))-1/9*(b^(2/3)*(2*b*c-5*a*f)+a^(2/3)*(4*b*e-7*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(1/3)*b^(10/3))+1/18*(b^(2/3)*(2*b*c-5*a*f)+a^(2/3)*(4*b*e-7*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)*b^(10/3))+1/3*(b*d-2*a*g)*log(a+b*x^3)/b^3-1/3*(2*b^(5/3)*c-4*a^(2/3)*b*e-5*a*b^(2/3)*f+7*a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(1/3)*b^(10/3)*sqrt(3))],
[x^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^2,x,11,f*x/b^2+1/2*g*x^2/b^2+1/3*h*x^3/b^2-1/3*x*(b*c-a*f+(b*d-a*g)*x+(b*e-a*h)*x^2)/(b^2*(a+b*x^3))+1/9*(b^(1/3)*(b*c-4*a*f)-a^(1/3)*(2*b*d-5*a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(8/3))-1/18*(b^(1/3)*(b*c-4*a*f)-a^(1/3)*(2*b*d-5*a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(8/3))+1/3*(b*e-2*a*h)*log(a+b*x^3)/b^3-1/3*(b^(4/3)*c+2*a^(1/3)*b*d-4*a*b^(1/3)*f-5*a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(8/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^2,x,11,4/3*g*x/b^2+5/6*h*x^2/b^2+1/3*(-c-d*x-e*x^2-f*x^3-g*x^4-h*x^5)/(b*(a+b*x^3))+1/9*(b^(1/3)*(b*d-4*a*g)-a^(1/3)*(2*b*e-5*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(2/3)*b^(8/3))-1/18*(b^(1/3)*(b*d-4*a*g)-a^(1/3)*(2*b*e-5*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(2/3)*b^(8/3))+1/3*f*log(a+b*x^3)/b^2-1/3*(b^(4/3)*d+2*a^(1/3)*b*e-4*a*b^(1/3)*g-5*a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(8/3)*sqrt(3))],
[x*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^2,x,11,h*x/b^2-1/3*x*(a*(b*e-a*h)-b*(b*c-a*f)*x-b*(b*d-a*g)*x^2)/(a*b^2*(a+b*x^3))-1/9*(b^(2/3)*(b*c+2*a*f)-a^(2/3)*(b*e-4*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(7/3))+1/18*(b^(2/3)*(b*c+2*a*f)-a^(2/3)*(b*e-4*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(7/3))+1/3*g*log(a+b*x^3)/b^2-1/3*(b^(5/3)*c+a^(2/3)*b*e+2*a*b^(2/3)*f-4*a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(7/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^2,x,9,1/3*x*(b*c-a*f+(b*d-a*g)*x+(b*e-a*h)*x^2)/(a*b*(a+b*x^3))+1/9*(b^(1/3)*(2*b*c+a*f)-a^(1/3)*(b*d+2*a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(5/3))-1/18*(b^(1/3)*(2*b*c+a*f)-a^(1/3)*(b*d+2*a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(5/3))+1/3*h*log(a+b*x^3)/b^2-1/3*(2*b^(4/3)*c+a^(1/3)*b*d+a*b^(1/3)*f+2*a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(5/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x*(a+b*x^3)^2),x,11,1/3*x*(a*(b*d-a*g)+a*(b*e-a*h)*x-b*(b*c-a*f)*x^2)/(a^2*b*(a+b*x^3))+c*log(x)/a^2+1/9*(b^(1/3)*(2*b*d+a*g)-a^(1/3)*(b*e+2*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(5/3))-1/18*(b^(1/3)*(2*b*d+a*g)-a^(1/3)*(b*e+2*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(5/3))-1/3*c*log(a+b*x^3)/a^2-1/3*(2*b^(4/3)*d+a^(1/3)*b*e+a*b^(1/3)*g+2*a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(5/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^2*(a+b*x^3)^2),x,11,-c/(a^2*x)+1/3*x*(a*(b*e-a*h)-b*(b*c-a*f)*x-b*(b*d-a*g)*x^2)/(a^2*b*(a+b*x^3))+d*log(x)/a^2+1/9*(b^(2/3)*(4*b*c-a*f)+a^(2/3)*(2*b*e+a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(4/3))-1/18*(b^(2/3)*(4*b*c-a*f)+a^(2/3)*(2*b*e+a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(4/3))-1/3*d*log(a+b*x^3)/a^2+1/3*(4*b^(5/3)*c-2*a^(2/3)*b*e-a*b^(2/3)*f-a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^3*(a+b*x^3)^2),x,11,-1/2*c/(a^2*x^2)-d/(a^2*x)-1/3*x*(b*c-a*f+(b*d-a*g)*x+(b*e-a*h)*x^2)/(a^2*(a+b*x^3))+e*log(x)/a^2-1/9*(b^(1/3)*(5*b*c-2*a*f)-a^(1/3)*(4*b*d-a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(2/3))+1/18*(b^(1/3)*(5*b*c-2*a*f)-a^(1/3)*(4*b*d-a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(2/3))-1/3*e*log(a+b*x^3)/a^2+1/3*(5*b^(4/3)*c+4*a^(1/3)*b*d-2*a*b^(1/3)*f-a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^4*(a+b*x^3)^2),x,11,-1/3*c/(a^2*x^3)-1/2*d/(a^2*x^2)-e/(a^2*x)-1/3*x*(b*d-a*g+(b*e-a*h)*x-b*(b*c/a-f)*x^2)/(a^2*(a+b*x^3))-(2*b*c-a*f)*log(x)/a^3-1/9*(b^(1/3)*(5*b*d-2*a*g)-a^(1/3)*(4*b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(2/3))+1/18*(b^(1/3)*(5*b*d-2*a*g)-a^(1/3)*(4*b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(2/3))+1/3*(2*b*c-a*f)*log(a+b*x^3)/a^3+1/3*(5*b^(4/3)*d+4*a^(1/3)*b*e-2*a*b^(1/3)*g-a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(2/3)*sqrt(3))],
[x^4*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^3,x,12,h*x/b^3+1/6*x*(a*(b*e-a*h)-b*(b*c-a*f)*x-b*(b*d-a*g)*x^2)/(b^3*(a+b*x^3)^2)-1/18*x*(a*(7*b*e-13*a*h)-2*b*(b*c-4*a*f)*x-3*b*(b*d-3*a*g)*x^2)/(a*b^3*(a+b*x^3))-1/27*(b^(2/3)*(b*c+5*a*f)-2*a^(2/3)*(b*e-7*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(4/3)*b^(10/3))+1/54*(b^(2/3)*(b*c+5*a*f)-2*a^(2/3)*(b*e-7*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(4/3)*b^(10/3))+1/3*g*log(a+b*x^3)/b^3-1/9*(b^(5/3)*c+2*a^(2/3)*b*e+5*a*b^(2/3)*f-14*a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(4/3)*b^(10/3)*sqrt(3))],
[x^3*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^3,x,10,-1/6*x*(b*c-a*f+(b*d-a*g)*x+(b*e-a*h)*x^2)/(b^2*(a+b*x^3)^2)+1/18*x*(b*c-7*a*f+2*(b*d-4*a*g)*x+3*(b*e-3*a*h)*x^2)/(a*b^2*(a+b*x^3))+1/27*(b^(1/3)*(b*c+2*a*f)-a^(1/3)*(b*d+5*a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(8/3))-1/54*(b^(1/3)*(b*c+2*a*f)-a^(1/3)*(b*d+5*a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(8/3))+1/3*h*log(a+b*x^3)/b^3-1/9*(b^(4/3)*c+a^(1/3)*b*d+2*a*b^(1/3)*f+5*a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(8/3)*sqrt(3))],
[x^2*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^3,x,8,1/18*x*(b*d-4*a*g+(2*b*e-5*a*h)*x+3*b*f*x^2)/(a*b^2*(a+b*x^3))+1/6*(-c-d*x-e*x^2-f*x^3-g*x^4-h*x^5)/(b*(a+b*x^3)^2)+1/27*(b^(1/3)*(b*d+2*a*g)-a^(1/3)*(b*e+5*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(5/3)*b^(8/3))-1/54*(b^(1/3)*(b*d+2*a*g)-a^(1/3)*(b*e+5*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(5/3)*b^(8/3))-1/9*(b^(4/3)*d+a^(1/3)*b*e+2*a*b^(1/3)*g+5*a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(5/3)*b^(8/3)*sqrt(3))],
[x*(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^3,x,8,-1/6*x*(a*(b*e-a*h)-b*(b*c-a*f)*x-b*(b*d-a*g)*x^2)/(a*b^2*(a+b*x^3)^2)+1/18*x*(a*(b*e-7*a*h)+2*b*(2*b*c+a*f)*x+3*b*(b*d+a*g)*x^2)/(a^2*b^2*(a+b*x^3))-1/27*(b^(2/3)*(2*b*c+a*f)-a^(2/3)*(b*e+2*a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(7/3)*b^(7/3))+1/54*(b^(2/3)*(2*b*c+a*f)-a^(2/3)*(b*e+2*a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(7/3)*b^(7/3))-1/9*(2*b^(5/3)*c+a^(2/3)*b*e+a*b^(2/3)*f+2*a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(7/3)*b^(7/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(a+b*x^3)^3,x,8,1/6*x*(b*c-a*f+(b*d-a*g)*x+(b*e-a*h)*x^2)/(a*b*(a+b*x^3)^2)+1/18*(-3*a*(b*e+a*h)+b*x*(5*b*c+a*f+2*(2*b*d+a*g)*x))/(a^2*b^2*(a+b*x^3))+1/27*(b^(1/3)*(5*b*c+a*f)-a^(1/3)*(2*b*d+a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(5/3))-1/54*(b^(1/3)*(5*b*c+a*f)-a^(1/3)*(2*b*d+a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(5/3))-1/9*(5*b^(4/3)*c+2*a^(1/3)*b*d+a*b^(1/3)*f+a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(5/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x*(a+b*x^3)^3),x,12,1/6*x*(a*(b*d-a*g)+a*(b*e-a*h)*x-b*(b*c-a*f)*x^2)/(a^2*b*(a+b*x^3)^2)+1/18*x*(a*(5*b*d+a*g)+2*a*(2*b*e+a*h)*x-3*b*(3*b*c-a*f)*x^2)/(a^3*b*(a+b*x^3))+c*log(x)/a^3+1/27*(b^(1/3)*(5*b*d+a*g)-a^(1/3)*(2*b*e+a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(8/3)*b^(5/3))-1/54*(b^(1/3)*(5*b*d+a*g)-a^(1/3)*(2*b*e+a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(8/3)*b^(5/3))-1/3*c*log(a+b*x^3)/a^3-1/9*(5*b^(4/3)*d+2*a^(1/3)*b*e+a*b^(1/3)*g+a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(8/3)*b^(5/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^2*(a+b*x^3)^3),x,12,-c/(a^3*x)+1/6*x*(a*(b*e-a*h)-b*(b*c-a*f)*x-b*(b*d-a*g)*x^2)/(a^2*b*(a+b*x^3)^2)+1/18*x*(a*(5*b*e+a*h)-2*b*(5*b*c-2*a*f)*x-3*b*(3*b*d-a*g)*x^2)/(a^3*b*(a+b*x^3))+d*log(x)/a^3+1/27*(2*b^(2/3)*(7*b*c-a*f)+a^(2/3)*(5*b*e+a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(10/3)*b^(4/3))-1/54*(2*b^(2/3)*(7*b*c-a*f)+a^(2/3)*(5*b*e+a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(10/3)*b^(4/3))-1/3*d*log(a+b*x^3)/a^3+1/9*(14*b^(5/3)*c-5*a^(2/3)*b*e-2*a*b^(2/3)*f-a^(5/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(10/3)*b^(4/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^3*(a+b*x^3)^3),x,12,-1/2*c/(a^3*x^2)-d/(a^3*x)-1/6*x*(b*c-a*f+(b*d-a*g)*x+(b*e-a*h)*x^2)/(a^2*(a+b*x^3)^2)-1/18*x*(11*b*c-5*a*f+2*(5*b*d-2*a*g)*x+3*(3*b*e-a*h)*x^2)/(a^3*(a+b*x^3))+e*log(x)/a^3-1/27*(5*b^(1/3)*(4*b*c-a*f)-2*a^(1/3)*(7*b*d-a*g))*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(2/3))+1/54*(5*b^(1/3)*(4*b*c-a*f)-2*a^(1/3)*(7*b*d-a*g))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(2/3))-1/3*e*log(a+b*x^3)/a^3+1/9*(20*b^(4/3)*c+14*a^(1/3)*b*d-5*a*b^(1/3)*f-2*a^(4/3)*g)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(2/3)*sqrt(3))],
[(c+d*x+e*x^2+f*x^3+g*x^4+h*x^5)/(x^4*(a+b*x^3)^3),x,12,-1/3*c/(a^3*x^3)-1/2*d/(a^3*x^2)-e/(a^3*x)-1/6*x*(b*d-a*g+(b*e-a*h)*x-b*(b*c/a-f)*x^2)/(a^2*(a+b*x^3)^2)-1/18*x*(11*b*d-5*a*g+2*(5*b*e-2*a*h)*x-3*b*(5*b*c/a-3*f)*x^2)/(a^3*(a+b*x^3))-(3*b*c-a*f)*log(x)/a^4-1/27*(5*b^(1/3)*(4*b*d-a*g)-2*a^(1/3)*(7*b*e-a*h))*log(a^(1/3)+b^(1/3)*x)/(a^(11/3)*b^(2/3))+1/54*(5*b^(1/3)*(4*b*d-a*g)-2*a^(1/3)*(7*b*e-a*h))*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(11/3)*b^(2/3))+1/3*(3*b*c-a*f)*log(a+b*x^3)/a^4+1/9*(20*b^(4/3)*d+14*a^(1/3)*b*e-5*a*b^(1/3)*g-2*a^(4/3)*h)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(11/3)*b^(2/3)*sqrt(3))],

