# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.2 Quadratic\1.1.2.4 (e x)^m (a+b x^2)^p (c+d x^2)^q.txt"

lst:=[

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

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

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

# p>0
[x^2*(a+b*x^2)*(A+B*x^2),x,2,1/3*a*A*x^3+1/5*(A*b+a*B)*x^5+1/7*b*B*x^7],
[x*(a+b*x^2)*(A+B*x^2),x,3,1/2*a*A*x^2+1/4*(A*b+a*B)*x^4+1/6*b*B*x^6],
[(a+b*x^2)*(A+B*x^2),x,2,a*A*x+1/3*(A*b+a*B)*x^3+1/5*b*B*x^5],
[(a+b*x^2)*(A+B*x^2)/x,x,3,1/2*(A*b+a*B)*x^2+1/4*b*B*x^4+a*A*log(x)],
[(a+b*x^2)*(A+B*x^2)/x^2,x,2,-a*A/x+(A*b+a*B)*x+1/3*b*B*x^3],
[(a+b*x^2)*(A+B*x^2)/x^3,x,3,-1/2*a*A/x^2+1/2*b*B*x^2+(A*b+a*B)*log(x)],
[(a+b*x^2)*(A+B*x^2)/x^4,x,2,-1/3*a*A/x^3+(-A*b-a*B)/x+b*B*x],
[(a+b*x^2)*(A+B*x^2)/x^5,x,3,-1/4*a*A/x^4+1/2*(-A*b-a*B)/x^2+b*B*log(x)],
[(a+b*x^2)*(A+B*x^2)/x^6,x,2,-1/5*a*A/x^5+1/3*(-A*b-a*B)/x^3-b*B/x],
[(a+b*x^2)*(A+B*x^2)/x^7,x,3,-1/6*a*A/x^6+1/4*(-A*b-a*B)/x^4-1/2*b*B/x^2],
[x^2*(a+b*x^2)^2*(A+B*x^2),x,2,1/3*a^2*A*x^3+1/5*a*(2*A*b+a*B)*x^5+1/7*b*(A*b+2*a*B)*x^7+1/9*b^2*B*x^9],
[x*(a+b*x^2)^2*(A+B*x^2),x,3,1/6*(A*b-a*B)*(a+b*x^2)^3/b^2+1/8*B*(a+b*x^2)^4/b^2],
[(a+b*x^2)^2*(A+B*x^2),x,2,a^2*A*x+1/3*a*(2*A*b+a*B)*x^3+1/5*b*(A*b+2*a*B)*x^5+1/7*b^2*B*x^7],
[(a+b*x^2)^2*(A+B*x^2)/x,x,4,a*A*b*x^2+1/4*A*b^2*x^4+1/6*B*(a+b*x^2)^3/b+a^2*A*log(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^2,x,2,-a^2*A/x+a*(2*A*b+a*B)*x+1/3*b*(A*b+2*a*B)*x^3+1/5*b^2*B*x^5],
[(a+b*x^2)^2*(A+B*x^2)/x^3,x,3,-1/2*a^2*A/x^2+1/2*b*(A*b+2*a*B)*x^2+1/4*b^2*B*x^4+a*(2*A*b+a*B)*log(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^4,x,2,-1/3*a^2*A/x^3-a*(2*A*b+a*B)/x+b*(A*b+2*a*B)*x+1/3*b^2*B*x^3],
[(a+b*x^2)^2*(A+B*x^2)/x^5,x,3,-1/4*a^2*A/x^4-1/2*a*(2*A*b+a*B)/x^2+1/2*b^2*B*x^2+b*(A*b+2*a*B)*log(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^6,x,2,-1/5*a^2*A/x^5-1/3*a*(2*A*b+a*B)/x^3-b*(A*b+2*a*B)/x+b^2*B*x],
[(a+b*x^2)^2*(A+B*x^2)/x^7,x,3,-1/6*a^2*A/x^6-1/4*a*(2*A*b+a*B)/x^4-1/2*b*(A*b+2*a*B)/x^2+b^2*B*log(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^8,x,2,-1/7*a^2*A/x^7-1/5*a*(2*A*b+a*B)/x^5-1/3*b*(A*b+2*a*B)/x^3-b^2*B/x],
[(a+b*x^2)^2*(A+B*x^2)/x^9,x,3,-1/8*A*(a+b*x^2)^3/(a*x^8)+1/24*(A*b-4*a*B)*(a+b*x^2)^3/(a^2*x^6)],
[x^9*(a+b*x^2)^5*(A+B*x^2),x,3,1/10*a^5*A*x^10+1/12*a^4*(5*A*b+a*B)*x^12+5/14*a^3*b*(2*A*b+a*B)*x^14+5/8*a^2*b^2*(A*b+a*B)*x^16+5/18*a*b^3*(A*b+2*a*B)*x^18+1/20*b^4*(A*b+5*a*B)*x^20+1/22*b^5*B*x^22],
[x^8*(a+b*x^2)^5*(A+B*x^2),x,2,1/9*a^5*A*x^9+1/11*a^4*(5*A*b+a*B)*x^11+5/13*a^3*b*(2*A*b+a*B)*x^13+2/3*a^2*b^2*(A*b+a*B)*x^15+5/17*a*b^3*(A*b+2*a*B)*x^17+1/19*b^4*(A*b+5*a*B)*x^19+1/21*b^5*B*x^21],
[x^7*(a+b*x^2)^5*(A+B*x^2),x,3,-1/12*a^3*(A*b-a*B)*(a+b*x^2)^6/b^5+1/14*a^2*(3*A*b-4*a*B)*(a+b*x^2)^7/b^5-3/16*a*(A*b-2*a*B)*(a+b*x^2)^8/b^5+1/18*(A*b-4*a*B)*(a+b*x^2)^9/b^5+1/20*B*(a+b*x^2)^10/b^5],
[x^6*(a+b*x^2)^5*(A+B*x^2),x,2,1/7*a^5*A*x^7+1/9*a^4*(5*A*b+a*B)*x^9+5/11*a^3*b*(2*A*b+a*B)*x^11+10/13*a^2*b^2*(A*b+a*B)*x^13+1/3*a*b^3*(A*b+2*a*B)*x^15+1/17*b^4*(A*b+5*a*B)*x^17+1/19*b^5*B*x^19],
[x^5*(a+b*x^2)^5*(A+B*x^2),x,3,1/12*a^2*(A*b-a*B)*(a+b*x^2)^6/b^4-1/14*a*(2*A*b-3*a*B)*(a+b*x^2)^7/b^4+1/16*(A*b-3*a*B)*(a+b*x^2)^8/b^4+1/18*B*(a+b*x^2)^9/b^4],
[x^4*(a+b*x^2)^5*(A+B*x^2),x,2,1/5*a^5*A*x^5+1/7*a^4*(5*A*b+a*B)*x^7+5/9*a^3*b*(2*A*b+a*B)*x^9+10/11*a^2*b^2*(A*b+a*B)*x^11+5/13*a*b^3*(A*b+2*a*B)*x^13+1/15*b^4*(A*b+5*a*B)*x^15+1/17*b^5*B*x^17],
[x^3*(a+b*x^2)^5*(A+B*x^2),x,3,-1/12*a*(A*b-a*B)*(a+b*x^2)^6/b^3+1/14*(A*b-2*a*B)*(a+b*x^2)^7/b^3+1/16*B*(a+b*x^2)^8/b^3],
[x^2*(a+b*x^2)^5*(A+B*x^2),x,2,1/3*a^5*A*x^3+1/5*a^4*(5*A*b+a*B)*x^5+5/7*a^3*b*(2*A*b+a*B)*x^7+10/9*a^2*b^2*(A*b+a*B)*x^9+5/11*a*b^3*(A*b+2*a*B)*x^11+1/13*b^4*(A*b+5*a*B)*x^13+1/15*b^5*B*x^15],
[x*(a+b*x^2)^5*(A+B*x^2),x,3,1/12*(A*b-a*B)*(a+b*x^2)^6/b^2+1/14*B*(a+b*x^2)^7/b^2],
[(a+b*x^2)^5*(A+B*x^2),x,2,a^5*A*x+1/3*a^4*(5*A*b+a*B)*x^3+a^3*b*(2*A*b+a*B)*x^5+10/7*a^2*b^2*(A*b+a*B)*x^7+5/9*a*b^3*(A*b+2*a*B)*x^9+1/11*b^4*(A*b+5*a*B)*x^11+1/13*b^5*B*x^13],
[(a+b*x^2)^5*(A+B*x^2)/x,x,4,5/2*a^4*A*b*x^2+5/2*a^3*A*b^2*x^4+5/3*a^2*A*b^3*x^6+5/8*a*A*b^4*x^8+1/10*A*b^5*x^10+1/12*B*(a+b*x^2)^6/b+a^5*A*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^2,x,2,-a^5*A/x+a^4*(5*A*b+a*B)*x+5/3*a^3*b*(2*A*b+a*B)*x^3+2*a^2*b^2*(A*b+a*B)*x^5+5/7*a*b^3*(A*b+2*a*B)*x^7+1/9*b^4*(A*b+5*a*B)*x^9+1/11*b^5*B*x^11],
[(a+b*x^2)^5*(A+B*x^2)/x^3,x,3,-1/2*a^5*A/x^2+5/2*a^3*b*(2*A*b+a*B)*x^2+5/2*a^2*b^2*(A*b+a*B)*x^4+5/6*a*b^3*(A*b+2*a*B)*x^6+1/8*b^4*(A*b+5*a*B)*x^8+1/10*b^5*B*x^10+a^4*(5*A*b+a*B)*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^4,x,2,-1/3*a^5*A/x^3-a^4*(5*A*b+a*B)/x+5*a^3*b*(2*A*b+a*B)*x+10/3*a^2*b^2*(A*b+a*B)*x^3+a*b^3*(A*b+2*a*B)*x^5+1/7*b^4*(A*b+5*a*B)*x^7+1/9*b^5*B*x^9],
[(a+b*x^2)^5*(A+B*x^2)/x^5,x,3,-1/4*a^5*A/x^4-1/2*a^4*(5*A*b+a*B)/x^2+5*a^2*b^2*(A*b+a*B)*x^2+5/4*a*b^3*(A*b+2*a*B)*x^4+1/6*b^4*(A*b+5*a*B)*x^6+1/8*b^5*B*x^8+5*a^3*b*(2*A*b+a*B)*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^6,x,2,-1/5*a^5*A/x^5-1/3*a^4*(5*A*b+a*B)/x^3-5*a^3*b*(2*A*b+a*B)/x+10*a^2*b^2*(A*b+a*B)*x+5/3*a*b^3*(A*b+2*a*B)*x^3+1/5*b^4*(A*b+5*a*B)*x^5+1/7*b^5*B*x^7],
[(a+b*x^2)^5*(A+B*x^2)/x^7,x,3,-1/6*a^5*A/x^6-1/4*a^4*(5*A*b+a*B)/x^4-5/2*a^3*b*(2*A*b+a*B)/x^2+5/2*a*b^3*(A*b+2*a*B)*x^2+1/4*b^4*(A*b+5*a*B)*x^4+1/6*b^5*B*x^6+10*a^2*b^2*(A*b+a*B)*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^8,x,2,-1/7*a^5*A/x^7-1/5*a^4*(5*A*b+a*B)/x^5-5/3*a^3*b*(2*A*b+a*B)/x^3-10*a^2*b^2*(A*b+a*B)/x+5*a*b^3*(A*b+2*a*B)*x+1/3*b^4*(A*b+5*a*B)*x^3+1/5*b^5*B*x^5],
[(a+b*x^2)^5*(A+B*x^2)/x^9,x,3,-1/8*a^5*A/x^8-1/6*a^4*(5*A*b+a*B)/x^6-5/4*a^3*b*(2*A*b+a*B)/x^4-5*a^2*b^2*(A*b+a*B)/x^2+1/2*b^4*(A*b+5*a*B)*x^2+1/4*b^5*B*x^4+5*a*b^3*(A*b+2*a*B)*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^10,x,2,-1/9*a^5*A/x^9-1/7*a^4*(5*A*b+a*B)/x^7-a^3*b*(2*A*b+a*B)/x^5-10/3*a^2*b^2*(A*b+a*B)/x^3-5*a*b^3*(A*b+2*a*B)/x+b^4*(A*b+5*a*B)*x+1/3*b^5*B*x^3],
[(a+b*x^2)^5*(A+B*x^2)/x^11,x,3,-1/10*a^5*A/x^10-1/8*a^4*(5*A*b+a*B)/x^8-5/6*a^3*b*(2*A*b+a*B)/x^6-5/2*a^2*b^2*(A*b+a*B)/x^4-5/2*a*b^3*(A*b+2*a*B)/x^2+1/2*b^5*B*x^2+b^4*(A*b+5*a*B)*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^12,x,2,-1/11*a^5*A/x^11-1/9*a^4*(5*A*b+a*B)/x^9-5/7*a^3*b*(2*A*b+a*B)/x^7-2*a^2*b^2*(A*b+a*B)/x^5-5/3*a*b^3*(A*b+2*a*B)/x^3-b^4*(A*b+5*a*B)/x+b^5*B*x],
[(a+b*x^2)^5*(A+B*x^2)/x^13,x,4,-1/10*a^5*B/x^10-5/8*a^4*b*B/x^8-5/3*a^3*b^2*B/x^6-5/2*a^2*b^3*B/x^4-5/2*a*b^4*B/x^2-1/12*A*(a+b*x^2)^6/(a*x^12)+b^5*B*log(x)],
[(a+b*x^2)^5*(A+B*x^2)/x^14,x,2,-1/13*a^5*A/x^13-1/11*a^4*(5*A*b+a*B)/x^11-5/9*a^3*b*(2*A*b+a*B)/x^9-10/7*a^2*b^2*(A*b+a*B)/x^7-a*b^3*(A*b+2*a*B)/x^5-1/3*b^4*(A*b+5*a*B)/x^3-b^5*B/x],
[(a+b*x^2)^5*(A+B*x^2)/x^15,x,3,-1/14*A*(a+b*x^2)^6/(a*x^14)+1/84*(A*b-7*a*B)*(a+b*x^2)^6/(a^2*x^12)],
[(a+b*x^2)^5*(A+B*x^2)/x^16,x,2,-1/15*a^5*A/x^15-1/13*a^4*(5*A*b+a*B)/x^13-5/11*a^3*b*(2*A*b+a*B)/x^11-10/9*a^2*b^2*(A*b+a*B)/x^9-5/7*a*b^3*(A*b+2*a*B)/x^7-1/5*b^4*(A*b+5*a*B)/x^5-1/3*b^5*B/x^3],
[(a+b*x^2)^5*(A+B*x^2)/x^17,x,4,-1/16*A*(a+b*x^2)^6/(a*x^16)+1/56*(A*b-4*a*B)*(a+b*x^2)^6/(a^2*x^14)-1/336*b*(A*b-4*a*B)*(a+b*x^2)^6/(a^3*x^12)],
[(a+b*x^2)^5*(A+B*x^2)/x^18,x,2,-1/17*a^5*A/x^17-1/15*a^4*(5*A*b+a*B)/x^15-5/13*a^3*b*(2*A*b+a*B)/x^13-10/11*a^2*b^2*(A*b+a*B)/x^11-5/9*a*b^3*(A*b+2*a*B)/x^9-1/7*b^4*(A*b+5*a*B)/x^7-1/5*b^5*B/x^5],
[(a+b*x^2)^5*(A+B*x^2)/x^19,x,3,-1/18*a^5*A/x^18-1/16*a^4*(5*A*b+a*B)/x^16-5/14*a^3*b*(2*A*b+a*B)/x^14-5/6*a^2*b^2*(A*b+a*B)/x^12-1/2*a*b^3*(A*b+2*a*B)/x^10-1/8*b^4*(A*b+5*a*B)/x^8-1/6*b^5*B/x^6],
[(a+b*x^2)^5*(A+B*x^2)/x^20,x,2,-1/19*a^5*A/x^19-1/17*a^4*(5*A*b+a*B)/x^17-1/3*a^3*b*(2*A*b+a*B)/x^15-10/13*a^2*b^2*(A*b+a*B)/x^13-5/11*a*b^3*(A*b+2*a*B)/x^11-1/9*b^4*(A*b+5*a*B)/x^9-1/7*b^5*B/x^7],
[(a+b*x^2)^5*(A+B*x^2)/x^21,x,3,-1/20*a^5*A/x^20-1/18*a^4*(5*A*b+a*B)/x^18-5/16*a^3*b*(2*A*b+a*B)/x^16-5/7*a^2*b^2*(A*b+a*B)/x^14-5/12*a*b^3*(A*b+2*a*B)/x^12-1/10*b^4*(A*b+5*a*B)/x^10-1/8*b^5*B/x^8],
[(a+b*x^2)^5*(A+B*x^2)/x^22,x,2,-1/21*a^5*A/x^21-1/19*a^4*(5*A*b+a*B)/x^19-5/17*a^3*b*(2*A*b+a*B)/x^17-2/3*a^2*b^2*(A*b+a*B)/x^15-5/13*a*b^3*(A*b+2*a*B)/x^13-1/11*b^4*(A*b+5*a*B)/x^11-1/9*b^5*B/x^9],
[(a+b*x^2)^5*(A+B*x^2)/x^23,x,3,-1/22*a^5*A/x^22-1/20*a^4*(5*A*b+a*B)/x^20-5/18*a^3*b*(2*A*b+a*B)/x^18-5/8*a^2*b^2*(A*b+a*B)/x^16-5/14*a*b^3*(A*b+2*a*B)/x^14-1/12*b^4*(A*b+5*a*B)/x^12-1/10*b^5*B/x^10],

# p<0
[x^6*(A+B*x^2)/(a+b*x^2),x,4,a^2*(A*b-a*B)*x/b^4-1/3*a*(A*b-a*B)*x^3/b^3+1/5*(A*b-a*B)*x^5/b^2+1/7*B*x^7/b-a^(5/2)*(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/b^(9/2)],
[x^5*(A+B*x^2)/(a+b*x^2),x,3,-1/2*a*(A*b-a*B)*x^2/b^3+1/4*(A*b-a*B)*x^4/b^2+1/6*B*x^6/b+1/2*a^2*(A*b-a*B)*log(a+b*x^2)/b^4],
[x^4*(A+B*x^2)/(a+b*x^2),x,4,-a*(A*b-a*B)*x/b^3+1/3*(A*b-a*B)*x^3/b^2+1/5*B*x^5/b+a^(3/2)*(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/b^(7/2)],
[x^3*(A+B*x^2)/(a+b*x^2),x,3,1/2*(A*b-a*B)*x^2/b^2+1/4*B*x^4/b-1/2*a*(A*b-a*B)*log(a+b*x^2)/b^3],
[x^2*(A+B*x^2)/(a+b*x^2),x,3,(A*b-a*B)*x/b^2+1/3*B*x^3/b-(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(5/2)],
[x*(A+B*x^2)/(a+b*x^2),x,3,1/2*B*x^2/b+1/2*(A*b-a*B)*log(a+b*x^2)/b^2],
[(A+B*x^2)/(a+b*x^2),x,2,B*x/b+(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))],
[(A+B*x^2)/(x*(a+b*x^2)),x,3,A*log(x)/a-1/2*(A*b-a*B)*log(a+b*x^2)/(a*b)],
[(A+B*x^2)/(x^2*(a+b*x^2)),x,2,-A/(a*x)-(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))],
[(A+B*x^2)/(x^3*(a+b*x^2)),x,3,-1/2*A/(a*x^2)-(A*b-a*B)*log(x)/a^2+1/2*(A*b-a*B)*log(a+b*x^2)/a^2],
[(A+B*x^2)/(x^4*(a+b*x^2)),x,3,-1/3*A/(a*x^3)+(A*b-a*B)/(a^2*x)+(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(5/2)],
[(A+B*x^2)/(x^5*(a+b*x^2)),x,3,-1/4*A/(a*x^4)+1/2*(A*b-a*B)/(a^2*x^2)+b*(A*b-a*B)*log(x)/a^3-1/2*b*(A*b-a*B)*log(a+b*x^2)/a^3],
[(A+B*x^2)/(x^6*(a+b*x^2)),x,4,-1/5*A/(a*x^5)+1/3*(A*b-a*B)/(a^2*x^3)-b*(A*b-a*B)/(a^3*x)-b^(3/2)*(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/a^(7/2)],
[(A+B*x^2)/(x^7*(a+b*x^2)),x,3,-1/6*A/(a*x^6)+1/4*(A*b-a*B)/(a^2*x^4)-1/2*b*(A*b-a*B)/(a^3*x^2)-b^2*(A*b-a*B)*log(x)/a^4+1/2*b^2*(A*b-a*B)*log(a+b*x^2)/a^4],
[(A+B*x^2)/(x^8*(a+b*x^2)),x,5,-1/7*A/(a*x^7)+1/5*(A*b-a*B)/(a^2*x^5)-1/3*b*(A*b-a*B)/(a^3*x^3)+b^2*(A*b-a*B)/(a^4*x)+b^(5/2)*(A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/a^(9/2)],
[x^9*(A+B*x^2)/(a+b*x^2)^2,x,3,1/2*a^2*(3*A*b-4*a*B)*x^2/b^5-1/4*a*(2*A*b-3*a*B)*x^4/b^4+1/6*(A*b-2*a*B)*x^6/b^3+1/8*B*x^8/b^2-1/2*a^4*(A*b-a*B)/(b^6*(a+b*x^2))-1/2*a^3*(4*A*b-5*a*B)*log(a+b*x^2)/b^6],
[x^8*(A+B*x^2)/(a+b*x^2)^2,x,4,a^2*(3*A*b-4*a*B)*x/b^5-1/3*a*(2*A*b-3*a*B)*x^3/b^4+1/5*(A*b-2*a*B)*x^5/b^3+1/7*B*x^7/b^2+1/2*a^3*(A*b-a*B)*x/(b^5*(a+b*x^2))-1/2*a^(5/2)*(7*A*b-9*a*B)*arctan(x*sqrt(b)/sqrt(a))/b^(11/2)],
[x^7*(A+B*x^2)/(a+b*x^2)^2,x,3,-1/2*a*(2*A*b-3*a*B)*x^2/b^4+1/4*(A*b-2*a*B)*x^4/b^3+1/6*B*x^6/b^2+1/2*a^3*(A*b-a*B)/(b^5*(a+b*x^2))+1/2*a^2*(3*A*b-4*a*B)*log(a+b*x^2)/b^5],
[x^6*(A+B*x^2)/(a+b*x^2)^2,x,4,-a*(2*A*b-3*a*B)*x/b^4+1/3*(A*b-2*a*B)*x^3/b^3+1/5*B*x^5/b^2-1/2*a^2*(A*b-a*B)*x/(b^4*(a+b*x^2))+1/2*a^(3/2)*(5*A*b-7*a*B)*arctan(x*sqrt(b)/sqrt(a))/b^(9/2)],
[x^5*(A+B*x^2)/(a+b*x^2)^2,x,3,1/2*(A*b-2*a*B)*x^2/b^3+1/4*B*x^4/b^2-1/2*a^2*(A*b-a*B)/(b^4*(a+b*x^2))-1/2*a*(2*A*b-3*a*B)*log(a+b*x^2)/b^4],
[x^4*(A+B*x^2)/(a+b*x^2)^2,x,4,(A*b-2*a*B)*x/b^3+1/3*B*x^3/b^2+1/2*a*(A*b-a*B)*x/(b^3*(a+b*x^2))-1/2*(3*A*b-5*a*B)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(7/2)],
[x^3*(A+B*x^2)/(a+b*x^2)^2,x,3,1/2*B*x^2/b^2+1/2*a*(A*b-a*B)/(b^3*(a+b*x^2))+1/2*(A*b-2*a*B)*log(a+b*x^2)/b^3],
[x^2*(A+B*x^2)/(a+b*x^2)^2,x,3,B*x/b^2-1/2*(A*b-a*B)*x/(b^2*(a+b*x^2))+1/2*(A*b-3*a*B)*arctan(x*sqrt(b)/sqrt(a))/(b^(5/2)*sqrt(a))],
[x*(A+B*x^2)/(a+b*x^2)^2,x,3,1/2*(-A*b+a*B)/(b^2*(a+b*x^2))+1/2*B*log(a+b*x^2)/b^2],
[(A+B*x^2)/(a+b*x^2)^2,x,2,1/2*(A*b-a*B)*x/(a*b*(a+b*x^2))+1/2*(A*b+a*B)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))],
[(A+B*x^2)/(x*(a+b*x^2)^2),x,3,1/2*(A*b-a*B)/(a*b*(a+b*x^2))+A*log(x)/a^2-1/2*A*log(a+b*x^2)/a^2],
[(A+B*x^2)/(x^2*(a+b*x^2)^2),x,3,-A/(a^2*x)-1/2*(A*b-a*B)*x/(a^2*(a+b*x^2))-1/2*(3*A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))],
[(A+B*x^2)/(x^3*(a+b*x^2)^2),x,3,-1/2*A/(a^2*x^2)+1/2*(-A*b+a*B)/(a^2*(a+b*x^2))-(2*A*b-a*B)*log(x)/a^3+1/2*(2*A*b-a*B)*log(a+b*x^2)/a^3],
[(A+B*x^2)/(x^4*(a+b*x^2)^2),x,4,-1/3*A/(a^2*x^3)+(2*A*b-a*B)/(a^3*x)+1/2*b*(A*b-a*B)*x/(a^3*(a+b*x^2))+1/2*(5*A*b-3*a*B)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(7/2)],
[(A+B*x^2)/(x^5*(a+b*x^2)^2),x,3,-1/4*A/(a^2*x^4)+1/2*(2*A*b-a*B)/(a^3*x^2)+1/2*b*(A*b-a*B)/(a^3*(a+b*x^2))+b*(3*A*b-2*a*B)*log(x)/a^4-1/2*b*(3*A*b-2*a*B)*log(a+b*x^2)/a^4],
[(A+B*x^2)/(x^6*(a+b*x^2)^2),x,4,-1/5*A/(a^2*x^5)+1/3*(2*A*b-a*B)/(a^3*x^3)-b*(3*A*b-2*a*B)/(a^4*x)-1/2*b^2*(A*b-a*B)*x/(a^4*(a+b*x^2))-1/2*b^(3/2)*(7*A*b-5*a*B)*arctan(x*sqrt(b)/sqrt(a))/a^(9/2)],
[(A+B*x^2)/(x^7*(a+b*x^2)^2),x,3,-1/6*A/(a^2*x^6)+1/4*(2*A*b-a*B)/(a^3*x^4)-1/2*b*(3*A*b-2*a*B)/(a^4*x^2)-1/2*b^2*(A*b-a*B)/(a^4*(a+b*x^2))-b^2*(4*A*b-3*a*B)*log(x)/a^5+1/2*b^2*(4*A*b-3*a*B)*log(a+b*x^2)/a^5],
[x^11*(A+B*x^2)/(a+b*x^2)^3,x,3,a^2*(3*A*b-5*a*B)*x^2/b^6-3/4*a*(A*b-2*a*B)*x^4/b^5+1/6*(A*b-3*a*B)*x^6/b^4+1/8*B*x^8/b^3+1/4*a^5*(A*b-a*B)/(b^7*(a+b*x^2)^2)-1/2*a^4*(5*A*b-6*a*B)/(b^7*(a+b*x^2))-5/2*a^3*(2*A*b-3*a*B)*log(a+b*x^2)/b^7],
[x^9*(A+B*x^2)/(a+b*x^2)^3,x,3,-3/2*a*(A*b-2*a*B)*x^2/b^5+1/4*(A*b-3*a*B)*x^4/b^4+1/6*B*x^6/b^3-1/4*a^4*(A*b-a*B)/(b^6*(a+b*x^2)^2)+1/2*a^3*(4*A*b-5*a*B)/(b^6*(a+b*x^2))+a^2*(3*A*b-5*a*B)*log(a+b*x^2)/b^6],
[x^7*(A+B*x^2)/(a+b*x^2)^3,x,3,1/2*(A*b-3*a*B)*x^2/b^4+1/4*B*x^4/b^3+1/4*a^3*(A*b-a*B)/(b^5*(a+b*x^2)^2)-1/2*a^2*(3*A*b-4*a*B)/(b^5*(a+b*x^2))-3/2*a*(A*b-2*a*B)*log(a+b*x^2)/b^5],
[x^5*(A+B*x^2)/(a+b*x^2)^3,x,3,1/2*B*x^2/b^3-1/4*a^2*(A*b-a*B)/(b^4*(a+b*x^2)^2)+1/2*a*(2*A*b-3*a*B)/(b^4*(a+b*x^2))+1/2*(A*b-3*a*B)*log(a+b*x^2)/b^4],
[x^3*(A+B*x^2)/(a+b*x^2)^3,x,3,1/4*a*(A*b-a*B)/(b^3*(a+b*x^2)^2)+1/2*(-A*b+2*a*B)/(b^3*(a+b*x^2))+1/2*B*log(a+b*x^2)/b^3],
[x*(A+B*x^2)/(a+b*x^2)^3,x,2,-1/4*(A+B*x^2)^2/((A*b-a*B)*(a+b*x^2)^2)],
[(A+B*x^2)/(x*(a+b*x^2)^3),x,3,1/4*(A*b-a*B)/(a*b*(a+b*x^2)^2)+1/2*A/(a^2*(a+b*x^2))+A*log(x)/a^3-1/2*A*log(a+b*x^2)/a^3],
[(A+B*x^2)/(x^3*(a+b*x^2)^3),x,3,-1/2*A/(a^3*x^2)+1/4*(-A*b+a*B)/(a^2*(a+b*x^2)^2)+1/2*(-2*A*b+a*B)/(a^3*(a+b*x^2))-(3*A*b-a*B)*log(x)/a^4+1/2*(3*A*b-a*B)*log(a+b*x^2)/a^4],
[(A+B*x^2)/(x^5*(a+b*x^2)^3),x,3,-1/4*A/(a^3*x^4)+1/2*(3*A*b-a*B)/(a^4*x^2)+1/4*b*(A*b-a*B)/(a^3*(a+b*x^2)^2)+1/2*b*(3*A*b-2*a*B)/(a^4*(a+b*x^2))+3*b*(2*A*b-a*B)*log(x)/a^5-3/2*b*(2*A*b-a*B)*log(a+b*x^2)/a^5],
[(A+B*x^2)/(x^7*(a+b*x^2)^3),x,3,-1/6*A/(a^3*x^6)+1/4*(3*A*b-a*B)/(a^4*x^4)-3/2*b*(2*A*b-a*B)/(a^5*x^2)-1/4*b^2*(A*b-a*B)/(a^4*(a+b*x^2)^2)-1/2*b^2*(4*A*b-3*a*B)/(a^5*(a+b*x^2))-2*b^2*(5*A*b-3*a*B)*log(x)/a^6+b^2*(5*A*b-3*a*B)*log(a+b*x^2)/a^6],
[x^10*(A+B*x^2)/(a+b*x^2)^3,x,5,2*a^2*(3*A*b-5*a*B)*x/b^6-a*(A*b-2*a*B)*x^3/b^5+1/5*(A*b-3*a*B)*x^5/b^4+1/7*B*x^7/b^3-1/4*a^4*(A*b-a*B)*x/(b^6*(a+b*x^2)^2)+1/8*a^3*(17*A*b-21*a*B)*x/(b^6*(a+b*x^2))-9/8*a^(5/2)*(7*A*b-11*a*B)*arctan(x*sqrt(b)/sqrt(a))/b^(13/2)],
[x^8*(A+B*x^2)/(a+b*x^2)^3,x,5,-3*a*(A*b-2*a*B)*x/b^5+1/3*(A*b-3*a*B)*x^3/b^4+1/5*B*x^5/b^3+1/4*a^3*(A*b-a*B)*x/(b^5*(a+b*x^2)^2)-1/8*a^2*(13*A*b-17*a*B)*x/(b^5*(a+b*x^2))+7/8*a^(3/2)*(5*A*b-9*a*B)*arctan(x*sqrt(b)/sqrt(a))/b^(11/2)],
[x^6*(A+B*x^2)/(a+b*x^2)^3,x,5,(A*b-3*a*B)*x/b^4+1/3*B*x^3/b^3-1/4*a^2*(A*b-a*B)*x/(b^4*(a+b*x^2)^2)+1/8*a*(9*A*b-13*a*B)*x/(b^4*(a+b*x^2))-5/8*(3*A*b-7*a*B)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(9/2)],
[x^4*(A+B*x^2)/(a+b*x^2)^3,x,4,B*x/b^3+1/4*a*(A*b-a*B)*x/(b^3*(a+b*x^2)^2)-1/8*(5*A*b-9*a*B)*x/(b^3*(a+b*x^2))+3/8*(A*b-5*a*B)*arctan(x*sqrt(b)/sqrt(a))/(b^(7/2)*sqrt(a))],
[x^2*(A+B*x^2)/(a+b*x^2)^3,x,3,-1/4*(A*b-a*B)*x/(b^2*(a+b*x^2)^2)+1/8*(A*b-5*a*B)*x/(a*b^2*(a+b*x^2))+1/8*(A*b+3*a*B)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(5/2))],
[(A+B*x^2)/(a+b*x^2)^3,x,3,1/4*(A*b-a*B)*x/(a*b*(a+b*x^2)^2)+1/8*(3*A*b+a*B)*x/(a^2*b*(a+b*x^2))+1/8*(3*A*b+a*B)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))],
[(A+B*x^2)/(x^2*(a+b*x^2)^3),x,4,-A/(a^3*x)-1/4*(A*b-a*B)*x/(a^2*(a+b*x^2)^2)-1/8*(7*A*b-3*a*B)*x/(a^3*(a+b*x^2))-3/8*(5*A*b-a*B)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))],
[(A+B*x^2)/(x^4*(a+b*x^2)^3),x,5,-1/3*A/(a^3*x^3)+(3*A*b-a*B)/(a^4*x)+1/4*b*(A*b-a*B)*x/(a^3*(a+b*x^2)^2)+1/8*b*(11*A*b-7*a*B)*x/(a^4*(a+b*x^2))+5/8*(7*A*b-3*a*B)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(9/2)],
[(A+B*x^2)/(x^6*(a+b*x^2)^3),x,5,-1/5*A/(a^3*x^5)+1/3*(3*A*b-a*B)/(a^4*x^3)-3*b*(2*A*b-a*B)/(a^5*x)-1/4*b^2*(A*b-a*B)*x/(a^4*(a+b*x^2)^2)-1/8*b^2*(15*A*b-11*a*B)*x/(a^5*(a+b*x^2))-7/8*b^(3/2)*(9*A*b-5*a*B)*arctan(x*sqrt(b)/sqrt(a))/a^(11/2)],
[(a+b*x^2)/(1+x^2),x,2,b*x+(a-b)*arctan(x)],
[(a+b*x^2)/(1-x^2),x,2,-b*x+(a+b)*arctanh(x)],
[(1+x^2)/(-1+x^2)^2,x,1,x/(1-x^2)],
[(1-x^2)/(1+x^2)^2,x,1,x/(1+x^2)],
[(3+2*x^2)/(1+x^2)^2,x,2,1/2*x/(1+x^2)+5/2*arctan(x)],
[(-2+x^2)/(1+x^2)^2,x,2,-3/2*x/(1+x^2)-1/2*arctan(x)],
[(3+x^2)/(1+x^2)^2,x,2,x/(1+x^2)+2*arctan(x)],
[(a+b*x^2)/(-a+b*x^2)^2,x,1,x/(a-b*x^2)],
[(a+b*x^2)/(a-b*x^2)^2,x,1,x/(a-b*x^2)],
[(A+B*x^2)/(a-b*x^2),x,2,-B*x/b+(A*b+a*B)*arctanh(x*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))],
[(1+x^2)/(16+x^2)^3,x,3,-15/64*x/(16+x^2)^2+19/2048*x/(16+x^2)+19/8192*arctan(1/4*x)],
[(1+2*x^2)/(x^5*(1+x^2)^3),x,2,(-1/4)/(x^4*(1+x^2)^2)],
[(1-x^2)^2/(-1+x^2)^2,x,2,x],
[x^3*(a*c+b*c*x^2)/(a+b*x^2),x,2,1/4*c*x^4],
[x^2*(a*c+b*c*x^2)/(a+b*x^2),x,2,1/3*c*x^3],
[x*(a*c+b*c*x^2)/(a+b*x^2),x,2,1/2*c*x^2],
[(a*c+b*c*x^2)/(a+b*x^2),x,2,c*x],
[(a*c+b*c*x^2)/(x*(a+b*x^2)),x,2,c*log(x)],
[(a*c+b*c*x^2)/(x^2*(a+b*x^2)),x,2,-c/x],
[(a*c+b*c*x^2)/(x^3*(a+b*x^2)),x,2,-1/2*c/x^2],
[x^3*(a*c+b*c*x^2)/(a+b*x^2)^2,x,4,1/2*c*x^2/b-1/2*a*c*log(a+b*x^2)/b^2],
[x^2*(a*c+b*c*x^2)/(a+b*x^2)^2,x,3,c*x/b-c*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(3/2)],
[x*(a*c+b*c*x^2)/(a+b*x^2)^2,x,2,1/2*c*log(a+b*x^2)/b],
[(a*c+b*c*x^2)/(a+b*x^2)^2,x,2,c*arctan(x*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],
[(a*c+b*c*x^2)/(x*(a+b*x^2)^2),x,5,c*log(x)/a-1/2*c*log(a+b*x^2)/a],
[(a*c+b*c*x^2)/(x^2*(a+b*x^2)^2),x,3,-c/(a*x)-c*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(3/2)],
[(a*c+b*c*x^2)/(x^3*(a+b*x^2)^2),x,4,-1/2*c/(a*x^2)-b*c*log(x)/a^2+1/2*b*c*log(a+b*x^2)/a^2],
[x^3*(a*c+b*c*x^2)/(a+b*x^2)^3,x,4,1/2*a*c/(b^2*(a+b*x^2))+1/2*c*log(a+b*x^2)/b^2],
[x^2*(a*c+b*c*x^2)/(a+b*x^2)^3,x,3,-1/2*c*x/(b*(a+b*x^2))+1/2*c*arctan(x*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))],
[x*(a*c+b*c*x^2)/(a+b*x^2)^3,x,2,-1/2*c/(b*(a+b*x^2))],
[(a*c+b*c*x^2)/(a+b*x^2)^3,x,3,1/2*c*x/(a*(a+b*x^2))+1/2*c*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))],
[(a*c+b*c*x^2)/(x*(a+b*x^2)^3),x,4,1/2*c/(a*(a+b*x^2))+c*log(x)/a^2-1/2*c*log(a+b*x^2)/a^2],
[(a*c+b*c*x^2)/(x^2*(a+b*x^2)^3),x,4,-3/2*c/(a^2*x)+1/2*c/(a*x*(a+b*x^2))-3/2*c*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(5/2)],
[(a*c+b*c*x^2)/(x^3*(a+b*x^2)^3),x,4,-1/2*c/(a^2*x^2)-1/2*b*c/(a^2*(a+b*x^2))-2*b*c*log(x)/a^3+b*c*log(a+b*x^2)/a^3],

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

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

# p<0
[x^4*(a+b*x^2)^2/(c+d*x^2),x,3,-c*(b*c-a*d)^2*x/d^4+1/3*(b*c-a*d)^2*x^3/d^3-1/5*b*(b*c-2*a*d)*x^5/d^2+1/7*b^2*x^7/d+c^(3/2)*(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))/d^(9/2)],
[x^3*(a+b*x^2)^2/(c+d*x^2),x,3,1/2*(b*c-a*d)^2*x^2/d^3-1/4*b*(b*c-2*a*d)*x^4/d^2+1/6*b^2*x^6/d-1/2*c*(b*c-a*d)^2*log(c+d*x^2)/d^4],
[x^2*(a+b*x^2)^2/(c+d*x^2),x,3,(b*c-a*d)^2*x/d^3-1/3*b*(b*c-2*a*d)*x^3/d^2+1/5*b^2*x^5/d-(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))*sqrt(c)/d^(7/2)],
[x*(a+b*x^2)^2/(c+d*x^2),x,3,-1/2*b*(b*c-a*d)*x^2/d^2+1/4*(a+b*x^2)^2/d+1/2*(b*c-a*d)^2*log(c+d*x^2)/d^3],
[(a+b*x^2)^2/(c+d*x^2),x,3,-b*(b*c-2*a*d)*x/d^2+1/3*b^2*x^3/d+(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))/(d^(5/2)*sqrt(c))],
[(a+b*x^2)^2/(x*(c+d*x^2)),x,3,1/2*b^2*x^2/d+a^2*log(x)/c-1/2*(b*c-a*d)^2*log(c+d*x^2)/(c*d^2)],
[(a+b*x^2)^2/(x^2*(c+d*x^2)),x,3,-a^2/(c*x)+b^2*x/d-(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*d^(3/2))],
[(a+b*x^2)^2/(x^3*(c+d*x^2)),x,3,-1/2*a^2/(c*x^2)+a*(2*b*c-a*d)*log(x)/c^2+1/2*(b*c-a*d)^2*log(c+d*x^2)/(c^2*d)],
[(a+b*x^2)^2/(x^4*(c+d*x^2)),x,3,-1/3*a^2/(c*x^3)-a*(2*b*c-a*d)/(c^2*x)+(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*sqrt(d))],
[(a+b*x^2)^2/(x^5*(c+d*x^2)),x,3,-1/4*a^2/(c*x^4)-1/2*a*(2*b*c-a*d)/(c^2*x^2)+(b*c-a*d)^2*log(x)/c^3-1/2*(b*c-a*d)^2*log(c+d*x^2)/c^3],
[(a+b*x^2)^2/(x^6*(c+d*x^2)),x,3,-1/5*a^2/(c*x^5)-1/3*a*(2*b*c-a*d)/(c^2*x^3)-(b*c-a*d)^2/(c^3*x)-(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/c^(7/2)],
[(a+b*x^2)^2/(x^7*(c+d*x^2)),x,3,-1/6*a^2/(c*x^6)-1/4*a*(2*b*c-a*d)/(c^2*x^4)-1/2*(b*c-a*d)^2/(c^3*x^2)-d*(b*c-a*d)^2*log(x)/c^4+1/2*d*(b*c-a*d)^2*log(c+d*x^2)/c^4],
[x^4*(a+b*x^2)^2/(c+d*x^2)^2,x,5,1/2*(7*b*c-3*a*d)*(b*c-a*d)*x/d^4-1/6*(7*b*c-3*a*d)*(b*c-a*d)*x^3/(c*d^3)+1/5*b^2*x^5/d^2+1/2*(b*c-a*d)^2*x^5/(c*d^2*(c+d*x^2))-1/2*(7*b*c-3*a*d)*(b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))*sqrt(c)/d^(9/2)],
[x^3*(a+b*x^2)^2/(c+d*x^2)^2,x,3,-b*(b*c-a*d)*x^2/d^3+1/4*b^2*x^4/d^2+1/2*c*(b*c-a*d)^2/(d^4*(c+d*x^2))+1/2*(b*c-a*d)*(3*b*c-a*d)*log(c+d*x^2)/d^4],
[x^2*(a+b*x^2)^2/(c+d*x^2)^2,x,4,-1/2*(b*c-a*d)*(5*b*c-a*d)*x/(c*d^3)+1/3*b^2*x^3/d^2+1/2*(b*c-a*d)^2*x^3/(c*d^2*(c+d*x^2))+1/2*(b*c-a*d)*(5*b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))/(d^(7/2)*sqrt(c))],
[x*(a+b*x^2)^2/(c+d*x^2)^2,x,3,1/2*b^2*x^2/d^2-1/2*(b*c-a*d)^2/(d^3*(c+d*x^2))-b*(b*c-a*d)*log(c+d*x^2)/d^3],
[(a+b*x^2)^2/(c+d*x^2)^2,x,4,b^2*x/d^2+1/2*(b*c-a*d)^2*x/(c*d^2*(c+d*x^2))-1/2*(b*c-a*d)*(3*b*c+a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*d^(5/2))],
[(a+b*x^2)^2/(x*(c+d*x^2)^2),x,3,1/2*(b*c-a*d)^2/(c*d^2*(c+d*x^2))+a^2*log(x)/c^2-1/2*(a^2/c^2-b^2/d^2)*log(c+d*x^2)],
[(a+b*x^2)^2/(x^2*(c+d*x^2)^2),x,3,-a^2/(c*x*(c+d*x^2))-1/2*(b^2*c^2-2*a*b*c*d+3*a^2*d^2)*x/(c^2*d*(c+d*x^2))+1/2*(b*c-a*d)*(b*c+3*a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*d^(3/2))],
[(a+b*x^2)^2/(x^3*(c+d*x^2)^2),x,3,-1/2*a^2/(c^2*x^2)-1/2*(b*c-a*d)^2/(c^2*d*(c+d*x^2))+2*a*(b*c-a*d)*log(x)/c^3-a*(b*c-a*d)*log(c+d*x^2)/c^3],
[(a+b*x^2)^2/(x^4*(c+d*x^2)^2),x,4,-1/3*a*(6*b*c-5*a*d)/(c^3*x)-1/3*a^2/(c*x^3*(c+d*x^2))+1/6*(3*b^2*c^2-6*a*b*c*d+5*a^2*d^2)*x/(c^3*(c+d*x^2))+1/2*(b*c-5*a*d)*(b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*sqrt(d))],
[x^4*(a+b*x^2)^2/(c+d*x^2)^3,x,5,-1/4*(13*b^2*c^2-10*a*b*c*d+a^2*d^2)*x/(c*d^4)+1/3*b^2*x^3/d^3+1/4*(b*c-a*d)^2*x^5/(c*d^2*(c+d*x^2)^2)-1/8*(b*c-a*d)*(9*b*c-a*d)*x/(d^4*(c+d*x^2))+1/8*(35*b^2*c^2-30*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(d^(9/2)*sqrt(c))],
[x^3*(a+b*x^2)^2/(c+d*x^2)^3,x,3,1/2*b^2*x^2/d^3+1/4*c*(b*c-a*d)^2/(d^4*(c+d*x^2)^2)-1/2*(b*c-a*d)*(3*b*c-a*d)/(d^4*(c+d*x^2))-1/2*b*(3*b*c-2*a*d)*log(c+d*x^2)/d^4],
[x^2*(a+b*x^2)^2/(c+d*x^2)^3,x,4,b^2*x/d^3+1/4*(b*c-a*d)^2*x^3/(c*d^2*(c+d*x^2)^2)+1/8*(b*c-a*d)*(7*b*c+a*d)*x/(c*d^3*(c+d*x^2))-1/8*(15*b^2*c^2-6*a*b*c*d-a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*d^(7/2))],
[x*(a+b*x^2)^2/(c+d*x^2)^3,x,3,-1/4*(b*c-a*d)^2/(d^3*(c+d*x^2)^2)+b*(b*c-a*d)/(d^3*(c+d*x^2))+1/2*b^2*log(c+d*x^2)/d^3],
[(a+b*x^2)^2/(c+d*x^2)^3,x,3,-1/4*(b*c-a*d)*x*(a+b*x^2)/(c*d*(c+d*x^2)^2)+3/8*(a^2/c^2-b^2/d^2)*x/(c+d*x^2)+1/8*(3*b^2*c^2+2*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*d^(5/2))],
[(a+b*x^2)^2/(x*(c+d*x^2)^3),x,3,1/4*(b*c-a*d)^2/(c*d^2*(c+d*x^2)^2)+1/2*(a^2/c^2-b^2/d^2)/(c+d*x^2)+a^2*log(x)/c^3-1/2*a^2*log(c+d*x^2)/c^3],
[(a+b*x^2)^2/(x^2*(c+d*x^2)^3),x,4,-a^2/(c*x*(c+d*x^2)^2)-1/4*(b^2*c^2-2*a*b*c*d+5*a^2*d^2)*x/(c^2*d*(c+d*x^2)^2)+1/8*(b^2*c^2+3*a*d*(2*b*c-5*a*d))*x/(c^3*d*(c+d*x^2))+1/8*(b^2*c^2+3*a*d*(2*b*c-5*a*d))*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*d^(3/2))],
[(a+b*x^2)^2/(x^3*(c+d*x^2)^3),x,3,-1/2*a^2/(c^3*x^2)-1/4*(b*c-a*d)^2/(c^2*d*(c+d*x^2)^2)+a*(b*c-a*d)/(c^3*(c+d*x^2))+a*(2*b*c-3*a*d)*log(x)/c^4-1/2*a*(2*b*c-3*a*d)*log(c+d*x^2)/c^4],
[(a+b*x^2)^2/(x^4*(c+d*x^2)^3),x,5,-1/3*a*(6*b*c-7*a*d)/(c^4*x)-1/3*a^2/(c*x^3*(c+d*x^2)^2)+1/12*(3*b^2*c^2-6*a*b*c*d+7*a^2*d^2)*x/(c^3*(c+d*x^2)^2)+1/24*(3*b*c-7*a*d)^2*x/(c^4*(c+d*x^2))+1/8*(3*b^2*c^2-30*a*b*c*d+35*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(9/2)*sqrt(d))],

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

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

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

# p<0
[x^5/((a+b*x^2)*(c+d*x^2)),x,3,1/2*x^2/(b*d)+1/2*a^2*log(a+b*x^2)/(b^2*(b*c-a*d))-1/2*c^2*log(c+d*x^2)/(d^2*(b*c-a*d))],
[x^4/((a+b*x^2)*(c+d*x^2)),x,4,x/(b*d)+a^(3/2)*arctan(x*sqrt(b)/sqrt(a))/(b^(3/2)*(b*c-a*d))-c^(3/2)*arctan(x*sqrt(d)/sqrt(c))/(d^(3/2)*(b*c-a*d))],
[x^3/((a+b*x^2)*(c+d*x^2)),x,3,-1/2*a*log(a+b*x^2)/(b*(b*c-a*d))+1/2*c*log(c+d*x^2)/(d*(b*c-a*d))],
[x^2/((a+b*x^2)*(c+d*x^2)),x,3,-arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/((b*c-a*d)*sqrt(b))+arctan(x*sqrt(d)/sqrt(c))*sqrt(c)/((b*c-a*d)*sqrt(d))],
[x/((a+b*x^2)*(c+d*x^2)),x,4,1/2*log(a+b*x^2)/(b*c-a*d)-1/2*log(c+d*x^2)/(b*c-a*d)],
[1/((a+b*x^2)*(c+d*x^2)),x,3,arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/((b*c-a*d)*sqrt(a))-arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/((b*c-a*d)*sqrt(c))],
[1/(x*(a+b*x^2)*(c+d*x^2)),x,3,log(x)/(a*c)-1/2*b*log(a+b*x^2)/(a*(b*c-a*d))+1/2*d*log(c+d*x^2)/(c*(b*c-a*d))],
[1/(x^2*(a+b*x^2)*(c+d*x^2)),x,4,(-1)/(a*c*x)-b^(3/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d))+d^(3/2)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*(b*c-a*d))],
[1/(x^3*(a+b*x^2)*(c+d*x^2)),x,3,(-1/2)/(a*c*x^2)-(b*c+a*d)*log(x)/(a^2*c^2)+1/2*b^2*log(a+b*x^2)/(a^2*(b*c-a*d))-1/2*d^2*log(c+d*x^2)/(c^2*(b*c-a*d))],
[1/(x^4*(a+b*x^2)*(c+d*x^2)),x,5,(-1/3)/(a*c*x^3)+(b*c+a*d)/(a^2*c^2*x)+b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d))-d^(5/2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d))],
[1/(x^5*(a+b*x^2)*(c+d*x^2)),x,3,(-1/4)/(a*c*x^4)+1/2*(b*c+a*d)/(a^2*c^2*x^2)+(b^2*c^2+a*b*c*d+a^2*d^2)*log(x)/(a^3*c^3)-1/2*b^3*log(a+b*x^2)/(a^3*(b*c-a*d))+1/2*d^3*log(c+d*x^2)/(c^3*(b*c-a*d))],
[1/(x^6*(a+b*x^2)*(c+d*x^2)),x,6,(-1/5)/(a*c*x^5)+1/3*(b*c+a*d)/(a^2*c^2*x^3)+(-b^2*c^2-a*b*c*d-a^2*d^2)/(a^3*c^3*x)-b^(7/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*(b*c-a*d))+d^(7/2)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*(b*c-a*d))],
[1/(x^7*(a+b*x^2)*(c+d*x^2)),x,3,(-1/6)/(a*c*x^6)+1/4*(b*c+a*d)/(a^2*c^2*x^4)+1/2*(-b^2*c^2-a*b*c*d-a^2*d^2)/(a^3*c^3*x^2)-(b*c+a*d)*(b^2*c^2+a^2*d^2)*log(x)/(a^4*c^4)+1/2*b^4*log(a+b*x^2)/(a^4*(b*c-a*d))-1/2*d^4*log(c+d*x^2)/(c^4*(b*c-a*d))],
[x^5/((a+b*x^2)^2*(c+d*x^2)),x,3,-1/2*a^2/(b^2*(b*c-a*d)*(a+b*x^2))-1/2*a*(2*b*c-a*d)*log(a+b*x^2)/(b^2*(b*c-a*d)^2)+1/2*c^2*log(c+d*x^2)/(d*(b*c-a*d)^2)],
[x^4/((a+b*x^2)*(c+d*x^2)^2),x,4,-1/2*c*x/(d*(b*c-a*d)*(c+d*x^2))+a^(3/2)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^2*sqrt(b))+1/2*(b*c-3*a*d)*arctan(x*sqrt(d)/sqrt(c))*sqrt(c)/(d^(3/2)*(b*c-a*d)^2)],
[x^3/((a+b*x^2)*(c+d*x^2)^2),x,3,-1/2*c/(d*(b*c-a*d)*(c+d*x^2))-1/2*a*log(a+b*x^2)/(b*c-a*d)^2+1/2*a*log(c+d*x^2)/(b*c-a*d)^2],
[x^2/((a+b*x^2)*(c+d*x^2)^2),x,4,1/2*x/((b*c-a*d)*(c+d*x^2))-arctan(x*sqrt(b)/sqrt(a))*sqrt(a)*sqrt(b)/(b*c-a*d)^2+1/2*(b*c+a*d)*arctan(x*sqrt(d)/sqrt(c))/((b*c-a*d)^2*sqrt(c)*sqrt(d))],
[x/((a+b*x^2)*(c+d*x^2)^2),x,3,1/2/((b*c-a*d)*(c+d*x^2))+1/2*b*log(a+b*x^2)/(b*c-a*d)^2-1/2*b*log(c+d*x^2)/(b*c-a*d)^2],
[1/((a+b*x^2)*(c+d*x^2)^2),x,4,-1/2*d*x/(c*(b*c-a*d)*(c+d*x^2))+b^(3/2)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^2*sqrt(a))-1/2*(3*b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/(c^(3/2)*(b*c-a*d)^2)],
[1/(x*(a+b*x^2)*(c+d*x^2)^2),x,3,-1/2*d/(c*(b*c-a*d)*(c+d*x^2))+log(x)/(a*c^2)-1/2*b^2*log(a+b*x^2)/(a*(b*c-a*d)^2)+1/2*d*(2*b*c-a*d)*log(c+d*x^2)/(c^2*(b*c-a*d)^2)],
[1/(x^2*(a+b*x^2)*(c+d*x^2)^2),x,5,1/2*(-2*b*c+3*a*d)/(a*c^2*(b*c-a*d)*x)-1/2*d/(c*(b*c-a*d)*x*(c+d*x^2))-b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d)^2)+1/2*d^(3/2)*(5*b*c-3*a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d)^2)],
[1/(x^3*(a+b*x^2)*(c+d*x^2)^2),x,3,(-1/2)/(a*c^2*x^2)+1/2*d^2/(c^2*(b*c-a*d)*(c+d*x^2))-(b*c+2*a*d)*log(x)/(a^2*c^3)+1/2*b^3*log(a+b*x^2)/(a^2*(b*c-a*d)^2)-1/2*d^2*(3*b*c-2*a*d)*log(c+d*x^2)/(c^3*(b*c-a*d)^2)],
[1/(x^4*(a+b*x^2)*(c+d*x^2)^2),x,6,1/6*(-2*b*c+5*a*d)/(a*c^2*(b*c-a*d)*x^3)+1/2*(2*b^2*c^2+2*a*b*c*d-5*a^2*d^2)/(a^2*c^3*(b*c-a*d)*x)-1/2*d/(c*(b*c-a*d)*x^3*(c+d*x^2))+b^(7/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^2)-1/2*d^(5/2)*(7*b*c-5*a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*(b*c-a*d)^2)],
[x^5/((a+b*x^2)^3*(c+d*x^2)),x,3,-1/4*a^2/(b^2*(b*c-a*d)*(a+b*x^2)^2)+1/2*a*(2*b*c-a*d)/(b^2*(b*c-a*d)^2*(a+b*x^2))+1/2*c^2*log(a+b*x^2)/(b*c-a*d)^3-1/2*c^2*log(c+d*x^2)/(b*c-a*d)^3],
[x^4/((a+b*x^2)*(c+d*x^2)^3),x,5,-1/4*c*x/(d*(b*c-a*d)*(c+d*x^2)^2)+1/8*(b*c-5*a*d)*x/(d*(b*c-a*d)^2*(c+d*x^2))+a^(3/2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(b*c-a*d)^3+1/8*(b^2*c^2-6*a*b*c*d-3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(d^(3/2)*(b*c-a*d)^3*sqrt(c))],
[x^3/((a+b*x^2)*(c+d*x^2)^3),x,3,-1/4*c/(d*(b*c-a*d)*(c+d*x^2)^2)-1/2*a/((b*c-a*d)^2*(c+d*x^2))-1/2*a*b*log(a+b*x^2)/(b*c-a*d)^3+1/2*a*b*log(c+d*x^2)/(b*c-a*d)^3],
[x^2/((a+b*x^2)*(c+d*x^2)^3),x,5,1/4*x/((b*c-a*d)*(c+d*x^2)^2)+1/8*(3*b*c+a*d)*x/(c*(b*c-a*d)^2*(c+d*x^2))-b^(3/2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/(b*c-a*d)^3+1/8*(3*b^2*c^2+6*a*b*c*d-a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*(b*c-a*d)^3*sqrt(d))],
[x/((a+b*x^2)*(c+d*x^2)^3),x,3,1/4/((b*c-a*d)*(c+d*x^2)^2)+1/2*b/((b*c-a*d)^2*(c+d*x^2))+1/2*b^2*log(a+b*x^2)/(b*c-a*d)^3-1/2*b^2*log(c+d*x^2)/(b*c-a*d)^3],
[1/((a+b*x^2)*(c+d*x^2)^3),x,5,-1/4*d*x/(c*(b*c-a*d)*(c+d*x^2)^2)-1/8*d*(7*b*c-3*a*d)*x/(c^2*(b*c-a*d)^2*(c+d*x^2))+b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^3*sqrt(a))-1/8*(15*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/(c^(5/2)*(b*c-a*d)^3)],
[1/(x*(a+b*x^2)*(c+d*x^2)^3),x,3,-1/4*d/(c*(b*c-a*d)*(c+d*x^2)^2)-1/2*d*(2*b*c-a*d)/(c^2*(b*c-a*d)^2*(c+d*x^2))+log(x)/(a*c^3)-1/2*b^3*log(a+b*x^2)/(a*(b*c-a*d)^3)+1/2*d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*log(c+d*x^2)/(c^3*(b*c-a*d)^3)],
[1/(x^2*(a+b*x^2)*(c+d*x^2)^3),x,6,1/8*(-8*b^2*c^2+27*a*b*c*d-15*a^2*d^2)/(a*c^3*(b*c-a*d)^2*x)-1/4*d/(c*(b*c-a*d)*x*(c+d*x^2)^2)-1/8*d*(9*b*c-5*a*d)/(c^2*(b*c-a*d)^2*x*(c+d*x^2))-b^(7/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d)^3)+1/8*d^(3/2)*(35*b^2*c^2-42*a*b*c*d+15*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*(b*c-a*d)^3)],
[1/(x^3*(a+b*x^2)*(c+d*x^2)^3),x,3,(-1/2)/(a*c^3*x^2)+1/4*d^2/(c^2*(b*c-a*d)*(c+d*x^2)^2)+1/2*d^2*(3*b*c-2*a*d)/(c^3*(b*c-a*d)^2*(c+d*x^2))-(b*c+3*a*d)*log(x)/(a^2*c^4)+1/2*b^4*log(a+b*x^2)/(a^2*(b*c-a*d)^3)-1/2*d^2*(6*b^2*c^2-8*a*b*c*d+3*a^2*d^2)*log(c+d*x^2)/(c^4*(b*c-a*d)^3)],
[1/(x^4*(a+b*x^2)*(c+d*x^2)^3),x,7,1/24*(-8*b^2*c^2+55*a*b*c*d-35*a^2*d^2)/(a*c^3*(b*c-a*d)^2*x^3)+1/8*(8*b^3*c^3+8*a*b^2*c^2*d-55*a^2*b*c*d^2+35*a^3*d^3)/(a^2*c^4*(b*c-a*d)^2*x)-1/4*d/(c*(b*c-a*d)*x^3*(c+d*x^2)^2)-1/8*d*(11*b*c-7*a*d)/(c^2*(b*c-a*d)^2*x^3*(c+d*x^2))+b^(9/2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^3)-1/8*d^(5/2)*(63*b^2*c^2-90*a*b*c*d+35*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(9/2)*(b*c-a*d)^3)],
[x/((1+x^2)*(4+x^2)),x,4,1/6*log(1+x^2)-1/6*log(4+x^2)],

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

# p>0
[x^4*(c+d*x^2)/(a+b*x^2)^2,x,4,(b*c-2*a*d)*x/b^3+1/3*d*x^3/b^2+1/2*a*(b*c-a*d)*x/(b^3*(a+b*x^2))-1/2*(3*b*c-5*a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(7/2)],
[x^3*(c+d*x^2)/(a+b*x^2)^2,x,3,1/2*d*x^2/b^2+1/2*a*(b*c-a*d)/(b^3*(a+b*x^2))+1/2*(b*c-2*a*d)*log(a+b*x^2)/b^3],
[x^2*(c+d*x^2)/(a+b*x^2)^2,x,3,d*x/b^2-1/2*(b*c-a*d)*x/(b^2*(a+b*x^2))+1/2*(b*c-3*a*d)*arctan(x*sqrt(b)/sqrt(a))/(b^(5/2)*sqrt(a))],
[x*(c+d*x^2)/(a+b*x^2)^2,x,3,1/2*(-b*c+a*d)/(b^2*(a+b*x^2))+1/2*d*log(a+b*x^2)/b^2],
[(c+d*x^2)/(a+b*x^2)^2,x,2,1/2*(b*c-a*d)*x/(a*b*(a+b*x^2))+1/2*(b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))],
[(c+d*x^2)/(x*(a+b*x^2)^2),x,3,1/2*(b*c-a*d)/(a*b*(a+b*x^2))+c*log(x)/a^2-1/2*c*log(a+b*x^2)/a^2],
[(c+d*x^2)/(x^2*(a+b*x^2)^2),x,3,-c/(a^2*x)-1/2*(b*c-a*d)*x/(a^2*(a+b*x^2))-1/2*(3*b*c-a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(b))],
[(c+d*x^2)/(x^3*(a+b*x^2)^2),x,3,-1/2*c/(a^2*x^2)+1/2*(-b*c+a*d)/(a^2*(a+b*x^2))-(2*b*c-a*d)*log(x)/a^3+1/2*(2*b*c-a*d)*log(a+b*x^2)/a^3],
[(c+d*x^2)/(x^4*(a+b*x^2)^2),x,4,-1/3*c/(a^2*x^3)+(2*b*c-a*d)/(a^3*x)+1/2*b*(b*c-a*d)*x/(a^3*(a+b*x^2))+1/2*(5*b*c-3*a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(7/2)],
[x^4*(c+d*x^2)^2/(a+b*x^2)^2,x,5,1/2*(3*b*c-7*a*d)*(b*c-a*d)*x/b^4-1/6*(3*b*c-7*a*d)*(b*c-a*d)*x^3/(a*b^3)+1/5*d^2*x^5/b^2+1/2*(b*c-a*d)^2*x^5/(a*b^2*(a+b*x^2))-1/2*(3*b*c-7*a*d)*(b*c-a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(9/2)],
[x^3*(c+d*x^2)^2/(a+b*x^2)^2,x,3,d*(b*c-a*d)*x^2/b^3+1/4*d^2*x^4/b^2+1/2*a*(b*c-a*d)^2/(b^4*(a+b*x^2))+1/2*(b*c-3*a*d)*(b*c-a*d)*log(a+b*x^2)/b^4],
[x^2*(c+d*x^2)^2/(a+b*x^2)^2,x,4,-1/2*(b*c-5*a*d)*(b*c-a*d)*x/(a*b^3)+1/3*d^2*x^3/b^2+1/2*(b*c-a*d)^2*x^3/(a*b^2*(a+b*x^2))+1/2*(b*c-5*a*d)*(b*c-a*d)*arctan(x*sqrt(b)/sqrt(a))/(b^(7/2)*sqrt(a))],
[x*(c+d*x^2)^2/(a+b*x^2)^2,x,3,1/2*d^2*x^2/b^2-1/2*(b*c-a*d)^2/(b^3*(a+b*x^2))+d*(b*c-a*d)*log(a+b*x^2)/b^3],
[(c+d*x^2)^2/(a+b*x^2)^2,x,4,d^2*x/b^2+1/2*(b*c-a*d)^2*x/(a*b^2*(a+b*x^2))+1/2*(b*c-a*d)*(b*c+3*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(5/2))],
[(c+d*x^2)^2/(x*(a+b*x^2)^2),x,3,1/2*(b*c-a*d)^2/(a*b^2*(a+b*x^2))+c^2*log(x)/a^2-1/2*(c^2/a^2-d^2/b^2)*log(a+b*x^2)],
[(c+d*x^2)^2/(x^2*(a+b*x^2)^2),x,3,-c^2/(a*x*(a+b*x^2))-1/2*(3*b*c^2/a-2*c*d+a*d^2/b)*x/(a*(a+b*x^2))-1/2*(b*c-a*d)*(3*b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))],
[(c+d*x^2)^2/(x^3*(a+b*x^2)^2),x,3,-1/2*c^2/(a^2*x^2)-1/2*(b*c-a*d)^2/(a^2*b*(a+b*x^2))-2*c*(b*c-a*d)*log(x)/a^3+c*(b*c-a*d)*log(a+b*x^2)/a^3],
[(c+d*x^2)^2/(x^4*(a+b*x^2)^2),x,4,1/3*c*(5*b*c-6*a*d)/(a^3*x)-1/3*c^2/(a*x^3*(a+b*x^2))+1/6*(5*b^2*c^2-6*a*b*c*d+3*a^2*d^2)*x/(a^3*(a+b*x^2))+1/2*(b*c-a*d)*(5*b*c-a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))],
[x^4*(c+d*x^2)^3/(a+b*x^2)^2,x,4,3/2*(b*c-3*a*d)*(b*c-a*d)^2*x/b^5+1/2*d*(5*b^2*c^2-7*a*b*c*d+3*a^2*d^2)*x^3/b^4+3/10*d^2*(7*b*c-3*a*d)*x^5/b^3+9/14*d^3*x^7/b^2-1/2*x^3*(c+d*x^2)^3/(b*(a+b*x^2))-3/2*(b*c-3*a*d)*(b*c-a*d)^2*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(11/2)],
[x^3*(c+d*x^2)^3/(a+b*x^2)^2,x,3,3/2*d*(b*c-a*d)^2*x^2/b^4+1/4*d^2*(3*b*c-2*a*d)*x^4/b^3+1/6*d^3*x^6/b^2+1/2*a*(b*c-a*d)^3/(b^5*(a+b*x^2))+1/2*(b*c-4*a*d)*(b*c-a*d)^2*log(a+b*x^2)/b^5],
[x^2*(c+d*x^2)^3/(a+b*x^2)^2,x,5,1/30*d*(81*b^2*c^2-190*a*b*c*d+105*a^2*d^2)*x/b^4+1/30*d*(33*b*c-35*a*d)*x*(c+d*x^2)/b^3+7/10*d*x*(c+d*x^2)^2/b^2-1/2*x*(c+d*x^2)^3/(b*(a+b*x^2))+1/2*(b*c-7*a*d)*(b*c-a*d)^2*arctan(x*sqrt(b)/sqrt(a))/(b^(9/2)*sqrt(a))],
[x*(c+d*x^2)^3/(a+b*x^2)^2,x,3,1/2*d^2*(3*b*c-2*a*d)*x^2/b^3+1/4*d^3*x^4/b^2-1/2*(b*c-a*d)^3/(b^4*(a+b*x^2))+3/2*d*(b*c-a*d)^2*log(a+b*x^2)/b^4],
[(c+d*x^2)^3/(a+b*x^2)^2,x,4,d^2*(3*b*c-2*a*d)*x/b^3+1/3*d^3*x^3/b^2+1/2*(b*c-a*d)^3*x/(a*b^3*(a+b*x^2))+1/2*(b*c-a*d)^2*(b*c+5*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(7/2))],
[(c+d*x^2)^3/(x*(a+b*x^2)^2),x,3,1/2*d^3*x^2/b^2+1/2*(b*c-a*d)^3/(a*b^3*(a+b*x^2))+c^3*log(x)/a^2-1/2*(b*c-a*d)^2*(b*c+2*a*d)*log(a+b*x^2)/(a^2*b^3)],
[(c+d*x^2)^3/(x^2*(a+b*x^2)^2),x,4,-1/2*c^2*(3*b*c-a*d)/(a^2*b*x)-1/2*d^2*(b*c-3*a*d)*x/(a*b^2)+1/2*(b*c-a*d)*(c+d*x^2)^2/(a*b*x*(a+b*x^2))-3/2*(b*c-a*d)^2*(b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(5/2))],
[(c+d*x^2)^3/(x^3*(a+b*x^2)^2),x,3,-1/2*c^3/(a^2*x^2)-1/2*(b*c-a*d)^3/(a^2*b^2*(a+b*x^2))-c^2*(2*b*c-3*a*d)*log(x)/a^3+1/2*(b*c-a*d)^2*(2*b*c+a*d)*log(a+b*x^2)/(a^3*b^2)],
[(c+d*x^2)^3/(x^4*(a+b*x^2)^2),x,4,-1/6*c^2*(5*b*c-3*a*d)/(a^2*b*x^3)+1/2*c*(5*b^2*c^2-9*a*b*c*d+2*a^2*d^2)/(a^3*b*x)+1/2*(b*c-a*d)*(c+d*x^2)^2/(a*b*x^3*(a+b*x^2))+1/2*(b*c-a*d)^2*(5*b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2))],

# p<0
[x^4/((a+b*x^2)^2*(c+d*x^2)),x,4,1/2*a*x/(b*(b*c-a*d)*(a+b*x^2))-1/2*(3*b*c-a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/(b^(3/2)*(b*c-a*d)^2)+c^(3/2)*arctan(x*sqrt(d)/sqrt(c))/((b*c-a*d)^2*sqrt(d))],
[x^3/((a+b*x^2)^2*(c+d*x^2)),x,3,1/2*a/(b*(b*c-a*d)*(a+b*x^2))+1/2*c*log(a+b*x^2)/(b*c-a*d)^2-1/2*c*log(c+d*x^2)/(b*c-a*d)^2],
[x^2/((a+b*x^2)^2*(c+d*x^2)),x,4,-1/2*x/((b*c-a*d)*(a+b*x^2))+1/2*(b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^2*sqrt(a)*sqrt(b))-arctan(x*sqrt(d)/sqrt(c))*sqrt(c)*sqrt(d)/(b*c-a*d)^2],
[x/((a+b*x^2)^2*(c+d*x^2)),x,3,(-1/2)/((b*c-a*d)*(a+b*x^2))-1/2*d*log(a+b*x^2)/(b*c-a*d)^2+1/2*d*log(c+d*x^2)/(b*c-a*d)^2],
[1/((a+b*x^2)^2*(c+d*x^2)),x,4,1/2*b*x/(a*(b*c-a*d)*(a+b*x^2))+1/2*(b*c-3*a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(3/2)*(b*c-a*d)^2)+d^(3/2)*arctan(x*sqrt(d)/sqrt(c))/((b*c-a*d)^2*sqrt(c))],
[1/(x*(a+b*x^2)^2*(c+d*x^2)),x,3,1/2*b/(a*(b*c-a*d)*(a+b*x^2))+log(x)/(a^2*c)-1/2*b*(b*c-2*a*d)*log(a+b*x^2)/(a^2*(b*c-a*d)^2)-1/2*d^2*log(c+d*x^2)/(c*(b*c-a*d)^2)],
[1/(x^2*(a+b*x^2)^2*(c+d*x^2)),x,5,1/2*(-3*b*c+2*a*d)/(a^2*c*(b*c-a*d)*x)+1/2*b/(a*(b*c-a*d)*x*(a+b*x^2))-1/2*b^(3/2)*(3*b*c-5*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^2)-d^(5/2)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*(b*c-a*d)^2)],
[1/(x^3*(a+b*x^2)^2*(c+d*x^2)),x,3,(-1/2)/(a^2*c*x^2)-1/2*b^2/(a^2*(b*c-a*d)*(a+b*x^2))-(2*b*c+a*d)*log(x)/(a^3*c^2)+1/2*b^2*(2*b*c-3*a*d)*log(a+b*x^2)/(a^3*(b*c-a*d)^2)+1/2*d^3*log(c+d*x^2)/(c^2*(b*c-a*d)^2)],
[1/(x^4*(a+b*x^2)^2*(c+d*x^2)),x,6,1/6*(-5*b*c+2*a*d)/(a^2*c*(b*c-a*d)*x^3)+1/2*(5*b^2*c^2-2*a*b*c*d-2*a^2*d^2)/(a^3*c^2*(b*c-a*d)*x)+1/2*b/(a*(b*c-a*d)*x^3*(a+b*x^2))+1/2*b^(5/2)*(5*b*c-7*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*(b*c-a*d)^2)+d^(7/2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d)^2)],
[1/(x^5*(a+b*x^2)^2*(c+d*x^2)),x,3,(-1/4)/(a^2*c*x^4)+1/2*(2*b*c+a*d)/(a^3*c^2*x^2)+1/2*b^3/(a^3*(b*c-a*d)*(a+b*x^2))+(3*b^2*c^2+2*a*b*c*d+a^2*d^2)*log(x)/(a^4*c^3)-1/2*b^3*(3*b*c-4*a*d)*log(a+b*x^2)/(a^4*(b*c-a*d)^2)-1/2*d^4*log(c+d*x^2)/(c^3*(b*c-a*d)^2)],
[1/(x^6*(a+b*x^2)^2*(c+d*x^2)),x,7,1/10*(-7*b*c+2*a*d)/(a^2*c*(b*c-a*d)*x^5)+1/6*(7*b^2*c^2-2*a*b*c*d-2*a^2*d^2)/(a^3*c^2*(b*c-a*d)*x^3)+1/2*(-7*b^3*c^3+2*a*b^2*c^2*d+2*a^2*b*c*d^2+2*a^3*d^3)/(a^4*c^3*(b*c-a*d)*x)+1/2*b/(a*(b*c-a*d)*x^5*(a+b*x^2))-1/2*b^(7/2)*(7*b*c-9*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(9/2)*(b*c-a*d)^2)-d^(9/2)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*(b*c-a*d)^2)],
[1/(x^7*(a+b*x^2)^2*(c+d*x^2)),x,3,(-1/6)/(a^2*c*x^6)+1/4*(2*b*c+a*d)/(a^3*c^2*x^4)+1/2*(-3*b^2*c^2-2*a*b*c*d-a^2*d^2)/(a^4*c^3*x^2)-1/2*b^4/(a^4*(b*c-a*d)*(a+b*x^2))-(4*b^3*c^3+3*a*b^2*c^2*d+2*a^2*b*c*d^2+a^3*d^3)*log(x)/(a^5*c^4)+1/2*b^4*(4*b*c-5*a*d)*log(a+b*x^2)/(a^5*(b*c-a*d)^2)+1/2*d^5*log(c+d*x^2)/(c^4*(b*c-a*d)^2)],
[x^4/((a+b*x^2)^2*(c+d*x^2)^2),x,5,1/2*(b*c+a*d)*x/(b*(b*c-a*d)^2*(c+d*x^2))+1/2*a*x/(b*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))-1/2*(3*b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/((b*c-a*d)^3*sqrt(b))+1/2*(b*c+3*a*d)*arctan(x*sqrt(d)/sqrt(c))*sqrt(c)/((b*c-a*d)^3*sqrt(d))],
[x^3/((a+b*x^2)^2*(c+d*x^2)^2),x,3,1/2*a/((b*c-a*d)^2*(a+b*x^2))+1/2*c/((b*c-a*d)^2*(c+d*x^2))+1/2*(b*c+a*d)*log(a+b*x^2)/(b*c-a*d)^3-1/2*(b*c+a*d)*log(c+d*x^2)/(b*c-a*d)^3],
[x^2/((a+b*x^2)^2*(c+d*x^2)^2),x,5,-d*x/((b*c-a*d)^2*(c+d*x^2))-1/2*x/((b*c-a*d)*(a+b*x^2)*(c+d*x^2))+1/2*(b*c+3*a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/((b*c-a*d)^3*sqrt(a))-1/2*(3*b*c+a*d)*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/((b*c-a*d)^3*sqrt(c))],
[x/((a+b*x^2)^2*(c+d*x^2)^2),x,3,-1/2*b/((b*c-a*d)^2*(a+b*x^2))-1/2*d/((b*c-a*d)^2*(c+d*x^2))-b*d*log(a+b*x^2)/(b*c-a*d)^3+b*d*log(c+d*x^2)/(b*c-a*d)^3],
[1/((a+b*x^2)^2*(c+d*x^2)^2),x,5,1/2*d*(b*c+a*d)*x/(a*c*(b*c-a*d)^2*(c+d*x^2))+1/2*b*x/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))+1/2*b^(3/2)*(b*c-5*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d)^3)+1/2*d^(3/2)*(5*b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*(b*c-a*d)^3)],
[1/(x*(a+b*x^2)^2*(c+d*x^2)^2),x,3,1/2*b^2/(a*(b*c-a*d)^2*(a+b*x^2))+1/2*d^2/(c*(b*c-a*d)^2*(c+d*x^2))+log(x)/(a^2*c^2)-1/2*b^2*(b*c-3*a*d)*log(a+b*x^2)/(a^2*(b*c-a*d)^3)-1/2*d^2*(3*b*c-a*d)*log(c+d*x^2)/(c^2*(b*c-a*d)^3)],
[1/(x^2*(a+b*x^2)^2*(c+d*x^2)^2),x,6,1/2*(-3*b^2*c^2+4*a*b*c*d-3*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*x)+1/2*d*(b*c+a*d)/(a*c*(b*c-a*d)^2*x*(c+d*x^2))+1/2*b/(a*(b*c-a*d)*x*(a+b*x^2)*(c+d*x^2))-1/2*b^(5/2)*(3*b*c-7*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^3)-1/2*d^(5/2)*(7*b*c-3*a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d)^3)],
[1/(x^3*(a+b*x^2)^2*(c+d*x^2)^2),x,3,(-1/2)/(a^2*c^2*x^2)-1/2*b^3/(a^2*(b*c-a*d)^2*(a+b*x^2))-1/2*d^3/(c^2*(b*c-a*d)^2*(c+d*x^2))-2*(b*c+a*d)*log(x)/(a^3*c^3)+b^3*(b*c-2*a*d)*log(a+b*x^2)/(a^3*(b*c-a*d)^3)+d^3*(2*b*c-a*d)*log(c+d*x^2)/(c^3*(b*c-a*d)^3)],
[1/(x^4*(a+b*x^2)^2*(c+d*x^2)^2),x,7,1/6*(-5*b^2*c^2+4*a*b*c*d-5*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*x^3)+1/2*(b*c+a*d)*(5*b^2*c^2-9*a*b*c*d+5*a^2*d^2)/(a^3*c^3*(b*c-a*d)^2*x)+1/2*d*(b*c+a*d)/(a*c*(b*c-a*d)^2*x^3*(c+d*x^2))+1/2*b/(a*(b*c-a*d)*x^3*(a+b*x^2)*(c+d*x^2))+1/2*b^(7/2)*(5*b*c-9*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*(b*c-a*d)^3)+1/2*d^(7/2)*(9*b*c-5*a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*(b*c-a*d)^3)],
[x^4/((a+b*x^2)^2*(c+d*x^2)^3),x,6,1/4*(b*c+2*a*d)*x/(b*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*a*x/(b*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+3/8*(b*c+3*a*d)*x/((b*c-a*d)^3*(c+d*x^2))-3/2*(b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)*sqrt(b)/(b*c-a*d)^4+3/8*(b^2*c^2+6*a*b*c*d+a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/((b*c-a*d)^4*sqrt(c)*sqrt(d))],
[x^3/((a+b*x^2)^2*(c+d*x^2)^3),x,3,1/2*a*b/((b*c-a*d)^3*(a+b*x^2))+1/4*c/((b*c-a*d)^2*(c+d*x^2)^2)+1/2*(b*c+a*d)/((b*c-a*d)^3*(c+d*x^2))+1/2*b*(b*c+2*a*d)*log(a+b*x^2)/(b*c-a*d)^4-1/2*b*(b*c+2*a*d)*log(c+d*x^2)/(b*c-a*d)^4],
[x^2/((a+b*x^2)^2*(c+d*x^2)^3),x,6,-3/4*d*x/((b*c-a*d)^2*(c+d*x^2)^2)-1/2*x/((b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)-1/8*d*(11*b*c+a*d)*x/(c*(b*c-a*d)^3*(c+d*x^2))+1/2*b^(3/2)*(b*c+5*a*d)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^4*sqrt(a))-1/8*(15*b^2*c^2+10*a*b*c*d-a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/(c^(3/2)*(b*c-a*d)^4)],
[x/((a+b*x^2)^2*(c+d*x^2)^3),x,3,-1/2*b^2/((b*c-a*d)^3*(a+b*x^2))-1/4*d/((b*c-a*d)^2*(c+d*x^2)^2)-b*d/((b*c-a*d)^3*(c+d*x^2))-3/2*b^2*d*log(a+b*x^2)/(b*c-a*d)^4+3/2*b^2*d*log(c+d*x^2)/(b*c-a*d)^4],
[1/((a+b*x^2)^2*(c+d*x^2)^3),x,6,1/4*d*(2*b*c+a*d)*x/(a*c*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*b*x/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+1/8*d*(4*b*c-a*d)*(b*c+3*a*d)*x/(a*c^2*(b*c-a*d)^3*(c+d*x^2))+1/2*b^(5/2)*(b*c-7*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d)^4)+1/8*d^(3/2)*(35*b^2*c^2-14*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d)^4)],
[1/(x*(a+b*x^2)^2*(c+d*x^2)^3),x,3,1/2*b^3/(a*(b*c-a*d)^3*(a+b*x^2))+1/4*d^2/(c*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*d^2*(3*b*c-a*d)/(c^2*(b*c-a*d)^3*(c+d*x^2))+log(x)/(a^2*c^3)-1/2*b^3*(b*c-4*a*d)*log(a+b*x^2)/(a^2*(b*c-a*d)^4)-1/2*d^2*(6*b^2*c^2-4*a*b*c*d+a^2*d^2)*log(c+d*x^2)/(c^3*(b*c-a*d)^4)],
[1/(x^2*(a+b*x^2)^2*(c+d*x^2)^3),x,7,-3/8*(2*b*c-a*d)*(2*b^2*c^2-3*a*b*c*d+5*a^2*d^2)/(a^2*c^3*(b*c-a*d)^3*x)+1/4*d*(2*b*c+a*d)/(a*c*(b*c-a*d)^2*x*(c+d*x^2)^2)+1/2*b/(a*(b*c-a*d)*x*(a+b*x^2)*(c+d*x^2)^2)+1/8*d*(4*b^2*c^2+13*a*b*c*d-5*a^2*d^2)/(a*c^2*(b*c-a*d)^3*x*(c+d*x^2))-3/2*b^(7/2)*(b*c-3*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^4)-3/8*d^(5/2)*(21*b^2*c^2-18*a*b*c*d+5*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(7/2)*(b*c-a*d)^4)],
[1/(x^3*(a+b*x^2)^2*(c+d*x^2)^3),x,3,(-1/2)/(a^2*c^3*x^2)-1/2*b^4/(a^2*(b*c-a*d)^3*(a+b*x^2))-1/4*d^3/(c^2*(b*c-a*d)^2*(c+d*x^2)^2)-d^3*(2*b*c-a*d)/(c^3*(b*c-a*d)^3*(c+d*x^2))-(2*b*c+3*a*d)*log(x)/(a^3*c^4)+1/2*b^4*(2*b*c-5*a*d)*log(a+b*x^2)/(a^3*(b*c-a*d)^4)+1/2*d^3*(10*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*log(c+d*x^2)/(c^4*(b*c-a*d)^4)],
[1/(x^4*(a+b*x^2)^2*(c+d*x^2)^3),x,8,1/24*(-20*b^3*c^3+24*a*b^2*c^2*d-75*a^2*b*c*d^2+35*a^3*d^3)/(a^2*c^3*(b*c-a*d)^3*x^3)+1/8*(20*b^4*c^4-24*a*b^3*c^3*d-24*a^2*b^2*c^2*d^2+75*a^3*b*c*d^3-35*a^4*d^4)/(a^3*c^4*(b*c-a*d)^3*x)+1/4*d*(2*b*c+a*d)/(a*c*(b*c-a*d)^2*x^3*(c+d*x^2)^2)+1/2*b/(a*(b*c-a*d)*x^3*(a+b*x^2)*(c+d*x^2)^2)+1/8*d*(4*b^2*c^2+15*a*b*c*d-7*a^2*d^2)/(a*c^2*(b*c-a*d)^3*x^3*(c+d*x^2))+1/2*b^(9/2)*(5*b*c-11*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*(b*c-a*d)^4)+1/8*d^(7/2)*(99*b^2*c^2-110*a*b*c*d+35*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(9/2)*(b*c-a*d)^4)],