# Integrands of the form x^m P2(x) (a+b x^3)^(p/2)

# p>0

# p<0
[x^3*(c+d*x+e*x^2)/sqrt(a+b*x^3),x,10,-4/9*a*e*sqrt(a+b*x^3)/b^2+2/5*c*x*sqrt(a+b*x^3)/b+2/7*d*x^2*sqrt(a+b*x^3)/b+2/9*e*x^3*sqrt(a+b*x^3)/b-8/7*a*d*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+4/7*3^(1/4)*a^(4/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-4/35*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*c-10*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^2*(c+d*x+e*x^2)/sqrt(a+b*x^3),x,8,2/3*c*sqrt(a+b*x^3)/b+2/5*d*x*sqrt(a+b*x^3)/b+2/7*e*x^2*sqrt(a+b*x^3)/b-8/7*a*e*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+4/7*3^(1/4)*a^(4/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-4/35*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*d-10*a^(1/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x*(c+d*x+e*x^2)/sqrt(a+b*x^3),x,6,2/3*d*sqrt(a+b*x^3)/b+2/5*e*x*sqrt(a+b*x^3)/b+2*c*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*c*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-2/5*3^(1/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(2*a^(2/3)*e+5*b^(2/3)*c*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(4/3)*sqrt(3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/sqrt(a+b*x^3),x,5,2/3*e*sqrt(a+b*x^3)/b+2*d*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*c-a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/(x*sqrt(a+b*x^3)),x,7,-2/3*c*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)+2*e*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3^(1/4)*a^(1/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*d-a^(1/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/(x^2*sqrt(a+b*x^3)),x,8,-2/3*d*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)-c*sqrt(a+b*x^3)/(a*x)+b^(1/3)*c*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-1/2*3^(1/4)*b^(1/3)*c*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(-2*a^(2/3)*e+b^(2/3)*c*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(2/3)*b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/(x^3*sqrt(a+b*x^3)),x,9,-2/3*e*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)-1/2*c*sqrt(a+b*x^3)/(a*x^2)-d*sqrt(a+b*x^3)/(a*x)+b^(1/3)*d*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-1/2*3^(1/4)*b^(1/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-1/2*3^(1/4)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*c+2*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^5*(c+d*x+e*x^2)/(a+b*x^3)^(3/2),x,8,2/3*x*(a*d+a*e*x-b*c*x^2)/(b^2*sqrt(a+b*x^3))+4/3*c*sqrt(a+b*x^3)/b^2+2/5*d*x*sqrt(a+b*x^3)/b^2+2/7*e*x^2*sqrt(a+b*x^3)/b^2-80/21*a*e*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+40/7*a^(4/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-16/105*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(14*b^(1/3)*d-25*a^(1/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^4*(c+d*x+e*x^2)/(a+b*x^3)^(3/2),x,7,2/3*x*(a*e-b*c*x-b*d*x^2)/(b^2*sqrt(a+b*x^3))+4/3*d*sqrt(a+b*x^3)/b^2+2/5*e*x*sqrt(a+b*x^3)/b^2+8/3*c*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-4*a^(1/3)*c*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-8/5*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(4*a^(2/3)*e+5*b^(2/3)*c*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(7/3)*sqrt(3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3*(c+d*x+e*x^2)/(a+b*x^3)^(3/2),x,6,-2/3*x*(c+d*x+e*x^2)/(b*sqrt(a+b*x^3))+4/3*e*sqrt(a+b*x^3)/b^2+8/3*d*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-4*a^(1/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+4/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*c-2*a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^2*(c+d*x+e*x^2)/(a+b*x^3)^(3/2),x,4,-2/3*(c+d*x+e*x^2)/(b*sqrt(a+b*x^3))+8/3*e*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-4*a^(1/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+4/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*d-2*a^(1/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x*(c+d*x+e*x^2)/(a+b*x^3)^(3/2),x,6,-2/3*x*(a*e-b*c*x-b*d*x^2)/(a*b*sqrt(a+b*x^3))-2/3*d*sqrt(a+b*x^3)/(a*b)-2/3*c*sqrt(a+b*x^3)/(a*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+c*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(2*a^(2/3)*e+b^(2/3)*(c-c*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(2/3)*b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/(a+b*x^3)^(3/2),x,4,-2/3*(a*e-b*x*(c+d*x))/(a*b*sqrt(a+b*x^3))-2/3*d*sqrt(a+b*x^3)/(a*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*c+a^(1/3)*d*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/(x*(a+b*x^3)^(3/2)),x,10,-2/3*c*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)+2/3*x*(a*d+a*e*x-b*c*x^2)/(a^2*sqrt(a+b*x^3))+2/3*c*sqrt(a+b*x^3)/a^2-2/3*e*sqrt(a+b*x^3)/(a*b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(2/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*d+a^(1/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2)/(x^2*(a+b*x^3)^(3/2)),x,11,-2/3*d*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)+2/3*x*(a*e-b*c*x-b*d*x^2)/(a^2*sqrt(a+b*x^3))+2/3*d*sqrt(a+b*x^3)/a^2-c*sqrt(a+b*x^3)/(a^2*x)+5/3*b^(1/3)*c*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-5/2*b^(1/3)*c*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(-2*a^(2/3)*e+5*b^(2/3)*c*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(5/3)*b^(1/3)*sqrt(3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],

# Integrands of the form x^m P4(x) (a+b x^3)^(p/2)

# p>0
[x^3*(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3),x,13,-4/45*a^2*e*sqrt(a+b*x^3)/b^2+6/935*a*(17*b*c-8*a*f)*x*sqrt(a+b*x^3)/b^2+6/1729*a*(19*b*d-10*a*g)*x^2*sqrt(a+b*x^3)/b^2+2/45*a*e*x^3*sqrt(a+b*x^3)/b+6/187*a*f*x^4*sqrt(a+b*x^3)/b+6/247*a*g*x^5*sqrt(a+b*x^3)/b+2/692835*x^3*(62985*c*x+53295*d*x^2+46189*e*x^3+40755*f*x^4+36465*g*x^5)*sqrt(a+b*x^3)-24/1729*a^2*(19*b*d-10*a*g)*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+12/1729*3^(1/4)*a^(7/3)*(19*b*d-10*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-4/1616615*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(1729*b^(1/3)*(17*b*c-8*a*f)-1870*a^(1/3)*(19*b*d-10*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^2*(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3),x,11,2/45*a*(5*b*c-2*a*f)*sqrt(a+b*x^3)/b^2+6/935*a*(17*b*d-8*a*g)*x*sqrt(a+b*x^3)/b^2+6/91*a*e*x^2*sqrt(a+b*x^3)/b+2/45*a*f*x^3*sqrt(a+b*x^3)/b+6/187*a*g*x^4*sqrt(a+b*x^3)/b+2/109395*x^2*(12155*c*x+9945*d*x^2+8415*e*x^3+7293*f*x^4+6435*g*x^5)*sqrt(a+b*x^3)-24/91*a^2*e*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+12/91*3^(1/4)*a^(7/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-4/85085*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(1547*b*d-728*a*g-1870*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(7/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x*(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3),x,9,2/45*a*(5*b*d-2*a*g)*sqrt(a+b*x^3)/b^2+6/55*a*e*x*sqrt(a+b*x^3)/b+6/91*a*f*x^2*sqrt(a+b*x^3)/b+2/45*a*g*x^3*sqrt(a+b*x^3)/b+2/45045*x*(6435*c*x+5005*d*x^2+4095*e*x^3+3465*f*x^4+3003*g*x^5)*sqrt(a+b*x^3)+6/91*a*(13*b*c-4*a*f)*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/91*3^(1/4)*a^(4/3)*(13*b*c-4*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-2/5005*3^(3/4)*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(182*a^(2/3)*b^(1/3)*e+55*(13*b*c-4*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3),x,8,2/9*a*e*sqrt(a+b*x^3)/b+6/55*a*f*x*sqrt(a+b*x^3)/b+6/91*a*g*x^2*sqrt(a+b*x^3)/b+2/45045*(9009*c*x+6435*d*x^2+5005*e*x^3+4095*f*x^4+3465*g*x^5)*sqrt(a+b*x^3)+6/91*a*(13*b*d-4*a*g)*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/91*3^(1/4)*a^(4/3)*(13*b*d-4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/5005*3^(3/4)*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(91*b^(1/3)*(11*b*c-2*a*f)-55*a^(1/3)*(13*b*d-4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x,x,11,-2/3*c*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+2/9*a*f*sqrt(a+b*x^3)/b+6/55*a*g*x*sqrt(a+b*x^3)/b+2/3465*(1155*c*x+693*d*x^2+495*e*x^3+385*f*x^4+315*g*x^5)*sqrt(a+b*x^3)/x+6/7*a*e*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/7*3^(1/4)*a^(4/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+2/385*3^(3/4)*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(77*b*d-14*a*g-55*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^2,x,11,-2/3*d*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+2/9*a*g*sqrt(a+b*x^3)/b-3*c*sqrt(a+b*x^3)/x+2/315*(315*c*x+105*d*x^2+63*e*x^3+45*f*x^4+35*g*x^5)*sqrt(a+b*x^3)/x^2+3/7*(7*b*c+2*a*f)*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/14*3^(1/4)*a^(1/3)*(7*b*c+2*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+1/35*3^(3/4)*a^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(14*a^(2/3)*b^(1/3)*e-5*(7*b*c+2*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^3,x,10,-2/3*e*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+3/2*c*sqrt(a+b*x^3)/x^2-3*d*sqrt(a+b*x^3)/x-2/105*(105*c*x-105*d*x^2-35*e*x^3-21*f*x^4-15*g*x^5)*sqrt(a+b*x^3)/x^3+3/7*(7*b*d+2*a*g)*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/14*3^(1/4)*a^(1/3)*(7*b*d+2*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+1/70*3^(3/4)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*(5*b*c+4*a*f)-10*a^(1/3)*(7*b*d+2*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^4,x,11,-1/3*(b*c+2*a*f)*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)+1/3*c*sqrt(a+b*x^3)/x^3+3/2*d*sqrt(a+b*x^3)/x^2-3*e*sqrt(a+b*x^3)/x-2/15*(5*c*x+15*d*x^2-15*e*x^3-5*f*x^4-3*g*x^5)*sqrt(a+b*x^3)/x^4+3*b^(1/3)*e*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))-3/2*3^(1/4)*a^(1/3)*b^(1/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+1/10*3^(3/4)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(5*b*d+4*a*g-10*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^5,x,12,-1/3*(b*d+2*a*g)*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)+3/20*c*sqrt(a+b*x^3)/x^4+1/3*d*sqrt(a+b*x^3)/x^3+3/2*e*sqrt(a+b*x^3)/x^2-3/8*(b*c+8*a*f)*sqrt(a+b*x^3)/(a*x)-2/15*(3*c*x+5*d*x^2+15*e*x^3-15*f*x^4-5*g*x^5)*sqrt(a+b*x^3)/x^5+3/8*b^(1/3)*(b*c+8*a*f)*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/16*3^(1/4)*b^(1/3)*(b*c+8*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+1/8*3^(3/4)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(4*a^(2/3)*b^(1/3)*e-(b*c+8*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^6,x,10,-1/3*b*e*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)-1/60*(12*c/x^5+15*d/x^4+20*e/x^3+30*f/x^2+60*g/x)*sqrt(a+b*x^3)-3/20*b*c*sqrt(a+b*x^3)/(a*x^2)-3/8*b*d*sqrt(a+b*x^3)/(a*x)+3/8*b^(1/3)*(b*d+8*a*g)*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/16*3^(1/4)*b^(1/3)*(b*d+8*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-1/40*3^(3/4)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(2*b^(1/3)*(b*c-10*a*f)+5*a^(1/3)*(b*d+8*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^7,x,11,1/12*b*(b*c-4*a*f)*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/60*(10*c/x^6+12*d/x^5+15*e/x^4+20*f/x^3+30*g/x^2)*sqrt(a+b*x^3)-1/12*b*c*sqrt(a+b*x^3)/(a*x^3)-3/20*b*d*sqrt(a+b*x^3)/(a*x^2)-3/8*b*e*sqrt(a+b*x^3)/(a*x)+3/8*b^(4/3)*e*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-3/16*3^(1/4)*b^(4/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-1/40*3^(3/4)*b^(2/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(2*b*d-20*a*g+5*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^8,x,12,1/12*b*(b*d-4*a*g)*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/420*(60*c/x^7+70*d/x^6+84*e/x^5+105*f/x^4+140*g/x^3)*sqrt(a+b*x^3)-3/56*b*c*sqrt(a+b*x^3)/(a*x^4)-1/12*b*d*sqrt(a+b*x^3)/(a*x^3)-3/20*b*e*sqrt(a+b*x^3)/(a*x^2)+3/112*b*(5*b*c-14*a*f)*sqrt(a+b*x^3)/(a^2*x)-3/112*b^(4/3)*(5*b*c-14*a*f)*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+3/224*3^(1/4)*b^(4/3)*(5*b*c-14*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-1/560*3^(3/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(28*a^(2/3)*b^(1/3)*e-5*(5*b*c-14*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(c+d*x+e*x^2+f*x^3+g*x^4)*sqrt(a+b*x^3)/x^9,x,13,1/12*b^2*e*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/840*(105*c/x^8+120*d/x^7+140*e/x^6+168*f/x^5+210*g/x^4)*sqrt(a+b*x^3)-3/80*b*c*sqrt(a+b*x^3)/(a*x^5)-3/56*b*d*sqrt(a+b*x^3)/(a*x^4)-1/12*b*e*sqrt(a+b*x^3)/(a*x^3)+3/320*b*(7*b*c-16*a*f)*sqrt(a+b*x^3)/(a^2*x^2)+3/112*b*(5*b*d-14*a*g)*sqrt(a+b*x^3)/(a^2*x)-3/112*b^(4/3)*(5*b*d-14*a*g)*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+3/224*3^(1/4)*b^(4/3)*(5*b*d-14*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+1/2240*3^(3/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*(7*b*c-16*a*f)+20*a^(1/3)*(5*b*d-14*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^2*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^3*(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4),x,14,2/3900225*x^3*(a+b*x^3)^(3/2)*(229425*c*x+205275*d*x^2+185725*e*x^3+169575*f*x^4+156009*g*x^5)-4/105*a^3*e*sqrt(a+b*x^3)/b^2+54/21505*a^2*(23*b*c-8*a*f)*x*sqrt(a+b*x^3)/b^2+54/8645*a^2*(5*b*d-2*a*g)*x^2*sqrt(a+b*x^3)/b^2+2/105*a^2*e*x^3*sqrt(a+b*x^3)/b+54/4301*a^2*f*x^4*sqrt(a+b*x^3)/b+54/6175*a^2*g*x^5*sqrt(a+b*x^3)/b+2/185910725*a*x^3*(8947575*c*x+6774075*d*x^2+5311735*e*x^3+4279275*f*x^4+3522519*g*x^5)*sqrt(a+b*x^3)-216/8645*a^3*(5*b*d-2*a*g)*sqrt(a+b*x^3)/(b^(8/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+108/8645*3^(1/4)*a^(10/3)*(5*b*d-2*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-36/37182145*3^(3/4)*a^3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(1729*b^(1/3)*(23*b*c-8*a*f)-8602*a^(1/3)*(5*b*d-2*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(8/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x^2*(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4),x,12,2/780045*x^2*(a+b*x^3)^(3/2)*(52003*c*x+45885*d*x^2+41055*e*x^3+37145*f*x^4+33915*g*x^5)+2/105*a^2*(7*b*c-2*a*f)*sqrt(a+b*x^3)/b^2+54/21505*a^2*(23*b*d-8*a*g)*x*sqrt(a+b*x^3)/b^2+54/1729*a^2*e*x^2*sqrt(a+b*x^3)/b+2/105*a^2*f*x^3*sqrt(a+b*x^3)/b+54/4301*a^2*g*x^4*sqrt(a+b*x^3)/b+2/111546435*a*x^2*(7436429*c*x+5368545*d*x^2+4064445*e*x^3+3187041*f*x^4+2567565*g*x^5)*sqrt(a+b*x^3)-216/1729*a^3*e*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+108/1729*3^(1/4)*a^(10/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+36/37182145*3^(3/4)*a^3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(-1729*(23*b*d-8*a*g)+43010*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(7/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x*(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4),x,10,2/440895*x*(a+b*x^3)^(3/2)*(33915*c*x+29393*d*x^2+25935*e*x^3+23205*f*x^4+20995*g*x^5)+2/105*a^2*(7*b*d-2*a*g)*sqrt(a+b*x^3)/b^2+54/935*a^2*e*x*sqrt(a+b*x^3)/b+54/1729*a^2*f*x^2*sqrt(a+b*x^3)/b+2/105*a^2*g*x^3*sqrt(a+b*x^3)/b+2/4849845*a*x*(479655*c*x+323323*d*x^2+233415*e*x^3+176715*f*x^4+138567*g*x^5)*sqrt(a+b*x^3)+54/1729*a^2*(19*b*c-4*a*f)*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/1729*3^(1/4)*a^(7/3)*(19*b*c-4*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-18/1616615*3^(3/4)*a^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(3458*a^(2/3)*b^(1/3)*e+935*(19*b*c-4*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4),x,9,2/692835*(a+b*x^3)^(3/2)*(62985*c*x+53295*d*x^2+46189*e*x^3+40755*f*x^4+36465*g*x^5)+2/15*a^2*e*sqrt(a+b*x^3)/b+54/935*a^2*f*x*sqrt(a+b*x^3)/b+54/1729*a^2*g*x^2*sqrt(a+b*x^3)/b+2/4849845*a*(793611*c*x+479655*d*x^2+323323*e*x^3+233415*f*x^4+176715*g*x^5)*sqrt(a+b*x^3)+54/1729*a^2*(19*b*d-4*a*g)*sqrt(a+b*x^3)/(b^(5/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/1729*3^(1/4)*a^(7/3)*(19*b*d-4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+18/1616615*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(1729*b^(1/3)*(17*b*c-2*a*f)-935*a^(1/3)*(19*b*d-4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x,x,12,2/109395*(a+b*x^3)^(3/2)*(12155*c*x+9945*d*x^2+8415*e*x^3+7293*f*x^4+6435*g*x^5)/x-2/3*a^(3/2)*c*arctanh(sqrt(a+b*x^3)/sqrt(a))+2/15*a^2*f*sqrt(a+b*x^3)/b+54/935*a^2*g*x*sqrt(a+b*x^3)/b+2/255255*a*(85085*c*x+41769*d*x^2+25245*e*x^3+17017*f*x^4+12285*g*x^5)*sqrt(a+b*x^3)/x+54/91*a^2*e*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/91*3^(1/4)*a^(7/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+18/85085*3^(3/4)*a^2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(1547*b*d-182*a*g-935*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(4/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^2,x,12,2/45045*(a+b*x^3)^(3/2)*(6435*c*x+5005*d*x^2+4095*e*x^3+3465*f*x^4+3003*g*x^5)/x^2-2/3*a^(3/2)*d*arctanh(sqrt(a+b*x^3)/sqrt(a))+2/15*a^2*g*sqrt(a+b*x^3)/b-27/7*a*c*sqrt(a+b*x^3)/x+2/15015*a*(19305*c*x+5005*d*x^2+2457*e*x^3+1485*f*x^4+1001*g*x^5)*sqrt(a+b*x^3)/x^2+27/91*a*(13*b*c+2*a*f)*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/182*3^(1/4)*a^(4/3)*(13*b*c+2*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/5005*3^(3/4)*a^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(182*a^(2/3)*b^(1/3)*e-55*(13*b*c+2*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^3,x,11,2/45045*(a+b*x^3)^(3/2)*(9009*c*x+6435*d*x^2+5005*e*x^3+4095*f*x^4+3465*g*x^5)/x^3-2/3*a^(3/2)*e*arctanh(sqrt(a+b*x^3)/sqrt(a))+27/10*a*c*sqrt(a+b*x^3)/x^2-27/7*a*d*sqrt(a+b*x^3)/x-2/15015*a*(27027*c*x-19305*d*x^2-5005*e*x^3-2457*f*x^4-1485*g*x^5)*sqrt(a+b*x^3)/x^3+27/91*a*(13*b*d+2*a*g)*sqrt(a+b*x^3)/(b^(2/3)*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/182*3^(1/4)*a^(4/3)*(13*b*d+2*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/10010*3^(3/4)*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(91*b^(1/3)*(11*b*c+4*a*f)-110*a^(1/3)*(13*b*d+2*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^4,x,12,2/3465*(a+b*x^3)^(3/2)*(1155*c*x+693*d*x^2+495*e*x^3+385*f*x^4+315*g*x^5)/x^4-1/3*(3*b*c+2*a*f)*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+a*c*sqrt(a+b*x^3)/x^3+27/10*a*d*sqrt(a+b*x^3)/x^2-27/7*a*e*sqrt(a+b*x^3)/x-2/1155*a*(1155*c*x+2079*d*x^2-1485*e*x^3-385*f*x^4-189*g*x^5)*sqrt(a+b*x^3)/x^4+27/7*a*b^(1/3)*e*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))-27/14*3^(1/4)*a^(4/3)*b^(1/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/770*3^(3/4)*a*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(77*b*d+28*a*g-110*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^5,x,13,2/315*(a+b*x^3)^(3/2)*(315*c*x+105*d*x^2+63*e*x^3+45*f*x^4+35*g*x^5)/x^5-1/3*(3*b*d+2*a*g)*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+27/20*a*c*sqrt(a+b*x^3)/x^4+a*d*sqrt(a+b*x^3)/x^3+27/10*a*e*sqrt(a+b*x^3)/x^2-27/56*(7*b*c+8*a*f)*sqrt(a+b*x^3)/x-2/105*a*(189*c*x+105*d*x^2+189*e*x^3-135*f*x^4-35*g*x^5)*sqrt(a+b*x^3)/x^5+27/56*b^(1/3)*(7*b*c+8*a*f)*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))-27/112*3^(1/4)*a^(1/3)*b^(1/3)*(7*b*c+8*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/280*3^(3/4)*a^(1/3)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(28*a^(2/3)*b^(1/3)*e-5*(7*b*c+8*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^6,x,11,-1/60*(12*c/x^5+15*d/x^4+20*e/x^3+30*f/x^2+60*g/x)*(a+b*x^3)^(3/2)-b*e*arctanh(sqrt(a+b*x^3)/sqrt(a))*sqrt(a)+27/20*b*c*sqrt(a+b*x^3)/x^2-27/8*b*d*sqrt(a+b*x^3)/x-1/140*b*(252*c*x-315*d*x^2-140*e*x^3-126*f*x^4-180*g*x^5)*sqrt(a+b*x^3)/x^3+27/56*b^(1/3)*(7*b*d+8*a*g)*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))-27/112*3^(1/4)*a^(1/3)*b^(1/3)*(7*b*d+8*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/280*3^(3/4)*b^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(14*b^(1/3)*(b*c+2*a*f)-5*a^(1/3)*(7*b*d+8*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^7,x,12,-1/60*(10*c/x^6+12*d/x^5+15*e/x^4+20*f/x^3+30*g/x^2)*(a+b*x^3)^(3/2)-1/4*b*(b*c+4*a*f)*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)+1/4*b*c*sqrt(a+b*x^3)/x^3+27/20*b*d*sqrt(a+b*x^3)/x^2-27/8*b*e*sqrt(a+b*x^3)/x-1/20*b*(10*c*x+36*d*x^2-45*e*x^3-20*f*x^4-18*g*x^5)*sqrt(a+b*x^3)/x^4+27/8*b^(4/3)*e*sqrt(a+b*x^3)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))-27/16*3^(1/4)*a^(1/3)*b^(4/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/40*3^(3/4)*b^(2/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(2*b*d+4*a*g-5*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^8,x,13,-1/420*(60*c/x^7+70*d/x^6+84*e/x^5+105*f/x^4+140*g/x^3)*(a+b*x^3)^(3/2)-1/4*b*(b*d+4*a*g)*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)+27/280*b*c*sqrt(a+b*x^3)/x^4+1/4*b*d*sqrt(a+b*x^3)/x^3+27/20*b*e*sqrt(a+b*x^3)/x^2-27/112*b*(b*c+14*a*f)*sqrt(a+b*x^3)/(a*x)-1/140*b*(36*c*x+70*d*x^2+252*e*x^3-315*f*x^4-140*g*x^5)*sqrt(a+b*x^3)/x^5+27/112*b^(4/3)*(b*c+14*a*f)*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/224*3^(1/4)*b^(4/3)*(b*c+14*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/560*3^(3/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(28*a^(2/3)*b^(1/3)*e-5*(b*c+14*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^9,x,11,-1/840*(105*c/x^8+120*d/x^7+140*e/x^6+168*f/x^5+210*g/x^4)*(a+b*x^3)^(3/2)-1/4*b^2*e*arctanh(sqrt(a+b*x^3)/sqrt(a))/sqrt(a)-1/560*b*(63*c/x^5+90*d/x^4+140*e/x^3+252*f/x^2+630*g/x)*sqrt(a+b*x^3)-27/320*b^2*c*sqrt(a+b*x^3)/(a*x^2)-27/112*b^2*d*sqrt(a+b*x^3)/(a*x)+27/112*b^(4/3)*(b*d+14*a*g)*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/224*3^(1/4)*b^(4/3)*(b*d+14*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-9/2240*3^(3/4)*b^(4/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*(b*c-16*a*f)+20*a^(1/3)*(b*d+14*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^10,x,12,-1/2520*(280*c/x^9+315*d/x^8+360*e/x^7+420*f/x^6+504*g/x^5)*(a+b*x^3)^(3/2)+1/24*b^2*(b*c-6*a*f)*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/1680*b*(140*c/x^6+189*d/x^5+270*e/x^4+420*f/x^3+756*g/x^2)*sqrt(a+b*x^3)-1/24*b^2*c*sqrt(a+b*x^3)/(a*x^3)-27/320*b^2*d*sqrt(a+b*x^3)/(a*x^2)-27/112*b^2*e*sqrt(a+b*x^3)/(a*x)+27/112*b^(7/3)*e*sqrt(a+b*x^3)/(a*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))-27/224*3^(1/4)*b^(7/3)*e*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-9/2240*3^(3/4)*b^(5/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b*d-112*a*g+20*a^(1/3)*b^(2/3)*e*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^11,x,13,-1/2520*(252*c/x^10+280*d/x^9+315*e/x^8+360*f/x^7+420*g/x^6)*(a+b*x^3)^(3/2)+1/24*b^2*(b*d-6*a*g)*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/1680*b*(108*c/x^7+140*d/x^6+189*e/x^5+270*f/x^4+420*g/x^3)*sqrt(a+b*x^3)-27/1120*b^2*c*sqrt(a+b*x^3)/(a*x^4)-1/24*b^2*d*sqrt(a+b*x^3)/(a*x^3)-27/320*b^2*e*sqrt(a+b*x^3)/(a*x^2)+27/448*b^2*(b*c-4*a*f)*sqrt(a+b*x^3)/(a^2*x)-27/448*b^(7/3)*(b*c-4*a*f)*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+27/896*3^(1/4)*b^(7/3)*(b*c-4*a*f)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))-9/2240*3^(3/4)*b^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*a^(2/3)*b^(1/3)*e-5*(b*c-4*a*f)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(a+b*x^3)^(3/2)*(c+d*x+e*x^2+f*x^3+g*x^4)/x^12,x,14,-1/27720*(2520*c/x^11+2772*d/x^10+3080*e/x^9+3465*f/x^8+3960*g/x^7)*(a+b*x^3)^(3/2)+1/24*b^3*e*arctanh(sqrt(a+b*x^3)/sqrt(a))/a^(3/2)-1/18480*b*(945*c/x^8+1188*d/x^7+1540*e/x^6+2079*f/x^5+2970*g/x^4)*sqrt(a+b*x^3)-27/1760*b^2*c*sqrt(a+b*x^3)/(a*x^5)-27/1120*b^2*d*sqrt(a+b*x^3)/(a*x^4)-1/24*b^2*e*sqrt(a+b*x^3)/(a*x^3)+27/7040*b^2*(7*b*c-22*a*f)*sqrt(a+b*x^3)/(a^2*x^2)+27/448*b^2*(b*d-4*a*g)*sqrt(a+b*x^3)/(a^2*x)-27/448*b^(7/3)*(b*d-4*a*g)*sqrt(a+b*x^3)/(a^2*(b^(1/3)*x+a^(1/3)*(1+sqrt(3))))+27/896*3^(1/4)*b^(7/3)*(b*d-4*a*g)*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^(5/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))+9/49280*3^(3/4)*b^(7/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(7*b^(1/3)*(7*b*c-22*a*f)+110*a^(1/3)*(b*d-4*a*g)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(a^2*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],