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

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

# q>0
[x^m*(a+b*x^2)^3*(A+B*x^2),x,2,a^3*A*x^(1+m)/(1+m)+a^2*(3*A*b+a*B)*x^(3+m)/(3+m)+3*a*b*(A*b+a*B)*x^(5+m)/(5+m)+b^2*(A*b+3*a*B)*x^(7+m)/(7+m)+b^3*B*x^(9+m)/(9+m)],
[x^m*(a+b*x^2)^2*(A+B*x^2),x,2,a^2*A*x^(1+m)/(1+m)+a*(2*A*b+a*B)*x^(3+m)/(3+m)+b*(A*b+2*a*B)*x^(5+m)/(5+m)+b^2*B*x^(7+m)/(7+m)],
[x^m*(a+b*x^2)*(A+B*x^2),x,2,a*A*x^(1+m)/(1+m)+(A*b+a*B)*x^(3+m)/(3+m)+b*B*x^(5+m)/(5+m)],
[x^m*(A+B*x^2)/(a+b*x^2),x,2,B*x^(1+m)/(b*(1+m))+(A*b-a*B)*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*b*(1+m))],
[x^m*(A+B*x^2)/(a+b*x^2)^2,x,2,1/2*(A*b-a*B)*x^(1+m)/(a*b*(a+b*x^2))+1/2*(a*B*(1+m)+A*(b-b*m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*b*(1+m))],
[x^m*(A+B*x^2)/(a+b*x^2)^3,x,2,1/4*(A*b-a*B)*x^(1+m)/(a*b*(a+b*x^2)^2)+1/4*(A*b*(3-m)+a*B*(1+m))*x^(1+m)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^3*b*(1+m))],
[x^m*(a+b*x^2)^2*(c+d*x^2)^3,x,2,a^2*c^3*x^(1+m)/(1+m)+a*c^2*(2*b*c+3*a*d)*x^(3+m)/(3+m)+c*(b^2*c^2+6*a*b*c*d+3*a^2*d^2)*x^(5+m)/(5+m)+d*(3*b^2*c^2+6*a*b*c*d+a^2*d^2)*x^(7+m)/(7+m)+b*d^2*(3*b*c+2*a*d)*x^(9+m)/(9+m)+b^2*d^3*x^(11+m)/(11+m)],
[x^m*(a+b*x^2)^2*(c+d*x^2)^2,x,2,a^2*c^2*x^(1+m)/(1+m)+2*a*c*(b*c+a*d)*x^(3+m)/(3+m)+(b^2*c^2+4*a*b*c*d+a^2*d^2)*x^(5+m)/(5+m)+2*b*d*(b*c+a*d)*x^(7+m)/(7+m)+b^2*d^2*x^(9+m)/(9+m)],
[x^m*(a+b*x^2)^2*(c+d*x^2),x,2,a^2*c*x^(1+m)/(1+m)+a*(2*b*c+a*d)*x^(3+m)/(3+m)+b*(b*c+2*a*d)*x^(5+m)/(5+m)+b^2*d*x^(7+m)/(7+m)],
[x^m*(a+b*x^2)^2/(c+d*x^2),x,3,-b*(b*c-2*a*d)*x^(1+m)/(d^2*(1+m))+b^2*x^(3+m)/(d*(3+m))+(b*c-a*d)^2*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c*d^2*(1+m))],
[x^m*(a+b*x^2)^2/(c+d*x^2)^2,x,3,b^2*x^(1+m)/(d^2*(1+m))+1/2*(b*c-a*d)^2*x^(1+m)/(c*d^2*(c+d*x^2))-1/2*(b*c-a*d)*(a*d*(1-m)+b*c*(3+m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c^2*d^2*(1+m))],
[x^m*(a+b*x^2)^2/(c+d*x^2)^3,x,3,1/4*(b*c-a*d)^2*x^(1+m)/(c*d^2*(c+d*x^2)^2)-1/8*(b*c-a*d)*(a*d*(3-m)+b*c*(5+m))*x^(1+m)/(c^2*d^2*(c+d*x^2))+1/8*(2*a*b*c*d*(1-m^2)+a^2*d^2*(3-4*m+m^2)+b^2*c^2*(3+4*m+m^2))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c^3*d^2*(1+m))],

# q<0
[x^m*(c+d*x^2)^3/(a+b*x^2),x,3,d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*x^(1+m)/(b^3*(1+m))+d^2*(3*b*c-a*d)*x^(3+m)/(b^2*(3+m))+d^3*x^(5+m)/(b*(5+m))+(b*c-a*d)^3*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*b^3*(1+m))],
[x^m*(c+d*x^2)^2/(a+b*x^2),x,3,d*(2*b*c-a*d)*x^(1+m)/(b^2*(1+m))+d^2*x^(3+m)/(b*(3+m))+(b*c-a*d)^2*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*b^2*(1+m))],
[x^m*(c+d*x^2)/(a+b*x^2),x,2,d*x^(1+m)/(b*(1+m))+(b*c-a*d)*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*b*(1+m))],
[x^m/((a+b*x^2)*(c+d*x^2)),x,3,b*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*(b*c-a*d)*(1+m))-d*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c*(b*c-a*d)*(1+m))],
[x^m/((a+b*x^2)^2*(c+d*x^2)),x,5,1/2*b*x^(1+m)/(a*(b*c-a*d)*(a+b*x^2))+1/2*b*(b*c*(1-m)-a*d*(3-m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*(b*c-a*d)^2*(1+m))+d^2*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c*(b*c-a*d)^2*(1+m))],
[x^m/((a+b*x^2)^3*(c+d*x^2)),x,6,1/4*b*x^(1+m)/(a*(b*c-a*d)*(a+b*x^2)^2)+1/8*b*(b*c*(3-m)-a*d*(7-m))*x^(1+m)/(a^2*(b*c-a*d)^2*(a+b*x^2))+1/8*b*(a^2*d^2*(15-8*m+m^2)-2*a*b*c*d*(5-6*m+m^2)+b^2*c^2*(3-4*m+m^2))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^3*(b*c-a*d)^3*(1+m))-d^3*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c*(b*c-a*d)^3*(1+m))],
[x^m*(c+d*x^2)^3/(a+b*x^2)^2,x,4,-1/2*d*(2*b^2*c^2*(1+m)-3*a*b*c*d*(3+m)+a^2*d^2*(5+m))*x^(1+m)/(a*b^3*(1+m))-1/2*d^2*(b*c*(3+m)-a*d*(5+m))*x^(3+m)/(a*b^2*(3+m))+1/2*(b*c-a*d)*x^(1+m)*(c+d*x^2)^2/(a*b*(a+b*x^2))+1/2*(b*c-a*d)^2*(a*d*(5+m)+b*(c-c*m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*b^3*(1+m))],
[x^m*(c+d*x^2)^2/(a+b*x^2)^2,x,3,d^2*x^(1+m)/(b^2*(1+m))+1/2*(b*c-a*d)^2*x^(1+m)/(a*b^2*(a+b*x^2))+1/2*(b*c-a*d)*(a*d*(3+m)+b*(c-c*m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*b^2*(1+m))],
[x^m*(c+d*x^2)/(a+b*x^2)^2,x,2,1/2*(b*c-a*d)*x^(1+m)/(a*b*(a+b*x^2))+1/2*(a*d*(1+m)+b*(c-c*m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*b*(1+m))],
[x^m/((a+b*x^2)^2*(c+d*x^2)),x,5,1/2*b*x^(1+m)/(a*(b*c-a*d)*(a+b*x^2))+1/2*b*(b*c*(1-m)-a*d*(3-m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*(b*c-a*d)^2*(1+m))+d^2*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c*(b*c-a*d)^2*(1+m))],
[x^m/((a+b*x^2)^2*(c+d*x^2)^2),x,6,1/2*d*(b*c+a*d)*x^(1+m)/(a*c*(b*c-a*d)^2*(c+d*x^2))+1/2*b*x^(1+m)/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))-1/2*b^2*(a*d*(5-m)-b*(c-c*m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*(b*c-a*d)^3*(1+m))-1/2*d^2*(a*d*(1-m)-b*c*(5-m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c^2*(b*c-a*d)^3*(1+m))],
[x^m/((a+b*x^2)^2*(c+d*x^2)^3),x,7,1/4*d*(2*b*c+a*d)*x^(1+m)/(a*c*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*b*x^(1+m)/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+1/8*d*(4*b^2*c^2-a^2*d^2*(3-m)+a*b*c*d*(11-m))*x^(1+m)/(a*c^2*(b*c-a*d)^3*(c+d*x^2))-1/2*b^3*(a*d*(7-m)-b*(c-c*m))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*(b*c-a*d)^4*(1+m))+1/8*d^2*(b^2*c^2*(35-12*m+m^2)-2*a*b*c*d*(7-8*m+m^2)+a^2*d^2*(3-4*m+m^2))*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-d*x^2/c)/(c^3*(b*c-a*d)^4*(1+m))],

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

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

# p>0
[x^(7/2)*(a+b*x^2)*(A+B*x^2),x,2,2/9*a*A*x^(9/2)+2/13*(A*b+a*B)*x^(13/2)+2/17*b*B*x^(17/2)],
[x^(5/2)*(a+b*x^2)*(A+B*x^2),x,2,2/7*a*A*x^(7/2)+2/11*(A*b+a*B)*x^(11/2)+2/15*b*B*x^(15/2)],
[x^(3/2)*(a+b*x^2)*(A+B*x^2),x,2,2/5*a*A*x^(5/2)+2/9*(A*b+a*B)*x^(9/2)+2/13*b*B*x^(13/2)],
[(a+b*x^2)*(A+B*x^2)*sqrt(x),x,2,2/3*a*A*x^(3/2)+2/7*(A*b+a*B)*x^(7/2)+2/11*b*B*x^(11/2)],
[(a+b*x^2)*(A+B*x^2)/sqrt(x),x,2,2/5*(A*b+a*B)*x^(5/2)+2/9*b*B*x^(9/2)+2*a*A*sqrt(x)],
[(a+b*x^2)*(A+B*x^2)/x^(3/2),x,2,2/3*(A*b+a*B)*x^(3/2)+2/7*b*B*x^(7/2)-2*a*A/sqrt(x)],
[(a+b*x^2)*(A+B*x^2)/x^(5/2),x,2,-2/3*a*A/x^(3/2)+2/5*b*B*x^(5/2)+2*(A*b+a*B)*sqrt(x)],
[(a+b*x^2)*(A+B*x^2)/x^(7/2),x,2,-2/5*a*A/x^(5/2)+2/3*b*B*x^(3/2)-2*(A*b+a*B)/sqrt(x)],
[x^(7/2)*(a+b*x^2)^2*(A+B*x^2),x,2,2/9*a^2*A*x^(9/2)+2/13*a*(2*A*b+a*B)*x^(13/2)+2/17*b*(A*b+2*a*B)*x^(17/2)+2/21*b^2*B*x^(21/2)],
[x^(5/2)*(a+b*x^2)^2*(A+B*x^2),x,2,2/7*a^2*A*x^(7/2)+2/11*a*(2*A*b+a*B)*x^(11/2)+2/15*b*(A*b+2*a*B)*x^(15/2)+2/19*b^2*B*x^(19/2)],
[x^(3/2)*(a+b*x^2)^2*(A+B*x^2),x,2,2/5*a^2*A*x^(5/2)+2/9*a*(2*A*b+a*B)*x^(9/2)+2/13*b*(A*b+2*a*B)*x^(13/2)+2/17*b^2*B*x^(17/2)],
[(a+b*x^2)^2*(A+B*x^2)*sqrt(x),x,2,2/3*a^2*A*x^(3/2)+2/7*a*(2*A*b+a*B)*x^(7/2)+2/11*b*(A*b+2*a*B)*x^(11/2)+2/15*b^2*B*x^(15/2)],
[(a+b*x^2)^2*(A+B*x^2)/sqrt(x),x,2,2/5*a*(2*A*b+a*B)*x^(5/2)+2/9*b*(A*b+2*a*B)*x^(9/2)+2/13*b^2*B*x^(13/2)+2*a^2*A*sqrt(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^(3/2),x,2,2/3*a*(2*A*b+a*B)*x^(3/2)+2/7*b*(A*b+2*a*B)*x^(7/2)+2/11*b^2*B*x^(11/2)-2*a^2*A/sqrt(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^(5/2),x,2,-2/3*a^2*A/x^(3/2)+2/5*b*(A*b+2*a*B)*x^(5/2)+2/9*b^2*B*x^(9/2)+2*a*(2*A*b+a*B)*sqrt(x)],
[(a+b*x^2)^2*(A+B*x^2)/x^(7/2),x,2,-2/5*a^2*A/x^(5/2)+2/3*b*(A*b+2*a*B)*x^(3/2)+2/7*b^2*B*x^(7/2)-2*a*(2*A*b+a*B)/sqrt(x)],
[x^(7/2)*(a+b*x^2)^3*(A+B*x^2),x,2,2/9*a^3*A*x^(9/2)+2/13*a^2*(3*A*b+a*B)*x^(13/2)+6/17*a*b*(A*b+a*B)*x^(17/2)+2/21*b^2*(A*b+3*a*B)*x^(21/2)+2/25*b^3*B*x^(25/2)],
[x^(5/2)*(a+b*x^2)^3*(A+B*x^2),x,2,2/7*a^3*A*x^(7/2)+2/11*a^2*(3*A*b+a*B)*x^(11/2)+2/5*a*b*(A*b+a*B)*x^(15/2)+2/19*b^2*(A*b+3*a*B)*x^(19/2)+2/23*b^3*B*x^(23/2)],
[x^(3/2)*(a+b*x^2)^3*(A+B*x^2),x,2,2/5*a^3*A*x^(5/2)+2/9*a^2*(3*A*b+a*B)*x^(9/2)+6/13*a*b*(A*b+a*B)*x^(13/2)+2/17*b^2*(A*b+3*a*B)*x^(17/2)+2/21*b^3*B*x^(21/2)],
[(a+b*x^2)^3*(A+B*x^2)*sqrt(x),x,2,2/3*a^3*A*x^(3/2)+2/7*a^2*(3*A*b+a*B)*x^(7/2)+6/11*a*b*(A*b+a*B)*x^(11/2)+2/15*b^2*(A*b+3*a*B)*x^(15/2)+2/19*b^3*B*x^(19/2)],
[(a+b*x^2)^3*(A+B*x^2)/sqrt(x),x,2,2/5*a^2*(3*A*b+a*B)*x^(5/2)+2/3*a*b*(A*b+a*B)*x^(9/2)+2/13*b^2*(A*b+3*a*B)*x^(13/2)+2/17*b^3*B*x^(17/2)+2*a^3*A*sqrt(x)],
[(a+b*x^2)^3*(A+B*x^2)/x^(3/2),x,2,2/3*a^2*(3*A*b+a*B)*x^(3/2)+6/7*a*b*(A*b+a*B)*x^(7/2)+2/11*b^2*(A*b+3*a*B)*x^(11/2)+2/15*b^3*B*x^(15/2)-2*a^3*A/sqrt(x)],
[(a+b*x^2)^3*(A+B*x^2)/x^(5/2),x,2,-2/3*a^3*A/x^(3/2)+6/5*a*b*(A*b+a*B)*x^(5/2)+2/9*b^2*(A*b+3*a*B)*x^(9/2)+2/13*b^3*B*x^(13/2)+2*a^2*(3*A*b+a*B)*sqrt(x)],
[(a+b*x^2)^3*(A+B*x^2)/x^(7/2),x,2,-2/5*a^3*A/x^(5/2)+2*a*b*(A*b+a*B)*x^(3/2)+2/7*b^2*(A*b+3*a*B)*x^(7/2)+2/11*b^3*B*x^(11/2)-2*a^2*(3*A*b+a*B)/sqrt(x)],

# p<0
[x^(7/2)*(A+B*x^2)/(a+b*x^2),x,13,2/5*(A*b-a*B)*x^(5/2)/b^2+2/9*B*x^(9/2)/b-a^(5/4)*(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(13/4)*sqrt(2))+a^(5/4)*(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(13/4)*sqrt(2))-1/2*a^(5/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))+1/2*a^(5/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))-2*a*(A*b-a*B)*sqrt(x)/b^3],
[x^(5/2)*(A+B*x^2)/(a+b*x^2),x,12,2/3*(A*b-a*B)*x^(3/2)/b^2+2/7*B*x^(7/2)/b+a^(3/4)*(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(11/4)*sqrt(2))-a^(3/4)*(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(11/4)*sqrt(2))-1/2*a^(3/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*sqrt(2))+1/2*a^(3/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*sqrt(2))],
[x^(3/2)*(A+B*x^2)/(a+b*x^2),x,12,2/5*B*x^(5/2)/b+a^(1/4)*(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(9/4)*sqrt(2))-a^(1/4)*(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(9/4)*sqrt(2))+1/2*a^(1/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*sqrt(2))-1/2*a^(1/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*sqrt(2))+2*(A*b-a*B)*sqrt(x)/b^2],
[(A+B*x^2)*sqrt(x)/(a+b*x^2),x,11,2/3*B*x^(3/2)/b-(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(7/4)*sqrt(2))+(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(7/4)*sqrt(2))+1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(7/4)*sqrt(2))-1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(7/4)*sqrt(2))],
[(A+B*x^2)/((a+b*x^2)*sqrt(x)),x,11,-(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(5/4)*sqrt(2))+(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(5/4)*sqrt(2))-1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(5/4)*sqrt(2))+1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(5/4)*sqrt(2))+2*B*sqrt(x)/b],
[(A+B*x^2)/(x^(3/2)*(a+b*x^2)),x,11,(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(3/4)*sqrt(2))-(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(3/4)*sqrt(2))-1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(3/4)*sqrt(2))+1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(3/4)*sqrt(2))-2*A/(a*sqrt(x))],
[(A+B*x^2)/(x^(5/2)*(a+b*x^2)),x,11,-2/3*A/(a*x^(3/2))+(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(1/4)*sqrt(2))-(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(1/4)*sqrt(2))+1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(1/4)*sqrt(2))-1/2*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(1/4)*sqrt(2))],
[(A+B*x^2)/(x^(7/2)*(a+b*x^2)),x,12,-2/5*A/(a*x^(5/2))-b^(1/4)*(A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*sqrt(2))+b^(1/4)*(A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*sqrt(2))+1/2*b^(1/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*sqrt(2))-1/2*b^(1/4)*(A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*sqrt(2))+2*(A*b-a*B)/(a^2*sqrt(x))],
[x^(7/2)*(A+B*x^2)/(a+b*x^2)^2,x,13,-1/10*(5*A*b-9*a*B)*x^(5/2)/(a*b^2)+1/2*(A*b-a*B)*x^(9/2)/(a*b*(a+b*x^2))+1/4*a^(1/4)*(5*A*b-9*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(13/4)*sqrt(2))-1/4*a^(1/4)*(5*A*b-9*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(13/4)*sqrt(2))+1/8*a^(1/4)*(5*A*b-9*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))-1/8*a^(1/4)*(5*A*b-9*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))+1/2*(5*A*b-9*a*B)*sqrt(x)/b^3],
[x^(5/2)*(A+B*x^2)/(a+b*x^2)^2,x,12,-1/6*(3*A*b-7*a*B)*x^(3/2)/(a*b^2)+1/2*(A*b-a*B)*x^(7/2)/(a*b*(a+b*x^2))-1/4*(3*A*b-7*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(11/4)*sqrt(2))+1/4*(3*A*b-7*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(11/4)*sqrt(2))+1/8*(3*A*b-7*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(11/4)*sqrt(2))-1/8*(3*A*b-7*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(11/4)*sqrt(2))],
[x^(3/2)*(A+B*x^2)/(a+b*x^2)^2,x,12,1/2*(A*b-a*B)*x^(5/2)/(a*b*(a+b*x^2))-1/4*(A*b-5*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(9/4)*sqrt(2))+1/4*(A*b-5*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(9/4)*sqrt(2))-1/8*(A*b-5*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(9/4)*sqrt(2))+1/8*(A*b-5*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(9/4)*sqrt(2))-1/2*(A*b-5*a*B)*sqrt(x)/(a*b^2)],
[(A+B*x^2)*sqrt(x)/(a+b*x^2)^2,x,11,1/2*(A*b-a*B)*x^(3/2)/(a*b*(a+b*x^2))-1/4*(A*b+3*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(7/4)*sqrt(2))+1/4*(A*b+3*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(7/4)*sqrt(2))+1/8*(A*b+3*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(7/4)*sqrt(2))-1/8*(A*b+3*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(7/4)*sqrt(2))],
[(A+B*x^2)/((a+b*x^2)^2*sqrt(x)),x,11,-1/4*(3*A*b+a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(5/4)*sqrt(2))+1/4*(3*A*b+a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(5/4)*sqrt(2))-1/8*(3*A*b+a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(5/4)*sqrt(2))+1/8*(3*A*b+a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(5/4)*sqrt(2))+1/2*(A*b-a*B)*sqrt(x)/(a*b*(a+b*x^2))],
[(A+B*x^2)/(x^(3/2)*(a+b*x^2)^2),x,12,1/4*(5*A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(3/4)*sqrt(2))-1/4*(5*A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(3/4)*sqrt(2))-1/8*(5*A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(3/4)*sqrt(2))+1/8*(5*A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(3/4)*sqrt(2))+1/2*(-5*A*b+a*B)/(a^2*b*sqrt(x))+1/2*(A*b-a*B)/(a*b*(a+b*x^2)*sqrt(x))],
[(A+B*x^2)/(x^(5/2)*(a+b*x^2)^2),x,12,1/6*(-7*A*b+3*a*B)/(a^2*b*x^(3/2))+1/2*(A*b-a*B)/(a*b*x^(3/2)*(a+b*x^2))+1/4*(7*A*b-3*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(1/4)*sqrt(2))-1/4*(7*A*b-3*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(1/4)*sqrt(2))+1/8*(7*A*b-3*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(1/4)*sqrt(2))-1/8*(7*A*b-3*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(1/4)*sqrt(2))],
[(A+B*x^2)/(x^(7/2)*(a+b*x^2)^2),x,13,1/10*(-9*A*b+5*a*B)/(a^2*b*x^(5/2))+1/2*(A*b-a*B)/(a*b*x^(5/2)*(a+b*x^2))-1/4*b^(1/4)*(9*A*b-5*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*sqrt(2))+1/4*b^(1/4)*(9*A*b-5*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*sqrt(2))+1/8*b^(1/4)*(9*A*b-5*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*sqrt(2))-1/8*b^(1/4)*(9*A*b-5*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*sqrt(2))+1/2*(9*A*b-5*a*B)/(a^3*sqrt(x))],
[x^(7/2)*(A+B*x^2)/(a+b*x^2)^3,x,13,1/4*(A*b-a*B)*x^(9/2)/(a*b*(a+b*x^2)^2)+1/16*(A*b-9*a*B)*x^(5/2)/(a*b^2*(a+b*x^2))-5/32*(A*b-9*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(13/4)*sqrt(2))+5/32*(A*b-9*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(13/4)*sqrt(2))-5/64*(A*b-9*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(13/4)*sqrt(2))+5/64*(A*b-9*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(13/4)*sqrt(2))-5/16*(A*b-9*a*B)*sqrt(x)/(a*b^3)],
[x^(5/2)*(A+B*x^2)/(a+b*x^2)^3,x,12,1/4*(A*b-a*B)*x^(7/2)/(a*b*(a+b*x^2)^2)-1/16*(A*b+7*a*B)*x^(3/2)/(a*b^2*(a+b*x^2))-3/32*(A*b+7*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(11/4)*sqrt(2))+3/32*(A*b+7*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(11/4)*sqrt(2))+3/64*(A*b+7*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(11/4)*sqrt(2))-3/64*(A*b+7*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(11/4)*sqrt(2))],
[x^(3/2)*(A+B*x^2)/(a+b*x^2)^3,x,12,1/4*(A*b-a*B)*x^(5/2)/(a*b*(a+b*x^2)^2)-1/32*(3*A*b+5*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(9/4)*sqrt(2))+1/32*(3*A*b+5*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(9/4)*sqrt(2))-1/64*(3*A*b+5*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(9/4)*sqrt(2))+1/64*(3*A*b+5*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(9/4)*sqrt(2))-1/16*(3*A*b+5*a*B)*sqrt(x)/(a*b^2*(a+b*x^2))],
[(A+B*x^2)*sqrt(x)/(a+b*x^2)^3,x,12,1/4*(A*b-a*B)*x^(3/2)/(a*b*(a+b*x^2)^2)+1/16*(5*A*b+3*a*B)*x^(3/2)/(a^2*b*(a+b*x^2))-1/32*(5*A*b+3*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(7/4)*sqrt(2))+1/32*(5*A*b+3*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(7/4)*sqrt(2))+1/64*(5*A*b+3*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(7/4)*sqrt(2))-1/64*(5*A*b+3*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(7/4)*sqrt(2))],
[(A+B*x^2)/((a+b*x^2)^3*sqrt(x)),x,12,-3/32*(7*A*b+a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(5/4)*sqrt(2))+3/32*(7*A*b+a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(5/4)*sqrt(2))-3/64*(7*A*b+a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(5/4)*sqrt(2))+3/64*(7*A*b+a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(5/4)*sqrt(2))+1/4*(A*b-a*B)*sqrt(x)/(a*b*(a+b*x^2)^2)+1/16*(7*A*b+a*B)*sqrt(x)/(a^2*b*(a+b*x^2))],
[(A+B*x^2)/(x^(3/2)*(a+b*x^2)^3),x,13,5/32*(9*A*b-a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*b^(3/4)*sqrt(2))-5/32*(9*A*b-a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*b^(3/4)*sqrt(2))-5/64*(9*A*b-a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*b^(3/4)*sqrt(2))+5/64*(9*A*b-a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*b^(3/4)*sqrt(2))-5/16*(9*A*b-a*B)/(a^3*b*sqrt(x))+1/4*(A*b-a*B)/(a*b*(a+b*x^2)^2*sqrt(x))+1/16*(9*A*b-a*B)/(a^2*b*(a+b*x^2)*sqrt(x))],
[(A+B*x^2)/(x^(5/2)*(a+b*x^2)^3),x,13,-7/48*(11*A*b-3*a*B)/(a^3*b*x^(3/2))+1/4*(A*b-a*B)/(a*b*x^(3/2)*(a+b*x^2)^2)+1/16*(11*A*b-3*a*B)/(a^2*b*x^(3/2)*(a+b*x^2))+7/32*(11*A*b-3*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(15/4)*b^(1/4)*sqrt(2))-7/32*(11*A*b-3*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(15/4)*b^(1/4)*sqrt(2))+7/64*(11*A*b-3*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(15/4)*b^(1/4)*sqrt(2))-7/64*(11*A*b-3*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(15/4)*b^(1/4)*sqrt(2))],
[(A+B*x^2)/(x^(7/2)*(a+b*x^2)^3),x,14,-9/80*(13*A*b-5*a*B)/(a^3*b*x^(5/2))+1/4*(A*b-a*B)/(a*b*x^(5/2)*(a+b*x^2)^2)+1/16*(13*A*b-5*a*B)/(a^2*b*x^(5/2)*(a+b*x^2))-9/32*b^(1/4)*(13*A*b-5*a*B)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(17/4)*sqrt(2))+9/32*b^(1/4)*(13*A*b-5*a*B)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(17/4)*sqrt(2))+9/64*b^(1/4)*(13*A*b-5*a*B)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(17/4)*sqrt(2))-9/64*b^(1/4)*(13*A*b-5*a*B)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(17/4)*sqrt(2))+9/16*(13*A*b-5*a*B)/(a^4*sqrt(x))],