# p<0

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

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

# Integrands of the form x^m P3(x) (a+b x^4)^p

# p>0
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4),x,2,a*c*x+1/2*a*d*x^2+1/3*a*e*x^3+1/4*a*f*x^4+1/5*b*c*x^5+1/6*b*d*x^6+1/7*b*e*x^7+1/8*b*f*x^8],
[x^3*(c+d*x+e*x^2+f*x^3)*(a+b*x^4),x,2,1/4*a*c*x^4+1/5*a*d*x^5+1/6*a*e*x^6+1/7*a*f*x^7+1/8*b*c*x^8+1/9*b*d*x^9+1/10*b*e*x^10+1/11*b*f*x^11],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^2,x,3,a^2*c*x+1/2*a^2*d*x^2+1/3*a^2*e*x^3+2/5*a*b*c*x^5+1/3*a*b*d*x^6+2/7*a*b*e*x^7+1/9*b^2*c*x^9+1/10*b^2*d*x^10+1/11*b^2*e*x^11+1/12*f*(a+b*x^4)^3/b],
[x^3*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^2,x,3,1/5*a^2*d*x^5+1/6*a^2*e*x^6+1/7*a^2*f*x^7+2/9*a*b*d*x^9+1/5*a*b*e*x^10+2/11*a*b*f*x^11+1/13*b^2*d*x^13+1/14*b^2*e*x^14+1/15*b^2*f*x^15+1/12*c*(a+b*x^4)^3/b],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^3,x,3,a^3*c*x+1/2*a^3*d*x^2+1/3*a^3*e*x^3+3/5*a^2*b*c*x^5+1/2*a^2*b*d*x^6+3/7*a^2*b*e*x^7+1/3*a*b^2*c*x^9+3/10*a*b^2*d*x^10+3/11*a*b^2*e*x^11+1/13*b^3*c*x^13+1/14*b^3*d*x^14+1/15*b^3*e*x^15+1/16*f*(a+b*x^4)^4/b],
[x^3*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^3,x,3,1/5*a^3*d*x^5+1/6*a^3*e*x^6+1/7*a^3*f*x^7+1/3*a^2*b*d*x^9+3/10*a^2*b*e*x^10+3/11*a^2*b*f*x^11+3/13*a*b^2*d*x^13+3/14*a*b^2*e*x^14+1/5*a*b^2*f*x^15+1/17*b^3*d*x^17+1/18*b^3*e*x^18+1/19*b^3*f*x^19+1/16*c*(a+b*x^4)^4/b],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^4,x,3,a^4*c*x+1/2*a^4*d*x^2+1/3*a^4*e*x^3+4/5*a^3*b*c*x^5+2/3*a^3*b*d*x^6+4/7*a^3*b*e*x^7+2/3*a^2*b^2*c*x^9+3/5*a^2*b^2*d*x^10+6/11*a^2*b^2*e*x^11+4/13*a*b^3*c*x^13+2/7*a*b^3*d*x^14+4/15*a*b^3*e*x^15+1/17*b^4*c*x^17+1/18*b^4*d*x^18+1/19*b^4*e*x^19+1/20*f*(a+b*x^4)^5/b],
[x^3*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^4,x,3,1/5*a^4*d*x^5+1/6*a^4*e*x^6+1/7*a^4*f*x^7+4/9*a^3*b*d*x^9+2/5*a^3*b*e*x^10+4/11*a^3*b*f*x^11+6/13*a^2*b^2*d*x^13+3/7*a^2*b^2*e*x^14+2/5*a^2*b^2*f*x^15+4/17*a*b^3*d*x^17+2/9*a*b^3*e*x^18+4/19*a*b^3*f*x^19+1/21*b^4*d*x^21+1/22*b^4*e*x^22+1/23*b^4*f*x^23+1/20*c*(a+b*x^4)^5/b],

# p<0
[(c+d*x+e*x^2+f*x^3)/(a-b*x^4),x,9,-1/4*f*log(a-b*x^4)/b+1/2*d*arctanh(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))+1/2*arctan(b^(1/4)*x/a^(1/4))*(-e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4))+1/2*arctanh(b^(1/4)*x/a^(1/4))*(e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4))],
[x^3*(c+d*x+e*x^2+f*x^3)/(a-b*x^4),x,12,-d*x/b-1/2*e*x^2/b-1/3*f*x^3/b-1/4*c*log(a-b*x^4)/b+1/2*e*arctanh(x^2*sqrt(b)/sqrt(a))*sqrt(a)/b^(3/2)+1/2*a^(1/4)*arctan(b^(1/4)*x/a^(1/4))*(-f*sqrt(a)+d*sqrt(b))/b^(7/4)+1/2*a^(1/4)*arctanh(b^(1/4)*x/a^(1/4))*(f*sqrt(a)+d*sqrt(b))/b^(7/4)],
[(c+d*x+e*x^2+f*x^3)/(a+b*x^4),x,15,1/4*f*log(a+b*x^4)/b+1/2*d*arctan(x^2*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))-1/4*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))-1/2*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))+1/2*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+c*sqrt(b))/(a^(3/4)*b^(3/4)*sqrt(2))],
[x^3*(c+d*x+e*x^2+f*x^3)/(a+b*x^4),x,18,d*x/b+1/2*e*x^2/b+1/3*f*x^3/b+1/4*c*log(a+b*x^4)/b-1/2*e*arctan(x^2*sqrt(b)/sqrt(a))*sqrt(a)/b^(3/2)+1/4*a^(1/4)*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-f*sqrt(a)+d*sqrt(b))/(b^(7/4)*sqrt(2))-1/4*a^(1/4)*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-f*sqrt(a)+d*sqrt(b))/(b^(7/4)*sqrt(2))+1/2*a^(1/4)*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(f*sqrt(a)+d*sqrt(b))/(b^(7/4)*sqrt(2))-1/2*a^(1/4)*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(f*sqrt(a)+d*sqrt(b))/(b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^2,x,14,1/4*(-a*f+b*x*(c+d*x+e*x^2))/(a*b*(a+b*x^4))+1/4*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+3*c*sqrt(b))/(a^(7/4)*b^(3/4)*sqrt(2))],
[x^3*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^2,x,14,1/4*(-c-d*x-e*x^2-f*x^3)/(b*(a+b*x^4))+1/4*e*arctan(x^2*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))-1/16*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-3*f*sqrt(a)+d*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))+1/16*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-3*f*sqrt(a)+d*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))-1/8*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(3*f*sqrt(a)+d*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))+1/8*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(3*f*sqrt(a)+d*sqrt(b))/(a^(3/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^3,x,15,1/32*x*(7*c+6*d*x+5*e*x^2)/(a^2*(a+b*x^4))+1/8*(-a*f+b*x*(c+d*x+e*x^2))/(a*b*(a+b*x^4)^2)+3/16*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))-1/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))+1/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))-1/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))+1/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(5*e*sqrt(a)+21*c*sqrt(b))/(a^(11/4)*b^(3/4)*sqrt(2))],
[x^3*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^3,x,15,1/8*(-c-d*x-e*x^2-f*x^3)/(b*(a+b*x^4)^2)+1/32*x*(d+2*e*x+3*f*x^2)/(a*b*(a+b*x^4))+1/16*e*arctan(x^2*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))-3/128*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-f*sqrt(a)+d*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))+3/128*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-f*sqrt(a)+d*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))-3/64*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(f*sqrt(a)+d*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))+3/64*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(f*sqrt(a)+d*sqrt(b))/(a^(7/4)*b^(7/4)*sqrt(2))],
[(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^4,x,16,1/96*x*(11*c+10*d*x+9*e*x^2)/(a^2*(a+b*x^4)^2)+1/384*x*(77*c+60*d*x+45*e*x^2)/(a^3*(a+b*x^4))+1/12*(-a*f+b*x*(c+d*x+e*x^2))/(a*b*(a+b*x^4)^3)+5/32*d*arctan(x^2*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(15*e*sqrt(a)+77*c*sqrt(b))/(a^(15/4)*b^(3/4)*sqrt(2))],
[x^3*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^4,x,16,1/12*(-c-d*x-e*x^2-f*x^3)/(b*(a+b*x^4)^3)+1/96*x*(d+2*e*x+3*f*x^2)/(a*b*(a+b*x^4)^2)+1/384*x*(7*d+12*e*x+15*f*x^2)/(a^2*b*(a+b*x^4))+1/32*e*arctan(x^2*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))-1/512*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*f*sqrt(a)+7*d*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))+1/512*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(-5*f*sqrt(a)+7*d*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))-1/256*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(5*f*sqrt(a)+7*d*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))+1/256*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(5*f*sqrt(a)+7*d*sqrt(b))/(a^(11/4)*b^(7/4)*sqrt(2))],