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

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

# p<0
[x^(7/2)*(a+b*x^2)^2/(c+d*x^2),x,14,2/5*(b*c-a*d)^2*x^(5/2)/d^3-2/9*b*(b*c-2*a*d)*x^(9/2)/d^2+2/13*b^2*x^(13/2)/d-c^(5/4)*(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(17/4)*sqrt(2))+c^(5/4)*(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(17/4)*sqrt(2))-1/2*c^(5/4)*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(17/4)*sqrt(2))+1/2*c^(5/4)*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(17/4)*sqrt(2))-2*c*(b*c-a*d)^2*sqrt(x)/d^4],
[x^(5/2)*(a+b*x^2)^2/(c+d*x^2),x,13,2/3*(b*c-a*d)^2*x^(3/2)/d^3-2/7*b*(b*c-2*a*d)*x^(7/2)/d^2+2/11*b^2*x^(11/2)/d+c^(3/4)*(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(15/4)*sqrt(2))-c^(3/4)*(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(15/4)*sqrt(2))-1/2*c^(3/4)*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(15/4)*sqrt(2))+1/2*c^(3/4)*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(15/4)*sqrt(2))],
[x^(3/2)*(a+b*x^2)^2/(c+d*x^2),x,13,-2/5*b*(b*c-2*a*d)*x^(5/2)/d^2+2/9*b^2*x^(9/2)/d+c^(1/4)*(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(13/4)*sqrt(2))-c^(1/4)*(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(13/4)*sqrt(2))+1/2*c^(1/4)*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(13/4)*sqrt(2))-1/2*c^(1/4)*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(13/4)*sqrt(2))+2*(b*c-a*d)^2*sqrt(x)/d^3],
[(a+b*x^2)^2*sqrt(x)/(c+d*x^2),x,12,-2/3*b*(b*c-2*a*d)*x^(3/2)/d^2+2/7*b^2*x^(7/2)/d-(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*d^(11/4)*sqrt(2))+(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*d^(11/4)*sqrt(2))+1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*d^(11/4)*sqrt(2))-1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*d^(11/4)*sqrt(2))],
[(a+b*x^2)^2/((c+d*x^2)*sqrt(x)),x,12,2/5*b^2*x^(5/2)/d-(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(9/4)*sqrt(2))+(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(9/4)*sqrt(2))-1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(9/4)*sqrt(2))+1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(9/4)*sqrt(2))-2*b*(b*c-2*a*d)*sqrt(x)/d^2],
[(a+b*x^2)^2/(x^(3/2)*(c+d*x^2)),x,12,2/3*b^2*x^(3/2)/d+(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(7/4)*sqrt(2))-(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(7/4)*sqrt(2))-1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(7/4)*sqrt(2))+1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(7/4)*sqrt(2))-2*a^2/(c*sqrt(x))],
[(a+b*x^2)^2/(x^(5/2)*(c+d*x^2)),x,12,-2/3*a^2/(c*x^(3/2))+(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(5/4)*sqrt(2))-(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(5/4)*sqrt(2))+1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(5/4)*sqrt(2))-1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(5/4)*sqrt(2))+2*b^2*sqrt(x)/d],
[(a+b*x^2)^2/(x^(7/2)*(c+d*x^2)),x,12,-2/5*a^2/(c*x^(5/2))-(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*d^(3/4)*sqrt(2))+(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*d^(3/4)*sqrt(2))+1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*d^(3/4)*sqrt(2))-1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*d^(3/4)*sqrt(2))-2*a*(2*b*c-a*d)/(c^2*sqrt(x))],
[(a+b*x^2)^2/(x^(9/2)*(c+d*x^2)),x,12,-2/7*a^2/(c*x^(7/2))-2/3*a*(2*b*c-a*d)/(c^2*x^(3/2))-(b*c-a*d)^2*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*d^(1/4)*sqrt(2))+(b*c-a*d)^2*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*d^(1/4)*sqrt(2))-1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*d^(1/4)*sqrt(2))+1/2*(b*c-a*d)^2*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*d^(1/4)*sqrt(2))],
[(c+d*x^2)^2/(x^(11/2)*(a+b*x^2)),x,13,-2/9*c^2/(a*x^(9/2))+2/5*c*(b*c-2*a*d)/(a^2*x^(5/2))+b^(1/4)*(b*c-a*d)^2*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*sqrt(2))-b^(1/4)*(b*c-a*d)^2*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*sqrt(2))-1/2*b^(1/4)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*sqrt(2))+1/2*b^(1/4)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*sqrt(2))-2*(b*c-a*d)^2/(a^3*sqrt(x))],
[x^(7/2)*(a+b*x^2)^2/(c+d*x^2)^2,x,14,-1/10*(13*b*c-5*a*d)*(b*c-a*d)*x^(5/2)/(c*d^3)+2/9*b^2*x^(9/2)/d^2+1/2*(b*c-a*d)^2*x^(9/2)/(c*d^2*(c+d*x^2))+1/4*c^(1/4)*(13*b*c-5*a*d)*(b*c-a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(17/4)*sqrt(2))-1/4*c^(1/4)*(13*b*c-5*a*d)*(b*c-a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(17/4)*sqrt(2))+1/8*c^(1/4)*(13*b*c-5*a*d)*(b*c-a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(17/4)*sqrt(2))-1/8*c^(1/4)*(13*b*c-5*a*d)*(b*c-a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(17/4)*sqrt(2))+1/2*(13*b*c-5*a*d)*(b*c-a*d)*sqrt(x)/d^4],
[x^(5/2)*(a+b*x^2)^2/(c+d*x^2)^2,x,13,-1/6*(11*b*c-3*a*d)*(b*c-a*d)*x^(3/2)/(c*d^3)+2/7*b^2*x^(7/2)/d^2+1/2*(b*c-a*d)^2*x^(7/2)/(c*d^2*(c+d*x^2))-1/4*(11*b*c-3*a*d)*(b*c-a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*d^(15/4)*sqrt(2))+1/4*(11*b*c-3*a*d)*(b*c-a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*d^(15/4)*sqrt(2))+1/8*(11*b*c-3*a*d)*(b*c-a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*d^(15/4)*sqrt(2))-1/8*(11*b*c-3*a*d)*(b*c-a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*d^(15/4)*sqrt(2))],
[x^(3/2)*(a+b*x^2)^2/(c+d*x^2)^2,x,13,2/5*b^2*x^(5/2)/d^2+1/2*(b*c-a*d)^2*x^(5/2)/(c*d^2*(c+d*x^2))-1/4*(b*c-a*d)*(9*b*c-a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(13/4)*sqrt(2))+1/4*(b*c-a*d)*(9*b*c-a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(13/4)*sqrt(2))-1/8*(b*c-a*d)*(9*b*c-a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(13/4)*sqrt(2))+1/8*(b*c-a*d)*(9*b*c-a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(13/4)*sqrt(2))-1/2*(b*c-a*d)*(9*b*c-a*d)*sqrt(x)/(c*d^3)],
[(a+b*x^2)^2*sqrt(x)/(c+d*x^2)^2,x,12,2/3*b^2*x^(3/2)/d^2+1/2*(b*c-a*d)^2*x^(3/2)/(c*d^2*(c+d*x^2))+1/4*(b*c-a*d)*(7*b*c+a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(11/4)*sqrt(2))-1/4*(b*c-a*d)*(7*b*c+a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(11/4)*sqrt(2))-1/8*(b*c-a*d)*(7*b*c+a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(11/4)*sqrt(2))+1/8*(b*c-a*d)*(7*b*c+a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(11/4)*sqrt(2))],
[(a+b*x^2)^2/((c+d*x^2)^2*sqrt(x)),x,12,1/4*(b*c-a*d)*(5*b*c+3*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(9/4)*sqrt(2))-1/4*(b*c-a*d)*(5*b*c+3*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(9/4)*sqrt(2))+1/8*(b*c-a*d)*(5*b*c+3*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(9/4)*sqrt(2))-1/8*(b*c-a*d)*(5*b*c+3*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(9/4)*sqrt(2))+2*b^2*sqrt(x)/d^2+1/2*(b*c-a*d)^2*sqrt(x)/(c*d^2*(c+d*x^2))],
[(a+b*x^2)^2/(x^(3/2)*(c+d*x^2)^2),x,12,-1/2*(b^2*c^2-2*a*b*c*d+5*a^2*d^2)*x^(3/2)/(c^2*d*(c+d*x^2))-1/4*(b*c-a*d)*(3*b*c+5*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*d^(7/4)*sqrt(2))+1/4*(b*c-a*d)*(3*b*c+5*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*d^(7/4)*sqrt(2))+1/8*(b*c-a*d)*(3*b*c+5*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*d^(7/4)*sqrt(2))-1/8*(b*c-a*d)*(3*b*c+5*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*d^(7/4)*sqrt(2))-2*a^2/(c*(c+d*x^2)*sqrt(x))],
[(a+b*x^2)^2/(x^(5/2)*(c+d*x^2)^2),x,12,-2/3*a^2/(c*x^(3/2)*(c+d*x^2))-1/4*(b*c-a*d)*(b*c+7*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*d^(5/4)*sqrt(2))+1/4*(b*c-a*d)*(b*c+7*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*d^(5/4)*sqrt(2))-1/8*(b*c-a*d)*(b*c+7*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*d^(5/4)*sqrt(2))+1/8*(b*c-a*d)*(b*c+7*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*d^(5/4)*sqrt(2))-1/6*(3*b^2*c^2-6*a*b*c*d+7*a^2*d^2)*sqrt(x)/(c^2*d*(c+d*x^2))],
[(a+b*x^2)^2/(x^(7/2)*(c+d*x^2)^2),x,13,-2/5*a^2/(c*x^(5/2)*(c+d*x^2))-1/4*(b*c-9*a*d)*(b*c-a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*d^(3/4)*sqrt(2))+1/4*(b*c-9*a*d)*(b*c-a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*d^(3/4)*sqrt(2))+1/8*(b*c-9*a*d)*(b*c-a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*d^(3/4)*sqrt(2))-1/8*(b*c-9*a*d)*(b*c-a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*d^(3/4)*sqrt(2))+1/2*(b*c-9*a*d)*(b*c-a*d)/(c^3*d*sqrt(x))+1/10*(-5*b^2*c^2+10*a*b*c*d-9*a^2*d^2)/(c^2*d*(c+d*x^2)*sqrt(x))],
[x^(7/2)*(a+b*x^2)^2/(c+d*x^2)^3,x,14,1/80*(117*b^2*c^2-90*a*b*c*d+5*a^2*d^2)*x^(5/2)/(c^2*d^3)+1/4*(b*c-a*d)^2*x^(9/2)/(c*d^2*(c+d*x^2)^2)-1/16*(b*c-a*d)*(17*b*c-a*d)*x^(9/2)/(c^2*d^2*(c+d*x^2))-1/32*(117*b^2*c^2-90*a*b*c*d+5*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(17/4)*sqrt(2))+1/32*(117*b^2*c^2-90*a*b*c*d+5*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(17/4)*sqrt(2))-1/64*(117*b^2*c^2-90*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(17/4)*sqrt(2))+1/64*(117*b^2*c^2-90*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(17/4)*sqrt(2))-1/16*(117*b^2*c^2-90*a*b*c*d+5*a^2*d^2)*sqrt(x)/(c*d^4)],
[x^(5/2)*(a+b*x^2)^2/(c+d*x^2)^3,x,13,-1/48*(42*a*b-77*b^2*c/d+3*a^2*d/c)*x^(3/2)/(c*d^2)+1/4*(b*c-a*d)^2*x^(7/2)/(c*d^2*(c+d*x^2)^2)-1/16*(b*c-a*d)*(15*b*c+a*d)*x^(7/2)/(c^2*d^2*(c+d*x^2))+1/32*(77*b^2*c^2-42*a*b*c*d-3*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(15/4)*sqrt(2))-1/32*(77*b^2*c^2-42*a*b*c*d-3*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(15/4)*sqrt(2))-1/64*(77*b^2*c^2-42*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(15/4)*sqrt(2))+1/64*(77*b^2*c^2-42*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(15/4)*sqrt(2))],
[x^(3/2)*(a+b*x^2)^2/(c+d*x^2)^3,x,13,1/4*(b*c-a*d)^2*x^(5/2)/(c*d^2*(c+d*x^2)^2)-1/16*(b*c-a*d)*(13*b*c+3*a*d)*x^(5/2)/(c^2*d^2*(c+d*x^2))+1/32*(45*b^2*c^2-10*a*b*c*d-3*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(13/4)*sqrt(2))-1/32*(45*b^2*c^2-10*a*b*c*d-3*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(13/4)*sqrt(2))+1/64*(45*b^2*c^2-10*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(13/4)*sqrt(2))-1/64*(45*b^2*c^2-10*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(13/4)*sqrt(2))-1/16*(10*a*b-45*b^2*c/d+3*a^2*d/c)*sqrt(x)/(c*d^2)],
[(a+b*x^2)^2*sqrt(x)/(c+d*x^2)^3,x,12,1/4*(b*c-a*d)^2*x^(3/2)/(c*d^2*(c+d*x^2)^2)-1/16*(b*c-a*d)*(11*b*c+5*a*d)*x^(3/2)/(c^2*d^2*(c+d*x^2))-1/32*(21*b^2*c^2+6*a*b*c*d+5*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*d^(11/4)*sqrt(2))+1/32*(21*b^2*c^2+6*a*b*c*d+5*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*d^(11/4)*sqrt(2))+1/64*(21*b^2*c^2+6*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*d^(11/4)*sqrt(2))-1/64*(21*b^2*c^2+6*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*d^(11/4)*sqrt(2))],
[(a+b*x^2)^2/((c+d*x^2)^3*sqrt(x)),x,12,-1/32*(5*b^2*c^2+6*a*b*c*d+21*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*d^(9/4)*sqrt(2))+1/32*(5*b^2*c^2+6*a*b*c*d+21*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*d^(9/4)*sqrt(2))-1/64*(5*b^2*c^2+6*a*b*c*d+21*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*d^(9/4)*sqrt(2))+1/64*(5*b^2*c^2+6*a*b*c*d+21*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*d^(9/4)*sqrt(2))+1/4*(b*c-a*d)^2*sqrt(x)/(c*d^2*(c+d*x^2)^2)-1/16*(b*c-a*d)*(9*b*c+7*a*d)*sqrt(x)/(c^2*d^2*(c+d*x^2))],
[(a+b*x^2)^2/(x^(3/2)*(c+d*x^2)^3),x,13,-1/4*(b^2*c^2-2*a*b*c*d+9*a^2*d^2)*x^(3/2)/(c^2*d*(c+d*x^2)^2)+1/16*(3*b^2*c^2+5*a*d*(2*b*c-9*a*d))*x^(3/2)/(c^3*d*(c+d*x^2))-1/32*(3*b^2*c^2+5*a*d*(2*b*c-9*a*d))*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*d^(7/4)*sqrt(2))+1/32*(3*b^2*c^2+5*a*d*(2*b*c-9*a*d))*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*d^(7/4)*sqrt(2))+1/64*(3*b^2*c^2+5*a*d*(2*b*c-9*a*d))*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*d^(7/4)*sqrt(2))-1/64*(3*b^2*c^2+5*a*d*(2*b*c-9*a*d))*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*d^(7/4)*sqrt(2))-2*a^2/(c*(c+d*x^2)^2*sqrt(x))],
[(a+b*x^2)^2/(x^(5/2)*(c+d*x^2)^3),x,13,-2/3*a^2/(c*x^(3/2)*(c+d*x^2)^2)-1/32*(3*b^2*c^2+7*a*d*(6*b*c-11*a*d))*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(15/4)*d^(5/4)*sqrt(2))+1/32*(3*b^2*c^2+7*a*d*(6*b*c-11*a*d))*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(15/4)*d^(5/4)*sqrt(2))-1/64*(3*b^2*c^2+7*a*d*(6*b*c-11*a*d))*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(15/4)*d^(5/4)*sqrt(2))+1/64*(3*b^2*c^2+7*a*d*(6*b*c-11*a*d))*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(15/4)*d^(5/4)*sqrt(2))-1/12*(3*b^2*c^2-6*a*b*c*d+11*a^2*d^2)*sqrt(x)/(c^2*d*(c+d*x^2)^2)+1/48*(3*b^2*c^2+7*a*d*(6*b*c-11*a*d))*sqrt(x)/(c^3*d*(c+d*x^2))],
[(a+b*x^2)^2/(x^(7/2)*(c+d*x^2)^3),x,14,-2/5*a^2/(c*x^(5/2)*(c+d*x^2)^2)-1/32*(5*b^2*c^2-9*a*d*(10*b*c-13*a*d))*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(17/4)*d^(3/4)*sqrt(2))+1/32*(5*b^2*c^2-9*a*d*(10*b*c-13*a*d))*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(17/4)*d^(3/4)*sqrt(2))+1/64*(5*b^2*c^2-9*a*d*(10*b*c-13*a*d))*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(17/4)*d^(3/4)*sqrt(2))-1/64*(5*b^2*c^2-9*a*d*(10*b*c-13*a*d))*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(17/4)*d^(3/4)*sqrt(2))+1/16*(5*b^2*c^2-9*a*d*(10*b*c-13*a*d))/(c^4*d*sqrt(x))+1/20*(-5*b^2*c^2+10*a*b*c*d-13*a^2*d^2)/(c^2*d*(c+d*x^2)^2*sqrt(x))+1/80*(-5*b^2*c^2+9*a*d*(10*b*c-13*a*d))/(c^3*d*(c+d*x^2)*sqrt(x))],

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

# p>0

# p<0
[x^(5/2)*(c+d*x^2)^3/(a+b*x^2),x,13,2/3*(b*c-a*d)^3*x^(3/2)/b^4+2/7*d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*x^(7/2)/b^3+2/11*d^2*(3*b*c-a*d)*x^(11/2)/b^2+2/15*d^3*x^(15/2)/b+a^(3/4)*(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(19/4)*sqrt(2))-a^(3/4)*(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(19/4)*sqrt(2))-1/2*a^(3/4)*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(19/4)*sqrt(2))+1/2*a^(3/4)*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(19/4)*sqrt(2))],
[x^(3/2)*(c+d*x^2)^3/(a+b*x^2),x,13,2/5*d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*x^(5/2)/b^3+2/9*d^2*(3*b*c-a*d)*x^(9/2)/b^2+2/13*d^3*x^(13/2)/b+a^(1/4)*(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(17/4)*sqrt(2))-a^(1/4)*(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(17/4)*sqrt(2))+1/2*a^(1/4)*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(17/4)*sqrt(2))-1/2*a^(1/4)*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(17/4)*sqrt(2))+2*(b*c-a*d)^3*sqrt(x)/b^4],
[(c+d*x^2)^3*sqrt(x)/(a+b*x^2),x,12,2/3*d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*x^(3/2)/b^3+2/7*d^2*(3*b*c-a*d)*x^(7/2)/b^2+2/11*d^3*x^(11/2)/b-(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(15/4)*sqrt(2))+(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(15/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(15/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(15/4)*sqrt(2))],
[(c+d*x^2)^3/((a+b*x^2)*sqrt(x)),x,12,2/5*d^2*(3*b*c-a*d)*x^(5/2)/b^2+2/9*d^3*x^(9/2)/b-(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(13/4)*sqrt(2))+(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(13/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(13/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(13/4)*sqrt(2))+2*d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*sqrt(x)/b^3],
[(c+d*x^2)^3/(x^(3/2)*(a+b*x^2)),x,12,2/3*d^2*(3*b*c-a*d)*x^(3/2)/b^2+2/7*d^3*x^(7/2)/b+(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(11/4)*sqrt(2))-(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(11/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(11/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(11/4)*sqrt(2))-2*c^3/(a*sqrt(x))],
[(c+d*x^2)^3/(x^(5/2)*(a+b*x^2)),x,12,-2/3*c^3/(a*x^(3/2))+2/5*d^3*x^(5/2)/b+(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(9/4)*sqrt(2))-(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(9/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(9/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(9/4)*sqrt(2))+2*d^2*(3*b*c-a*d)*sqrt(x)/b^2],
[(c+d*x^2)^3/(x^(7/2)*(a+b*x^2)),x,12,-2/5*c^3/(a*x^(5/2))+2/3*d^3*x^(3/2)/b-(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(7/4)*sqrt(2))+(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(7/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(7/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(7/4)*sqrt(2))+2*c^2*(b*c-3*a*d)/(a^2*sqrt(x))],
[(c+d*x^2)^3/(x^(9/2)*(a+b*x^2)),x,12,-2/7*c^3/(a*x^(7/2))+2/3*c^2*(b*c-3*a*d)/(a^2*x^(3/2))-(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(5/4)*sqrt(2))+(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(5/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(5/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(5/4)*sqrt(2))+2*d^3*sqrt(x)/b],
[(c+d*x^2)^3/(x^(11/2)*(a+b*x^2)),x,12,-2/9*c^3/(a*x^(9/2))+2/5*c^2*(b*c-3*a*d)/(a^2*x^(5/2))+(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*b^(3/4)*sqrt(2))-(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*b^(3/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*b^(3/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*b^(3/4)*sqrt(2))-2*c*(b^2*c^2-3*a*b*c*d+3*a^2*d^2)/(a^3*sqrt(x))],
[(c+d*x^2)^3/(x^(13/2)*(a+b*x^2)),x,12,-2/11*c^3/(a*x^(11/2))+2/7*c^2*(b*c-3*a*d)/(a^2*x^(7/2))-2/3*c*(b^2*c^2-3*a*b*c*d+3*a^2*d^2)/(a^3*x^(3/2))+(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(15/4)*b^(1/4)*sqrt(2))-(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(15/4)*b^(1/4)*sqrt(2))+1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(15/4)*b^(1/4)*sqrt(2))-1/2*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(15/4)*b^(1/4)*sqrt(2))],
[(c+d*x^2)^3/(x^(15/2)*(a+b*x^2)),x,12,-2/13*c^3/(a*x^(13/2))+2/9*c^2*(b*c-3*a*d)/(a^2*x^(9/2))-2/5*c*(b^2*c^2-3*a*b*c*d+3*a^2*d^2)/(a^3*x^(5/2))-b^(1/4)*(b*c-a*d)^3*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(17/4)*sqrt(2))+b^(1/4)*(b*c-a*d)^3*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(17/4)*sqrt(2))+1/2*b^(1/4)*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(17/4)*sqrt(2))-1/2*b^(1/4)*(b*c-a*d)^3*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(17/4)*sqrt(2))+2*(b*c-a*d)^3/(a^4*sqrt(x))],
[x^(7/2)*(c+d*x^2)^3/(a+b*x^2)^2,x,13,1/10*d*(27*b^2*c^2-39*a*b*c*d+17*a^2*d^2)*x^(5/2)/b^4+1/18*d^2*(39*b*c-17*a*d)*x^(9/2)/b^3+17/26*d^3*x^(13/2)/b^2-1/2*x^(5/2)*(c+d*x^2)^3/(b*(a+b*x^2))+1/4*a^(1/4)*(5*b*c-17*a*d)*(b*c-a*d)^2*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(21/4)*sqrt(2))-1/4*a^(1/4)*(5*b*c-17*a*d)*(b*c-a*d)^2*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(21/4)*sqrt(2))+1/8*a^(1/4)*(5*b*c-17*a*d)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(21/4)*sqrt(2))-1/8*a^(1/4)*(5*b*c-17*a*d)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(21/4)*sqrt(2))+1/2*(5*b*c-17*a*d)*(b*c-a*d)^2*sqrt(x)/b^5],
[x^(5/2)*(c+d*x^2)^3/(a+b*x^2)^2,x,13,1/2*d*(7*b^2*c^2-11*a*b*c*d+5*a^2*d^2)*x^(3/2)/b^4+3/14*d^2*(11*b*c-5*a*d)*x^(7/2)/b^3+15/22*d^3*x^(11/2)/b^2-1/2*x^(3/2)*(c+d*x^2)^3/(b*(a+b*x^2))-3/4*(b*c-5*a*d)*(b*c-a*d)^2*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(19/4)*sqrt(2))+3/4*(b*c-5*a*d)*(b*c-a*d)^2*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*b^(19/4)*sqrt(2))+3/8*(b*c-5*a*d)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(19/4)*sqrt(2))-3/8*(b*c-5*a*d)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*b^(19/4)*sqrt(2))],
[x^(3/2)*(c+d*x^2)^3/(a+b*x^2)^2,x,14,-1/4*(b*c-13*a*d)*(b*c-a*d)^2*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(17/4)*sqrt(2))+1/4*(b*c-13*a*d)*(b*c-a*d)^2*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*b^(17/4)*sqrt(2))-1/8*(b*c-13*a*d)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(17/4)*sqrt(2))+1/8*(b*c-13*a*d)*(b*c-a*d)^2*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*b^(17/4)*sqrt(2))+1/90*d*(497*b^2*c^2-1098*a*b*c*d+585*a^2*d^2)*sqrt(x)/b^4+1/90*d*(113*b*c-117*a*d)*(c+d*x^2)*sqrt(x)/b^3+13/18*d*(c+d*x^2)^2*sqrt(x)/b^2-1/2*(c+d*x^2)^3*sqrt(x)/(b*(a+b*x^2))],
[(c+d*x^2)^3*sqrt(x)/(a+b*x^2)^2,x,13,-1/6*d*(6*b^2*c^2-21*a*b*c*d+11*a^2*d^2)*x^(3/2)/(a*b^3)-1/14*d^2*(7*b*c-11*a*d)*x^(7/2)/(a*b^2)+1/2*(b*c-a*d)*x^(3/2)*(c+d*x^2)^2/(a*b*(a+b*x^2))-1/4*(b*c-a*d)^2*(b*c+11*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(15/4)*sqrt(2))+1/4*(b*c-a*d)^2*(b*c+11*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*b^(15/4)*sqrt(2))+1/8*(b*c-a*d)^2*(b*c+11*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(15/4)*sqrt(2))-1/8*(b*c-a*d)^2*(b*c+11*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*b^(15/4)*sqrt(2))],
[(c+d*x^2)^3/((a+b*x^2)^2*sqrt(x)),x,13,2/5*d^3*x^(5/2)/b^2-3/4*(b*c-a*d)^2*(b*c+3*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(13/4)*sqrt(2))+3/4*(b*c-a*d)^2*(b*c+3*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*b^(13/4)*sqrt(2))-3/8*(b*c-a*d)^2*(b*c+3*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(13/4)*sqrt(2))+3/8*(b*c-a*d)^2*(b*c+3*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*b^(13/4)*sqrt(2))+2*d^2*(3*b*c-2*a*d)*sqrt(x)/b^3+1/2*(b*c-a*d)^3*sqrt(x)/(a*b^3*(a+b*x^2))],
[(c+d*x^2)^3/(x^(3/2)*(a+b*x^2)^2),x,13,-1/6*d^2*(3*b*c-7*a*d)*x^(3/2)/(a*b^2)+1/4*(b*c-a*d)^2*(5*b*c+7*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(11/4)*sqrt(2))-1/4*(b*c-a*d)^2*(5*b*c+7*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*b^(11/4)*sqrt(2))-1/8*(b*c-a*d)^2*(5*b*c+7*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(11/4)*sqrt(2))+1/8*(b*c-a*d)^2*(5*b*c+7*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*b^(11/4)*sqrt(2))-1/2*c^2*(5*b*c-a*d)/(a^2*b*sqrt(x))+1/2*(b*c-a*d)*(c+d*x^2)^2/(a*b*(a+b*x^2)*sqrt(x))],
[(c+d*x^2)^3/(x^(5/2)*(a+b*x^2)^2),x,13,-1/6*c^2*(7*b*c-3*a*d)/(a^2*b*x^(3/2))+1/2*(b*c-a*d)*(c+d*x^2)^2/(a*b*x^(3/2)*(a+b*x^2))+1/4*(b*c-a*d)^2*(7*b*c+5*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(9/4)*sqrt(2))-1/4*(b*c-a*d)^2*(7*b*c+5*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*b^(9/4)*sqrt(2))+1/8*(b*c-a*d)^2*(7*b*c+5*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(9/4)*sqrt(2))-1/8*(b*c-a*d)^2*(7*b*c+5*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*b^(9/4)*sqrt(2))-1/2*d^2*(b*c-5*a*d)*sqrt(x)/(a*b^2)],
[(c+d*x^2)^3/(x^(7/2)*(a+b*x^2)^2),x,13,-1/10*c^2*(9*b*c-5*a*d)/(a^2*b*x^(5/2))+1/2*(b*c-a*d)*(c+d*x^2)^2/(a*b*x^(5/2)*(a+b*x^2))-3/4*(b*c-a*d)^2*(3*b*c+a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*b^(7/4)*sqrt(2))+3/4*(b*c-a*d)^2*(3*b*c+a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*b^(7/4)*sqrt(2))+3/8*(b*c-a*d)^2*(3*b*c+a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*b^(7/4)*sqrt(2))-3/8*(b*c-a*d)^2*(3*b*c+a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*b^(7/4)*sqrt(2))+1/2*c*(9*b^2*c^2-15*a*b*c*d+2*a^2*d^2)/(a^3*b*sqrt(x))],
[(c+d*x^2)^3/(x^(9/2)*(a+b*x^2)^2),x,13,-1/14*c^2*(11*b*c-7*a*d)/(a^2*b*x^(7/2))+1/6*c*(11*b^2*c^2-21*a*b*c*d+6*a^2*d^2)/(a^3*b*x^(3/2))+1/2*(b*c-a*d)*(c+d*x^2)^2/(a*b*x^(7/2)*(a+b*x^2))-1/4*(b*c-a*d)^2*(11*b*c+a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(15/4)*b^(5/4)*sqrt(2))+1/4*(b*c-a*d)^2*(11*b*c+a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(15/4)*b^(5/4)*sqrt(2))-1/8*(b*c-a*d)^2*(11*b*c+a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(15/4)*b^(5/4)*sqrt(2))+1/8*(b*c-a*d)^2*(11*b*c+a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(15/4)*b^(5/4)*sqrt(2))],

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

# p<0
[x^(9/2)/((a+b*x^2)*(c+d*x^2)),x,22,2/3*x^(3/2)/(b*d)-a^(7/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(7/4)*(b*c-a*d)*sqrt(2))+a^(7/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(7/4)*(b*c-a*d)*sqrt(2))+c^(7/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(7/4)*(b*c-a*d)*sqrt(2))-c^(7/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(7/4)*(b*c-a*d)*sqrt(2))+1/2*a^(7/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*(b*c-a*d)*sqrt(2))-1/2*a^(7/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*(b*c-a*d)*sqrt(2))-1/2*c^(7/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(7/4)*(b*c-a*d)*sqrt(2))+1/2*c^(7/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(7/4)*(b*c-a*d)*sqrt(2))],
[x^(7/2)/((a+b*x^2)*(c+d*x^2)),x,21,-a^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(5/4)*(b*c-a*d)*sqrt(2))+a^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(5/4)*(b*c-a*d)*sqrt(2))+c^(5/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(5/4)*(b*c-a*d)*sqrt(2))-c^(5/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(5/4)*(b*c-a*d)*sqrt(2))-1/2*a^(5/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*(b*c-a*d)*sqrt(2))+1/2*a^(5/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*(b*c-a*d)*sqrt(2))+1/2*c^(5/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(5/4)*(b*c-a*d)*sqrt(2))-1/2*c^(5/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(5/4)*(b*c-a*d)*sqrt(2))+2*sqrt(x)/(b*d)],
[x^(5/2)/((a+b*x^2)*(c+d*x^2)),x,20,a^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(3/4)*(b*c-a*d)*sqrt(2))-a^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(3/4)*(b*c-a*d)*sqrt(2))-c^(3/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(3/4)*(b*c-a*d)*sqrt(2))+c^(3/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(3/4)*(b*c-a*d)*sqrt(2))-1/2*a^(3/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*(b*c-a*d)*sqrt(2))+1/2*a^(3/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*(b*c-a*d)*sqrt(2))+1/2*c^(3/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(3/4)*(b*c-a*d)*sqrt(2))-1/2*c^(3/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(3/4)*(b*c-a*d)*sqrt(2))],
[x^(3/2)/((a+b*x^2)*(c+d*x^2)),x,20,a^(1/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(1/4)*(b*c-a*d)*sqrt(2))-a^(1/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(1/4)*(b*c-a*d)*sqrt(2))-c^(1/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(1/4)*(b*c-a*d)*sqrt(2))+c^(1/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(1/4)*(b*c-a*d)*sqrt(2))+1/2*a^(1/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*(b*c-a*d)*sqrt(2))-1/2*a^(1/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*(b*c-a*d)*sqrt(2))-1/2*c^(1/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(1/4)*(b*c-a*d)*sqrt(2))+1/2*c^(1/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(1/4)*(b*c-a*d)*sqrt(2))],
[sqrt(x)/((a+b*x^2)*(c+d*x^2)),x,20,-b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)*sqrt(2))+b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)*sqrt(2))+d^(1/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*(b*c-a*d)*sqrt(2))-d^(1/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*(b*c-a*d)*sqrt(2))+1/2*b^(1/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)*sqrt(2))-1/2*b^(1/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)*sqrt(2))-1/2*d^(1/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*(b*c-a*d)*sqrt(2))+1/2*d^(1/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*(b*c-a*d)*sqrt(2))],
[1/((a+b*x^2)*(c+d*x^2)*sqrt(x)),x,20,-b^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)*sqrt(2))+b^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)*sqrt(2))+d^(3/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*(b*c-a*d)*sqrt(2))-d^(3/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*(b*c-a*d)*sqrt(2))-1/2*b^(3/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)*sqrt(2))+1/2*b^(3/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)*sqrt(2))+1/2*d^(3/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*(b*c-a*d)*sqrt(2))-1/2*d^(3/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*(b*c-a*d)*sqrt(2))],
[1/(x^(3/2)*(a+b*x^2)*(c+d*x^2)),x,22,b^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)*sqrt(2))-b^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)*sqrt(2))-d^(5/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)*sqrt(2))+d^(5/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)*sqrt(2))-1/2*b^(5/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)*sqrt(2))+1/2*b^(5/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)*sqrt(2))+1/2*d^(5/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)*sqrt(2))-1/2*d^(5/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)*sqrt(2))+(-2)/(a*c*sqrt(x))],
[1/(x^(5/2)*(a+b*x^2)*(c+d*x^2)),x,21,(-2/3)/(a*c*x^(3/2))+b^(7/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)*sqrt(2))-b^(7/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)*sqrt(2))-d^(7/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)*sqrt(2))+d^(7/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)*sqrt(2))+1/2*b^(7/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)*sqrt(2))-1/2*b^(7/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)*sqrt(2))-1/2*d^(7/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)*sqrt(2))+1/2*d^(7/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)*sqrt(2))],
[1/(x^(7/2)*(a+b*x^2)*(c+d*x^2)),x,23,(-2/5)/(a*c*x^(5/2))-b^(9/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)*sqrt(2))+b^(9/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)*sqrt(2))+d^(9/4)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)*sqrt(2))-d^(9/4)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)*sqrt(2))+1/2*b^(9/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)*sqrt(2))-1/2*b^(9/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)*sqrt(2))-1/2*d^(9/4)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)*sqrt(2))+1/2*d^(9/4)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)*sqrt(2))+2*(b*c+a*d)/(a^2*c^2*sqrt(x))],
[x^(11/2)/((a+b*x^2)*(c+d*x^2)^2),x,22,-1/2*c*x^(5/2)/(d*(b*c-a*d)*(c+d*x^2))+a^(9/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(5/4)*(b*c-a*d)^2*sqrt(2))-a^(9/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(5/4)*(b*c-a*d)^2*sqrt(2))+1/4*c^(5/4)*(5*b*c-9*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(9/4)*(b*c-a*d)^2*sqrt(2))-1/4*c^(5/4)*(5*b*c-9*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(9/4)*(b*c-a*d)^2*sqrt(2))+1/2*a^(9/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*(b*c-a*d)^2*sqrt(2))-1/2*a^(9/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*(b*c-a*d)^2*sqrt(2))+1/8*c^(5/4)*(5*b*c-9*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(9/4)*(b*c-a*d)^2*sqrt(2))-1/8*c^(5/4)*(5*b*c-9*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(9/4)*(b*c-a*d)^2*sqrt(2))+1/2*(5*b*c-4*a*d)*sqrt(x)/(b*d^2*(b*c-a*d))],
[x^(9/2)/((a+b*x^2)*(c+d*x^2)^2),x,22,-1/2*c*x^(3/2)/(d*(b*c-a*d)*(c+d*x^2))-a^(7/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(3/4)*(b*c-a*d)^2*sqrt(2))+a^(7/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(3/4)*(b*c-a*d)^2*sqrt(2))-1/4*c^(3/4)*(3*b*c-7*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(7/4)*(b*c-a*d)^2*sqrt(2))+1/4*c^(3/4)*(3*b*c-7*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(7/4)*(b*c-a*d)^2*sqrt(2))+1/2*a^(7/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*(b*c-a*d)^2*sqrt(2))-1/2*a^(7/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*(b*c-a*d)^2*sqrt(2))+1/8*c^(3/4)*(3*b*c-7*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(7/4)*(b*c-a*d)^2*sqrt(2))-1/8*c^(3/4)*(3*b*c-7*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(7/4)*(b*c-a*d)^2*sqrt(2))],
[x^(7/2)/((a+b*x^2)*(c+d*x^2)^2),x,21,-a^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(1/4)*(b*c-a*d)^2*sqrt(2))+a^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(1/4)*(b*c-a*d)^2*sqrt(2))-1/4*c^(1/4)*(b*c-5*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(5/4)*(b*c-a*d)^2*sqrt(2))+1/4*c^(1/4)*(b*c-5*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(5/4)*(b*c-a*d)^2*sqrt(2))-1/2*a^(5/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*(b*c-a*d)^2*sqrt(2))+1/2*a^(5/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*(b*c-a*d)^2*sqrt(2))-1/8*c^(1/4)*(b*c-5*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(5/4)*(b*c-a*d)^2*sqrt(2))+1/8*c^(1/4)*(b*c-5*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(5/4)*(b*c-a*d)^2*sqrt(2))-1/2*c*sqrt(x)/(d*(b*c-a*d)*(c+d*x^2))],
[x^(5/2)/((a+b*x^2)*(c+d*x^2)^2),x,22,1/2*x^(3/2)/((b*c-a*d)*(c+d*x^2))+a^(3/4)*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^2*sqrt(2))-a^(3/4)*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^2*sqrt(2))-1/4*(b*c+3*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*d^(3/4)*(b*c-a*d)^2*sqrt(2))+1/4*(b*c+3*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*d^(3/4)*(b*c-a*d)^2*sqrt(2))-1/2*a^(3/4)*b^(1/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^2*sqrt(2))+1/2*a^(3/4)*b^(1/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^2*sqrt(2))+1/8*(b*c+3*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*d^(3/4)*(b*c-a*d)^2*sqrt(2))-1/8*(b*c+3*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*d^(3/4)*(b*c-a*d)^2*sqrt(2))],
[x^(3/2)/((a+b*x^2)*(c+d*x^2)^2),x,21,a^(1/4)*b^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^2*sqrt(2))-a^(1/4)*b^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^2*sqrt(2))-1/4*(3*b*c+a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(1/4)*(b*c-a*d)^2*sqrt(2))+1/4*(3*b*c+a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(1/4)*(b*c-a*d)^2*sqrt(2))+1/2*a^(1/4)*b^(3/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^2*sqrt(2))-1/2*a^(1/4)*b^(3/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^2*sqrt(2))-1/8*(3*b*c+a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(1/4)*(b*c-a*d)^2*sqrt(2))+1/8*(3*b*c+a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(1/4)*(b*c-a*d)^2*sqrt(2))+1/2*sqrt(x)/((b*c-a*d)*(c+d*x^2))],
[sqrt(x)/((a+b*x^2)*(c+d*x^2)^2),x,22,-1/2*d*x^(3/2)/(c*(b*c-a*d)*(c+d*x^2))-b^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^2*sqrt(2))+b^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^2*sqrt(2))+1/4*d^(1/4)*(5*b*c-a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)^2*sqrt(2))-1/4*d^(1/4)*(5*b*c-a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)^2*sqrt(2))+1/2*b^(5/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^2*sqrt(2))-1/2*b^(5/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^2*sqrt(2))-1/8*d^(1/4)*(5*b*c-a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)^2*sqrt(2))+1/8*d^(1/4)*(5*b*c-a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)^2*sqrt(2))],
[1/((a+b*x^2)*(c+d*x^2)^2*sqrt(x)),x,21,-b^(7/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^2*sqrt(2))+b^(7/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^2*sqrt(2))+1/4*d^(3/4)*(7*b*c-3*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)^2*sqrt(2))-1/4*d^(3/4)*(7*b*c-3*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)^2*sqrt(2))-1/2*b^(7/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^2*sqrt(2))+1/2*b^(7/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^2*sqrt(2))+1/8*d^(3/4)*(7*b*c-3*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)^2*sqrt(2))-1/8*d^(3/4)*(7*b*c-3*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)^2*sqrt(2))-1/2*d*sqrt(x)/(c*(b*c-a*d)*(c+d*x^2))],
[1/(x^(3/2)*(a+b*x^2)*(c+d*x^2)^2),x,23,b^(9/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^2*sqrt(2))-b^(9/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^2*sqrt(2))-1/4*d^(5/4)*(9*b*c-5*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^2*sqrt(2))+1/4*d^(5/4)*(9*b*c-5*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^2*sqrt(2))-1/2*b^(9/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^2*sqrt(2))+1/2*b^(9/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^2*sqrt(2))+1/8*d^(5/4)*(9*b*c-5*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^2*sqrt(2))-1/8*d^(5/4)*(9*b*c-5*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^2*sqrt(2))+1/2*(-4*b*c+5*a*d)/(a*c^2*(b*c-a*d)*sqrt(x))-1/2*d/(c*(b*c-a*d)*(c+d*x^2)*sqrt(x))],
[1/(x^(5/2)*(a+b*x^2)*(c+d*x^2)^2),x,22,1/6*(-4*b*c+7*a*d)/(a*c^2*(b*c-a*d)*x^(3/2))-1/2*d/(c*(b*c-a*d)*x^(3/2)*(c+d*x^2))+b^(11/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^2*sqrt(2))-b^(11/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^2*sqrt(2))-1/4*d^(7/4)*(11*b*c-7*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^2*sqrt(2))+1/4*d^(7/4)*(11*b*c-7*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^2*sqrt(2))+1/2*b^(11/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^2*sqrt(2))-1/2*b^(11/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^2*sqrt(2))-1/8*d^(7/4)*(11*b*c-7*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^2*sqrt(2))+1/8*d^(7/4)*(11*b*c-7*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^2*sqrt(2))],
[1/(x^(7/2)*(a+b*x^2)*(c+d*x^2)^2),x,24,1/10*(-4*b*c+9*a*d)/(a*c^2*(b*c-a*d)*x^(5/2))-1/2*d/(c*(b*c-a*d)*x^(5/2)*(c+d*x^2))-b^(13/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^2*sqrt(2))+b^(13/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^2*sqrt(2))+1/4*d^(9/4)*(13*b*c-9*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^2*sqrt(2))-1/4*d^(9/4)*(13*b*c-9*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^2*sqrt(2))+1/2*b^(13/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^2*sqrt(2))-1/2*b^(13/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^2*sqrt(2))-1/8*d^(9/4)*(13*b*c-9*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^2*sqrt(2))+1/8*d^(9/4)*(13*b*c-9*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^2*sqrt(2))+1/2*(4*b^2*c^2+4*a*b*c*d-9*a^2*d^2)/(a^2*c^3*(b*c-a*d)*sqrt(x))],
[x^(7/2)/((a+b*x^2)*(c+d*x^2)^3),x,22,-a^(5/4)*b^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^3*sqrt(2))+a^(5/4)*b^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^3*sqrt(2))-1/32*(3*b^2*c^2-30*a*b*c*d-5*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(5/4)*(b*c-a*d)^3*sqrt(2))+1/32*(3*b^2*c^2-30*a*b*c*d-5*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(5/4)*(b*c-a*d)^3*sqrt(2))-1/2*a^(5/4)*b^(3/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^3*sqrt(2))+1/2*a^(5/4)*b^(3/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^3*sqrt(2))-1/64*(3*b^2*c^2-30*a*b*c*d-5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(5/4)*(b*c-a*d)^3*sqrt(2))+1/64*(3*b^2*c^2-30*a*b*c*d-5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(5/4)*(b*c-a*d)^3*sqrt(2))-1/4*c*sqrt(x)/(d*(b*c-a*d)*(c+d*x^2)^2)+1/16*(b*c-9*a*d)*sqrt(x)/(d*(b*c-a*d)^2*(c+d*x^2))],
[x^(5/2)/((a+b*x^2)*(c+d*x^2)^3),x,23,1/4*x^(3/2)/((b*c-a*d)*(c+d*x^2)^2)+1/16*(5*b*c+3*a*d)*x^(3/2)/(c*(b*c-a*d)^2*(c+d*x^2))+a^(3/4)*b^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^3*sqrt(2))-a^(3/4)*b^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^3*sqrt(2))-1/32*(5*b^2*c^2+30*a*b*c*d-3*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(3/4)*(b*c-a*d)^3*sqrt(2))+1/32*(5*b^2*c^2+30*a*b*c*d-3*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*d^(3/4)*(b*c-a*d)^3*sqrt(2))-1/2*a^(3/4)*b^(5/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^3*sqrt(2))+1/2*a^(3/4)*b^(5/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^3*sqrt(2))+1/64*(5*b^2*c^2+30*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(3/4)*(b*c-a*d)^3*sqrt(2))-1/64*(5*b^2*c^2+30*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*d^(3/4)*(b*c-a*d)^3*sqrt(2))],
[x^(3/2)/((a+b*x^2)*(c+d*x^2)^3),x,22,a^(1/4)*b^(7/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^3*sqrt(2))-a^(1/4)*b^(7/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^3*sqrt(2))-1/32*(21*b^2*c^2+14*a*b*c*d-3*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/32*(21*b^2*c^2+14*a*b*c*d-3*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/2*a^(1/4)*b^(7/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^3*sqrt(2))-1/2*a^(1/4)*b^(7/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^3*sqrt(2))-1/64*(21*b^2*c^2+14*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/64*(21*b^2*c^2+14*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/4*sqrt(x)/((b*c-a*d)*(c+d*x^2)^2)+1/16*(7*b*c+a*d)*sqrt(x)/(c*(b*c-a*d)^2*(c+d*x^2))],
[sqrt(x)/((a+b*x^2)*(c+d*x^2)^3),x,23,-1/4*d*x^(3/2)/(c*(b*c-a*d)*(c+d*x^2)^2)-1/16*d*(13*b*c-5*a*d)*x^(3/2)/(c^2*(b*c-a*d)^2*(c+d*x^2))-b^(9/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))+b^(9/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))+1/32*d^(1/4)*(45*b^2*c^2-18*a*b*c*d+5*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))-1/32*d^(1/4)*(45*b^2*c^2-18*a*b*c*d+5*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))+1/2*b^(9/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))-1/2*b^(9/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))-1/64*d^(1/4)*(45*b^2*c^2-18*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))+1/64*d^(1/4)*(45*b^2*c^2-18*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))],
[1/((a+b*x^2)*(c+d*x^2)^3*sqrt(x)),x,22,-b^(11/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+b^(11/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/32*d^(3/4)*(77*b^2*c^2-66*a*b*c*d+21*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))-1/32*d^(3/4)*(77*b^2*c^2-66*a*b*c*d+21*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))-1/2*b^(11/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/2*b^(11/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/64*d^(3/4)*(77*b^2*c^2-66*a*b*c*d+21*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))-1/64*d^(3/4)*(77*b^2*c^2-66*a*b*c*d+21*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))-1/4*d*sqrt(x)/(c*(b*c-a*d)*(c+d*x^2)^2)-1/16*d*(15*b*c-7*a*d)*sqrt(x)/(c^2*(b*c-a*d)^2*(c+d*x^2))],
[1/(x^(3/2)*(a+b*x^2)*(c+d*x^2)^3),x,24,b^(13/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))-b^(13/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))-1/32*d^(5/4)*(117*b^2*c^2-130*a*b*c*d+45*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))+1/32*d^(5/4)*(117*b^2*c^2-130*a*b*c*d+45*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))-1/2*b^(13/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))+1/2*b^(13/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))+1/64*d^(5/4)*(117*b^2*c^2-130*a*b*c*d+45*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))-1/64*d^(5/4)*(117*b^2*c^2-130*a*b*c*d+45*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))+1/16*(-32*b^2*c^2+85*a*b*c*d-45*a^2*d^2)/(a*c^3*(b*c-a*d)^2*sqrt(x))-1/4*d/(c*(b*c-a*d)*(c+d*x^2)^2*sqrt(x))-1/16*d*(17*b*c-9*a*d)/(c^2*(b*c-a*d)^2*(c+d*x^2)*sqrt(x))],
[1/(x^(5/2)*(a+b*x^2)*(c+d*x^2)^3),x,23,1/48*(-32*b^2*c^2+133*a*b*c*d-77*a^2*d^2)/(a*c^3*(b*c-a*d)^2*x^(3/2))-1/4*d/(c*(b*c-a*d)*x^(3/2)*(c+d*x^2)^2)-1/16*d*(19*b*c-11*a*d)/(c^2*(b*c-a*d)^2*x^(3/2)*(c+d*x^2))+b^(15/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))-b^(15/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))-1/32*d^(7/4)*(165*b^2*c^2-210*a*b*c*d+77*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(15/4)*(b*c-a*d)^3*sqrt(2))+1/32*d^(7/4)*(165*b^2*c^2-210*a*b*c*d+77*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(15/4)*(b*c-a*d)^3*sqrt(2))+1/2*b^(15/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))-1/2*b^(15/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))-1/64*d^(7/4)*(165*b^2*c^2-210*a*b*c*d+77*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(15/4)*(b*c-a*d)^3*sqrt(2))+1/64*d^(7/4)*(165*b^2*c^2-210*a*b*c*d+77*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(15/4)*(b*c-a*d)^3*sqrt(2))],
[1/(x^(7/2)*(a+b*x^2)*(c+d*x^2)^3),x,25,1/80*(-32*b^2*c^2+189*a*b*c*d-117*a^2*d^2)/(a*c^3*(b*c-a*d)^2*x^(5/2))-1/4*d/(c*(b*c-a*d)*x^(5/2)*(c+d*x^2)^2)-1/16*d*(21*b*c-13*a*d)/(c^2*(b*c-a*d)^2*x^(5/2)*(c+d*x^2))-b^(17/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))+b^(17/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))+1/32*d^(9/4)*(221*b^2*c^2-306*a*b*c*d+117*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(17/4)*(b*c-a*d)^3*sqrt(2))-1/32*d^(9/4)*(221*b^2*c^2-306*a*b*c*d+117*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(17/4)*(b*c-a*d)^3*sqrt(2))+1/2*b^(17/4)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))-1/2*b^(17/4)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))-1/64*d^(9/4)*(221*b^2*c^2-306*a*b*c*d+117*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(17/4)*(b*c-a*d)^3*sqrt(2))+1/64*d^(9/4)*(221*b^2*c^2-306*a*b*c*d+117*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(17/4)*(b*c-a*d)^3*sqrt(2))+1/16*(32*b^3*c^3+32*a*b^2*c^2*d-189*a^2*b*c*d^2+117*a^3*d^3)/(a^2*c^4*(b*c-a*d)^2*sqrt(x))],