# Integrands of the form x^m P3(x) (a+b x^4)^(p/2)

# p>0
[x^4*(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4),x,14,1/10*f*x^4*(a+b*x^4)^(3/2)/b-1/120*(8*a*f-15*b*d*x^2)*(a+b*x^4)^(3/2)/b^2-1/16*a^2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+2/21*a*c*x*sqrt(a+b*x^4)/b-1/16*a*d*x^2*sqrt(a+b*x^4)/b+2/45*a*e*x^3*sqrt(a+b*x^4)/b+1/63*x^5*(9*c+7*e*x^2)*sqrt(a+b*x^4)-2/15*a^2*e*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+2/15*a^(9/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-1/105*a^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(7*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^3*(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4),x,13,1/24*(4*c+3*e*x^2)*(a+b*x^4)^(3/2)/b-1/16*a^2*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+2/21*a*d*x*sqrt(a+b*x^4)/b-1/16*a*e*x^2*sqrt(a+b*x^4)/b+2/45*a*f*x^3*sqrt(a+b*x^4)/b+1/63*x^5*(9*d+7*f*x^2)*sqrt(a+b*x^4)-2/15*a^2*f*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+2/15*a^(9/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-1/105*a^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(7*f*sqrt(a)+5*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^2*(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4),x,12,1/24*(4*d+3*f*x^2)*(a+b*x^4)^(3/2)/b-1/16*a^2*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+2/21*a*e*x*sqrt(a+b*x^4)/b-1/16*a*f*x^2*sqrt(a+b*x^4)/b+1/35*x^3*(7*c+5*e*x^2)*sqrt(a+b*x^4)+2/5*a*c*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-2/5*a^(5/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/105*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-5*e*sqrt(a)+21*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[x*(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4),x,12,1/6*e*(a+b*x^4)^(3/2)/b+1/4*a*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+2/21*a*f*x*sqrt(a+b*x^4)/b+1/4*c*x^2*sqrt(a+b*x^4)+1/35*x^3*(7*d+5*f*x^2)*sqrt(a+b*x^4)+2/5*a*d*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-2/5*a^(5/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/105*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-5*f*sqrt(a)+21*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4),x,11,1/6*f*(a+b*x^4)^(3/2)/b+1/4*a*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+1/4*d*x^2*sqrt(a+b*x^4)+1/15*x*(5*c+3*e*x^2)*sqrt(a+b*x^4)+2/5*a*e*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-2/5*a^(5/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/15*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x,x,14,-1/2*c*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)+1/4*a*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+1/4*(2*c+e*x^2)*sqrt(a+b*x^4)+1/15*x*(5*d+3*f*x^2)*sqrt(a+b*x^4)+2/5*a*f*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-2/5*a^(5/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/15*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*f*sqrt(a)+5*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^2,x,14,-1/2*d*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)+1/4*a*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)-1/3*(3*c-e*x^2)*sqrt(a+b*x^4)/x+1/4*(2*d+f*x^2)*sqrt(a+b*x^4)+2*c*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-2*a^(1/4)*b^(1/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+1/3*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(e*sqrt(a)+3*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^3,x,14,-1/2*e*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)+1/2*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)-1/2*(c-e*x^2)*sqrt(a+b*x^4)/x^2-1/3*(3*d-f*x^2)*sqrt(a+b*x^4)/x+2*d*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-2*a^(1/4)*b^(1/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+1/3*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(f*sqrt(a)+3*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^4,x,15,-1/2*f*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)+1/2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)-2*e*sqrt(a+b*x^4)/x-1/3*(c-3*e*x^2)*sqrt(a+b*x^4)/x^3-1/2*(d-f*x^2)*sqrt(a+b*x^4)/x^2+2*e*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-2*a^(1/4)*b^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+1/3*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*e*sqrt(a)+c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^5,x,13,-1/4*b*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)+1/2*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)-1/12*(3*c/x^4+4*d/x^3+6*e/x^2+12*f/x)*sqrt(a+b*x^4)+2*f*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-2*a^(1/4)*b^(1/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+1/3*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*f*sqrt(a)+d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^6,x,14,-1/4*b*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)+1/2*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)-1/60*(12*c/x^5+15*d/x^4+20*e/x^3+30*f/x^2)*sqrt(a+b*x^4)-2/5*b*c*sqrt(a+b*x^4)/(a*x)+2/5*b^(3/2)*c*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-2/5*b^(5/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+1/15*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*e*sqrt(a)+3*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^7,x,12,-1/4*b*e*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/60*(10*c/x^6+12*d/x^5+15*e/x^4+20*f/x^3)*sqrt(a+b*x^4)-1/6*b*c*sqrt(a+b*x^4)/(a*x^2)-2/5*b*d*sqrt(a+b*x^4)/(a*x)+2/5*b^(3/2)*d*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-2/5*b^(5/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+1/15*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*f*sqrt(a)+3*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^8,x,13,-1/4*b*f*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/420*(60*c/x^7+70*d/x^6+84*e/x^5+105*f/x^4)*sqrt(a+b*x^4)-2/21*b*c*sqrt(a+b*x^4)/(a*x^3)-1/6*b*d*sqrt(a+b*x^4)/(a*x^2)-2/5*b*e*sqrt(a+b*x^4)/(a*x)+2/5*b^(3/2)*e*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-2/5*b^(5/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))-1/105*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-21*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^9,x,14,1/16*b^2*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/840*(105*c/x^8+120*d/x^7+140*e/x^6+168*f/x^5)*sqrt(a+b*x^4)-1/16*b*c*sqrt(a+b*x^4)/(a*x^4)-2/21*b*d*sqrt(a+b*x^4)/(a*x^3)-1/6*b*e*sqrt(a+b*x^4)/(a*x^2)-2/5*b*f*sqrt(a+b*x^4)/(a*x)+2/5*b^(3/2)*f*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-2/5*b^(5/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))-1/105*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-21*f*sqrt(a)+5*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*sqrt(a+b*x^4)/x^10,x,15,1/16*b^2*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/504*(56*c/x^9+63*d/x^8+72*e/x^7+84*f/x^6)*sqrt(a+b*x^4)-2/45*b*c*sqrt(a+b*x^4)/(a*x^5)-1/16*b*d*sqrt(a+b*x^4)/(a*x^4)-2/21*b*e*sqrt(a+b*x^4)/(a*x^3)-1/6*b*f*sqrt(a+b*x^4)/(a*x^2)+2/15*b^2*c*sqrt(a+b*x^4)/(a^2*x)-2/15*b^(5/2)*c*x*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))+2/15*b^(9/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))-1/105*b^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*e*sqrt(a)+7*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))],
[x^4*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2),x,16,-1/48*a*d*x^2*(a+b*x^4)^(3/2)/b+1/143*x^5*(13*c+11*e*x^2)*(a+b*x^4)^(3/2)+1/14*f*x^4*(a+b*x^4)^(5/2)/b-1/420*(12*a*f-35*b*d*x^2)*(a+b*x^4)^(5/2)/b^2-1/32*a^3*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+4/77*a^2*c*x*sqrt(a+b*x^4)/b-1/32*a^2*d*x^2*sqrt(a+b*x^4)/b+4/195*a^2*e*x^3*sqrt(a+b*x^4)/b+2/3003*a*x^5*(117*c+77*e*x^2)*sqrt(a+b*x^4)-4/65*a^3*e*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+4/65*a^(13/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-2/5005*a^(11/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(77*e*sqrt(a)+65*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^3*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2),x,15,-1/48*a*e*x^2*(a+b*x^4)^(3/2)/b+1/143*x^5*(13*d+11*f*x^2)*(a+b*x^4)^(3/2)+1/60*(6*c+5*e*x^2)*(a+b*x^4)^(5/2)/b-1/32*a^3*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+4/77*a^2*d*x*sqrt(a+b*x^4)/b-1/32*a^2*e*x^2*sqrt(a+b*x^4)/b+4/195*a^2*f*x^3*sqrt(a+b*x^4)/b+2/3003*a*x^5*(117*d+77*f*x^2)*sqrt(a+b*x^4)-4/65*a^3*f*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+4/65*a^(13/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-2/5005*a^(11/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(77*f*sqrt(a)+65*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^2*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2),x,14,-1/48*a*f*x^2*(a+b*x^4)^(3/2)/b+1/99*x^3*(11*c+9*e*x^2)*(a+b*x^4)^(3/2)+1/60*(6*d+5*f*x^2)*(a+b*x^4)^(5/2)/b-1/32*a^3*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+4/77*a^2*e*x*sqrt(a+b*x^4)/b-1/32*a^2*f*x^2*sqrt(a+b*x^4)/b+2/1155*a*x^3*(77*c+45*e*x^2)*sqrt(a+b*x^4)+4/15*a^2*c*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-4/15*a^(9/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+2/1155*a^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-15*e*sqrt(a)+77*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[x*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2),x,14,1/8*c*x^2*(a+b*x^4)^(3/2)+1/99*x^3*(11*d+9*f*x^2)*(a+b*x^4)^(3/2)+1/10*e*(a+b*x^4)^(5/2)/b+3/16*a^2*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+4/77*a^2*f*x*sqrt(a+b*x^4)/b+3/16*a*c*x^2*sqrt(a+b*x^4)+2/1155*a*x^3*(77*d+45*f*x^2)*sqrt(a+b*x^4)+4/15*a^2*d*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-4/15*a^(9/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+2/1155*a^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-15*f*sqrt(a)+77*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2),x,13,1/8*d*x^2*(a+b*x^4)^(3/2)+1/63*x*(9*c+7*e*x^2)*(a+b*x^4)^(3/2)+1/10*f*(a+b*x^4)^(5/2)/b+3/16*a^2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+3/16*a*d*x^2*sqrt(a+b*x^4)+2/105*a*x*(15*c+7*e*x^2)*sqrt(a+b*x^4)+4/15*a^2*e*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-4/15*a^(9/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+2/105*a^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(7*e*sqrt(a)+15*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x,x,16,1/24*(4*c+3*e*x^2)*(a+b*x^4)^(3/2)+1/63*x*(9*d+7*f*x^2)*(a+b*x^4)^(3/2)-1/2*a^(3/2)*c*arctanh(sqrt(a+b*x^4)/sqrt(a))+3/16*a^2*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+1/16*a*(8*c+3*e*x^2)*sqrt(a+b*x^4)+2/105*a*x*(15*d+7*f*x^2)*sqrt(a+b*x^4)+4/15*a^2*f*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-4/15*a^(9/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+2/105*a^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(7*f*sqrt(a)+15*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^2,x,16,-1/7*(7*c-e*x^2)*(a+b*x^4)^(3/2)/x+1/24*(4*d+3*f*x^2)*(a+b*x^4)^(3/2)-1/2*a^(3/2)*d*arctanh(sqrt(a+b*x^4)/sqrt(a))+3/16*a^2*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+2/35*x*(5*a*e+21*b*c*x^2)*sqrt(a+b*x^4)+1/16*a*(8*d+3*f*x^2)*sqrt(a+b*x^4)+12/5*a*c*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(5/4)*b^(1/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/35*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*e*sqrt(a)+21*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^3,x,16,-1/6*(3*c-e*x^2)*(a+b*x^4)^(3/2)/x^2-1/7*(7*d-f*x^2)*(a+b*x^4)^(3/2)/x-1/2*a^(3/2)*e*arctanh(sqrt(a+b*x^4)/sqrt(a))+3/4*a*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)+1/4*(2*a*e+3*b*c*x^2)*sqrt(a+b*x^4)+2/35*x*(5*a*f+21*b*d*x^2)*sqrt(a+b*x^4)+12/5*a*d*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(5/4)*b^(1/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/35*a^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*f*sqrt(a)+21*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^4,x,16,-1/15*(5*c-3*e*x^2)*(a+b*x^4)^(3/2)/x^3-1/6*(3*d-f*x^2)*(a+b*x^4)^(3/2)/x^2-1/2*a^(3/2)*f*arctanh(sqrt(a+b*x^4)/sqrt(a))+3/4*a*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)-2/15*(9*a*e-5*b*c*x^2)*sqrt(a+b*x^4)/x+1/4*(2*a*f+3*b*d*x^2)*sqrt(a+b*x^4)+12/5*a*e*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(5/4)*b^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/15*a^(3/4)*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(9*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^5,x,15,-1/12*(3*c/x^4+4*d/x^3+6*e/x^2+12*f/x)*(a+b*x^4)^(3/2)-3/4*b*c*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)+3/4*a*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)+3/4*b*(c+e*x^2)*sqrt(a+b*x^4)+2/15*b*x*(5*d+9*f*x^2)*sqrt(a+b*x^4)+12/5*a*f*x*sqrt(b)*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(5/4)*b^(1/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/15*a^(3/4)*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(9*f*sqrt(a)+5*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^6,x,15,-1/60*(12*c/x^5+15*d/x^4+20*e/x^3+30*f/x^2)*(a+b*x^4)^(3/2)-3/4*b*d*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)+3/4*a*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))*sqrt(b)-2/15*b*(9*c-5*e*x^2)*sqrt(a+b*x^4)/x+3/4*b*(d+f*x^2)*sqrt(a+b*x^4)+12/5*b^(3/2)*c*x*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(1/4)*b^(5/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/15*a^(1/4)*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*e*sqrt(a)+9*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^7,x,15,-1/60*(10*c/x^6+12*d/x^5+15*e/x^4+20*f/x^3)*(a+b*x^4)^(3/2)+1/2*b^(3/2)*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))-3/4*b*e*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)-1/4*b*(2*c-3*e*x^2)*sqrt(a+b*x^4)/x^2-2/15*b*(9*d-5*f*x^2)*sqrt(a+b*x^4)/x+12/5*b^(3/2)*d*x*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(1/4)*b^(5/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/15*a^(1/4)*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*f*sqrt(a)+9*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^8,x,16,-1/420*(60*c/x^7+70*d/x^6+84*e/x^5+105*f/x^4)*(a+b*x^4)^(3/2)+1/2*b^(3/2)*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))-3/4*b*f*arctanh(sqrt(a+b*x^4)/sqrt(a))*sqrt(a)-12/5*b*e*sqrt(a+b*x^4)/x-2/35*b*(5*c-21*e*x^2)*sqrt(a+b*x^4)/x^3-1/4*b*(2*d-3*f*x^2)*sqrt(a+b*x^4)/x^2+12/5*b^(3/2)*e*x*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(1/4)*b^(5/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/35*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(21*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^9,x,14,-1/840*(105*c/x^8+120*d/x^7+140*e/x^6+168*f/x^5)*(a+b*x^4)^(3/2)+1/2*b^(3/2)*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))-3/16*b^2*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/560*b*(105*c/x^4+160*d/x^3+280*e/x^2+672*f/x)*sqrt(a+b*x^4)+12/5*b^(3/2)*f*x*sqrt(a+b*x^4)/(sqrt(a)+x^2*sqrt(b))-12/5*a^(1/4)*b^(5/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/sqrt(a+b*x^4)+2/35*b^(5/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(21*f*sqrt(a)+5*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^10,x,15,-1/504*(56*c/x^9+63*d/x^8+72*e/x^7+84*f/x^6)*(a+b*x^4)^(3/2)+1/2*b^(3/2)*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))-3/16*b^2*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/1680*b*(224*c/x^5+315*d/x^4+480*e/x^3+840*f/x^2)*sqrt(a+b*x^4)-4/15*b^2*c*sqrt(a+b*x^4)/(a*x)+4/15*b^(5/2)*c*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-4/15*b^(9/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+2/105*b^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(15*e*sqrt(a)+7*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^11,x,13,-1/2520*(252*c/x^10+280*d/x^9+315*e/x^8+360*f/x^7)*(a+b*x^4)^(3/2)-3/16*b^2*e*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/1680*b*(168*c/x^6+224*d/x^5+315*e/x^4+480*f/x^3)*sqrt(a+b*x^4)-1/10*b^2*c*sqrt(a+b*x^4)/(a*x^2)-4/15*b^2*d*sqrt(a+b*x^4)/(a*x)+4/15*b^(5/2)*d*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-4/15*b^(9/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+2/105*b^(7/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(15*f*sqrt(a)+7*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^12,x,14,-1/3960*(360*c/x^11+396*d/x^10+440*e/x^9+495*f/x^8)*(a+b*x^4)^(3/2)-3/16*b^2*f*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/18480*b*(1440*c/x^7+1848*d/x^6+2464*e/x^5+3465*f/x^4)*sqrt(a+b*x^4)-4/77*b^2*c*sqrt(a+b*x^4)/(a*x^3)-1/10*b^2*d*sqrt(a+b*x^4)/(a*x^2)-4/15*b^2*e*sqrt(a+b*x^4)/(a*x)+4/15*b^(5/2)*e*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-4/15*b^(9/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))-2/1155*b^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-77*e*sqrt(a)+15*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^13,x,15,-1/1980*(165*c/x^12+180*d/x^11+198*e/x^10+220*f/x^9)*(a+b*x^4)^(3/2)+1/32*b^3*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/18480*b*(1155*c/x^8+1440*d/x^7+1848*e/x^6+2464*f/x^5)*sqrt(a+b*x^4)-1/32*b^2*c*sqrt(a+b*x^4)/(a*x^4)-4/77*b^2*d*sqrt(a+b*x^4)/(a*x^3)-1/10*b^2*e*sqrt(a+b*x^4)/(a*x^2)-4/15*b^2*f*sqrt(a+b*x^4)/(a*x)+4/15*b^(5/2)*f*x*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-4/15*b^(9/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))-2/1155*b^(9/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-77*f*sqrt(a)+15*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^(3/2)/x^14,x,16,-1/8580*(660*c/x^13+715*d/x^12+780*e/x^11+858*f/x^10)*(a+b*x^4)^(3/2)+1/32*b^3*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/240240*b*(12320*c/x^9+15015*d/x^8+18720*e/x^7+24024*f/x^6)*sqrt(a+b*x^4)-4/195*b^2*c*sqrt(a+b*x^4)/(a*x^5)-1/32*b^2*d*sqrt(a+b*x^4)/(a*x^4)-4/77*b^2*e*sqrt(a+b*x^4)/(a*x^3)-1/10*b^2*f*sqrt(a+b*x^4)/(a*x^2)+4/65*b^3*c*sqrt(a+b*x^4)/(a^2*x)-4/65*b^(7/2)*c*x*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))+4/65*b^(13/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))-2/5005*b^(11/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(65*e*sqrt(a)+77*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))],

# p<0
[x^4*(c+d*x+e*x^2+f*x^3)/sqrt(a+b*x^4),x,12,-1/4*a*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/3*c*x*sqrt(a+b*x^4)/b+1/5*e*x^3*sqrt(a+b*x^4)/b+1/6*f*x^4*sqrt(a+b*x^4)/b-1/12*(4*a*f-3*b*d*x^2)*sqrt(a+b*x^4)/b^2-3/5*a*e*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+3/5*a^(5/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-1/30*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(9*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^3*(c+d*x+e*x^2+f*x^3)/sqrt(a+b*x^4),x,11,-1/4*a*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/3*d*x*sqrt(a+b*x^4)/b+1/5*f*x^3*sqrt(a+b*x^4)/b+1/4*(2*c+e*x^2)*sqrt(a+b*x^4)/b-3/5*a*f*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))+3/5*a^(5/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))-1/30*a^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(9*f*sqrt(a)+5*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))],
[x^2*(c+d*x+e*x^2+f*x^3)/sqrt(a+b*x^4),x,10,-1/4*a*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/3*e*x*sqrt(a+b*x^4)/b+1/4*(2*d+f*x^2)*sqrt(a+b*x^4)/b+c*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-a^(1/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/6*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-e*sqrt(a)+3*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[x*(c+d*x+e*x^2+f*x^3)/sqrt(a+b*x^4),x,10,1/2*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+1/2*e*sqrt(a+b*x^4)/b+1/3*f*x*sqrt(a+b*x^4)/b+d*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-a^(1/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/6*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-f*sqrt(a)+3*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/sqrt(a+b*x^4),x,9,1/2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+1/2*f*sqrt(a+b*x^4)/b+e*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-a^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/2*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(e+c*sqrt(b)/sqrt(a))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x*sqrt(a+b*x^4)),x,12,-1/2*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)+1/2*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)+f*x*sqrt(a+b*x^4)/(sqrt(b)*(sqrt(a)+x^2*sqrt(b)))-a^(1/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))+1/2*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(f+d*sqrt(b)/sqrt(a))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^2*sqrt(a+b*x^4)),x,13,-1/2*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)+1/2*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/sqrt(b)-c*sqrt(a+b*x^4)/(a*x)+c*x*sqrt(b)*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-b^(1/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+1/2*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(e*sqrt(a)+c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^3*sqrt(a+b*x^4)),x,11,-1/2*e*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/2*c*sqrt(a+b*x^4)/(a*x^2)-d*sqrt(a+b*x^4)/(a*x)+d*x*sqrt(b)*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-b^(1/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))+1/2*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(f*sqrt(a)+d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^4*sqrt(a+b*x^4)),x,12,-1/2*f*arctanh(sqrt(a+b*x^4)/sqrt(a))/sqrt(a)-1/3*c*sqrt(a+b*x^4)/(a*x^3)-1/2*d*sqrt(a+b*x^4)/(a*x^2)-e*sqrt(a+b*x^4)/(a*x)+e*x*sqrt(b)*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-b^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))-1/6*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-3*e*sqrt(a)+c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^5*sqrt(a+b*x^4)),x,13,1/4*b*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/4*c*sqrt(a+b*x^4)/(a*x^4)-1/3*d*sqrt(a+b*x^4)/(a*x^3)-1/2*e*sqrt(a+b*x^4)/(a*x^2)-f*sqrt(a+b*x^4)/(a*x)+f*x*sqrt(b)*sqrt(a+b*x^4)/(a*(sqrt(a)+x^2*sqrt(b)))-b^(1/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*sqrt(a+b*x^4))-1/6*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-3*f*sqrt(a)+d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^6*sqrt(a+b*x^4)),x,14,1/4*b*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/5*c*sqrt(a+b*x^4)/(a*x^5)-1/4*d*sqrt(a+b*x^4)/(a*x^4)-1/3*e*sqrt(a+b*x^4)/(a*x^3)-1/2*f*sqrt(a+b*x^4)/(a*x^2)+3/5*b*c*sqrt(a+b*x^4)/(a^2*x)-3/5*b^(3/2)*c*x*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))+3/5*b^(5/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))-1/30*b^(3/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(5*e*sqrt(a)+9*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))],
[x^6*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,12,-3/4*a*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(5/2)+1/2*x*(a*e+a*f*x-b*c*x^2-b*d*x^3)/(b^2*sqrt(a+b*x^4))+d*sqrt(a+b*x^4)/b^2+1/3*e*x*sqrt(a+b*x^4)/b^2+1/4*f*x^2*sqrt(a+b*x^4)/b^2+3/2*c*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))-3/2*a^(1/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))+1/12*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-5*e*sqrt(a)+9*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^4))],
[x^5*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,11,1/2*c*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/2*x*(a*f-b*c*x-b*d*x^2-b*e*x^3)/(b^2*sqrt(a+b*x^4))+e*sqrt(a+b*x^4)/b^2+1/3*f*x*sqrt(a+b*x^4)/b^2+3/2*d*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))-3/2*a^(1/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))+1/12*a^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-5*f*sqrt(a)+9*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^4))],
[x^4*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,10,1/2*d*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)-1/2*x*(c+d*x+e*x^2+f*x^3)/(b*sqrt(a+b*x^4))+f*sqrt(a+b*x^4)/b^2+3/2*e*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))-3/2*a^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*e*sqrt(a)+c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(7/4)*sqrt(a+b*x^4))],
[x^3*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,9,1/2*e*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)+1/2*(-c-d*x-e*x^2-f*x^3)/(b*sqrt(a+b*x^4))+3/2*f*x*sqrt(a+b*x^4)/(b^(3/2)*(sqrt(a)+x^2*sqrt(b)))-3/2*a^(1/4)*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*f*sqrt(a)+d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(7/4)*sqrt(a+b*x^4))],
[x^2*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,10,1/2*f*arctanh(x^2*sqrt(b)/sqrt(a+b*x^4))/b^(3/2)-1/2*x*(a*e+a*f*x-b*c*x^2-b*d*x^3)/(a*b*sqrt(a+b*x^4))-1/2*d*sqrt(a+b*x^4)/(a*b)-1/2*c*x*sqrt(a+b*x^4)/(a*sqrt(b)*(sqrt(a)+x^2*sqrt(b)))+1/2*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(3/4)*sqrt(a+b*x^4))-1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-e*sqrt(a)+c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(5/4)*sqrt(a+b*x^4))],
[x*(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,7,-1/2*x*(a*f-b*c*x-b*d*x^2-b*e*x^3)/(a*b*sqrt(a+b*x^4))-1/2*e*sqrt(a+b*x^4)/(a*b)-1/2*d*x*sqrt(a+b*x^4)/(a*sqrt(b)*(sqrt(a)+x^2*sqrt(b)))+1/2*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(3/4)*sqrt(a+b*x^4))-1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-f*sqrt(a)+d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(5/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(a+b*x^4)^(3/2),x,4,1/2*(-a*f+b*x*(c+d*x+e*x^2))/(a*b*sqrt(a+b*x^4))-1/2*e*x*sqrt(a+b*x^4)/(a*sqrt(b)*(sqrt(a)+x^2*sqrt(b)))+1/2*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(3/4)*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-e*sqrt(a)+c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x*(a+b*x^4)^(3/2)),x,11,-1/2*c*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)+1/2*x*(a*d+a*e*x+a*f*x^2-b*c*x^3)/(a^2*sqrt(a+b*x^4))+1/2*c*sqrt(a+b*x^4)/a^2-1/2*f*x*sqrt(a+b*x^4)/(a*sqrt(b)*(sqrt(a)+x^2*sqrt(b)))+1/2*f*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(3/4)*b^(3/4)*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-f*sqrt(a)+d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(5/4)*b^(3/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^2*(a+b*x^4)^(3/2)),x,13,-1/2*d*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)+1/2*x*(a*e+a*f*x-b*c*x^2-b*d*x^3)/(a^2*sqrt(a+b*x^4))+1/2*d*sqrt(a+b*x^4)/a^2-c*sqrt(a+b*x^4)/(a^2*x)+3/2*c*x*sqrt(b)*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))-3/2*b^(1/4)*c*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(e*sqrt(a)+3*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^3*(a+b*x^4)^(3/2)),x,15,-1/2*e*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)+1/2*x*(a*f-b*c*x-b*d*x^2-b*e*x^3)/(a^2*sqrt(a+b*x^4))+1/2*e*sqrt(a+b*x^4)/a^2-1/2*c*sqrt(a+b*x^4)/(a^2*x^2)-d*sqrt(a+b*x^4)/(a^2*x)+3/2*d*x*sqrt(b)*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))-3/2*b^(1/4)*d*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))+1/4*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(f*sqrt(a)+3*d*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*b^(1/4)*sqrt(a+b*x^4))],
[(c+d*x+e*x^2+f*x^3)/(x^4*(a+b*x^4)^(3/2)),x,17,-1/2*f*arctanh(sqrt(a+b*x^4)/sqrt(a))/a^(3/2)-1/2*x*(b*c+b*d*x+b*e*x^2+b*f*x^3)/(a^2*sqrt(a+b*x^4))+1/2*f*sqrt(a+b*x^4)/a^2-1/3*c*sqrt(a+b*x^4)/(a^2*x^3)-1/2*d*sqrt(a+b*x^4)/(a^2*x^2)-e*sqrt(a+b*x^4)/(a^2*x)+3/2*e*x*sqrt(b)*sqrt(a+b*x^4)/(a^2*(sqrt(a)+x^2*sqrt(b)))-3/2*b^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^4))-1/12*b^(1/4)*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(-9*e*sqrt(a)+5*c*sqrt(b))*(sqrt(a)+x^2*sqrt(b))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(9/4)*sqrt(a+b*x^4))],

# Integrands of the form (c x)^m Pq(x) (a+b x^4)^p with p symbolic
[(g*x)^m*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^p,x,14,c*(g*x)^(1+m)*(a+b*x^4)^p*hypergeom([1/4*(1+m),-p],[1/4*(5+m)],-b*x^4/a)/(g*(1+m)*(1+b*x^4/a)^p)+d*(g*x)^(2+m)*(a+b*x^4)^p*hypergeom([1/4*(2+m),-p],[1/4*(6+m)],-b*x^4/a)/(g^2*(2+m)*(1+b*x^4/a)^p)+e*(g*x)^(3+m)*(a+b*x^4)^p*hypergeom([1/4*(3+m),-p],[1/4*(7+m)],-b*x^4/a)/(g^3*(3+m)*(1+b*x^4/a)^p)+f*(g*x)^(4+m)*(a+b*x^4)^p*hypergeom([1/4*(4+m),-p],[1/4*(8+m)],-b*x^4/a)/(g^4*(4+m)*(1+b*x^4/a)^p)],
[(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^p,x,12,1/4*f*(a+b*x^4)^(1+p)/(b*(1+p))+c*x*(a+b*x^4)^(1+p)*hypergeom([1,5/4+p],[5/4],-b*x^4/a)/a+1/2*d*x^2*(a+b*x^4)^(1+p)*hypergeom([1,3/2+p],[3/2],-b*x^4/a)/a+1/3*e*x^3*(a+b*x^4)^(1+p)*hypergeom([1,7/4+p],[7/4],-b*x^4/a)/a,1/4*f*(a+b*x^4)^(1+p)/(b*(1+p))+c*x*(a+b*x^4)^p*hypergeom([1/4,-p],[5/4],-b*x^4/a)/(1+b*x^4/a)^p+1/2*d*x^2*(a+b*x^4)^p*hypergeom([1/2,-p],[3/2],-b*x^4/a)/(1+b*x^4/a)^p+1/3*e*x^3*(a+b*x^4)^p*hypergeom([3/4,-p],[7/4],-b*x^4/a)/(1+b*x^4/a)^p],
[x^3*(c+d*x+e*x^2+f*x^3)*(a+b*x^4)^p,x,13,1/4*c*(a+b*x^4)^(1+p)/(b*(1+p))+1/5*d*x^5*(a+b*x^4)^p*hypergeom([5/4,-p],[9/4],-b*x^4/a)/(1+b*x^4/a)^p+1/6*e*x^6*(a+b*x^4)^p*hypergeom([3/2,-p],[5/2],-b*x^4/a)/(1+b*x^4/a)^p+1/7*f*x^7*(a+b*x^4)^p*hypergeom([7/4,-p],[11/4],-b*x^4/a)/(1+b*x^4/a)^p],