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

# p<0
[x^(7/2)/((a+b*x^2)^2*(c+d*x^2)^2),x,22,1/4*a^(1/4)*(5*b*c+3*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(1/4)*(b*c-a*d)^3*sqrt(2))-1/4*a^(1/4)*(5*b*c+3*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(b^(1/4)*(b*c-a*d)^3*sqrt(2))-1/4*c^(1/4)*(3*b*c+5*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/4*c^(1/4)*(3*b*c+5*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/8*a^(1/4)*(5*b*c+3*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*(b*c-a*d)^3*sqrt(2))-1/8*a^(1/4)*(5*b*c+3*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*(b*c-a*d)^3*sqrt(2))-1/8*c^(1/4)*(3*b*c+5*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/8*c^(1/4)*(3*b*c+5*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(d^(1/4)*(b*c-a*d)^3*sqrt(2))+1/2*(b*c+a*d)*sqrt(x)/(b*(b*c-a*d)^2*(c+d*x^2))+1/2*a*sqrt(x)/(b*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))],
[x^(5/2)/((a+b*x^2)^2*(c+d*x^2)^2),x,23,-d*x^(3/2)/((b*c-a*d)^2*(c+d*x^2))-1/2*x^(3/2)/((b*c-a*d)*(a+b*x^2)*(c+d*x^2))-1/4*b^(1/4)*(3*b*c+5*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))+1/4*b^(1/4)*(3*b*c+5*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(1/4)*(5*b*c+3*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(1/4)*(5*b*c+3*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(1/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(1/4)*(3*b*c+5*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(1/4)*(3*b*c+5*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(1/4)*(5*b*c+3*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(1/4)*(5*b*c+3*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(1/4)*(b*c-a*d)^3*sqrt(2))],
[x^(3/2)/((a+b*x^2)^2*(c+d*x^2)^2),x,22,-1/4*b^(3/4)*(b*c+7*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/4*b^(3/4)*(b*c+7*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(3/4)*(7*b*c+a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(3/4)*(7*b*c+a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(3/4)*(b*c+7*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(3/4)*(b*c+7*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(3/4)*(7*b*c+a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(3/4)*(7*b*c+a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*(b*c-a*d)^3*sqrt(2))-d*sqrt(x)/((b*c-a*d)^2*(c+d*x^2))-1/2*sqrt(x)/((b*c-a*d)*(a+b*x^2)*(c+d*x^2))],
[sqrt(x)/((a+b*x^2)^2*(c+d*x^2)^2),x,23,1/2*d*(b*c+a*d)*x^(3/2)/(a*c*(b*c-a*d)^2*(c+d*x^2))+1/2*b*x^(3/2)/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))-1/4*b^(5/4)*(b*c-9*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))+1/4*b^(5/4)*(b*c-9*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(5/4)*(9*b*c-a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(5/4)*(9*b*c-a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(5/4)*(b*c-9*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(5/4)*(b*c-9*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(5/4)*(9*b*c-a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(5/4)*(9*b*c-a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)^3*sqrt(2))],
[1/((a+b*x^2)^2*(c+d*x^2)^2*sqrt(x)),x,22,-1/4*b^(7/4)*(3*b*c-11*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))+1/4*b^(7/4)*(3*b*c-11*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(7/4)*(11*b*c-3*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(7/4)*(11*b*c-3*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(7/4)*(3*b*c-11*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(7/4)*(3*b*c-11*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(7/4)*(11*b*c-3*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(7/4)*(11*b*c-3*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)^3*sqrt(2))+1/2*d*(b*c+a*d)*sqrt(x)/(a*c*(b*c-a*d)^2*(c+d*x^2))+1/2*b*sqrt(x)/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))],
[1/(x^(3/2)*(a+b*x^2)^2*(c+d*x^2)^2),x,24,1/4*b^(9/4)*(5*b*c-13*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))-1/4*b^(9/4)*(5*b*c-13*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(9/4)*(13*b*c-5*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(9/4)*(13*b*c-5*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(9/4)*(5*b*c-13*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(9/4)*(5*b*c-13*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(9/4)*(13*b*c-5*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(9/4)*(13*b*c-5*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^3*sqrt(2))+1/2*(-5*b^2*c^2+8*a*b*c*d-5*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*sqrt(x))+1/2*d*(b*c+a*d)/(a*c*(b*c-a*d)^2*(c+d*x^2)*sqrt(x))+1/2*b/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)*sqrt(x))],
[1/(x^(5/2)*(a+b*x^2)^2*(c+d*x^2)^2),x,23,1/6*(-7*b^2*c^2+8*a*b*c*d-7*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*x^(3/2))+1/2*d*(b*c+a*d)/(a*c*(b*c-a*d)^2*x^(3/2)*(c+d*x^2))+1/2*b/(a*(b*c-a*d)*x^(3/2)*(a+b*x^2)*(c+d*x^2))+1/4*b^(11/4)*(7*b*c-15*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*(b*c-a*d)^3*sqrt(2))-1/4*b^(11/4)*(7*b*c-15*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(11/4)*(15*b*c-7*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(11/4)*(15*b*c-7*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(11/4)*(7*b*c-15*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(11/4)*(7*b*c-15*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(11/4)*(15*b*c-7*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(11/4)*(15*b*c-7*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^3*sqrt(2))],
[1/(x^(7/2)*(a+b*x^2)^2*(c+d*x^2)^2),x,25,1/10*(-9*b^2*c^2+8*a*b*c*d-9*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*x^(5/2))+1/2*d*(b*c+a*d)/(a*c*(b*c-a*d)^2*x^(5/2)*(c+d*x^2))+1/2*b/(a*(b*c-a*d)*x^(5/2)*(a+b*x^2)*(c+d*x^2))-1/4*b^(13/4)*(9*b*c-17*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*(b*c-a*d)^3*sqrt(2))+1/4*b^(13/4)*(9*b*c-17*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*(b*c-a*d)^3*sqrt(2))-1/4*d^(13/4)*(17*b*c-9*a*d)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))+1/4*d^(13/4)*(17*b*c-9*a*d)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))+1/8*b^(13/4)*(9*b*c-17*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*(b*c-a*d)^3*sqrt(2))-1/8*b^(13/4)*(9*b*c-17*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*(b*c-a*d)^3*sqrt(2))+1/8*d^(13/4)*(17*b*c-9*a*d)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))-1/8*d^(13/4)*(17*b*c-9*a*d)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^3*sqrt(2))+1/2*(b*c+a*d)*(9*b^2*c^2-17*a*b*c*d+9*a^2*d^2)/(a^3*c^3*(b*c-a*d)^2*sqrt(x))],
[x^(7/2)/((a+b*x^2)^2*(c+d*x^2)^3),x,23,1/4*a^(1/4)*b^(3/4)*(5*b*c+7*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^4*sqrt(2))-1/4*a^(1/4)*b^(3/4)*(5*b*c+7*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/((b*c-a*d)^4*sqrt(2))-1/32*(21*b^2*c^2+70*a*b*c*d+5*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(1/4)*(b*c-a*d)^4*sqrt(2))+1/32*(21*b^2*c^2+70*a*b*c*d+5*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(3/4)*d^(1/4)*(b*c-a*d)^4*sqrt(2))+1/8*a^(1/4)*b^(3/4)*(5*b*c+7*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^4*sqrt(2))-1/8*a^(1/4)*b^(3/4)*(5*b*c+7*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/((b*c-a*d)^4*sqrt(2))-1/64*(21*b^2*c^2+70*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(1/4)*(b*c-a*d)^4*sqrt(2))+1/64*(21*b^2*c^2+70*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(3/4)*d^(1/4)*(b*c-a*d)^4*sqrt(2))+1/4*(b*c+2*a*d)*sqrt(x)/(b*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*a*sqrt(x)/(b*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+1/16*(7*b*c+17*a*d)*sqrt(x)/((b*c-a*d)^3*(c+d*x^2))],
[x^(5/2)/((a+b*x^2)^2*(c+d*x^2)^3),x,24,-3/4*d*x^(3/2)/((b*c-a*d)^2*(c+d*x^2)^2)-1/2*x^(3/2)/((b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)-3/16*d*(7*b*c+a*d)*x^(3/2)/(c*(b*c-a*d)^3*(c+d*x^2))-3/4*b^(5/4)*(b*c+3*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^4*sqrt(2))+3/4*b^(5/4)*(b*c+3*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(1/4)*(b*c-a*d)^4*sqrt(2))+3/32*d^(1/4)*(15*b^2*c^2+18*a*b*c*d-a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)^4*sqrt(2))-3/32*d^(1/4)*(15*b^2*c^2+18*a*b*c*d-a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(5/4)*(b*c-a*d)^4*sqrt(2))+3/8*b^(5/4)*(b*c+3*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^4*sqrt(2))-3/8*b^(5/4)*(b*c+3*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(1/4)*(b*c-a*d)^4*sqrt(2))-3/64*d^(1/4)*(15*b^2*c^2+18*a*b*c*d-a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)^4*sqrt(2))+3/64*d^(1/4)*(15*b^2*c^2+18*a*b*c*d-a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*(b*c-a*d)^4*sqrt(2))],
[x^(3/2)/((a+b*x^2)^2*(c+d*x^2)^3),x,23,-1/4*b^(7/4)*(b*c+11*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^4*sqrt(2))+1/4*b^(7/4)*(b*c+11*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(3/4)*(b*c-a*d)^4*sqrt(2))+1/32*d^(3/4)*(77*b^2*c^2+22*a*b*c*d-3*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)^4*sqrt(2))-1/32*d^(3/4)*(77*b^2*c^2+22*a*b*c*d-3*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(7/4)*(b*c-a*d)^4*sqrt(2))-1/8*b^(7/4)*(b*c+11*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^4*sqrt(2))+1/8*b^(7/4)*(b*c+11*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(3/4)*(b*c-a*d)^4*sqrt(2))+1/64*d^(3/4)*(77*b^2*c^2+22*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)^4*sqrt(2))-1/64*d^(3/4)*(77*b^2*c^2+22*a*b*c*d-3*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*(b*c-a*d)^4*sqrt(2))-3/4*d*sqrt(x)/((b*c-a*d)^2*(c+d*x^2)^2)-1/2*sqrt(x)/((b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)-1/16*d*(23*b*c+a*d)*sqrt(x)/(c*(b*c-a*d)^3*(c+d*x^2))],
[sqrt(x)/((a+b*x^2)^2*(c+d*x^2)^3),x,24,1/4*d*(2*b*c+a*d)*x^(3/2)/(a*c*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*b*x^(3/2)/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+1/16*d*(8*b^2*c^2+21*a*b*c*d-5*a^2*d^2)*x^(3/2)/(a*c^2*(b*c-a*d)^3*(c+d*x^2))-1/4*b^(9/4)*(b*c-13*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^4*sqrt(2))+1/4*b^(9/4)*(b*c-13*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(5/4)*(b*c-a*d)^4*sqrt(2))-1/32*d^(5/4)*(117*b^2*c^2-26*a*b*c*d+5*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^4*sqrt(2))+1/32*d^(5/4)*(117*b^2*c^2-26*a*b*c*d+5*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(9/4)*(b*c-a*d)^4*sqrt(2))+1/8*b^(9/4)*(b*c-13*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^4*sqrt(2))-1/8*b^(9/4)*(b*c-13*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(5/4)*(b*c-a*d)^4*sqrt(2))+1/64*d^(5/4)*(117*b^2*c^2-26*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^4*sqrt(2))-1/64*d^(5/4)*(117*b^2*c^2-26*a*b*c*d+5*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*(b*c-a*d)^4*sqrt(2))],
[1/((a+b*x^2)^2*(c+d*x^2)^3*sqrt(x)),x,23,-3/4*b^(11/4)*(b*c-5*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^4*sqrt(2))+3/4*b^(11/4)*(b*c-5*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(7/4)*(b*c-a*d)^4*sqrt(2))-3/32*d^(7/4)*(55*b^2*c^2-30*a*b*c*d+7*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^4*sqrt(2))+3/32*d^(7/4)*(55*b^2*c^2-30*a*b*c*d+7*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(11/4)*(b*c-a*d)^4*sqrt(2))-3/8*b^(11/4)*(b*c-5*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^4*sqrt(2))+3/8*b^(11/4)*(b*c-5*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(7/4)*(b*c-a*d)^4*sqrt(2))-3/64*d^(7/4)*(55*b^2*c^2-30*a*b*c*d+7*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^4*sqrt(2))+3/64*d^(7/4)*(55*b^2*c^2-30*a*b*c*d+7*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*(b*c-a*d)^4*sqrt(2))+1/4*d*(2*b*c+a*d)*sqrt(x)/(a*c*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*b*sqrt(x)/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+1/16*d*(8*b^2*c^2+23*a*b*c*d-7*a^2*d^2)*sqrt(x)/(a*c^2*(b*c-a*d)^3*(c+d*x^2))],
[1/(x^(3/2)*(a+b*x^2)^2*(c+d*x^2)^3),x,25,1/4*b^(13/4)*(5*b*c-17*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^4*sqrt(2))-1/4*b^(13/4)*(5*b*c-17*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(9/4)*(b*c-a*d)^4*sqrt(2))+1/32*d^(9/4)*(221*b^2*c^2-170*a*b*c*d+45*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^4*sqrt(2))-1/32*d^(9/4)*(221*b^2*c^2-170*a*b*c*d+45*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(13/4)*(b*c-a*d)^4*sqrt(2))-1/8*b^(13/4)*(5*b*c-17*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^4*sqrt(2))+1/8*b^(13/4)*(5*b*c-17*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(9/4)*(b*c-a*d)^4*sqrt(2))-1/64*d^(9/4)*(221*b^2*c^2-170*a*b*c*d+45*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^4*sqrt(2))+1/64*d^(9/4)*(221*b^2*c^2-170*a*b*c*d+45*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*(b*c-a*d)^4*sqrt(2))+1/16*(-40*b^3*c^3+96*a*b^2*c^2*d-125*a^2*b*c*d^2+45*a^3*d^3)/(a^2*c^3*(b*c-a*d)^3*sqrt(x))+1/4*d*(2*b*c+a*d)/(a*c*(b*c-a*d)^2*(c+d*x^2)^2*sqrt(x))+1/2*b/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2*sqrt(x))+1/16*d*(8*b^2*c^2+25*a*b*c*d-9*a^2*d^2)/(a*c^2*(b*c-a*d)^3*(c+d*x^2)*sqrt(x))],
[1/(x^(5/2)*(a+b*x^2)^2*(c+d*x^2)^3),x,24,1/48*(-56*b^3*c^3+96*a*b^2*c^2*d-189*a^2*b*c*d^2+77*a^3*d^3)/(a^2*c^3*(b*c-a*d)^3*x^(3/2))+1/4*d*(2*b*c+a*d)/(a*c*(b*c-a*d)^2*x^(3/2)*(c+d*x^2)^2)+1/2*b/(a*(b*c-a*d)*x^(3/2)*(a+b*x^2)*(c+d*x^2)^2)+1/16*d*(8*b^2*c^2+27*a*b*c*d-11*a^2*d^2)/(a*c^2*(b*c-a*d)^3*x^(3/2)*(c+d*x^2))+1/4*b^(15/4)*(7*b*c-19*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*(b*c-a*d)^4*sqrt(2))-1/4*b^(15/4)*(7*b*c-19*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(11/4)*(b*c-a*d)^4*sqrt(2))+1/32*d^(11/4)*(285*b^2*c^2-266*a*b*c*d+77*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(15/4)*(b*c-a*d)^4*sqrt(2))-1/32*d^(11/4)*(285*b^2*c^2-266*a*b*c*d+77*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(15/4)*(b*c-a*d)^4*sqrt(2))+1/8*b^(15/4)*(7*b*c-19*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*(b*c-a*d)^4*sqrt(2))-1/8*b^(15/4)*(7*b*c-19*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(11/4)*(b*c-a*d)^4*sqrt(2))+1/64*d^(11/4)*(285*b^2*c^2-266*a*b*c*d+77*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(15/4)*(b*c-a*d)^4*sqrt(2))-1/64*d^(11/4)*(285*b^2*c^2-266*a*b*c*d+77*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(15/4)*(b*c-a*d)^4*sqrt(2))],
[1/(x^(7/2)*(a+b*x^2)^2*(c+d*x^2)^3),x,26,-3/80*(24*b^3*c^3-32*a*b^2*c^2*d+87*a^2*b*c*d^2-39*a^3*d^3)/(a^2*c^3*(b*c-a*d)^3*x^(5/2))+1/4*d*(2*b*c+a*d)/(a*c*(b*c-a*d)^2*x^(5/2)*(c+d*x^2)^2)+1/2*b/(a*(b*c-a*d)*x^(5/2)*(a+b*x^2)*(c+d*x^2)^2)+1/16*d*(8*b^2*c^2+29*a*b*c*d-13*a^2*d^2)/(a*c^2*(b*c-a*d)^3*x^(5/2)*(c+d*x^2))-3/4*b^(17/4)*(3*b*c-7*a*d)*arctan(1-b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*(b*c-a*d)^4*sqrt(2))+3/4*b^(17/4)*(3*b*c-7*a*d)*arctan(1+b^(1/4)*sqrt(2)*sqrt(x)/a^(1/4))/(a^(13/4)*(b*c-a*d)^4*sqrt(2))-3/32*d^(13/4)*(119*b^2*c^2-126*a*b*c*d+39*a^2*d^2)*arctan(1-d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(17/4)*(b*c-a*d)^4*sqrt(2))+3/32*d^(13/4)*(119*b^2*c^2-126*a*b*c*d+39*a^2*d^2)*arctan(1+d^(1/4)*sqrt(2)*sqrt(x)/c^(1/4))/(c^(17/4)*(b*c-a*d)^4*sqrt(2))+3/8*b^(17/4)*(3*b*c-7*a*d)*log(sqrt(a)+x*sqrt(b)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*(b*c-a*d)^4*sqrt(2))-3/8*b^(17/4)*(3*b*c-7*a*d)*log(sqrt(a)+x*sqrt(b)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(x))/(a^(13/4)*(b*c-a*d)^4*sqrt(2))+3/64*d^(13/4)*(119*b^2*c^2-126*a*b*c*d+39*a^2*d^2)*log(sqrt(c)+x*sqrt(d)-c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(17/4)*(b*c-a*d)^4*sqrt(2))-3/64*d^(13/4)*(119*b^2*c^2-126*a*b*c*d+39*a^2*d^2)*log(sqrt(c)+x*sqrt(d)+c^(1/4)*d^(1/4)*sqrt(2)*sqrt(x))/(c^(17/4)*(b*c-a*d)^4*sqrt(2))+3/16*(24*b^4*c^4-32*a*b^3*c^3*d-32*a^2*b^2*c^2*d^2+87*a^3*b*c*d^3-39*a^4*d^4)/(a^3*c^4*(b*c-a*d)^3*sqrt(x))],

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

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

# p>0
[x^5*(A+B*x^2)*sqrt(a+b*x^2),x,3,1/3*a^2*(A*b-a*B)*(a+b*x^2)^(3/2)/b^4-1/5*a*(2*A*b-3*a*B)*(a+b*x^2)^(5/2)/b^4+1/7*(A*b-3*a*B)*(a+b*x^2)^(7/2)/b^4+1/9*B*(a+b*x^2)^(9/2)/b^4],
[x^4*(A+B*x^2)*sqrt(a+b*x^2),x,6,1/8*B*x^5*(a+b*x^2)^(3/2)/b+1/128*a^3*(8*A*b-5*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)-1/128*a^2*(8*A*b-5*a*B)*x*sqrt(a+b*x^2)/b^3+1/192*a*(8*A*b-5*a*B)*x^3*sqrt(a+b*x^2)/b^2+1/48*(8*A*b-5*a*B)*x^5*sqrt(a+b*x^2)/b],
[x^3*(A+B*x^2)*sqrt(a+b*x^2),x,3,-1/3*a*(A*b-a*B)*(a+b*x^2)^(3/2)/b^3+1/5*(A*b-2*a*B)*(a+b*x^2)^(5/2)/b^3+1/7*B*(a+b*x^2)^(7/2)/b^3],
[x^2*(A+B*x^2)*sqrt(a+b*x^2),x,5,1/6*B*x^3*(a+b*x^2)^(3/2)/b-1/16*a^2*(2*A*b-a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/16*a*(2*A*b-a*B)*x*sqrt(a+b*x^2)/b^2+1/8*(2*A*b-a*B)*x^3*sqrt(a+b*x^2)/b],
[x*(A+B*x^2)*sqrt(a+b*x^2),x,3,1/3*(A*b-a*B)*(a+b*x^2)^(3/2)/b^2+1/5*B*(a+b*x^2)^(5/2)/b^2],
[(A+B*x^2)*sqrt(a+b*x^2),x,4,1/4*B*x*(a+b*x^2)^(3/2)/b+1/8*a*(4*A*b-a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/8*(4*A*b-a*B)*x*sqrt(a+b*x^2)/b],
[(A+B*x^2)*sqrt(a+b*x^2)/x,x,5,1/3*B*(a+b*x^2)^(3/2)/b-A*arctanh(sqrt(a+b*x^2)/sqrt(a))*sqrt(a)+A*sqrt(a+b*x^2)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^2,x,4,-A*(a+b*x^2)^(3/2)/(a*x)+1/2*(2*A*b+a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)+1/2*(2*A*b+a*B)*x*sqrt(a+b*x^2)/a],
[(A+B*x^2)*sqrt(a+b*x^2)/x^3,x,5,-1/2*A*(a+b*x^2)^(3/2)/(a*x^2)-1/2*(A*b+2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/sqrt(a)+1/2*(A*b+2*a*B)*sqrt(a+b*x^2)/a],
[(A+B*x^2)*sqrt(a+b*x^2)/x^4,x,4,-1/3*A*(a+b*x^2)^(3/2)/(a*x^3)+B*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(b)-B*sqrt(a+b*x^2)/x],
[(A+B*x^2)*sqrt(a+b*x^2)/x^5,x,5,-1/4*A*(a+b*x^2)^(3/2)/(a*x^4)+1/8*b*(A*b-4*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(3/2)+1/8*(A*b-4*a*B)*sqrt(a+b*x^2)/(a*x^2)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^6,x,2,-1/5*A*(a+b*x^2)^(3/2)/(a*x^5)+1/15*(2*A*b-5*a*B)*(a+b*x^2)^(3/2)/(a^2*x^3)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^7,x,6,-1/6*A*(a+b*x^2)^(3/2)/(a*x^6)-1/16*b^2*(A*b-2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(5/2)+1/8*(A*b-2*a*B)*sqrt(a+b*x^2)/(a*x^4)+1/16*b*(A*b-2*a*B)*sqrt(a+b*x^2)/(a^2*x^2)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^8,x,3,-1/7*A*(a+b*x^2)^(3/2)/(a*x^7)+1/35*(4*A*b-7*a*B)*(a+b*x^2)^(3/2)/(a^2*x^5)-2/105*b*(4*A*b-7*a*B)*(a+b*x^2)^(3/2)/(a^3*x^3)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^9,x,7,-1/8*A*(a+b*x^2)^(3/2)/(a*x^8)+1/128*b^3*(5*A*b-8*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(7/2)+1/48*(5*A*b-8*a*B)*sqrt(a+b*x^2)/(a*x^6)+1/192*b*(5*A*b-8*a*B)*sqrt(a+b*x^2)/(a^2*x^4)-1/128*b^2*(5*A*b-8*a*B)*sqrt(a+b*x^2)/(a^3*x^2)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^10,x,4,-1/9*A*(a+b*x^2)^(3/2)/(a*x^9)+1/21*(2*A*b-3*a*B)*(a+b*x^2)^(3/2)/(a^2*x^7)-4/105*b*(2*A*b-3*a*B)*(a+b*x^2)^(3/2)/(a^3*x^5)+8/315*b^2*(2*A*b-3*a*B)*(a+b*x^2)^(3/2)/(a^4*x^3)],
[(A+B*x^2)*sqrt(a+b*x^2)/x^11,x,8,-1/10*A*(a+b*x^2)^(3/2)/(a*x^10)-1/256*b^4*(7*A*b-10*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(9/2)+1/80*(7*A*b-10*a*B)*sqrt(a+b*x^2)/(a*x^8)+1/480*b*(7*A*b-10*a*B)*sqrt(a+b*x^2)/(a^2*x^6)-1/384*b^2*(7*A*b-10*a*B)*sqrt(a+b*x^2)/(a^3*x^4)+1/256*b^3*(7*A*b-10*a*B)*sqrt(a+b*x^2)/(a^4*x^2)],
[x^5*(a+b*x^2)^(3/2)*(A+B*x^2),x,3,1/5*a^2*(A*b-a*B)*(a+b*x^2)^(5/2)/b^4-1/7*a*(2*A*b-3*a*B)*(a+b*x^2)^(7/2)/b^4+1/9*(A*b-3*a*B)*(a+b*x^2)^(9/2)/b^4+1/11*B*(a+b*x^2)^(11/2)/b^4],
[x^4*(a+b*x^2)^(3/2)*(A+B*x^2),x,7,1/16*(2*A*b-a*B)*x^5*(a+b*x^2)^(3/2)/b+1/10*B*x^5*(a+b*x^2)^(5/2)/b+3/256*a^4*(2*A*b-a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)-3/256*a^3*(2*A*b-a*B)*x*sqrt(a+b*x^2)/b^3+1/128*a^2*(2*A*b-a*B)*x^3*sqrt(a+b*x^2)/b^2+1/32*a*(2*A*b-a*B)*x^5*sqrt(a+b*x^2)/b],
[x^3*(a+b*x^2)^(3/2)*(A+B*x^2),x,3,-1/5*a*(A*b-a*B)*(a+b*x^2)^(5/2)/b^3+1/7*(A*b-2*a*B)*(a+b*x^2)^(7/2)/b^3+1/9*B*(a+b*x^2)^(9/2)/b^3],
[x^2*(a+b*x^2)^(3/2)*(A+B*x^2),x,6,1/48*(8*A*b-3*a*B)*x^3*(a+b*x^2)^(3/2)/b+1/8*B*x^3*(a+b*x^2)^(5/2)/b-1/128*a^3*(8*A*b-3*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/128*a^2*(8*A*b-3*a*B)*x*sqrt(a+b*x^2)/b^2+1/64*a*(8*A*b-3*a*B)*x^3*sqrt(a+b*x^2)/b],
[x*(a+b*x^2)^(3/2)*(A+B*x^2),x,3,1/5*(A*b-a*B)*(a+b*x^2)^(5/2)/b^2+1/7*B*(a+b*x^2)^(7/2)/b^2],
[(a+b*x^2)^(3/2)*(A+B*x^2),x,5,1/24*(6*A*b-a*B)*x*(a+b*x^2)^(3/2)/b+1/6*B*x*(a+b*x^2)^(5/2)/b+1/16*a^2*(6*A*b-a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/16*a*(6*A*b-a*B)*x*sqrt(a+b*x^2)/b],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x,x,6,1/3*A*(a+b*x^2)^(3/2)+1/5*B*(a+b*x^2)^(5/2)/b-a^(3/2)*A*arctanh(sqrt(a+b*x^2)/sqrt(a))+a*A*sqrt(a+b*x^2)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^2,x,5,1/4*(4*A*b+a*B)*x*(a+b*x^2)^(3/2)/a-A*(a+b*x^2)^(5/2)/(a*x)+3/8*a*(4*A*b+a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)+3/8*(4*A*b+a*B)*x*sqrt(a+b*x^2)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^3,x,6,1/6*(3*A*b+2*a*B)*(a+b*x^2)^(3/2)/a-1/2*A*(a+b*x^2)^(5/2)/(a*x^2)-1/2*(3*A*b+2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))*sqrt(a)+1/2*(3*A*b+2*a*B)*sqrt(a+b*x^2)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^4,x,5,-1/3*(2*A*b+3*a*B)*(a+b*x^2)^(3/2)/(a*x)-1/3*A*(a+b*x^2)^(5/2)/(a*x^3)+1/2*(2*A*b+3*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(b)+1/2*b*(2*A*b+3*a*B)*x*sqrt(a+b*x^2)/a],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^5,x,6,-1/8*(A*b+4*a*B)*(a+b*x^2)^(3/2)/(a*x^2)-1/4*A*(a+b*x^2)^(5/2)/(a*x^4)-3/8*b*(A*b+4*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/sqrt(a)+3/8*b*(A*b+4*a*B)*sqrt(a+b*x^2)/a],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^6,x,5,-1/3*B*(a+b*x^2)^(3/2)/x^3-1/5*A*(a+b*x^2)^(5/2)/(a*x^5)+b^(3/2)*B*arctanh(x*sqrt(b)/sqrt(a+b*x^2))-b*B*sqrt(a+b*x^2)/x],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^7,x,6,1/24*(A*b-6*a*B)*(a+b*x^2)^(3/2)/(a*x^4)-1/6*A*(a+b*x^2)^(5/2)/(a*x^6)+1/16*b^2*(A*b-6*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(3/2)+1/16*b*(A*b-6*a*B)*sqrt(a+b*x^2)/(a*x^2)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^8,x,2,-1/7*A*(a+b*x^2)^(5/2)/(a*x^7)+1/35*(2*A*b-7*a*B)*(a+b*x^2)^(5/2)/(a^2*x^5)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^9,x,7,1/48*(3*A*b-8*a*B)*(a+b*x^2)^(3/2)/(a*x^6)-1/8*A*(a+b*x^2)^(5/2)/(a*x^8)-1/128*b^3*(3*A*b-8*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(5/2)+1/64*b*(3*A*b-8*a*B)*sqrt(a+b*x^2)/(a*x^4)+1/128*b^2*(3*A*b-8*a*B)*sqrt(a+b*x^2)/(a^2*x^2)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^10,x,3,-1/9*A*(a+b*x^2)^(5/2)/(a*x^9)+1/63*(4*A*b-9*a*B)*(a+b*x^2)^(5/2)/(a^2*x^7)-2/315*b*(4*A*b-9*a*B)*(a+b*x^2)^(5/2)/(a^3*x^5)],
[(a+b*x^2)^(3/2)*(A+B*x^2)/x^11,x,8,1/16*(A*b-2*a*B)*(a+b*x^2)^(3/2)/(a*x^8)-1/10*A*(a+b*x^2)^(5/2)/(a*x^10)+3/256*b^4*(A*b-2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(7/2)+1/32*b*(A*b-2*a*B)*sqrt(a+b*x^2)/(a*x^6)+1/128*b^2*(A*b-2*a*B)*sqrt(a+b*x^2)/(a^2*x^4)-3/256*b^3*(A*b-2*a*B)*sqrt(a+b*x^2)/(a^3*x^2)],
[x^5*(a+b*x^2)^(5/2)*(A+B*x^2),x,3,1/7*a^2*(A*b-a*B)*(a+b*x^2)^(7/2)/b^4-1/9*a*(2*A*b-3*a*B)*(a+b*x^2)^(9/2)/b^4+1/11*(A*b-3*a*B)*(a+b*x^2)^(11/2)/b^4+1/13*B*(a+b*x^2)^(13/2)/b^4],
[x^4*(a+b*x^2)^(5/2)*(A+B*x^2),x,8,1/192*a*(12*A*b-5*a*B)*x^5*(a+b*x^2)^(3/2)/b+1/120*(12*A*b-5*a*B)*x^5*(a+b*x^2)^(5/2)/b+1/12*B*x^5*(a+b*x^2)^(7/2)/b+1/1024*a^5*(12*A*b-5*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)-1/1024*a^4*(12*A*b-5*a*B)*x*sqrt(a+b*x^2)/b^3+1/1536*a^3*(12*A*b-5*a*B)*x^3*sqrt(a+b*x^2)/b^2+1/384*a^2*(12*A*b-5*a*B)*x^5*sqrt(a+b*x^2)/b],
[x^3*(a+b*x^2)^(5/2)*(A+B*x^2),x,3,-1/7*a*(A*b-a*B)*(a+b*x^2)^(7/2)/b^3+1/9*(A*b-2*a*B)*(a+b*x^2)^(9/2)/b^3+1/11*B*(a+b*x^2)^(11/2)/b^3],
[x^2*(a+b*x^2)^(5/2)*(A+B*x^2),x,7,1/96*a*(10*A*b-3*a*B)*x^3*(a+b*x^2)^(3/2)/b+1/80*(10*A*b-3*a*B)*x^3*(a+b*x^2)^(5/2)/b+1/10*B*x^3*(a+b*x^2)^(7/2)/b-1/256*a^4*(10*A*b-3*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/256*a^3*(10*A*b-3*a*B)*x*sqrt(a+b*x^2)/b^2+1/128*a^2*(10*A*b-3*a*B)*x^3*sqrt(a+b*x^2)/b],
[x*(a+b*x^2)^(5/2)*(A+B*x^2),x,3,1/7*(A*b-a*B)*(a+b*x^2)^(7/2)/b^2+1/9*B*(a+b*x^2)^(9/2)/b^2],
[(a+b*x^2)^(5/2)*(A+B*x^2),x,6,5/192*a*(8*A*b-a*B)*x*(a+b*x^2)^(3/2)/b+1/48*(8*A*b-a*B)*x*(a+b*x^2)^(5/2)/b+1/8*B*x*(a+b*x^2)^(7/2)/b+5/128*a^3*(8*A*b-a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+5/128*a^2*(8*A*b-a*B)*x*sqrt(a+b*x^2)/b],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x,x,7,1/3*a*A*(a+b*x^2)^(3/2)+1/5*A*(a+b*x^2)^(5/2)+1/7*B*(a+b*x^2)^(7/2)/b-a^(5/2)*A*arctanh(sqrt(a+b*x^2)/sqrt(a))+a^2*A*sqrt(a+b*x^2)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^2,x,6,5/24*(6*A*b+a*B)*x*(a+b*x^2)^(3/2)+1/6*(6*A*b+a*B)*x*(a+b*x^2)^(5/2)/a-A*(a+b*x^2)^(7/2)/(a*x)+5/16*a^2*(6*A*b+a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)+5/16*a*(6*A*b+a*B)*x*sqrt(a+b*x^2)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^3,x,7,1/6*(5*A*b+2*a*B)*(a+b*x^2)^(3/2)+1/10*(5*A*b+2*a*B)*(a+b*x^2)^(5/2)/a-1/2*A*(a+b*x^2)^(7/2)/(a*x^2)-1/2*a^(3/2)*(5*A*b+2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))+1/2*a*(5*A*b+2*a*B)*sqrt(a+b*x^2)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^4,x,6,5/12*b*(4*A*b+3*a*B)*x*(a+b*x^2)^(3/2)/a-1/3*(4*A*b+3*a*B)*(a+b*x^2)^(5/2)/(a*x)-1/3*A*(a+b*x^2)^(7/2)/(a*x^3)+5/8*a*(4*A*b+3*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(b)+5/8*b*(4*A*b+3*a*B)*x*sqrt(a+b*x^2)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^5,x,7,5/24*b*(3*A*b+4*a*B)*(a+b*x^2)^(3/2)/a-1/8*(3*A*b+4*a*B)*(a+b*x^2)^(5/2)/(a*x^2)-1/4*A*(a+b*x^2)^(7/2)/(a*x^4)-5/8*b*(3*A*b+4*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))*sqrt(a)+5/8*b*(3*A*b+4*a*B)*sqrt(a+b*x^2)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^6,x,6,-1/3*b*(2*A*b+5*a*B)*(a+b*x^2)^(3/2)/(a*x)-1/15*(2*A*b+5*a*B)*(a+b*x^2)^(5/2)/(a*x^3)-1/5*A*(a+b*x^2)^(7/2)/(a*x^5)+1/2*b^(3/2)*(2*A*b+5*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))+1/2*b^2*(2*A*b+5*a*B)*x*sqrt(a+b*x^2)/a],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^7,x,7,-5/48*b*(A*b+6*a*B)*(a+b*x^2)^(3/2)/(a*x^2)-1/24*(A*b+6*a*B)*(a+b*x^2)^(5/2)/(a*x^4)-1/6*A*(a+b*x^2)^(7/2)/(a*x^6)-5/16*b^2*(A*b+6*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/sqrt(a)+5/16*b^2*(A*b+6*a*B)*sqrt(a+b*x^2)/a],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^8,x,6,-1/3*b*B*(a+b*x^2)^(3/2)/x^3-1/5*B*(a+b*x^2)^(5/2)/x^5-1/7*A*(a+b*x^2)^(7/2)/(a*x^7)+b^(5/2)*B*arctanh(x*sqrt(b)/sqrt(a+b*x^2))-b^2*B*sqrt(a+b*x^2)/x],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^9,x,7,5/192*b*(A*b-8*a*B)*(a+b*x^2)^(3/2)/(a*x^4)+1/48*(A*b-8*a*B)*(a+b*x^2)^(5/2)/(a*x^6)-1/8*A*(a+b*x^2)^(7/2)/(a*x^8)+5/128*b^3*(A*b-8*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(3/2)+5/128*b^2*(A*b-8*a*B)*sqrt(a+b*x^2)/(a*x^2)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^10,x,2,-1/9*A*(a+b*x^2)^(7/2)/(a*x^9)+1/63*(2*A*b-9*a*B)*(a+b*x^2)^(7/2)/(a^2*x^7)],
[(a+b*x^2)^(5/2)*(A+B*x^2)/x^11,x,8,1/96*b*(3*A*b-10*a*B)*(a+b*x^2)^(3/2)/(a*x^6)+1/80*(3*A*b-10*a*B)*(a+b*x^2)^(5/2)/(a*x^8)-1/10*A*(a+b*x^2)^(7/2)/(a*x^10)-1/256*b^4*(3*A*b-10*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(5/2)+1/128*b^2*(3*A*b-10*a*B)*sqrt(a+b*x^2)/(a*x^4)+1/256*b^3*(3*A*b-10*a*B)*sqrt(a+b*x^2)/(a^2*x^2)],

# p<0
[x^5*(A+B*x^2)/sqrt(a+b*x^2),x,3,-1/3*a*(2*A*b-3*a*B)*(a+b*x^2)^(3/2)/b^4+1/5*(A*b-3*a*B)*(a+b*x^2)^(5/2)/b^4+1/7*B*(a+b*x^2)^(7/2)/b^4+a^2*(A*b-a*B)*sqrt(a+b*x^2)/b^4],
[x^4*(A+B*x^2)/sqrt(a+b*x^2),x,5,1/16*a^2*(6*A*b-5*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)-1/16*a*(6*A*b-5*a*B)*x*sqrt(a+b*x^2)/b^3+1/24*(6*A*b-5*a*B)*x^3*sqrt(a+b*x^2)/b^2+1/6*B*x^5*sqrt(a+b*x^2)/b],
[x^3*(A+B*x^2)/sqrt(a+b*x^2),x,3,1/3*(A*b-2*a*B)*(a+b*x^2)^(3/2)/b^3+1/5*B*(a+b*x^2)^(5/2)/b^3-a*(A*b-a*B)*sqrt(a+b*x^2)/b^3],
[x^2*(A+B*x^2)/sqrt(a+b*x^2),x,4,-1/8*a*(4*A*b-3*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/8*(4*A*b-3*a*B)*x*sqrt(a+b*x^2)/b^2+1/4*B*x^3*sqrt(a+b*x^2)/b],
[x*(A+B*x^2)/sqrt(a+b*x^2),x,3,1/3*B*(a+b*x^2)^(3/2)/b^2+(A*b-a*B)*sqrt(a+b*x^2)/b^2],
[(A+B*x^2)/sqrt(a+b*x^2),x,3,1/2*(2*A*b-a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/2*B*x*sqrt(a+b*x^2)/b],
[(A+B*x^2)/(x*sqrt(a+b*x^2)),x,4,-A*arctanh(sqrt(a+b*x^2)/sqrt(a))/sqrt(a)+B*sqrt(a+b*x^2)/b],
[(A+B*x^2)/(x^2*sqrt(a+b*x^2)),x,3,B*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)-A*sqrt(a+b*x^2)/(a*x)],
[(A+B*x^2)/(x^3*sqrt(a+b*x^2)),x,4,1/2*(A*b-2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(3/2)-1/2*A*sqrt(a+b*x^2)/(a*x^2)],
[(A+B*x^2)/(x^4*sqrt(a+b*x^2)),x,2,-1/3*A*sqrt(a+b*x^2)/(a*x^3)+1/3*(2*A*b-3*a*B)*sqrt(a+b*x^2)/(a^2*x)],
[(A+B*x^2)/(x^5*sqrt(a+b*x^2)),x,5,-1/8*b*(3*A*b-4*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(5/2)-1/4*A*sqrt(a+b*x^2)/(a*x^4)+1/8*(3*A*b-4*a*B)*sqrt(a+b*x^2)/(a^2*x^2)],
[(A+B*x^2)/(x^6*sqrt(a+b*x^2)),x,3,-1/5*A*sqrt(a+b*x^2)/(a*x^5)+1/15*(4*A*b-5*a*B)*sqrt(a+b*x^2)/(a^2*x^3)-2/15*b*(4*A*b-5*a*B)*sqrt(a+b*x^2)/(a^3*x)],
[(A+B*x^2)/(x^7*sqrt(a+b*x^2)),x,6,1/16*b^2*(5*A*b-6*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(7/2)-1/6*A*sqrt(a+b*x^2)/(a*x^6)+1/24*(5*A*b-6*a*B)*sqrt(a+b*x^2)/(a^2*x^4)-1/16*b*(5*A*b-6*a*B)*sqrt(a+b*x^2)/(a^3*x^2)],
[(A+B*x^2)/(x^8*sqrt(a+b*x^2)),x,4,-1/7*A*sqrt(a+b*x^2)/(a*x^7)+1/35*(6*A*b-7*a*B)*sqrt(a+b*x^2)/(a^2*x^5)-4/105*b*(6*A*b-7*a*B)*sqrt(a+b*x^2)/(a^3*x^3)+8/105*b^2*(6*A*b-7*a*B)*sqrt(a+b*x^2)/(a^4*x)],
[x^6*(A+B*x^2)/(a+b*x^2)^(3/2),x,6,5/16*a^2*(6*A*b-7*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(9/2)-1/6*(6*A*b-7*a*B)*x^5/(b^2*sqrt(a+b*x^2))+1/6*B*x^7/(b*sqrt(a+b*x^2))-5/16*a*(6*A*b-7*a*B)*x*sqrt(a+b*x^2)/b^4+5/24*(6*A*b-7*a*B)*x^3*sqrt(a+b*x^2)/b^3],
[x^5*(A+B*x^2)/(a+b*x^2)^(3/2),x,3,1/3*(A*b-3*a*B)*(a+b*x^2)^(3/2)/b^4+1/5*B*(a+b*x^2)^(5/2)/b^4-a^2*(A*b-a*B)/(b^4*sqrt(a+b*x^2))-a*(2*A*b-3*a*B)*sqrt(a+b*x^2)/b^4],
[x^4*(A+B*x^2)/(a+b*x^2)^(3/2),x,5,-3/8*a*(4*A*b-5*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)-1/4*(4*A*b-5*a*B)*x^3/(b^2*sqrt(a+b*x^2))+1/4*B*x^5/(b*sqrt(a+b*x^2))+3/8*(4*A*b-5*a*B)*x*sqrt(a+b*x^2)/b^3],
[x^3*(A+B*x^2)/(a+b*x^2)^(3/2),x,3,1/3*B*(a+b*x^2)^(3/2)/b^3+a*(A*b-a*B)/(b^3*sqrt(a+b*x^2))+(A*b-2*a*B)*sqrt(a+b*x^2)/b^3],
[x^2*(A+B*x^2)/(a+b*x^2)^(3/2),x,4,1/2*(2*A*b-3*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)-(A*b-a*B)*x/(b^2*sqrt(a+b*x^2))+1/2*B*x*sqrt(a+b*x^2)/b^2],
[x*(A+B*x^2)/(a+b*x^2)^(3/2),x,3,(-A*b+a*B)/(b^2*sqrt(a+b*x^2))+B*sqrt(a+b*x^2)/b^2],
[(A+B*x^2)/(a+b*x^2)^(3/2),x,3,B*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+(A*b-a*B)*x/(a*b*sqrt(a+b*x^2))],
[(A+B*x^2)/(x*(a+b*x^2)^(3/2)),x,4,-A*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(3/2)+(A*b-a*B)/(a*b*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^2*(a+b*x^2)^(3/2)),x,2,-A/(a*x*sqrt(a+b*x^2))-(2*A*b-a*B)*x/(a^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^3*(a+b*x^2)^(3/2)),x,5,1/2*(3*A*b-2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(5/2)+1/2*(-3*A*b+2*a*B)/(a^2*sqrt(a+b*x^2))-1/2*A/(a*x^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^4*(a+b*x^2)^(3/2)),x,3,-1/3*A/(a*x^3*sqrt(a+b*x^2))+1/3*(4*A*b-3*a*B)/(a^2*x*sqrt(a+b*x^2))+2/3*b*(4*A*b-3*a*B)*x/(a^3*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^5*(a+b*x^2)^(3/2)),x,6,-3/8*b*(5*A*b-4*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(7/2)+3/8*b*(5*A*b-4*a*B)/(a^3*sqrt(a+b*x^2))-1/4*A/(a*x^4*sqrt(a+b*x^2))+1/8*(5*A*b-4*a*B)/(a^2*x^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^6*(a+b*x^2)^(3/2)),x,4,-1/5*A/(a*x^5*sqrt(a+b*x^2))+1/15*(6*A*b-5*a*B)/(a^2*x^3*sqrt(a+b*x^2))-4/15*b*(6*A*b-5*a*B)/(a^3*x*sqrt(a+b*x^2))-8/15*b^2*(6*A*b-5*a*B)*x/(a^4*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^7*(a+b*x^2)^(3/2)),x,7,5/16*b^2*(7*A*b-6*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(9/2)-5/16*b^2*(7*A*b-6*a*B)/(a^4*sqrt(a+b*x^2))-1/6*A/(a*x^6*sqrt(a+b*x^2))+1/24*(7*A*b-6*a*B)/(a^2*x^4*sqrt(a+b*x^2))-5/48*b*(7*A*b-6*a*B)/(a^3*x^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^8*(a+b*x^2)^(3/2)),x,5,-1/7*A/(a*x^7*sqrt(a+b*x^2))+1/35*(8*A*b-7*a*B)/(a^2*x^5*sqrt(a+b*x^2))-2/35*b*(8*A*b-7*a*B)/(a^3*x^3*sqrt(a+b*x^2))+8/35*b^2*(8*A*b-7*a*B)/(a^4*x*sqrt(a+b*x^2))+16/35*b^3*(8*A*b-7*a*B)*x/(a^5*sqrt(a+b*x^2))],
[x^7*(A+B*x^2)/(a+b*x^2)^(5/2),x,3,1/3*a^3*(A*b-a*B)/(b^5*(a+b*x^2)^(3/2))+1/3*(A*b-4*a*B)*(a+b*x^2)^(3/2)/b^5+1/5*B*(a+b*x^2)^(5/2)/b^5-a^2*(3*A*b-4*a*B)/(b^5*sqrt(a+b*x^2))-3*a*(A*b-2*a*B)*sqrt(a+b*x^2)/b^5],
[x^6*(A+B*x^2)/(a+b*x^2)^(5/2),x,6,-1/12*(4*A*b-7*a*B)*x^5/(b^2*(a+b*x^2)^(3/2))+1/4*B*x^7/(b*(a+b*x^2)^(3/2))-5/8*a*(4*A*b-7*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(9/2)-5/12*(4*A*b-7*a*B)*x^3/(b^3*sqrt(a+b*x^2))+5/8*(4*A*b-7*a*B)*x*sqrt(a+b*x^2)/b^4],
[x^5*(A+B*x^2)/(a+b*x^2)^(5/2),x,3,-1/3*a^2*(A*b-a*B)/(b^4*(a+b*x^2)^(3/2))+1/3*B*(a+b*x^2)^(3/2)/b^4+a*(2*A*b-3*a*B)/(b^4*sqrt(a+b*x^2))+(A*b-3*a*B)*sqrt(a+b*x^2)/b^4],
[x^4*(A+B*x^2)/(a+b*x^2)^(5/2),x,5,1/3*a*(A*b-a*B)*x/(b^3*(a+b*x^2)^(3/2))+1/2*(2*A*b-5*a*B)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)-1/3*(4*A*b-7*a*B)*x/(b^3*sqrt(a+b*x^2))+1/2*B*x*sqrt(a+b*x^2)/b^3],
[x^3*(A+B*x^2)/(a+b*x^2)^(5/2),x,3,1/3*a*(A*b-a*B)/(b^3*(a+b*x^2)^(3/2))+(-A*b+2*a*B)/(b^3*sqrt(a+b*x^2))+B*sqrt(a+b*x^2)/b^3],
[x^2*(A+B*x^2)/(a+b*x^2)^(5/2),x,4,1/3*(A*b-a*B)*x^3/(a*b*(a+b*x^2)^(3/2))+B*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)-B*x/(b^2*sqrt(a+b*x^2))],
[x*(A+B*x^2)/(a+b*x^2)^(5/2),x,3,1/3*(-A*b+a*B)/(b^2*(a+b*x^2)^(3/2))-B/(b^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(a+b*x^2)^(5/2),x,2,1/3*x*(A+B*x^2)/(a*(a+b*x^2)^(3/2))+2/3*A*x/(a^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(x*(a+b*x^2)^(5/2)),x,5,1/3*(A*b-a*B)/(a*b*(a+b*x^2)^(3/2))-A*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(5/2)+A/(a^2*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^2*(a+b*x^2)^(5/2)),x,3,-A/(a*x*(a+b*x^2)^(3/2))-1/3*(4*A*b-a*B)*x/(a^2*(a+b*x^2)^(3/2))-2/3*(4*A*b-a*B)*x/(a^3*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^3*(a+b*x^2)^(5/2)),x,6,1/6*(-5*A*b+2*a*B)/(a^2*(a+b*x^2)^(3/2))-1/2*A/(a*x^2*(a+b*x^2)^(3/2))+1/2*(5*A*b-2*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(7/2)+1/2*(-5*A*b+2*a*B)/(a^3*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^4*(a+b*x^2)^(5/2)),x,4,-1/3*A/(a*x^3*(a+b*x^2)^(3/2))+(2*A*b-a*B)/(a^2*x*(a+b*x^2)^(3/2))+4/3*b*(2*A*b-a*B)*x/(a^3*(a+b*x^2)^(3/2))+8/3*b*(2*A*b-a*B)*x/(a^4*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^5*(a+b*x^2)^(5/2)),x,7,5/24*b*(7*A*b-4*a*B)/(a^3*(a+b*x^2)^(3/2))-1/4*A/(a*x^4*(a+b*x^2)^(3/2))+1/8*(7*A*b-4*a*B)/(a^2*x^2*(a+b*x^2)^(3/2))-5/8*b*(7*A*b-4*a*B)*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(9/2)+5/8*b*(7*A*b-4*a*B)/(a^4*sqrt(a+b*x^2))],
[(A+B*x^2)/(x^6*(a+b*x^2)^(5/2)),x,5,-1/5*A/(a*x^5*(a+b*x^2)^(3/2))+1/15*(8*A*b-5*a*B)/(a^2*x^3*(a+b*x^2)^(3/2))-2/5*b*(8*A*b-5*a*B)/(a^3*x*(a+b*x^2)^(3/2))-8/15*b^2*(8*A*b-5*a*B)*x/(a^4*(a+b*x^2)^(3/2))-16/15*b^2*(8*A*b-5*a*B)*x/(a^5*sqrt(a+b*x^2))],

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

# p>0
[x^5*(a+b*x^2)^2*sqrt(c+d*x^2),x,3,1/3*c^2*(b*c-a*d)^2*(c+d*x^2)^(3/2)/d^5-2/5*c*(b*c-a*d)*(2*b*c-a*d)*(c+d*x^2)^(5/2)/d^5+1/7*(6*b^2*c^2-6*a*b*c*d+a^2*d^2)*(c+d*x^2)^(7/2)/d^5-2/9*b*(2*b*c-a*d)*(c+d*x^2)^(9/2)/d^5+1/11*b^2*(c+d*x^2)^(11/2)/d^5],
[x^3*(a+b*x^2)^2*sqrt(c+d*x^2),x,3,-1/3*c*(b*c-a*d)^2*(c+d*x^2)^(3/2)/d^4+1/5*(b*c-a*d)*(3*b*c-a*d)*(c+d*x^2)^(5/2)/d^4-1/7*b*(3*b*c-2*a*d)*(c+d*x^2)^(7/2)/d^4+1/9*b^2*(c+d*x^2)^(9/2)/d^4],
[x*(a+b*x^2)^2*sqrt(c+d*x^2),x,3,1/3*(b*c-a*d)^2*(c+d*x^2)^(3/2)/d^3-2/5*b*(b*c-a*d)*(c+d*x^2)^(5/2)/d^3+1/7*b^2*(c+d*x^2)^(7/2)/d^3],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x,x,6,-1/3*b*(b*c-2*a*d)*(c+d*x^2)^(3/2)/d^2+1/5*b^2*(c+d*x^2)^(5/2)/d^2-a^2*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)+a^2*sqrt(c+d*x^2)],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^3,x,6,1/3*b^2*(c+d*x^2)^(3/2)/d-1/2*a^2*(c+d*x^2)^(3/2)/(c*x^2)-1/2*a*(4*b*c+a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/sqrt(c)+1/2*a*(4*b*c+a*d)*sqrt(c+d*x^2)/c],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^5,x,6,-1/4*a^2*(c+d*x^2)^(3/2)/(c*x^4)-1/8*a*(8*b*c-a*d)*(c+d*x^2)^(3/2)/(c^2*x^2)-1/8*(8*b^2*c^2+a*d*(8*b*c-a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(3/2)+1/8*(8*b^2*c^2+a*d*(8*b*c-a*d))*sqrt(c+d*x^2)/c^2],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^7,x,6,-1/6*a^2*(c+d*x^2)^(3/2)/(c*x^6)-1/8*a*(4*b*c-a*d)*(c+d*x^2)^(3/2)/(c^2*x^4)-1/16*d*(8*b^2*c^2-4*a*b*c*d+a^2*d^2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(5/2)-1/16*(8*b^2*c^2-4*a*b*c*d+a^2*d^2)*sqrt(c+d*x^2)/(c^2*x^2)],
[x^2*(a+b*x^2)^2*sqrt(c+d*x^2),x,6,-1/48*b*(5*b*c-16*a*d)*x^3*(c+d*x^2)^(3/2)/d^2+1/8*b^2*x^5*(c+d*x^2)^(3/2)/d-1/128*c^2*(16*a^2*d^2+b*c*(5*b*c-16*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(7/2)+1/128*c*(16*a^2*d^2+b*c*(5*b*c-16*a*d))*x*sqrt(c+d*x^2)/d^3+1/64*(16*a^2*d^2+b*c*(5*b*c-16*a*d))*x^3*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^2*sqrt(c+d*x^2),x,5,-1/24*b*(3*b*c-8*a*d)*x*(c+d*x^2)^(3/2)/d^2+1/6*b*x*(a+b*x^2)*(c+d*x^2)^(3/2)/d+1/16*c*(b^2*c^2-4*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(5/2)+1/16*(b^2*c^2-4*a*b*c*d+8*a^2*d^2)*x*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^2,x,5,-a^2*(c+d*x^2)^(3/2)/(c*x)+1/4*b^2*x*(c+d*x^2)^(3/2)/d-1/8*(b^2*c^2-8*a*d*(b*c+a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(3/2)-1/8*(b^2*c^2-8*a*d*(b*c+a*d))*x*sqrt(c+d*x^2)/(c*d)],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^4,x,5,-1/3*a^2*(c+d*x^2)^(3/2)/(c*x^3)-2*a*b*(c+d*x^2)^(3/2)/(c*x)+1/2*b*(b*c+4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/sqrt(d)+1/2*b*(b*c+4*a*d)*x*sqrt(c+d*x^2)/c],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^6,x,5,-1/5*a^2*(c+d*x^2)^(3/2)/(c*x^5)-2/15*a*(5*b*c-a*d)*(c+d*x^2)^(3/2)/(c^2*x^3)+b^2*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)-b^2*sqrt(c+d*x^2)/x],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^8,x,3,-1/7*a^2*(c+d*x^2)^(3/2)/(c*x^7)-2/35*a*(7*b*c-2*a*d)*(c+d*x^2)^(3/2)/(c^2*x^5)-1/105*(35*b^2*c^2-4*a*d*(7*b*c-2*a*d))*(c+d*x^2)^(3/2)/(c^3*x^3)],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^10,x,4,-1/9*a^2*(c+d*x^2)^(3/2)/(c*x^9)-2/21*a*(3*b*c-a*d)*(c+d*x^2)^(3/2)/(c^2*x^7)-1/105*(21*b^2*c^2-8*a*d*(3*b*c-a*d))*(c+d*x^2)^(3/2)/(c^3*x^5)+2/315*d*(21*b^2*c^2-8*a*d*(3*b*c-a*d))*(c+d*x^2)^(3/2)/(c^4*x^3)],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^12,x,5,-1/11*a^2*(c+d*x^2)^(3/2)/(c*x^11)-2/99*a*(11*b*c-4*a*d)*(c+d*x^2)^(3/2)/(c^2*x^9)-1/231*(33*b^2*c^2-4*a*d*(11*b*c-4*a*d))*(c+d*x^2)^(3/2)/(c^3*x^7)+4/1155*d*(33*b^2*c^2-4*a*d*(11*b*c-4*a*d))*(c+d*x^2)^(3/2)/(c^4*x^5)-8/3465*d^2*(33*b^2*c^2-4*a*d*(11*b*c-4*a*d))*(c+d*x^2)^(3/2)/(c^5*x^3)],
[x^4*(a+b*x^2)^2*(c+d*x^2)^(3/2),x,8,1/192*(24*a^2*d^2+b*c*(7*b*c-24*a*d))*x^5*(c+d*x^2)^(3/2)/d^2-1/120*b*(7*b*c-24*a*d)*x^5*(c+d*x^2)^(5/2)/d^2+1/12*b^2*x^7*(c+d*x^2)^(5/2)/d+1/1024*c^4*(24*a^2*d^2+b*c*(7*b*c-24*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(9/2)-1/1024*c^3*(24*a^2*d^2+b*c*(7*b*c-24*a*d))*x*sqrt(c+d*x^2)/d^4+1/1536*c^2*(24*a^2*d^2+b*c*(7*b*c-24*a*d))*x^3*sqrt(c+d*x^2)/d^3+1/384*c*(24*a^2*d^2+b*c*(7*b*c-24*a*d))*x^5*sqrt(c+d*x^2)/d^2],
[x^3*(a+b*x^2)^2*(c+d*x^2)^(3/2),x,3,-1/5*c*(b*c-a*d)^2*(c+d*x^2)^(5/2)/d^4+1/7*(b*c-a*d)*(3*b*c-a*d)*(c+d*x^2)^(7/2)/d^4-1/9*b*(3*b*c-2*a*d)*(c+d*x^2)^(9/2)/d^4+1/11*b^2*(c+d*x^2)^(11/2)/d^4],
[x^2*(a+b*x^2)^2*(c+d*x^2)^(3/2),x,7,1/96*(16*a^2*d^2+3*b*c*(b*c-4*a*d))*x^3*(c+d*x^2)^(3/2)/d^2-1/16*b*(b*c-4*a*d)*x^3*(c+d*x^2)^(5/2)/d^2+1/10*b^2*x^5*(c+d*x^2)^(5/2)/d-1/256*c^3*(16*a^2*d^2+3*b*c*(b*c-4*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(7/2)+1/256*c^2*(16*a^2*d^2+3*b*c*(b*c-4*a*d))*x*sqrt(c+d*x^2)/d^3+1/128*c*(16*a^2*d^2+3*b*c*(b*c-4*a*d))*x^3*sqrt(c+d*x^2)/d^2],
[x*(a+b*x^2)^2*(c+d*x^2)^(3/2),x,3,1/5*(b*c-a*d)^2*(c+d*x^2)^(5/2)/d^3-2/7*b*(b*c-a*d)*(c+d*x^2)^(7/2)/d^3+1/9*b^2*(c+d*x^2)^(9/2)/d^3],
[(a+b*x^2)^2*(c+d*x^2)^(3/2),x,6,1/192*(3*b^2*c^2-16*a*b*c*d+48*a^2*d^2)*x*(c+d*x^2)^(3/2)/d^2-1/48*b*(3*b*c-10*a*d)*x*(c+d*x^2)^(5/2)/d^2+1/8*b*x*(a+b*x^2)*(c+d*x^2)^(5/2)/d+1/128*c^2*(3*b^2*c^2-16*a*b*c*d+48*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(5/2)+1/128*c*(3*b^2*c^2-16*a*b*c*d+48*a^2*d^2)*x*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x,x,7,1/3*a^2*(c+d*x^2)^(3/2)-1/5*b*(b*c-2*a*d)*(c+d*x^2)^(5/2)/d^2+1/7*b^2*(c+d*x^2)^(7/2)/d^2-a^2*c^(3/2)*arctanh(sqrt(c+d*x^2)/sqrt(c))+a^2*c*sqrt(c+d*x^2)],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x^2,x,6,-1/24*(b^2*c^2-12*a*d*(b*c+2*a*d))*x*(c+d*x^2)^(3/2)/(c*d)-a^2*(c+d*x^2)^(5/2)/(c*x)+1/6*b^2*x*(c+d*x^2)^(5/2)/d-1/16*c*(b^2*c^2-12*a*d*(b*c+2*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(3/2)-1/16*(b^2*c^2-12*a*d*(b*c+2*a*d))*x*sqrt(c+d*x^2)/d],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x^3,x,7,1/6*a*(4*b*c+3*a*d)*(c+d*x^2)^(3/2)/c+1/5*b^2*(c+d*x^2)^(5/2)/d-1/2*a^2*(c+d*x^2)^(5/2)/(c*x^2)-1/2*a*(4*b*c+3*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)+1/2*a*(4*b*c+3*a*d)*sqrt(c+d*x^2)],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x^4,x,6,1/12*(3*b^2*c^2+8*a*d*(3*b*c+a*d))*x*(c+d*x^2)^(3/2)/c^2-1/3*a^2*(c+d*x^2)^(5/2)/(c*x^3)-2/3*a*(3*b*c+a*d)*(c+d*x^2)^(5/2)/(c^2*x)+1/8*(3*b^2*c^2+8*a*d*(3*b*c+a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/sqrt(d)+1/8*(3*b^2*c^2+8*a*d*(3*b*c+a*d))*x*sqrt(c+d*x^2)/c],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x^5,x,7,1/24*(8*b^2*c^2+3*a*d*(8*b*c+a*d))*(c+d*x^2)^(3/2)/c^2-1/4*a^2*(c+d*x^2)^(5/2)/(c*x^4)-1/8*a*(8*b*c+a*d)*(c+d*x^2)^(5/2)/(c^2*x^2)-1/8*(8*b^2*c^2+3*a*d*(8*b*c+a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/sqrt(c)+1/8*(8*b^2*c^2+3*a*d*(8*b*c+a*d))*sqrt(c+d*x^2)/c],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x^6,x,6,-1/3*b*(3*b*c+4*a*d)*(c+d*x^2)^(3/2)/(c*x)-1/5*a^2*(c+d*x^2)^(5/2)/(c*x^5)-2/3*a*b*(c+d*x^2)^(5/2)/(c*x^3)+1/2*b*(3*b*c+4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)+1/2*b*d*(3*b*c+4*a*d)*x*sqrt(c+d*x^2)/c],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/x^7,x,7,-1/48*(24*b^2*c^2+a*d*(12*b*c-a*d))*(c+d*x^2)^(3/2)/(c^2*x^2)-1/6*a^2*(c+d*x^2)^(5/2)/(c*x^6)-1/24*a*(12*b*c-a*d)*(c+d*x^2)^(5/2)/(c^2*x^4)-1/16*d*(24*b^2*c^2+a*d*(12*b*c-a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(3/2)+1/16*d*(24*b^2*c^2+a*d*(12*b*c-a*d))*sqrt(c+d*x^2)/c^2],
[x^3*(a+b*x^2)^2*(c+d*x^2)^(5/2),x,3,-1/7*c*(b*c-a*d)^2*(c+d*x^2)^(7/2)/d^4+1/9*(b*c-a*d)*(3*b*c-a*d)*(c+d*x^2)^(9/2)/d^4-1/11*b*(3*b*c-2*a*d)*(c+d*x^2)^(11/2)/d^4+1/13*b^2*(c+d*x^2)^(13/2)/d^4],
[x^2*(a+b*x^2)^2*(c+d*x^2)^(5/2),x,8,1/384*c*(40*a^2*d^2+b*c*(5*b*c-24*a*d))*x^3*(c+d*x^2)^(3/2)/d^2+1/320*(40*a^2*d^2+b*c*(5*b*c-24*a*d))*x^3*(c+d*x^2)^(5/2)/d^2-1/120*b*(5*b*c-24*a*d)*x^3*(c+d*x^2)^(7/2)/d^2+1/12*b^2*x^5*(c+d*x^2)^(7/2)/d-1/1024*c^4*(40*a^2*d^2+b*c*(5*b*c-24*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(7/2)+1/1024*c^3*(40*a^2*d^2+b*c*(5*b*c-24*a*d))*x*sqrt(c+d*x^2)/d^3+1/512*c^2*(40*a^2*d^2+b*c*(5*b*c-24*a*d))*x^3*sqrt(c+d*x^2)/d^2],
[x*(a+b*x^2)^2*(c+d*x^2)^(5/2),x,3,1/7*(b*c-a*d)^2*(c+d*x^2)^(7/2)/d^3-2/9*b*(b*c-a*d)*(c+d*x^2)^(9/2)/d^3+1/11*b^2*(c+d*x^2)^(11/2)/d^3],
[(a+b*x^2)^2*(c+d*x^2)^(5/2),x,7,1/384*c*(3*b^2*c^2-20*a*b*c*d+80*a^2*d^2)*x*(c+d*x^2)^(3/2)/d^2+1/480*(3*b^2*c^2-20*a*b*c*d+80*a^2*d^2)*x*(c+d*x^2)^(5/2)/d^2-3/80*b*(b*c-4*a*d)*x*(c+d*x^2)^(7/2)/d^2+1/10*b*x*(a+b*x^2)*(c+d*x^2)^(7/2)/d+1/256*c^3*(3*b^2*c^2-20*a*b*c*d+80*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(5/2)+1/256*c^2*(3*b^2*c^2-20*a*b*c*d+80*a^2*d^2)*x*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x,x,8,1/3*a^2*c*(c+d*x^2)^(3/2)+1/5*a^2*(c+d*x^2)^(5/2)-1/7*b*(b*c-2*a*d)*(c+d*x^2)^(7/2)/d^2+1/9*b^2*(c+d*x^2)^(9/2)/d^2-a^2*c^(5/2)*arctanh(sqrt(c+d*x^2)/sqrt(c))+a^2*c^2*sqrt(c+d*x^2)],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x^2,x,7,-5/192*(b^2*c^2-16*a*d*(b*c+3*a*d))*x*(c+d*x^2)^(3/2)/d-1/48*(b^2*c^2-16*a*d*(b*c+3*a*d))*x*(c+d*x^2)^(5/2)/(c*d)-a^2*(c+d*x^2)^(7/2)/(c*x)+1/8*b^2*x*(c+d*x^2)^(7/2)/d-5/128*c^2*(b^2*c^2-16*a*d*(b*c+3*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(3/2)-5/128*c*(b^2*c^2-16*a*d*(b*c+3*a*d))*x*sqrt(c+d*x^2)/d],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x^3,x,8,1/6*a*(4*b*c+5*a*d)*(c+d*x^2)^(3/2)+1/10*a*(4*b*c+5*a*d)*(c+d*x^2)^(5/2)/c+1/7*b^2*(c+d*x^2)^(7/2)/d-1/2*a^2*(c+d*x^2)^(7/2)/(c*x^2)-1/2*a*c^(3/2)*(4*b*c+5*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))+1/2*a*c*(4*b*c+5*a*d)*sqrt(c+d*x^2)],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x^4,x,7,5/24*(b^2*c^2+4*a*d*(3*b*c+2*a*d))*x*(c+d*x^2)^(3/2)/c+1/6*(b^2*c^2+4*a*d*(3*b*c+2*a*d))*x*(c+d*x^2)^(5/2)/c^2-1/3*a^2*(c+d*x^2)^(7/2)/(c*x^3)-2/3*a*(3*b*c+2*a*d)*(c+d*x^2)^(7/2)/(c^2*x)+5/16*c*(b^2*c^2+4*a*d*(3*b*c+2*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/sqrt(d)+5/16*(b^2*c^2+4*a*d*(3*b*c+2*a*d))*x*sqrt(c+d*x^2)],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x^5,x,8,1/24*(8*b^2*c^2+5*a*d*(8*b*c+3*a*d))*(c+d*x^2)^(3/2)/c+1/40*(8*b^2*c^2+5*a*d*(8*b*c+3*a*d))*(c+d*x^2)^(5/2)/c^2-1/4*a^2*(c+d*x^2)^(7/2)/(c*x^4)-1/8*a*(8*b*c+3*a*d)*(c+d*x^2)^(7/2)/(c^2*x^2)-1/8*(8*b^2*c^2+5*a*d*(8*b*c+3*a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)+1/8*(8*b^2*c^2+5*a*d*(8*b*c+3*a*d))*sqrt(c+d*x^2)],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x^6,x,7,1/12*d*(15*b^2*c^2+8*a*d*(5*b*c+a*d))*x*(c+d*x^2)^(3/2)/c^2-1/15*(15*b^2*c^2+8*a*d*(5*b*c+a*d))*(c+d*x^2)^(5/2)/(c^2*x)-1/5*a^2*(c+d*x^2)^(7/2)/(c*x^5)-2/15*a*(5*b*c+a*d)*(c+d*x^2)^(7/2)/(c^2*x^3)+1/8*(15*b^2*c^2+8*a*d*(5*b*c+a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)+1/8*d*(15*b^2*c^2+8*a*d*(5*b*c+a*d))*x*sqrt(c+d*x^2)/c],
[(a+b*x^2)^2*(c+d*x^2)^(5/2)/x^7,x,8,5/48*d*(8*b^2*c^2+a*d*(12*b*c+a*d))*(c+d*x^2)^(3/2)/c^2-1/16*(8*b^2*c^2+a*d*(12*b*c+a*d))*(c+d*x^2)^(5/2)/(c^2*x^2)-1/6*a^2*(c+d*x^2)^(7/2)/(c*x^6)-1/24*a*(12*b*c+a*d)*(c+d*x^2)^(7/2)/(c^2*x^4)-5/16*d*(8*b^2*c^2+a*d*(12*b*c+a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/sqrt(c)+5/16*d*(8*b^2*c^2+a*d*(12*b*c+a*d))*sqrt(c+d*x^2)/c],

# p<0
[x^4*(a+b*x^2)^2/sqrt(c+d*x^2),x,6,1/128*c^2*(48*a^2*d^2+5*b*c*(7*b*c-16*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(9/2)-1/128*c*(48*a^2*d^2+5*b*c*(7*b*c-16*a*d))*x*sqrt(c+d*x^2)/d^4+1/192*(48*a^2*d^2+5*b*c*(7*b*c-16*a*d))*x^3*sqrt(c+d*x^2)/d^3-1/48*b*(7*b*c-16*a*d)*x^5*sqrt(c+d*x^2)/d^2+1/8*b^2*x^7*sqrt(c+d*x^2)/d],
[x^3*(a+b*x^2)^2/sqrt(c+d*x^2),x,3,1/3*(b*c-a*d)*(3*b*c-a*d)*(c+d*x^2)^(3/2)/d^4-1/5*b*(3*b*c-2*a*d)*(c+d*x^2)^(5/2)/d^4+1/7*b^2*(c+d*x^2)^(7/2)/d^4-c*(b*c-a*d)^2*sqrt(c+d*x^2)/d^4],
[x^2*(a+b*x^2)^2/sqrt(c+d*x^2),x,5,-1/16*c*(8*a^2*d^2+b*c*(5*b*c-12*a*d))*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(7/2)+1/16*(8*a^2*d^2+b*c*(5*b*c-12*a*d))*x*sqrt(c+d*x^2)/d^3-1/24*b*(5*b*c-12*a*d)*x^3*sqrt(c+d*x^2)/d^2+1/6*b^2*x^5*sqrt(c+d*x^2)/d],
[x*(a+b*x^2)^2/sqrt(c+d*x^2),x,3,-2/3*b*(b*c-a*d)*(c+d*x^2)^(3/2)/d^3+1/5*b^2*(c+d*x^2)^(5/2)/d^3+(b*c-a*d)^2*sqrt(c+d*x^2)/d^3],
[(a+b*x^2)^2/sqrt(c+d*x^2),x,4,1/8*(3*b^2*c^2-8*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(5/2)-3/8*b*(b*c-2*a*d)*x*sqrt(c+d*x^2)/d^2+1/4*b*x*(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^2/(x*sqrt(c+d*x^2)),x,5,1/3*b^2*(c+d*x^2)^(3/2)/d^2-a^2*arctanh(sqrt(c+d*x^2)/sqrt(c))/sqrt(c)-b*(b*c-2*a*d)*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^2/(x^2*sqrt(c+d*x^2)),x,4,-1/2*b*(b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(3/2)-a^2*sqrt(c+d*x^2)/(c*x)+1/2*b^2*x*sqrt(c+d*x^2)/d],
[(a+b*x^2)^2/(x^3*sqrt(c+d*x^2)),x,5,-1/2*a*(4*b*c-a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(3/2)+b^2*sqrt(c+d*x^2)/d-1/2*a^2*sqrt(c+d*x^2)/(c*x^2)],
[(a+b*x^2)^2/(x^4*sqrt(c+d*x^2)),x,4,b^2*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/sqrt(d)-1/3*a^2*sqrt(c+d*x^2)/(c*x^3)-2/3*a*(3*b*c-a*d)*sqrt(c+d*x^2)/(c^2*x)],
[(a+b*x^2)^2/(x^5*sqrt(c+d*x^2)),x,5,-1/8*(8*b^2*c^2-8*a*b*c*d+3*a^2*d^2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(5/2)-1/4*a^2*sqrt(c+d*x^2)/(c*x^4)-1/8*a*(8*b*c-3*a*d)*sqrt(c+d*x^2)/(c^2*x^2)],
[(a+b*x^2)^2/(x^6*sqrt(c+d*x^2)),x,3,-1/5*a^2*sqrt(c+d*x^2)/(c*x^5)-2/15*a*(5*b*c-2*a*d)*sqrt(c+d*x^2)/(c^2*x^3)-1/15*(15*b^2*c^2-4*a*d*(5*b*c-2*a*d))*sqrt(c+d*x^2)/(c^3*x)],
[(a+b*x^2)^2/(x^7*sqrt(c+d*x^2)),x,6,1/16*d*(8*b^2*c^2-12*a*b*c*d+5*a^2*d^2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(7/2)-1/6*a^2*sqrt(c+d*x^2)/(c*x^6)-1/24*a*(12*b*c-5*a*d)*sqrt(c+d*x^2)/(c^2*x^4)-1/16*(8*b^2*c^2-12*a*b*c*d+5*a^2*d^2)*sqrt(c+d*x^2)/(c^3*x^2)],
[x^4*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,6,-1/16*c*(35*b^2*c^2-60*a*b*c*d+24*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(9/2)+(b*c-a*d)^2*x^5/(c*d^2*sqrt(c+d*x^2))+1/16*(35*b^2*c^2-60*a*b*c*d+24*a^2*d^2)*x*sqrt(c+d*x^2)/d^4-1/24*(35*b^2*c^2-60*a*b*c*d+24*a^2*d^2)*x^3*sqrt(c+d*x^2)/(c*d^3)+1/6*b^2*x^5*sqrt(c+d*x^2)/d^2],
[x^3*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,3,-1/3*b*(3*b*c-2*a*d)*(c+d*x^2)^(3/2)/d^4+1/5*b^2*(c+d*x^2)^(5/2)/d^4+c*(b*c-a*d)^2/(d^4*sqrt(c+d*x^2))+(b*c-a*d)*(3*b*c-a*d)*sqrt(c+d*x^2)/d^4],
[x^2*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,5,1/8*(15*b^2*c^2-24*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(7/2)+(b*c-a*d)^2*x^3/(c*d^2*sqrt(c+d*x^2))-1/8*(15*b^2*c^2-24*a*b*c*d+8*a^2*d^2)*x*sqrt(c+d*x^2)/(c*d^3)+1/4*b^2*x^3*sqrt(c+d*x^2)/d^2],
[x*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,3,1/3*b^2*(c+d*x^2)^(3/2)/d^3-(b*c-a*d)^2/(d^3*sqrt(c+d*x^2))-2*b*(b*c-a*d)*sqrt(c+d*x^2)/d^3],
[(a+b*x^2)^2/(c+d*x^2)^(3/2),x,4,-1/2*b*(3*b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(5/2)-(b*c-a*d)*x*(a+b*x^2)/(c*d*sqrt(c+d*x^2))+1/2*b*(3*b*c-2*a*d)*x*sqrt(c+d*x^2)/(c*d^2)],
[(a+b*x^2)^2/(x*(c+d*x^2)^(3/2)),x,5,-a^2*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(3/2)+(b*c-a*d)^2/(c*d^2*sqrt(c+d*x^2))+b^2*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^2/(x^2*(c+d*x^2)^(3/2)),x,4,b^2*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(3/2)-a^2/(c*x*sqrt(c+d*x^2))-(b^2*c^2-2*a*d*(b*c-a*d))*x/(c^2*d*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^3*(c+d*x^2)^(3/2)),x,5,-1/2*a*(4*b*c-3*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(5/2)+1/2*(4*a*b-2*b^2*c/d-3*a^2*d/c)/(c*sqrt(c+d*x^2))-1/2*a^2/(c*x^2*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^4*(c+d*x^2)^(3/2)),x,3,-1/3*a^2/(c*x^3*sqrt(c+d*x^2))-2/3*a*(3*b*c-2*a*d)/(c^2*x*sqrt(c+d*x^2))+1/3*(3*b^2*c^2-4*a*d*(3*b*c-2*a*d))*x/(c^3*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^5*(c+d*x^2)^(3/2)),x,6,-1/8*(8*b^2*c^2-3*a*d*(8*b*c-5*a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(7/2)+1/8*(8*b^2*c^2-3*a*d*(8*b*c-5*a*d))/(c^3*sqrt(c+d*x^2))-1/4*a^2/(c*x^4*sqrt(c+d*x^2))-1/8*a*(8*b*c-5*a*d)/(c^2*x^2*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^6*(c+d*x^2)^(3/2)),x,4,-1/5*a^2/(c*x^5*sqrt(c+d*x^2))-2/15*a*(5*b*c-3*a*d)/(c^2*x^3*sqrt(c+d*x^2))+1/15*(-15*b^2*c^2+8*a*d*(5*b*c-3*a*d))/(c^3*x*sqrt(c+d*x^2))-2/15*d*(15*b^2*c^2-8*a*d*(5*b*c-3*a*d))*x/(c^4*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^7*(c+d*x^2)^(3/2)),x,7,1/16*d*(24*b^2*c^2-5*a*d*(12*b*c-7*a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(9/2)-1/16*d*(24*b^2*c^2-5*a*d*(12*b*c-7*a*d))/(c^4*sqrt(c+d*x^2))-1/6*a^2/(c*x^6*sqrt(c+d*x^2))-1/24*a*(12*b*c-7*a*d)/(c^2*x^4*sqrt(c+d*x^2))+1/48*(-24*b^2*c^2+5*a*d*(12*b*c-7*a*d))/(c^3*x^2*sqrt(c+d*x^2))],
[x^4*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,6,1/3*(b*c-a*d)^2*x^5/(c*d^2*(c+d*x^2)^(3/2))+1/8*(35*b^2*c^2-40*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(9/2)+1/12*(35*b^2*c^2-40*a*b*c*d+8*a^2*d^2)*x^3/(c*d^3*sqrt(c+d*x^2))+1/4*b^2*x^5/(d^2*sqrt(c+d*x^2))-1/8*(35*b^2*c^2-40*a*b*c*d+8*a^2*d^2)*x*sqrt(c+d*x^2)/(c*d^4)],
[x^3*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,3,1/3*c*(b*c-a*d)^2/(d^4*(c+d*x^2)^(3/2))+1/3*b^2*(c+d*x^2)^(3/2)/d^4-(b*c-a*d)*(3*b*c-a*d)/(d^4*sqrt(c+d*x^2))-b*(3*b*c-2*a*d)*sqrt(c+d*x^2)/d^4],
[x^2*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,5,1/3*(b*c-a*d)^2*x^3/(c*d^2*(c+d*x^2)^(3/2))-1/2*b*(5*b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(7/2)+2*b*(b*c-a*d)*x/(d^3*sqrt(c+d*x^2))+1/2*b^2*x*sqrt(c+d*x^2)/d^3],
[x*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,3,-1/3*(b*c-a*d)^2/(d^3*(c+d*x^2)^(3/2))+2*b*(b*c-a*d)/(d^3*sqrt(c+d*x^2))+b^2*sqrt(c+d*x^2)/d^3],
[(a+b*x^2)^2/(c+d*x^2)^(5/2),x,4,-1/3*(b*c-a*d)*x*(a+b*x^2)/(c*d*(c+d*x^2)^(3/2))+b^2*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/d^(5/2)-1/3*(b*c-a*d)*(3*b*c+2*a*d)*x/(c^2*d^2*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x*(c+d*x^2)^(5/2)),x,5,1/3*(b*c-a*d)^2/(c*d^2*(c+d*x^2)^(3/2))-a^2*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(5/2)+(a^2/c^2-b^2/d^2)/sqrt(c+d*x^2)],
[(a+b*x^2)^2/(x^2*(c+d*x^2)^(5/2)),x,3,-a^2/(c*x*(c+d*x^2)^(3/2))+1/3*x*(2*a*(b*c-2*a*d)+b^2*c*x^2)/(c^2*(c+d*x^2)^(3/2))+4/3*a*(b*c-2*a*d)*x/(c^3*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^3*(c+d*x^2)^(5/2)),x,6,1/6*(4*a*b-2*b^2*c/d-5*a^2*d/c)/(c*(c+d*x^2)^(3/2))-1/2*a^2/(c*x^2*(c+d*x^2)^(3/2))-1/2*a*(4*b*c-5*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(7/2)+1/2*a*(4*b*c-5*a*d)/(c^3*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^4*(c+d*x^2)^(5/2)),x,4,-1/3*a^2/(c*x^3*(c+d*x^2)^(3/2))-2*a*(b*c-a*d)/(c^2*x*(c+d*x^2)^(3/2))+1/3*(b^2*c^2-8*a*d*(b*c-a*d))*x/(c^3*(c+d*x^2)^(3/2))+2/3*(b^2*c^2-8*a*d*(b*c-a*d))*x/(c^4*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^5*(c+d*x^2)^(5/2)),x,7,1/24*(8*b^2*c^2-5*a*d*(8*b*c-7*a*d))/(c^3*(c+d*x^2)^(3/2))-1/4*a^2/(c*x^4*(c+d*x^2)^(3/2))-1/8*a*(8*b*c-7*a*d)/(c^2*x^2*(c+d*x^2)^(3/2))-1/8*(8*b^2*c^2-5*a*d*(8*b*c-7*a*d))*arctanh(sqrt(c+d*x^2)/sqrt(c))/c^(9/2)+1/8*(8*b^2*c^2-5*a*d*(8*b*c-7*a*d))/(c^4*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(x^6*(c+d*x^2)^(5/2)),x,5,-1/5*a^2/(c*x^5*(c+d*x^2)^(3/2))-2/15*a*(5*b*c-4*a*d)/(c^2*x^3*(c+d*x^2)^(3/2))+1/5*(-5*b^2*c^2+4*a*d*(5*b*c-4*a*d))/(c^3*x*(c+d*x^2)^(3/2))-4/15*d*(5*b^2*c^2-4*a*d*(5*b*c-4*a*d))*x/(c^4*(c+d*x^2)^(3/2))-8/15*d*(5*b^2*c^2-4*a*d*(5*b*c-4*a*d))*x/(c^5*sqrt(c+d*x^2))],

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

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

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

# p>0
[x^4*sqrt(c+d*x^2)/(a+b*x^2),x,7,-1/8*(b^2*c^2+4*a*b*c*d-8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^3*d^(3/2))+a^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/b^3+1/8*(b*c-4*a*d)*x*sqrt(c+d*x^2)/(b^2*d)+1/4*x^3*sqrt(c+d*x^2)/b],
[x^3*sqrt(c+d*x^2)/(a+b*x^2),x,5,1/3*(c+d*x^2)^(3/2)/(b*d)+a*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/b^(5/2)-a*sqrt(c+d*x^2)/b^2],
[x^2*sqrt(c+d*x^2)/(a+b*x^2),x,6,1/2*(b*c-2*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^2*sqrt(d))-arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)*sqrt(b*c-a*d)/b^2+1/2*x*sqrt(c+d*x^2)/b],
[x*sqrt(c+d*x^2)/(a+b*x^2),x,4,-arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/b^(3/2)+sqrt(c+d*x^2)/b],
[sqrt(c+d*x^2)/(a+b*x^2),x,5,arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)/b+arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/(b*sqrt(a))],
[sqrt(c+d*x^2)/(x*(a+b*x^2)),x,6,-arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)/a+arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(a*sqrt(b))],
[sqrt(c+d*x^2)/(x^2*(a+b*x^2)),x,4,-arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/a^(3/2)-sqrt(c+d*x^2)/(a*x)],
[sqrt(c+d*x^2)/(x^3*(a+b*x^2)),x,7,1/2*(2*b*c-a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^2*sqrt(c))-arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b)*sqrt(b*c-a*d)/a^2-1/2*sqrt(c+d*x^2)/(a*x^2)],
[sqrt(c+d*x^2)/(x^4*(a+b*x^2)),x,5,b*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/a^(5/2)-1/3*sqrt(c+d*x^2)/(a*x^3)+1/3*(3*b*c-a*d)*sqrt(c+d*x^2)/(a^2*c*x)],
[x^4*(c+d*x^2)^(3/2)/(a+b*x^2),x,8,a^(3/2)*(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/b^4-1/16*(b*c-2*a*d)*(b^2*c^2+8*a*b*c*d-8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^4*d^(3/2))+1/16*(b^2*c^2-10*a*b*c*d+8*a^2*d^2)*x*sqrt(c+d*x^2)/(b^3*d)+1/24*(7*b*c-6*a*d)*x^3*sqrt(c+d*x^2)/b^2+1/6*d*x^5*sqrt(c+d*x^2)/b],
[x^3*(c+d*x^2)^(3/2)/(a+b*x^2),x,6,-1/3*a*(c+d*x^2)^(3/2)/b^2+1/5*(c+d*x^2)^(5/2)/(b*d)+a*(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/b^(7/2)-a*(b*c-a*d)*sqrt(c+d*x^2)/b^3],
[x^2*(c+d*x^2)^(3/2)/(a+b*x^2),x,7,-(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/b^3+1/8*(3*b^2*c^2-12*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^3*sqrt(d))+1/8*(5*b*c-4*a*d)*x*sqrt(c+d*x^2)/b^2+1/4*d*x^3*sqrt(c+d*x^2)/b],
[x*(c+d*x^2)^(3/2)/(a+b*x^2),x,5,1/3*(c+d*x^2)^(3/2)/b-(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/b^(5/2)+(b*c-a*d)*sqrt(c+d*x^2)/b^2],
[(c+d*x^2)^(3/2)/(a+b*x^2),x,6,(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(b^2*sqrt(a))+1/2*(3*b*c-2*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)/b^2+1/2*d*x*sqrt(c+d*x^2)/b],
[(c+d*x^2)^(3/2)/(x*(a+b*x^2)),x,7,-c^(3/2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/a+(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a*b^(3/2))+d*sqrt(c+d*x^2)/b],
[(c+d*x^2)^(3/2)/(x^2*(a+b*x^2)),x,6,-(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*b)+d^(3/2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/b-c*sqrt(c+d*x^2)/(a*x)],
[(c+d*x^2)^(3/2)/(x^3*(a+b*x^2)),x,7,-(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^2*sqrt(b))+1/2*(2*b*c-3*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)/a^2-1/2*c*sqrt(c+d*x^2)/(a*x^2)],
[(c+d*x^2)^(3/2)/(x^4*(a+b*x^2)),x,5,(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/a^(5/2)-1/3*c*sqrt(c+d*x^2)/(a*x^3)+1/3*(3*b*c-4*a*d)*sqrt(c+d*x^2)/(a^2*x)],
[x^4*(c+d*x^2)^(5/2)/(a+b*x^2),x,9,1/8*d*x^5*(c+d*x^2)^(3/2)/b+a^(3/2)*(b*c-a*d)^(5/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/b^5-1/128*(5*b^4*c^4+40*a*b^3*c^3*d-240*a^2*b^2*c^2*d^2+320*a^3*b*c*d^3-128*a^4*d^4)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^5*d^(3/2))+1/128*(5*b^3*c^3-88*a*b^2*c^2*d+144*a^2*b*c*d^2-64*a^3*d^3)*x*sqrt(c+d*x^2)/(b^4*d)+1/192*(59*b^2*c^2-104*a*b*c*d+48*a^2*d^2)*x^3*sqrt(c+d*x^2)/b^3+1/48*d*(11*b*c-8*a*d)*x^5*sqrt(c+d*x^2)/b^2],
[x^3*(c+d*x^2)^(5/2)/(a+b*x^2),x,7,-1/3*a*(b*c-a*d)*(c+d*x^2)^(3/2)/b^3-1/5*a*(c+d*x^2)^(5/2)/b^2+1/7*(c+d*x^2)^(7/2)/(b*d)+a*(b*c-a*d)^(5/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/b^(9/2)-a*(b*c-a*d)^2*sqrt(c+d*x^2)/b^4],
[x^2*(c+d*x^2)^(5/2)/(a+b*x^2),x,8,1/6*d*x^3*(c+d*x^2)^(3/2)/b-(b*c-a*d)^(5/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/b^4+1/16*(5*b^3*c^3-30*a*b^2*c^2*d+40*a^2*b*c*d^2-16*a^3*d^3)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^4*sqrt(d))+1/16*(11*b^2*c^2-18*a*b*c*d+8*a^2*d^2)*x*sqrt(c+d*x^2)/b^3+1/8*d*(3*b*c-2*a*d)*x^3*sqrt(c+d*x^2)/b^2],
[x*(c+d*x^2)^(5/2)/(a+b*x^2),x,6,1/3*(b*c-a*d)*(c+d*x^2)^(3/2)/b^2+1/5*(c+d*x^2)^(5/2)/b-(b*c-a*d)^(5/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/b^(7/2)+(b*c-a*d)^2*sqrt(c+d*x^2)/b^3],
[(c+d*x^2)^(5/2)/(a+b*x^2),x,7,1/4*d*x*(c+d*x^2)^(3/2)/b+(b*c-a*d)^(5/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(b^3*sqrt(a))+1/8*(15*b^2*c^2-20*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)/b^3+1/8*d*(7*b*c-4*a*d)*x*sqrt(c+d*x^2)/b^2],
[(c+d*x^2)^(5/2)/(x*(a+b*x^2)),x,8,1/3*d*(c+d*x^2)^(3/2)/b-c^(5/2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/a+(b*c-a*d)^(5/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a*b^(5/2))+d*(2*b*c-a*d)*sqrt(c+d*x^2)/b^2],
[(c+d*x^2)^(5/2)/(x^2*(a+b*x^2)),x,7,-c*(c+d*x^2)^(3/2)/(a*x)-(b*c-a*d)^(5/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*b^2)+1/2*d^(3/2)*(5*b*c-2*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/b^2+1/2*d*(2*b*c+a*d)*x*sqrt(c+d*x^2)/(a*b)],
[(c+d*x^2)^(5/2)/(x^3*(a+b*x^2)),x,8,-1/2*c*(c+d*x^2)^(3/2)/(a*x^2)+1/2*c^(3/2)*(2*b*c-5*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/a^2-(b*c-a*d)^(5/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^2*b^(3/2))+1/2*d*(b*c+2*a*d)*sqrt(c+d*x^2)/(a*b)],
[(c+d*x^2)^(5/2)/(x^4*(a+b*x^2)),x,7,-1/3*c*(c+d*x^2)^(3/2)/(a*x^3)+(b*c-a*d)^(5/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*b)+d^(5/2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/b+c*(b*c-2*a*d)*sqrt(c+d*x^2)/(a^2*x)],

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

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

# p>0
[x^4*sqrt(c+d*x^2)/(a+b*x^2)^2,x,7,1/2*(b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^3*sqrt(d))-1/2*(3*b*c-4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/(b^3*sqrt(b*c-a*d))+x*sqrt(c+d*x^2)/b^2-1/2*x^3*sqrt(c+d*x^2)/(b*(a+b*x^2))],
[x^3*sqrt(c+d*x^2)/(a+b*x^2)^2,x,5,1/2*a*(c+d*x^2)^(3/2)/(b*(b*c-a*d)*(a+b*x^2))-1/2*(2*b*c-3*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(b^(5/2)*sqrt(b*c-a*d))+1/2*(2*b*c-3*a*d)*sqrt(c+d*x^2)/(b^2*(b*c-a*d))],
[x^2*sqrt(c+d*x^2)/(a+b*x^2)^2,x,6,arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)/b^2+1/2*(b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(b^2*sqrt(a)*sqrt(b*c-a*d))-1/2*x*sqrt(c+d*x^2)/(b*(a+b*x^2))],
[x*sqrt(c+d*x^2)/(a+b*x^2)^2,x,4,-1/2*d*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(b^(3/2)*sqrt(b*c-a*d))-1/2*sqrt(c+d*x^2)/(b*(a+b*x^2))],
[sqrt(c+d*x^2)/(a+b*x^2)^2,x,3,1/2*c*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*sqrt(b*c-a*d))+1/2*x*sqrt(c+d*x^2)/(a*(a+b*x^2))],
[sqrt(c+d*x^2)/(x*(a+b*x^2)^2),x,7,-arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)/a^2+1/2*(2*b*c-a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^2*sqrt(b)*sqrt(b*c-a*d))+1/2*sqrt(c+d*x^2)/(a*(a+b*x^2))],
[sqrt(c+d*x^2)/(x^2*(a+b*x^2)^2),x,5,-1/2*(3*b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*sqrt(b*c-a*d))-3/2*sqrt(c+d*x^2)/(a^2*x)+1/2*sqrt(c+d*x^2)/(a*x*(a+b*x^2))],
[sqrt(c+d*x^2)/(x^3*(a+b*x^2)^2),x,8,1/2*(4*b*c-a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^3*sqrt(c))-1/2*(4*b*c-3*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b)/(a^3*sqrt(b*c-a*d))-b*sqrt(c+d*x^2)/(a^2*(a+b*x^2))-1/2*sqrt(c+d*x^2)/(a*x^2*(a+b*x^2))],
[sqrt(c+d*x^2)/(x^4*(a+b*x^2)^2),x,6,1/2*b*(5*b*c-4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(7/2)*sqrt(b*c-a*d))-5/6*sqrt(c+d*x^2)/(a^2*x^3)+1/6*(15*b*c-2*a*d)*sqrt(c+d*x^2)/(a^3*c*x)+1/2*sqrt(c+d*x^2)/(a*x^3*(a+b*x^2))],
[x^4*(c+d*x^2)^(3/2)/(a+b*x^2)^2,x,8,-1/2*x^3*(c+d*x^2)^(3/2)/(b*(a+b*x^2))+3/8*(b^2*c^2-8*a*b*c*d+8*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^4*sqrt(d))-3/2*(b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)*sqrt(b*c-a*d)/b^4+3/8*(3*b*c-4*a*d)*x*sqrt(c+d*x^2)/b^3+3/4*d*x^3*sqrt(c+d*x^2)/b^2],
[x^3*(c+d*x^2)^(3/2)/(a+b*x^2)^2,x,6,1/6*(2*b*c-5*a*d)*(c+d*x^2)^(3/2)/(b^2*(b*c-a*d))+1/2*a*(c+d*x^2)^(5/2)/(b*(b*c-a*d)*(a+b*x^2))-1/2*(2*b*c-5*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/b^(7/2)+1/2*(2*b*c-5*a*d)*sqrt(c+d*x^2)/b^3],
[x^2*(c+d*x^2)^(3/2)/(a+b*x^2)^2,x,7,-1/2*x*(c+d*x^2)^(3/2)/(b*(a+b*x^2))+1/2*(3*b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)/b^3+1/2*(b*c-4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/(b^3*sqrt(a))+d*x*sqrt(c+d*x^2)/b^2],
[x*(c+d*x^2)^(3/2)/(a+b*x^2)^2,x,5,-1/2*(c+d*x^2)^(3/2)/(b*(a+b*x^2))-3/2*d*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/b^(5/2)+3/2*d*sqrt(c+d*x^2)/b^2],
[(c+d*x^2)^(3/2)/(a+b*x^2)^2,x,6,d^(3/2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/b^2+1/2*(b*c+2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/(a^(3/2)*b^2)+1/2*(b*c-a*d)*x*sqrt(c+d*x^2)/(a*b*(a+b*x^2))],
[(c+d*x^2)^(3/2)/(x*(a+b*x^2)^2),x,7,-c^(3/2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/a^2+1/2*(2*b*c+a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(a^2*b^(3/2))+1/2*(b*c-a*d)*sqrt(c+d*x^2)/(a*b*(a+b*x^2))],
[(c+d*x^2)^(3/2)/(x^2*(a+b*x^2)^2),x,5,-3/2*c*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/a^(5/2)-1/2*(3*b*c-a*d)*sqrt(c+d*x^2)/(a^2*b*x)+1/2*(b*c-a*d)*sqrt(c+d*x^2)/(a*b*x*(a+b*x^2))],
[(c+d*x^2)^(3/2)/(x^3*(a+b*x^2)^2),x,8,1/2*(4*b*c-3*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)/a^3-1/2*(4*b*c-a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b*c-a*d)/(a^3*sqrt(b))-1/2*(2*b*c-a*d)*sqrt(c+d*x^2)/(a^2*(a+b*x^2))-1/2*c*sqrt(c+d*x^2)/(a*x^2*(a+b*x^2))],
[(c+d*x^2)^(3/2)/(x^4*(a+b*x^2)^2),x,6,1/2*(5*b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b*c-a*d)/a^(7/2)-1/6*(5*b*c-3*a*d)*sqrt(c+d*x^2)/(a^2*b*x^3)+1/6*(15*b*c-11*a*d)*sqrt(c+d*x^2)/(a^3*x)+1/2*(b*c-a*d)*sqrt(c+d*x^2)/(a*b*x^3*(a+b*x^2))],
[x^4*(c+d*x^2)^(5/2)/(a+b*x^2)^2,x,9,2/3*d*x^3*(c+d*x^2)^(3/2)/b^2-1/2*x^3*(c+d*x^2)^(5/2)/(b*(a+b*x^2))-1/2*(3*b*c-8*a*d)*(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/b^5+1/16*(5*b^3*c^3-60*a*b^2*c^2*d+120*a^2*b*c*d^2-64*a^3*d^3)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^5*sqrt(d))+1/16*(19*b^2*c^2-52*a*b*c*d+32*a^2*d^2)*x*sqrt(c+d*x^2)/b^4+1/8*d*(7*b*c-8*a*d)*x^3*sqrt(c+d*x^2)/b^3],
[x^3*(c+d*x^2)^(5/2)/(a+b*x^2)^2,x,7,1/6*(2*b*c-7*a*d)*(c+d*x^2)^(3/2)/b^3+1/10*(2*b*c-7*a*d)*(c+d*x^2)^(5/2)/(b^2*(b*c-a*d))+1/2*a*(c+d*x^2)^(7/2)/(b*(b*c-a*d)*(a+b*x^2))-1/2*(2*b*c-7*a*d)*(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/b^(9/2)+1/2*(2*b*c-7*a*d)*(b*c-a*d)*sqrt(c+d*x^2)/b^4],
[x^2*(c+d*x^2)^(5/2)/(a+b*x^2)^2,x,8,3/4*d*x*(c+d*x^2)^(3/2)/b^2-1/2*x*(c+d*x^2)^(5/2)/(b*(a+b*x^2))+1/2*(b*c-6*a*d)*(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(b^4*sqrt(a))+1/8*(15*b^2*c^2-40*a*b*c*d+24*a^2*d^2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(d)/b^4+1/8*d*(11*b*c-12*a*d)*x*sqrt(c+d*x^2)/b^3],
[x*(c+d*x^2)^(5/2)/(a+b*x^2)^2,x,6,5/6*d*(c+d*x^2)^(3/2)/b^2-1/2*(c+d*x^2)^(5/2)/(b*(a+b*x^2))-5/2*d*(b*c-a*d)^(3/2)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/b^(7/2)+5/2*d*(b*c-a*d)*sqrt(c+d*x^2)/b^3],
[(c+d*x^2)^(5/2)/(a+b*x^2)^2,x,7,1/2*(b*c-a*d)*x*(c+d*x^2)^(3/2)/(a*b*(a+b*x^2))+1/2*(b*c-a*d)^(3/2)*(b*c+4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*b^3)+1/2*d^(3/2)*(5*b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/b^3-1/2*d*(b*c-2*a*d)*x*sqrt(c+d*x^2)/(a*b^2)],
[(c+d*x^2)^(5/2)/(x*(a+b*x^2)^2),x,8,1/2*(b*c-a*d)*(c+d*x^2)^(3/2)/(a*b*(a+b*x^2))-c^(5/2)*arctanh(sqrt(c+d*x^2)/sqrt(c))/a^2+1/2*(b*c-a*d)^(3/2)*(2*b*c+3*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^2*b^(5/2))-1/2*d*(b*c-3*a*d)*sqrt(c+d*x^2)/(a*b^2)],
[(c+d*x^2)^(5/2)/(x^2*(a+b*x^2)^2),x,7,1/2*(b*c-a*d)*(c+d*x^2)^(3/2)/(a*b*x*(a+b*x^2))-1/2*(b*c-a*d)^(3/2)*(3*b*c+2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*b^2)+d^(5/2)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))/b^2-1/2*c*(3*b*c-a*d)*sqrt(c+d*x^2)/(a^2*b*x)],
[(c+d*x^2)^(5/2)/(x^3*(a+b*x^2)^2),x,8,-1/2*c*(c+d*x^2)^(3/2)/(a*x^2*(a+b*x^2))+1/2*c^(3/2)*(4*b*c-5*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/a^3-1/2*(b*c-a*d)^(3/2)*(4*b*c+a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^3*b^(3/2))-1/2*(b*c-a*d)*(2*b*c-a*d)*sqrt(c+d*x^2)/(a^2*b*(a+b*x^2))],
[(c+d*x^2)^(5/2)/(x^4*(a+b*x^2)^2),x,6,1/2*(b*c-a*d)*(c+d*x^2)^(3/2)/(a*b*x^3*(a+b*x^2))+5/2*c*(b*c-a*d)^(3/2)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/a^(7/2)-1/6*c*(5*b*c-3*a*d)*sqrt(c+d*x^2)/(a^2*b*x^3)+1/6*(15*b^2*c^2-20*a*b*c*d+3*a^2*d^2)*sqrt(c+d*x^2)/(a^3*b*x)],

# p<0
[x^4/((a+b*x^2)^2*sqrt(c+d*x^2)),x,6,-1/2*(3*b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/(b^2*(b*c-a*d)^(3/2))+arctanh(x*sqrt(d)/sqrt(c+d*x^2))/(b^2*sqrt(d))+1/2*a*x*sqrt(c+d*x^2)/(b*(b*c-a*d)*(a+b*x^2))],
[x^3/((a+b*x^2)^2*sqrt(c+d*x^2)),x,4,-1/2*(2*b*c-a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(b^(3/2)*(b*c-a*d)^(3/2))+1/2*a*sqrt(c+d*x^2)/(b*(b*c-a*d)*(a+b*x^2))],
[x^2/((a+b*x^2)^2*sqrt(c+d*x^2)),x,4,1/2*c*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/((b*c-a*d)^(3/2)*sqrt(a))-1/2*x*sqrt(c+d*x^2)/((b*c-a*d)*(a+b*x^2))],
[x/((a+b*x^2)^2*sqrt(c+d*x^2)),x,4,1/2*d*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/((b*c-a*d)^(3/2)*sqrt(b))-1/2*sqrt(c+d*x^2)/((b*c-a*d)*(a+b*x^2))],
[1/((a+b*x^2)^2*sqrt(c+d*x^2)),x,3,1/2*(b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*(b*c-a*d)^(3/2))+1/2*b*x*sqrt(c+d*x^2)/(a*(b*c-a*d)*(a+b*x^2))],
[1/(x*(a+b*x^2)^2*sqrt(c+d*x^2)),x,7,1/2*(2*b*c-3*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b)/(a^2*(b*c-a*d)^(3/2))-arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^2*sqrt(c))+1/2*b*sqrt(c+d*x^2)/(a*(b*c-a*d)*(a+b*x^2))],
[1/(x^2*(a+b*x^2)^2*sqrt(c+d*x^2)),x,5,-1/2*b*(3*b*c-4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*(b*c-a*d)^(3/2))-1/2*(3*b*c-2*a*d)*sqrt(c+d*x^2)/(a^2*c*(b*c-a*d)*x)+1/2*b*sqrt(c+d*x^2)/(a*(b*c-a*d)*x*(a+b*x^2))],
[1/(x^3*(a+b*x^2)^2*sqrt(c+d*x^2)),x,8,1/2*(4*b*c+a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^3*c^(3/2))-1/2*b^(3/2)*(4*b*c-5*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^3*(b*c-a*d)^(3/2))-1/2*b*(2*b*c-a*d)*sqrt(c+d*x^2)/(a^2*c*(b*c-a*d)*(a+b*x^2))-1/2*sqrt(c+d*x^2)/(a*c*x^2*(a+b*x^2))],
[1/(x^4*(a+b*x^2)^2*sqrt(c+d*x^2)),x,6,1/2*b^2*(5*b*c-6*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(7/2)*(b*c-a*d)^(3/2))-1/6*(5*b*c-2*a*d)*sqrt(c+d*x^2)/(a^2*c*(b*c-a*d)*x^3)+1/6*(15*b^2*c^2-8*a*b*c*d-4*a^2*d^2)*sqrt(c+d*x^2)/(a^3*c^2*(b*c-a*d)*x)+1/2*b*sqrt(c+d*x^2)/(a*(b*c-a*d)*x^3*(a+b*x^2))],
[x^4/((a+b*x^2)^2*(c+d*x^2)^(3/2)),x,5,-3/2*c*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/(b*c-a*d)^(5/2)+1/2*(2*b*c+a*d)*x/(b*(b*c-a*d)^2*sqrt(c+d*x^2))+1/2*a*x/(b*(b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))],
[x^3/((a+b*x^2)^2*(c+d*x^2)^(3/2)),x,5,-1/2*(2*b*c+a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/((b*c-a*d)^(5/2)*sqrt(b))+1/2*(2*b*c+a*d)/(b*(b*c-a*d)^2*sqrt(c+d*x^2))+1/2*a/(b*(b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))],
[x^2/((a+b*x^2)^2*(c+d*x^2)^(3/2)),x,5,1/2*(b*c+2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/((b*c-a*d)^(5/2)*sqrt(a))-3/2*d*x/((b*c-a*d)^2*sqrt(c+d*x^2))-1/2*x/((b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))],
[x/((a+b*x^2)^2*(c+d*x^2)^(3/2)),x,5,3/2*d*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b)/(b*c-a*d)^(5/2)-3/2*d/((b*c-a*d)^2*sqrt(c+d*x^2))+(-1/2)/((b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))],
[1/((a+b*x^2)^2*(c+d*x^2)^(3/2)),x,5,1/2*b*(b*c-4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*(b*c-a*d)^(5/2))+1/2*d*(b*c+2*a*d)*x/(a*c*(b*c-a*d)^2*sqrt(c+d*x^2))+1/2*b*x/(a*(b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))],
[1/(x*(a+b*x^2)^2*(c+d*x^2)^(3/2)),x,8,-arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^2*c^(3/2))+1/2*b^(3/2)*(2*b*c-5*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^2*(b*c-a*d)^(5/2))+1/2*d*(b*c+2*a*d)/(a*c*(b*c-a*d)^2*sqrt(c+d*x^2))+1/2*b/(a*(b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))],
[1/(x^2*(a+b*x^2)^2*(c+d*x^2)^(3/2)),x,6,-3/2*b^2*(b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*(b*c-a*d)^(5/2))+1/2*d*(b*c+2*a*d)/(a*c*(b*c-a*d)^2*x*sqrt(c+d*x^2))+1/2*b/(a*(b*c-a*d)*x*(a+b*x^2)*sqrt(c+d*x^2))-1/2*(3*b^2*c^2-4*a*b*c*d+4*a^2*d^2)*sqrt(c+d*x^2)/(a^2*c^2*(b*c-a*d)^2*x)],
[1/(x^3*(a+b*x^2)^2*(c+d*x^2)^(3/2)),x,9,1/2*(4*b*c+3*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^3*c^(5/2))-1/2*b^(5/2)*(4*b*c-7*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^3*(b*c-a*d)^(5/2))-1/2*d*(2*b^2*c^2-2*a*b*c*d+3*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*sqrt(c+d*x^2))-1/2*b*(2*b*c-a*d)/(a^2*c*(b*c-a*d)*(a+b*x^2)*sqrt(c+d*x^2))+(-1/2)/(a*c*x^2*(a+b*x^2)*sqrt(c+d*x^2))],
[1/(x^4*(a+b*x^2)^2*(c+d*x^2)^(3/2)),x,7,1/2*b^3*(5*b*c-8*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(7/2)*(b*c-a*d)^(5/2))+1/2*d*(b*c+2*a*d)/(a*c*(b*c-a*d)^2*x^3*sqrt(c+d*x^2))+1/2*b/(a*(b*c-a*d)*x^3*(a+b*x^2)*sqrt(c+d*x^2))-1/6*(5*b^2*c^2-4*a*b*c*d+8*a^2*d^2)*sqrt(c+d*x^2)/(a^2*c^2*(b*c-a*d)^2*x^3)+1/6*(15*b^3*c^3-14*a*b^2*c^2*d-8*a^2*b*c*d^2+16*a^3*d^3)*sqrt(c+d*x^2)/(a^3*c^3*(b*c-a*d)^2*x)],
[x^4/((a+b*x^2)^2*(c+d*x^2)^(5/2)),x,6,1/6*(2*b*c+3*a*d)*x/(b*(b*c-a*d)^2*(c+d*x^2)^(3/2))+1/2*a*x/(b*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))-1/2*(3*b*c+2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/(b*c-a*d)^(7/2)+1/6*(4*b*c+11*a*d)*x/((b*c-a*d)^3*sqrt(c+d*x^2))],
[x^3/((a+b*x^2)^2*(c+d*x^2)^(5/2)),x,6,1/6*(2*b*c+3*a*d)/(b*(b*c-a*d)^2*(c+d*x^2)^(3/2))+1/2*a/(b*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))-1/2*(2*b*c+3*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))*sqrt(b)/(b*c-a*d)^(7/2)+1/2*(2*b*c+3*a*d)/((b*c-a*d)^3*sqrt(c+d*x^2))],
[x^2/((a+b*x^2)^2*(c+d*x^2)^(5/2)),x,6,-5/6*d*x/((b*c-a*d)^2*(c+d*x^2)^(3/2))-1/2*x/((b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))+1/2*b*(b*c+4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/((b*c-a*d)^(7/2)*sqrt(a))-1/6*d*(13*b*c+2*a*d)*x/(c*(b*c-a*d)^3*sqrt(c+d*x^2))],
[x/((a+b*x^2)^2*(c+d*x^2)^(5/2)),x,6,-5/6*d/((b*c-a*d)^2*(c+d*x^2)^(3/2))+(-1/2)/((b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))+5/2*b^(3/2)*d*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(b*c-a*d)^(7/2)-5/2*b*d/((b*c-a*d)^3*sqrt(c+d*x^2))],
[1/((a+b*x^2)^2*(c+d*x^2)^(5/2)),x,6,1/6*d*(3*b*c+2*a*d)*x/(a*c*(b*c-a*d)^2*(c+d*x^2)^(3/2))+1/2*b*x/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))+1/2*b^2*(b*c-6*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*(b*c-a*d)^(7/2))+1/6*d*(3*b^2*c^2+16*a*b*c*d-4*a^2*d^2)*x/(a*c^2*(b*c-a*d)^3*sqrt(c+d*x^2))],
[1/(x*(a+b*x^2)^2*(c+d*x^2)^(5/2)),x,9,1/6*d*(3*b*c+2*a*d)/(a*c*(b*c-a*d)^2*(c+d*x^2)^(3/2))+1/2*b/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))-arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^2*c^(5/2))+1/2*b^(5/2)*(2*b*c-7*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^2*(b*c-a*d)^(7/2))+1/2*d*(b^2*c^2+6*a*b*c*d-2*a^2*d^2)/(a*c^2*(b*c-a*d)^3*sqrt(c+d*x^2))],
[1/(x^2*(a+b*x^2)^2*(c+d*x^2)^(5/2)),x,7,1/6*d*(3*b*c+2*a*d)/(a*c*(b*c-a*d)^2*x*(c+d*x^2)^(3/2))+1/2*b/(a*(b*c-a*d)*x*(a+b*x^2)*(c+d*x^2)^(3/2))-1/2*b^3*(3*b*c-8*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*(b*c-a*d)^(7/2))+1/6*d*(3*b^2*c^2+20*a*b*c*d-8*a^2*d^2)/(a*c^2*(b*c-a*d)^3*x*sqrt(c+d*x^2))-1/6*(9*b^3*c^3-18*a*b^2*c^2*d+40*a^2*b*c*d^2-16*a^3*d^3)*sqrt(c+d*x^2)/(a^2*c^3*(b*c-a*d)^3*x)],
[1/(x^3*(a+b*x^2)^2*(c+d*x^2)^(5/2)),x,10,-1/6*d*(6*b^2*c^2-6*a*b*c*d+5*a^2*d^2)/(a^2*c^2*(b*c-a*d)^2*(c+d*x^2)^(3/2))-1/2*b*(2*b*c-a*d)/(a^2*c*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^(3/2))+(-1/2)/(a*c*x^2*(a+b*x^2)*(c+d*x^2)^(3/2))+1/2*(4*b*c+5*a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))/(a^3*c^(7/2))-1/2*b^(7/2)*(4*b*c-9*a*d)*arctanh(sqrt(b)*sqrt(c+d*x^2)/sqrt(b*c-a*d))/(a^3*(b*c-a*d)^(7/2))-1/2*d*(2*b*c-a*d)*(b^2*c^2-a*b*c*d+5*a^2*d^2)/(a^2*c^3*(b*c-a*d)^3*sqrt(c+d*x^2))],
[1/(x^4*(a+b*x^2)^2*(c+d*x^2)^(5/2)),x,8,1/6*d*(3*b*c+2*a*d)/(a*c*(b*c-a*d)^2*x^3*(c+d*x^2)^(3/2))+1/2*b/(a*(b*c-a*d)*x^3*(a+b*x^2)*(c+d*x^2)^(3/2))+5/2*b^4*(b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(7/2)*(b*c-a*d)^(7/2))+1/2*d*(b^2*c^2+8*a*b*c*d-4*a^2*d^2)/(a*c^2*(b*c-a*d)^3*x^3*sqrt(c+d*x^2))-1/6*(5*b^3*c^3-6*a*b^2*c^2*d+32*a^2*b*c*d^2-16*a^3*d^3)*sqrt(c+d*x^2)/(a^2*c^3*(b*c-a*d)^3*x^3)+1/6*(15*b^4*c^4-20*a*b^3*c^3*d-12*a^2*b^2*c^2*d^2+64*a^3*b*c*d^3-32*a^4*d^4)*sqrt(c+d*x^2)/(a^3*c^4*(b*c-a*d)^3*x)],

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

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

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

# p>0
[(e*x)^(3/2)*(A+B*x^2)*sqrt(a+b*x^2),x,5,2/11*B*(e*x)^(5/2)*(a+b*x^2)^(3/2)/(b*e)+2/77*(11*A*b-5*a*B)*(e*x)^(5/2)*sqrt(a+b*x^2)/(b*e)+4/231*a*(11*A*b-5*a*B)*e*sqrt(e*x)*sqrt(a+b*x^2)/b^2-2/231*a^(7/4)*(11*A*b-5*a*B)*e^(3/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(e*x)*sqrt(a+b*x^2),x,6,2/9*B*(e*x)^(3/2)*(a+b*x^2)^(3/2)/(b*e)+2/15*(3*A*b-a*B)*(e*x)^(3/2)*sqrt(a+b*x^2)/(b*e)+4/15*a*(3*A*b-a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(b^(3/2)*(sqrt(a)+x*sqrt(b)))-4/15*a^(5/4)*(3*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^2))+2/15*a^(5/4)*(3*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/sqrt(e*x),x,4,2/7*B*(a+b*x^2)^(3/2)*sqrt(e*x)/(b*e)+2/21*(7*A*b-a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(b*e)+2/21*a^(3/4)*(7*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(5/4)*sqrt(e)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/(e*x)^(3/2),x,6,-2*A*(a+b*x^2)^(3/2)/(a*e*sqrt(e*x))+2/5*(5*A*b+a*B)*(e*x)^(3/2)*sqrt(a+b*x^2)/(a*e^3)+4/5*(5*A*b+a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(e^2*sqrt(b)*(sqrt(a)+x*sqrt(b)))-4/5*a^(1/4)*(5*A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(3/4)*e^(3/2)*sqrt(a+b*x^2))+2/5*a^(1/4)*(5*A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(3/4)*e^(3/2)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/(e*x)^(5/2),x,4,-2/3*A*(a+b*x^2)^(3/2)/(a*e*(e*x)^(3/2))+2/3*(A*b+a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(a*e^3)+2/3*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(1/4)*b^(1/4)*e^(5/2)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/(e*x)^(7/2),x,6,-2/5*A*(a+b*x^2)^(3/2)/(a*e*(e*x)^(5/2))-2/5*(A*b+5*a*B)*sqrt(a+b*x^2)/(a*e^3*sqrt(e*x))+4/5*(A*b+5*a*B)*sqrt(b)*sqrt(e*x)*sqrt(a+b*x^2)/(a*e^4*(sqrt(a)+x*sqrt(b)))-4/5*b^(1/4)*(A*b+5*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*e^(7/2)*sqrt(a+b*x^2))+2/5*b^(1/4)*(A*b+5*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*e^(7/2)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/x^(9/2),x,4,-2/7*A*(a+b*x^2)^(3/2)/(a*x^(7/2))+2/21*(A*b-7*a*B)*sqrt(a+b*x^2)/(a*x^(3/2))-2/21*b^(3/4)*(A*b-7*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(5/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/x^(11/2),x,7,-2/9*A*(a+b*x^2)^(3/2)/(a*x^(9/2))+2/15*(A*b-3*a*B)*sqrt(a+b*x^2)/(a*x^(5/2))+4/15*b*(A*b-3*a*B)*sqrt(a+b*x^2)/(a^2*sqrt(x))-4/15*b^(3/2)*(A*b-3*a*B)*sqrt(x)*sqrt(a+b*x^2)/(a^2*(sqrt(a)+x*sqrt(b)))+4/15*b^(5/4)*(A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^2))-2/15*b^(5/4)*(A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(a+b*x^2)/x^(13/2),x,5,-2/11*A*(a+b*x^2)^(3/2)/(a*x^(11/2))+2/77*(5*A*b-11*a*B)*sqrt(a+b*x^2)/(a*x^(7/2))+4/231*b*(5*A*b-11*a*B)*sqrt(a+b*x^2)/(a^2*x^(3/2))+2/231*b^(7/4)*(5*A*b-11*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*sqrt(x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(9/4)*sqrt(a+b*x^2))],
[(e*x)^(3/2)*(a+b*x^2)^(3/2)*(A+B*x^2),x,6,2/33*(3*A*b-a*B)*(e*x)^(5/2)*(a+b*x^2)^(3/2)/(b*e)+2/15*B*(e*x)^(5/2)*(a+b*x^2)^(5/2)/(b*e)+4/77*a*(3*A*b-a*B)*(e*x)^(5/2)*sqrt(a+b*x^2)/(b*e)+8/231*a^2*(3*A*b-a*B)*e*sqrt(e*x)*sqrt(a+b*x^2)/b^2-4/231*a^(11/4)*(3*A*b-a*B)*e^(3/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^2))],
[(a+b*x^2)^(3/2)*(A+B*x^2)*sqrt(e*x),x,7,2/117*(13*A*b-3*a*B)*(e*x)^(3/2)*(a+b*x^2)^(3/2)/(b*e)+2/13*B*(e*x)^(3/2)*(a+b*x^2)^(5/2)/(b*e)+4/195*a*(13*A*b-3*a*B)*(e*x)^(3/2)*sqrt(a+b*x^2)/(b*e)+8/195*a^2*(13*A*b-3*a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(b^(3/2)*(sqrt(a)+x*sqrt(b)))-8/195*a^(9/4)*(13*A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^2))+4/195*a^(9/4)*(13*A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^2))],
[(a+b*x^2)^(3/2)*(A+B*x^2)/sqrt(e*x),x,5,2/77*(11*A*b-a*B)*(a+b*x^2)^(3/2)*sqrt(e*x)/(b*e)+2/11*B*(a+b*x^2)^(5/2)*sqrt(e*x)/(b*e)+4/77*a*(11*A*b-a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(b*e)+4/77*a^(7/4)*(11*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(5/4)*sqrt(e)*sqrt(a+b*x^2))],
[(a+b*x^2)^(3/2)*(A+B*x^2)/(e*x)^(3/2),x,7,2/9*(9*A*b+a*B)*(e*x)^(3/2)*(a+b*x^2)^(3/2)/(a*e^3)-2*A*(a+b*x^2)^(5/2)/(a*e*sqrt(e*x))+4/15*(9*A*b+a*B)*(e*x)^(3/2)*sqrt(a+b*x^2)/e^3+8/15*a*(9*A*b+a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(e^2*sqrt(b)*(sqrt(a)+x*sqrt(b)))-8/15*a^(5/4)*(9*A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(3/4)*e^(3/2)*sqrt(a+b*x^2))+4/15*a^(5/4)*(9*A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(3/4)*e^(3/2)*sqrt(a+b*x^2))],
[(a+b*x^2)^(3/2)*(A+B*x^2)/(e*x)^(5/2),x,5,-2/3*A*(a+b*x^2)^(5/2)/(a*e*(e*x)^(3/2))+2/21*(7*A*b+3*a*B)*(a+b*x^2)^(3/2)*sqrt(e*x)/(a*e^3)+4/21*(7*A*b+3*a*B)*sqrt(e*x)*sqrt(a+b*x^2)/e^3+4/21*a^(3/4)*(7*A*b+3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(1/4)*e^(5/2)*sqrt(a+b*x^2))],
[(a+b*x^2)^(3/2)*(A+B*x^2)/(e*x)^(7/2),x,7,-2/5*A*(a+b*x^2)^(5/2)/(a*e*(e*x)^(5/2))-2*(A*b+a*B)*(a+b*x^2)^(3/2)/(a*e^3*sqrt(e*x))+12/5*b*(A*b+a*B)*(e*x)^(3/2)*sqrt(a+b*x^2)/(a*e^5)+24/5*(A*b+a*B)*sqrt(b)*sqrt(e*x)*sqrt(a+b*x^2)/(e^4*(sqrt(a)+x*sqrt(b)))-24/5*a^(1/4)*b^(1/4)*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(e^(7/2)*sqrt(a+b*x^2))+12/5*a^(1/4)*b^(1/4)*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(e^(7/2)*sqrt(a+b*x^2))],

# p<0
[(e*x)^(5/2)*(A+B*x^2)/sqrt(a+b*x^2),x,6,2/45*(9*A*b-7*a*B)*e*(e*x)^(3/2)*sqrt(a+b*x^2)/b^2+2/9*B*(e*x)^(7/2)*sqrt(a+b*x^2)/(b*e)-2/15*a*(9*A*b-7*a*B)*e^2*sqrt(e*x)*sqrt(a+b*x^2)/(b^(5/2)*(sqrt(a)+x*sqrt(b)))+2/15*a^(5/4)*(9*A*b-7*a*B)*e^(5/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^2))-1/15*a^(5/4)*(9*A*b-7*a*B)*e^(5/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^2))],
[(e*x)^(3/2)*(A+B*x^2)/sqrt(a+b*x^2),x,4,2/7*B*(e*x)^(5/2)*sqrt(a+b*x^2)/(b*e)+2/21*(7*A*b-5*a*B)*e*sqrt(e*x)*sqrt(a+b*x^2)/b^2-1/21*a^(3/4)*(7*A*b-5*a*B)*e^(3/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(9/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(e*x)/sqrt(a+b*x^2),x,5,2/5*B*(e*x)^(3/2)*sqrt(a+b*x^2)/(b*e)+2/5*(5*A*b-3*a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(b^(3/2)*(sqrt(a)+x*sqrt(b)))-2/5*a^(1/4)*(5*A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^2))+1/5*a^(1/4)*(5*A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(7/4)*sqrt(a+b*x^2))],
[(A+B*x^2)/(sqrt(e*x)*sqrt(a+b*x^2)),x,3,2/3*B*sqrt(e*x)*sqrt(a+b*x^2)/(b*e)+1/3*(3*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(1/4)*b^(5/4)*sqrt(e)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(3/2)*sqrt(a+b*x^2)),x,5,-2*A*sqrt(a+b*x^2)/(a*e*sqrt(e*x))+2*(A*b+a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(a*e^2*sqrt(b)*(sqrt(a)+x*sqrt(b)))-2*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*b^(3/4)*e^(3/2)*sqrt(a+b*x^2))+(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*b^(3/4)*e^(3/2)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(5/2)*sqrt(a+b*x^2)),x,3,-2/3*A*sqrt(a+b*x^2)/(a*e*(e*x)^(3/2))-1/3*(A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(5/4)*b^(1/4)*e^(5/2)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(7/2)*sqrt(a+b*x^2)),x,6,-2/5*A*sqrt(a+b*x^2)/(a*e*(e*x)^(5/2))+2/5*(3*A*b-5*a*B)*sqrt(a+b*x^2)/(a^2*e^3*sqrt(e*x))-2/5*(3*A*b-5*a*B)*sqrt(b)*sqrt(e*x)*sqrt(a+b*x^2)/(a^2*e^4*(sqrt(a)+x*sqrt(b)))+2/5*b^(1/4)*(3*A*b-5*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*e^(7/2)*sqrt(a+b*x^2))-1/5*b^(1/4)*(3*A*b-5*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*e^(7/2)*sqrt(a+b*x^2))],
[(e*x)^(7/2)*(A+B*x^2)/(a+b*x^2)^(3/2),x,5,-1/7*(7*A*b-9*a*B)*e*(e*x)^(5/2)/(b^2*sqrt(a+b*x^2))+2/7*B*(e*x)^(9/2)/(b*e*sqrt(a+b*x^2))+5/21*(7*A*b-9*a*B)*e^3*sqrt(e*x)*sqrt(a+b*x^2)/b^3-5/42*a^(3/4)*(7*A*b-9*a*B)*e^(7/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(13/4)*sqrt(a+b*x^2))],
[(e*x)^(5/2)*(A+B*x^2)/(a+b*x^2)^(3/2),x,6,-1/5*(5*A*b-7*a*B)*e*(e*x)^(3/2)/(b^2*sqrt(a+b*x^2))+2/5*B*(e*x)^(7/2)/(b*e*sqrt(a+b*x^2))+3/5*(5*A*b-7*a*B)*e^2*sqrt(e*x)*sqrt(a+b*x^2)/(b^(5/2)*(sqrt(a)+x*sqrt(b)))-3/5*a^(1/4)*(5*A*b-7*a*B)*e^(5/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^2))+3/10*a^(1/4)*(5*A*b-7*a*B)*e^(5/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(b^(11/4)*sqrt(a+b*x^2))],
[(e*x)^(3/2)*(A+B*x^2)/(a+b*x^2)^(3/2),x,4,2/3*B*(e*x)^(5/2)/(b*e*sqrt(a+b*x^2))-1/3*(3*A*b-5*a*B)*e*sqrt(e*x)/(b^2*sqrt(a+b*x^2))+1/6*(3*A*b-5*a*B)*e^(3/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(1/4)*b^(9/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(e*x)/(a+b*x^2)^(3/2),x,5,(A*b-a*B)*(e*x)^(3/2)/(a*b*e*sqrt(a+b*x^2))-(A*b-3*a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(a*b^(3/2)*(sqrt(a)+x*sqrt(b)))+(A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*b^(7/4)*sqrt(a+b*x^2))-1/2*(A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*b^(7/4)*sqrt(a+b*x^2))],
[(A+B*x^2)/((a+b*x^2)^(3/2)*sqrt(e*x)),x,3,(A*b-a*B)*sqrt(e*x)/(a*b*e*sqrt(a+b*x^2))+1/2*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(5/4)*b^(5/4)*sqrt(e)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(3/2)*(a+b*x^2)^(3/2)),x,6,-(3*A*b-a*B)*(e*x)^(3/2)/(a^2*e^3*sqrt(a+b*x^2))-2*A/(a*e*sqrt(e*x)*sqrt(a+b*x^2))+(3*A*b-a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(a^2*e^2*sqrt(b)*(sqrt(a)+x*sqrt(b)))-(3*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*b^(3/4)*e^(3/2)*sqrt(a+b*x^2))+1/2*(3*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*b^(3/4)*e^(3/2)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(5/2)*(a+b*x^2)^(3/2)),x,4,-2/3*A/(a*e*(e*x)^(3/2)*sqrt(a+b*x^2))-1/3*(5*A*b-3*a*B)*sqrt(e*x)/(a^2*e^3*sqrt(a+b*x^2))-1/6*(5*A*b-3*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(9/4)*b^(1/4)*e^(5/2)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(7/2)*(a+b*x^2)^(3/2)),x,7,-2/5*A/(a*e*(e*x)^(5/2)*sqrt(a+b*x^2))+1/5*(-7*A*b+5*a*B)/(a^2*e^3*sqrt(e*x)*sqrt(a+b*x^2))+3/5*(7*A*b-5*a*B)*sqrt(a+b*x^2)/(a^3*e^3*sqrt(e*x))-3/5*(7*A*b-5*a*B)*sqrt(b)*sqrt(e*x)*sqrt(a+b*x^2)/(a^3*e^4*(sqrt(a)+x*sqrt(b)))+3/5*b^(1/4)*(7*A*b-5*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(11/4)*e^(7/2)*sqrt(a+b*x^2))-3/10*b^(1/4)*(7*A*b-5*a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(11/4)*e^(7/2)*sqrt(a+b*x^2))],
[(e*x)^(7/2)*(A+B*x^2)/(a+b*x^2)^(5/2),x,5,-1/3*(A*b-3*a*B)*e*(e*x)^(5/2)/(b^2*(a+b*x^2)^(3/2))+2/3*B*(e*x)^(9/2)/(b*e*(a+b*x^2)^(3/2))-5/6*(A*b-3*a*B)*e^3*sqrt(e*x)/(b^3*sqrt(a+b*x^2))+5/12*(A*b-3*a*B)*e^(7/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(1/4)*b^(13/4)*sqrt(a+b*x^2))],
[(e*x)^(5/2)*(A+B*x^2)/(a+b*x^2)^(5/2),x,6,1/3*(A*b-a*B)*(e*x)^(7/2)/(a*b*e*(a+b*x^2)^(3/2))+1/6*(A*b-7*a*B)*e*(e*x)^(3/2)/(a*b^2*sqrt(a+b*x^2))-1/2*(A*b-7*a*B)*e^2*sqrt(e*x)*sqrt(a+b*x^2)/(a*b^(5/2)*(sqrt(a)+x*sqrt(b)))+1/2*(A*b-7*a*B)*e^(5/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*b^(11/4)*sqrt(a+b*x^2))-1/4*(A*b-7*a*B)*e^(5/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(3/4)*b^(11/4)*sqrt(a+b*x^2))],
[(e*x)^(3/2)*(A+B*x^2)/(a+b*x^2)^(5/2),x,4,1/3*(A*b-a*B)*(e*x)^(5/2)/(a*b*e*(a+b*x^2)^(3/2))-1/6*(A*b+5*a*B)*e*sqrt(e*x)/(a*b^2*sqrt(a+b*x^2))+1/12*(A*b+5*a*B)*e^(3/2)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(5/4)*b^(9/4)*sqrt(a+b*x^2))],
[(A+B*x^2)*sqrt(e*x)/(a+b*x^2)^(5/2),x,6,1/3*(A*b-a*B)*(e*x)^(3/2)/(a*b*e*(a+b*x^2)^(3/2))+1/2*(A*b+a*B)*(e*x)^(3/2)/(a^2*b*e*sqrt(a+b*x^2))-1/2*(A*b+a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(a^2*b^(3/2)*(sqrt(a)+x*sqrt(b)))+1/2*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*b^(7/4)*sqrt(a+b*x^2))-1/4*(A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt(e)*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(7/4)*b^(7/4)*sqrt(a+b*x^2))],
[(A+B*x^2)/((a+b*x^2)^(5/2)*sqrt(e*x)),x,4,1/3*(A*b-a*B)*sqrt(e*x)/(a*b*e*(a+b*x^2)^(3/2))+1/6*(5*A*b+a*B)*sqrt(e*x)/(a^2*b*e*sqrt(a+b*x^2))+1/12*(5*A*b+a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(9/4)*b^(5/4)*sqrt(e)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(3/2)*(a+b*x^2)^(5/2)),x,7,-1/3*(7*A*b-a*B)*(e*x)^(3/2)/(a^2*e^3*(a+b*x^2)^(3/2))-2*A/(a*e*(a+b*x^2)^(3/2)*sqrt(e*x))-1/2*(7*A*b-a*B)*(e*x)^(3/2)/(a^3*e^3*sqrt(a+b*x^2))+1/2*(7*A*b-a*B)*sqrt(e*x)*sqrt(a+b*x^2)/(a^3*e^2*sqrt(b)*(sqrt(a)+x*sqrt(b)))-1/2*(7*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(11/4)*b^(3/4)*e^(3/2)*sqrt(a+b*x^2))+1/4*(7*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(11/4)*b^(3/4)*e^(3/2)*sqrt(a+b*x^2))],
[(A+B*x^2)/((e*x)^(5/2)*(a+b*x^2)^(5/2)),x,5,-2/3*A/(a*e*(e*x)^(3/2)*(a+b*x^2)^(3/2))-1/3*(3*A*b-a*B)*sqrt(e*x)/(a^2*e^3*(a+b*x^2)^(3/2))-5/6*(3*A*b-a*B)*sqrt(e*x)/(a^3*e^3*sqrt(a+b*x^2))-5/12*(3*A*b-a*B)*sqrt(cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))^2)/cos(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(b^(1/4)*sqrt(e*x)/(a^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(a)+x*sqrt(b))*sqrt((a+b*x^2)/(sqrt(a)+x*sqrt(b))^2)/(a^(13/4)*b^(1/4)*e^(5/2)*sqrt(a+b*x^2))],

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

# p>0
[(e*x)^(3/2)*(a+b*x^2)^2*sqrt(c+d*x^2),x,6,-2/55*b*(3*b*c-10*a*d)*(e*x)^(5/2)*(c+d*x^2)^(3/2)/(d^2*e)+2/15*b^2*(e*x)^(9/2)*(c+d*x^2)^(3/2)/(d*e^3)+2/77*(11*a^2*d^2+b*c*(3*b*c-10*a*d))*(e*x)^(5/2)*sqrt(c+d*x^2)/(d^2*e)+4/231*c*(11*a^2*d^2+b*c*(3*b*c-10*a*d))*e*sqrt(e*x)*sqrt(c+d*x^2)/d^3-2/231*c^(7/4)*(11*a^2*d^2+b*c*(3*b*c-10*a*d))*e^(3/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(13/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(e*x)*sqrt(c+d*x^2),x,7,-2/117*b*(7*b*c-26*a*d)*(e*x)^(3/2)*(c+d*x^2)^(3/2)/(d^2*e)+2/13*b^2*(e*x)^(7/2)*(c+d*x^2)^(3/2)/(d*e^3)+2/195*(39*a^2*d^2+b*c*(7*b*c-26*a*d))*(e*x)^(3/2)*sqrt(c+d*x^2)/(d^2*e)+4/195*c*(39*a^2*d^2+b*c*(7*b*c-26*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d^(5/2)*(sqrt(c)+x*sqrt(d)))-4/195*c^(5/4)*(39*a^2*d^2+b*c*(7*b*c-26*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(11/4)*sqrt(c+d*x^2))+2/195*c^(5/4)*(39*a^2*d^2+b*c*(7*b*c-26*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(11/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/sqrt(e*x),x,5,2/11*b^2*(e*x)^(5/2)*(c+d*x^2)^(3/2)/(d*e^3)-2/77*b*(5*b*c-22*a*d)*(c+d*x^2)^(3/2)*sqrt(e*x)/(d^2*e)+2/231*(5*b^2*c^2-22*a*b*c*d+77*a^2*d^2)*sqrt(e*x)*sqrt(c+d*x^2)/(d^2*e)+2/231*c^(3/4)*(5*b^2*c^2-22*a*b*c*d+77*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(9/4)*sqrt(e)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/(e*x)^(3/2),x,7,2/9*b^2*(e*x)^(3/2)*(c+d*x^2)^(3/2)/(d*e^3)-2*a^2*(c+d*x^2)^(3/2)/(c*e*sqrt(e*x))-2/15*(b^2*c^2-3*a*d*(2*b*c+5*a*d))*(e*x)^(3/2)*sqrt(c+d*x^2)/(c*d*e^3)-4/15*(b^2*c^2-3*a*d*(2*b*c+5*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d^(3/2)*e^2*(sqrt(c)+x*sqrt(d)))+4/15*c^(1/4)*(b^2*c^2-3*a*d*(2*b*c+5*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(7/4)*e^(3/2)*sqrt(c+d*x^2))-2/15*c^(1/4)*(b^2*c^2-3*a*d*(2*b*c+5*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(7/4)*e^(3/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/(e*x)^(5/2),x,5,-2/3*a^2*(c+d*x^2)^(3/2)/(c*e*(e*x)^(3/2))+2/7*b^2*(c+d*x^2)^(3/2)*sqrt(e*x)/(d*e^3)-2/21*(b^2*c^2-7*a*d*(2*b*c+a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(c*d*e^3)-2/21*(b^2*c^2-7*a*d*(2*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(1/4)*d^(5/4)*e^(5/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/(e*x)^(7/2),x,7,-2/5*a^2*(c+d*x^2)^(3/2)/(c*e*(e*x)^(5/2))-2/5*a*(10*b*c+a*d)*(c+d*x^2)^(3/2)/(c^2*e^3*sqrt(e*x))+2/5*(b^2*c^2+a*d*(10*b*c+a*d))*(e*x)^(3/2)*sqrt(c+d*x^2)/(c^2*e^5)+4/5*(b^2*c^2+a*d*(10*b*c+a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(c*e^4*sqrt(d)*(sqrt(c)+x*sqrt(d)))-4/5*(b^2*c^2+a*d*(10*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))+2/5*(b^2*c^2+a*d*(10*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^(9/2),x,5,-2/7*a^2*(c+d*x^2)^(3/2)/(c*x^(7/2))-2/21*a*(14*b*c-a*d)*(c+d*x^2)^(3/2)/(c^2*x^(3/2))+2/21*(7*b^2*c^2+a*d*(14*b*c-a*d))*sqrt(x)*sqrt(c+d*x^2)/c^2+2/21*(7*b^2*c^2+a*d*(14*b*c-a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))^2)/cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(x)/c^(1/4))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(5/4)*d^(1/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^(11/2),x,7,-2/9*a^2*(c+d*x^2)^(3/2)/(c*x^(9/2))-2/15*a*(6*b*c-a*d)*(c+d*x^2)^(3/2)/(c^2*x^(5/2))-2/15*(15*b^2*c^2+a*d*(6*b*c-a*d))*sqrt(c+d*x^2)/(c^2*sqrt(x))+4/15*(15*b^2*c^2+a*d*(6*b*c-a*d))*sqrt(d)*sqrt(x)*sqrt(c+d*x^2)/(c^2*(sqrt(c)+x*sqrt(d)))-4/15*d^(1/4)*(15*b^2*c^2+a*d*(6*b*c-a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))^2)/cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(x)/c^(1/4))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*sqrt(c+d*x^2))+2/15*d^(1/4)*(15*b^2*c^2+a*d*(6*b*c-a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))^2)/cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(x)/c^(1/4))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^(13/2),x,5,-2/11*a^2*(c+d*x^2)^(3/2)/(c*x^(11/2))-2/77*a*(22*b*c-5*a*d)*(c+d*x^2)^(3/2)/(c^2*x^(7/2))-2/231*(77*b^2*c^2-22*a*b*c*d+5*a^2*d^2)*sqrt(c+d*x^2)/(c^2*x^(3/2))+2/231*d^(3/4)*(77*b^2*c^2-22*a*b*c*d+5*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))^2)/cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(x)/c^(1/4))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(9/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(c+d*x^2)/x^(15/2),x,8,-2/13*a^2*(c+d*x^2)^(3/2)/(c*x^(13/2))-2/117*a*(26*b*c-7*a*d)*(c+d*x^2)^(3/2)/(c^2*x^(9/2))-2/195*(39*b^2*c^2-26*a*b*c*d+7*a^2*d^2)*sqrt(c+d*x^2)/(c^2*x^(5/2))-4/195*d*(39*b^2*c^2-26*a*b*c*d+7*a^2*d^2)*sqrt(c+d*x^2)/(c^3*sqrt(x))+4/195*d^(3/2)*(39*b^2*c^2-26*a*b*c*d+7*a^2*d^2)*sqrt(x)*sqrt(c+d*x^2)/(c^3*(sqrt(c)+x*sqrt(d)))-4/195*d^(5/4)*(39*b^2*c^2-26*a*b*c*d+7*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))^2)/cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(x)/c^(1/4))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*sqrt(c+d*x^2))+2/195*d^(5/4)*(39*b^2*c^2-26*a*b*c*d+7*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))^2)/cos(2*arctan(d^(1/4)*sqrt(x)/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(x)/c^(1/4))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*sqrt(c+d*x^2))],
[(e*x)^(5/2)*(a+b*x^2)^2*(c+d*x^2)^(3/2),x,9,2/663*(51*a^2*d^2+b*c*(11*b*c-42*a*d))*(e*x)^(7/2)*(c+d*x^2)^(3/2)/(d^2*e)-2/357*b*(11*b*c-42*a*d)*(e*x)^(7/2)*(c+d*x^2)^(5/2)/(d^2*e)+2/21*b^2*(e*x)^(11/2)*(c+d*x^2)^(5/2)/(d*e^3)+8/9945*c^2*(51*a^2*d^2+b*c*(11*b*c-42*a*d))*e*(e*x)^(3/2)*sqrt(c+d*x^2)/d^3+4/1989*c*(51*a^2*d^2+b*c*(11*b*c-42*a*d))*(e*x)^(7/2)*sqrt(c+d*x^2)/(d^2*e)-8/3315*c^3*(51*a^2*d^2+b*c*(11*b*c-42*a*d))*e^2*sqrt(e*x)*sqrt(c+d*x^2)/(d^(7/2)*(sqrt(c)+x*sqrt(d)))+8/3315*c^(13/4)*(51*a^2*d^2+b*c*(11*b*c-42*a*d))*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(15/4)*sqrt(c+d*x^2))-4/3315*c^(13/4)*(51*a^2*d^2+b*c*(11*b*c-42*a*d))*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(15/4)*sqrt(c+d*x^2))],
[(e*x)^(3/2)*(a+b*x^2)^2*(c+d*x^2)^(3/2),x,7,2/627*(57*a^2*d^2+b*c*(9*b*c-38*a*d))*(e*x)^(5/2)*(c+d*x^2)^(3/2)/(d^2*e)-2/285*b*(9*b*c-38*a*d)*(e*x)^(5/2)*(c+d*x^2)^(5/2)/(d^2*e)+2/19*b^2*(e*x)^(9/2)*(c+d*x^2)^(5/2)/(d*e^3)+4/1463*c*(57*a^2*d^2+b*c*(9*b*c-38*a*d))*(e*x)^(5/2)*sqrt(c+d*x^2)/(d^2*e)+8/4389*c^2*(57*a^2*d^2+b*c*(9*b*c-38*a*d))*e*sqrt(e*x)*sqrt(c+d*x^2)/d^3-4/4389*c^(11/4)*(57*a^2*d^2+b*c*(9*b*c-38*a*d))*e^(3/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(13/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)*sqrt(e*x),x,8,2/1989*(221*a^2*d^2+3*b*c*(7*b*c-34*a*d))*(e*x)^(3/2)*(c+d*x^2)^(3/2)/(d^2*e)-2/221*b*(7*b*c-34*a*d)*(e*x)^(3/2)*(c+d*x^2)^(5/2)/(d^2*e)+2/17*b^2*(e*x)^(7/2)*(c+d*x^2)^(5/2)/(d*e^3)+4/3315*c*(221*a^2*d^2+3*b*c*(7*b*c-34*a*d))*(e*x)^(3/2)*sqrt(c+d*x^2)/(d^2*e)+8/3315*c^2*(221*a^2*d^2+3*b*c*(7*b*c-34*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d^(5/2)*(sqrt(c)+x*sqrt(d)))-8/3315*c^(9/4)*(221*a^2*d^2+3*b*c*(7*b*c-34*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(11/4)*sqrt(c+d*x^2))+4/3315*c^(9/4)*(221*a^2*d^2+3*b*c*(7*b*c-34*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(11/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/sqrt(e*x),x,6,2/15*b^2*(e*x)^(5/2)*(c+d*x^2)^(5/2)/(d*e^3)+2/231*(33*a^2*d^2+b*c*(b*c-6*a*d))*(c+d*x^2)^(3/2)*sqrt(e*x)/(d^2*e)-2/33*b*(b*c-6*a*d)*(c+d*x^2)^(5/2)*sqrt(e*x)/(d^2*e)+4/231*c*(33*a^2*d^2+b*c*(b*c-6*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d^2*e)+4/231*c^(7/4)*(33*a^2*d^2+b*c*(b*c-6*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(9/4)*sqrt(e)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/(e*x)^(3/2),x,8,-2/117*(3*b^2*c^2-13*a*d*(2*b*c+9*a*d))*(e*x)^(3/2)*(c+d*x^2)^(3/2)/(c*d*e^3)+2/13*b^2*(e*x)^(3/2)*(c+d*x^2)^(5/2)/(d*e^3)-2*a^2*(c+d*x^2)^(5/2)/(c*e*sqrt(e*x))-4/195*(3*b^2*c^2-13*a*d*(2*b*c+9*a*d))*(e*x)^(3/2)*sqrt(c+d*x^2)/(d*e^3)-8/195*c*(3*b^2*c^2-13*a*d*(2*b*c+9*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d^(3/2)*e^2*(sqrt(c)+x*sqrt(d)))+8/195*c^(5/4)*(3*b^2*c^2-13*a*d*(2*b*c+9*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(7/4)*e^(3/2)*sqrt(c+d*x^2))-4/195*c^(5/4)*(3*b^2*c^2-13*a*d*(2*b*c+9*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(7/4)*e^(3/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/(e*x)^(5/2),x,6,-2/3*a^2*(c+d*x^2)^(5/2)/(c*e*(e*x)^(3/2))-2/231*(3*b^2*c^2-11*a*d*(6*b*c+7*a*d))*(c+d*x^2)^(3/2)*sqrt(e*x)/(c*d*e^3)+2/11*b^2*(c+d*x^2)^(5/2)*sqrt(e*x)/(d*e^3)-4/231*(3*b^2*c^2-11*a*d*(6*b*c+7*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d*e^3)-4/231*c^(3/4)*(3*b^2*c^2-11*a*d*(6*b*c+7*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(5/4)*e^(5/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*(c+d*x^2)^(3/2)/(e*x)^(7/2),x,8,2/9*(b^2*c^2+9*a*d*(2*b*c+a*d))*(e*x)^(3/2)*(c+d*x^2)^(3/2)/(c^2*e^5)-2/5*a^2*(c+d*x^2)^(5/2)/(c*e*(e*x)^(5/2))-2*a*(2*b*c+a*d)*(c+d*x^2)^(5/2)/(c^2*e^3*sqrt(e*x))+4/15*(b^2*c^2+9*a*d*(2*b*c+a*d))*(e*x)^(3/2)*sqrt(c+d*x^2)/(c*e^5)+8/15*(b^2*c^2+9*a*d*(2*b*c+a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(e^4*sqrt(d)*(sqrt(c)+x*sqrt(d)))-8/15*c^(1/4)*(b^2*c^2+9*a*d*(2*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(3/4)*e^(7/2)*sqrt(c+d*x^2))+4/15*c^(1/4)*(b^2*c^2+9*a*d*(2*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(3/4)*e^(7/2)*sqrt(c+d*x^2))],

# p<0
[(e*x)^(5/2)*(a+b*x^2)^2/sqrt(c+d*x^2),x,7,2/585*(117*a^2*d^2+7*b*c*(11*b*c-26*a*d))*e*(e*x)^(3/2)*sqrt(c+d*x^2)/d^3-2/117*b*(11*b*c-26*a*d)*(e*x)^(7/2)*sqrt(c+d*x^2)/(d^2*e)+2/13*b^2*(e*x)^(11/2)*sqrt(c+d*x^