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

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

# p>0

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

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

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

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

# p>0

# p<0
[(243-162*x+108*x^2-72*x^3+48*x^4-32*x^5)/(729-64*x^6),x,2,1/2*log(3+2*x)],
[(243+162*x+108*x^2+72*x^3+48*x^4+32*x^5)/(729-64*x^6),x,2,-1/2*log(3-2*x)],
[(81+36*x^2+16*x^4)/(729-64*x^6),x,2,1/6*arctanh(2/3*x)],
[(81+54*x-24*x^3-16*x^4)/(729-64*x^6),x,3,-1/3*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)],
[(3-2*x)/(729-64*x^6),x,6,1/486*log(3+2*x)-1/972*log(9-6*x+4*x^2)+1/162*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(3+2*x)/(729-64*x^6),x,6,-1/486*log(3-2*x)+1/972*log(9+6*x+4*x^2)-1/162*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)],
[(9-6*x+4*x^2)/(729-64*x^6),x,7,-1/324*log(3-2*x)+1/108*log(3+2*x)-1/324*log(9+6*x+4*x^2)+1/54*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(9+6*x+4*x^2)/(729-64*x^6),x,7,-1/108*log(3-2*x)+1/324*log(3+2*x)+1/324*log(9-6*x+4*x^2)-1/54*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)],
[(27-8*x^3)/(729-64*x^6),x,7,1/54*log(3+2*x)-1/108*log(9-6*x+4*x^2)-1/18*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)],
[(27+36*x+24*x^2+8*x^3)/(729-64*x^6),x,7,-1/18*log(3-2*x)+1/36*log(9-6*x+4*x^2)-1/18*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)],
[(243-162*x+108*x^2-72*x^3+48*x^4-32*x^5)/(729-64*x^6)^2,x,11,(-1/2916)/(3+2*x)-1/17496*log(3-2*x)+5/17496*log(3+2*x)-1/17496*log(9-6*x+4*x^2)-1/17496*log(9+6*x+4*x^2)-1/8748*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/2916*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(243+162*x+108*x^2+72*x^3+48*x^4+32*x^5)/(729-64*x^6)^2,x,11,1/2916/(3-2*x)-5/17496*log(3-2*x)+1/17496*log(3+2*x)+1/17496*log(9-6*x+4*x^2)+1/17496*log(9+6*x+4*x^2)-1/2916*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/8748*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(81+36*x^2+16*x^4)/(729-64*x^6)^2,x,8,1/17496/(3-2*x)+(-1/17496)/(3+2*x)+1/8748*arctanh(2/3*x)-1/13122*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/13122*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(81+54*x-24*x^3-16*x^4)/(729-64*x^6)^2,x,11,1/4374*x/(9-6*x+4*x^2)-1/26244*log(3-2*x)+1/78732*log(3+2*x)-1/157464*log(9-6*x+4*x^2)+1/52488*log(9+6*x+4*x^2)-1/4374*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)],
[(3-2*x)/(729-64*x^6)^2,x,17,(-1/708588)/(3+2*x)+1/708588*(3-x)/(9-6*x+4*x^2)+1/236196*x/(9+6*x+4*x^2)-1/4251528*log(3-2*x)+1/472392*log(3+2*x)-1/944784*log(9-6*x+4*x^2)+1/8503056*log(9+6*x+4*x^2)-1/1417176*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/157464*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(3+2*x)/(729-64*x^6)^2,x,17,1/708588/(3-2*x)+1/236196*x/(9-6*x+4*x^2)+1/708588*(-3-x)/(9+6*x+4*x^2)-1/472392*log(3-2*x)+1/4251528*log(3+2*x)-1/8503056*log(9-6*x+4*x^2)+1/944784*log(9+6*x+4*x^2)-1/157464*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/1417176*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(9-6*x+4*x^2)/(729-64*x^6)^2,x,14,1/472392/(3-2*x)+(-1/157464)/(3+2*x)+1/236196*(3+4*x)/(9+6*x+4*x^2)-1/354294*log(3-2*x)+1/118098*log(3+2*x)-1/944784*log(9-6*x+4*x^2)-5/2834352*log(9+6*x+4*x^2)-1/472392*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/52488*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(9+6*x+4*x^2)/(729-64*x^6)^2,x,14,1/157464/(3-2*x)+(-1/472392)/(3+2*x)+1/236196*(-3+4*x)/(9-6*x+4*x^2)-1/118098*log(3-2*x)+1/354294*log(3+2*x)+5/2834352*log(9-6*x+4*x^2)+1/944784*log(9+6*x+4*x^2)-1/52488*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/472392*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(27-8*x^3)/(729-64*x^6)^2,x,15,1/4374*x/(27+8*x^3)-1/157464*log(3-2*x)+7/472392*log(3+2*x)-7/944784*log(9-6*x+4*x^2)+1/314928*log(9+6*x+4*x^2)-7/157464*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)+1/52488*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[(27+36*x+24*x^2+8*x^3)/(729-64*x^6)^2,x,14,1/26244/(3-2*x)+1/26244*(-3+2*x)/(9-6*x+4*x^2)-7/157464*log(3-2*x)+1/472392*log(3+2*x)+17/944784*log(9-6*x+4*x^2)+1/314928*log(9+6*x+4*x^2)-11/157464*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)-1/157464*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],
[x*(27-2*x^3)/(729-64*x^6),x,13,-1/96*log(3-2*x)-5/288*log(3+2*x)+5/576*log(9-6*x+4*x^2)+1/192*log(9+6*x+4*x^2)-5/96*arctan(1/3*(3-4*x)/sqrt(3))/sqrt(3)-1/32*arctan(1/3*(3+4*x)/sqrt(3))/sqrt(3)],

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

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

# Integrands of the form (c x)^m Pq(x) (a+b x^n)^p
[(c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x,7,(b*f-a*g)*x^(1+n)*(c*x)^m/(b^2*(1+m+n))+g*x^(1+2*n)*(c*x)^m/(b*(1+m+2*n))+(b^2*e-a*b*f+a^2*g)*(c*x)^(1+m)/(b^3*c*(1+m))+(b^3*d-a*b^2*e+a^2*b*f-a^3*g)*(c*x)^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-b*x^n/a)/(a*b^3*c*(1+m))],
[(c+d*x^(-1+n))*(a+b*x^n)^3,x,4,a^3*c*x+3*a^2*b*c*x^(1+n)/(1+n)+3*a*b^2*c*x^(1+2*n)/(1+2*n)+b^3*c*x^(1+3*n)/(1+3*n)+1/4*d*(a+b*x^n)^4/(b*n)],
[(c+d*x^(-1+n))*(a+b*x^n)^2,x,4,a^2*c*x+2*a*b*c*x^(1+n)/(1+n)+b^2*c*x^(1+2*n)/(1+2*n)+1/3*d*(a+b*x^n)^3/(b*n)],
[(c+d*x^(-1+n))*(a+b*x^n),x,4,a*c*x+a*d*x^n/n+1/2*b*d*x^(2*n)/n+b*c*x^(1+n)/(1+n)],
[c+d*x^(-1+n),x,1,c*x+d*x^n/n],
[(c+d*x^(-1+n))/(a+b*x^n),x,3,c*x*hypergeom([1,1/n],[1+1/n],-b*x^n/a)/a+d*log(a+b*x^n)/(b*n)],
[(c+d*x^(-1+n))/(a+b*x^n)^2,x,3,-d/(b*n*(a+b*x^n))+c*x*hypergeom([2,1/n],[1+1/n],-b*x^n/a)/a^2],
[(c+d*x^(-1+n))/(a+b*x^n)^3,x,3,-1/2*d/(b*n*(a+b*x^n)^2)+c*x*hypergeom([3,1/n],[1+1/n],-b*x^n/a)/a^3],

# Integrands of the form (c x)^m Pq(x) (a+b x^n)^(p/2)
[(c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/sqrt(a+b*x^n),x,13,d*(c*x)^(1+m)*hypergeom([1/2,(1+m)/n],[(1+m+n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/(c*(1+m)*sqrt(a+b*x^n))+e*x^(1+n)*(c*x)^m*hypergeom([1/2,(1+m+n)/n],[(1+m+2*n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m+n)*sqrt(a+b*x^n))+f*x^(1+2*n)*(c*x)^m*hypergeom([1/2,(1+m+2*n)/n],[(1+m+3*n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m+2*n)*sqrt(a+b*x^n))+g*x^(1+3*n)*(c*x)^m*hypergeom([1/2,(1+m+3*n)/n],[(1+m+4*n)/n],-b*x^n/a)*sqrt(1+b*x^n/a)/((1+m+3*n)*sqrt(a+b*x^n))],
[(-a*h*x^(-1+1/4*n)+b*f*x^(-1+1/2*n)+b*g*x^(-1+n)+b*h*x^(-1+5/4*n))/(a+b*x^n)^(3/2),x,2,-2*(a*g+2*a*h*x^(1/4*n)-b*f*x^(1/2*n))/(a*n*sqrt(a+b*x^n))],

# Integrands of the form (c x)^m Pq(x) (a+b x^n)^p with p symbolic
[(c*x)^m*(d+e*x+f*x^2+g*x^3)*(a+b*x^n)^p,x,10,d*(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([(1+m)/n,-p],[(1+m+n)/n],-b*x^n/a)/(c*(1+m)*(1+b*x^n/a)^p)+e*(c*x)^(2+m)*(a+b*x^n)^p*hypergeom([(2+m)/n,-p],[(2+m+n)/n],-b*x^n/a)/(c^2*(2+m)*(1+b*x^n/a)^p)+f*(c*x)^(3+m)*(a+b*x^n)^p*hypergeom([(3+m)/n,-p],[(3+m+n)/n],-b*x^n/a)/(c^3*(3+m)*(1+b*x^n/a)^p)+g*(c*x)^(4+m)*(a+b*x^n)^p*hypergeom([(4+m)/n,-p],[(4+m+n)/n],-b*x^n/a)/(c^4*(4+m)*(1+b*x^n/a)^p)],
[(c*x)^m*(a+b*x^n)^p*(d+e*x^n+f*x^(2*n)+g*x^(3*n)),x,13,d*(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([(1+m)/n,-p],[(1+m+n)/n],-b*x^n/a)/(c*(1+m)*(1+b*x^n/a)^p)+e*x^(1+n)*(c*x)^m*(a+b*x^n)^p*hypergeom([(1+m+n)/n,-p],[(1+m+2*n)/n],-b*x^n/a)/((1+m+n)*(1+b*x^n/a)^p)+f*x^(1+2*n)*(c*x)^m*(a+b*x^n)^p*hypergeom([(1+m+2*n)/n,-p],[(1+m+3*n)/n],-b*x^n/a)/((1+m+2*n)*(1+b*x^n/a)^p)+g*x^(1+3*n)*(c*x)^m*(a+b*x^n)^p*hypergeom([(1+m+3*n)/n,-p],[(1+m+4*n)/n],-b*x^n/a)/((1+m+3*n)*(1+b*x^n/a)^p)],

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

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

# Integrands of the form (e x)^m Pq(x) (a+b x^n)^p (c+d x^n)^q
[(a*c+2*(b*c+a*d)*x^2+3*b*d*x^4)/(sqrt(a+b*x^2)*sqrt(c+d*x^2)),x,1,x*sqrt(a+b*x^2)*sqrt(c+d*x^2)],
[(1+x^3)/((1-x^4)*(1+x^4)^(1/4)),x,10,1/2*arctan(2^(1/4)*x/(1+x^4)^(1/4))/2^(1/4)-1/2*arctan((1+x^4)^(1/4)/2^(1/4))/2^(1/4)+1/2*arctanh(2^(1/4)*x/(1+x^4)^(1/4))/2^(1/4)+1/2*arctanh((1+x^4)^(1/4)/2^(1/4))/2^(1/4)],

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

# Integrands of the form (h x)^m (e+f x^n+g x^(2 n)) (a+b x^n)^p (c+d x^n)^q
[(a+b*x^n)^p*(c+d*x^n)^p*(e+(b*c+a*d)*e*(1+n+n*p)*x^n/(a*c)+b*d*e*(1+2*n+2*n*p)*x^(2*n)/(a*c)),x,1,e*x*(a+b*x^n)^(1+p)*(c+d*x^n)^(1+p)/(a*c)],
[(h*x)^m*(a+b*x^n)^p*(c+d*x^n)^p*(e+(b*c+a*d)*e*(1+m+n+n*p)*x^n/(a*c*(1+m))+b*d*e*(1+m+2*n+2*n*p)*x^(2*n)/(a*c*(1+m))),x,1,e*(h*x)^(1+m)*(a+b*x^n)^(1+p)*(c+d*x^n)^(1+p)/(a*c*h*(1+m))]]:
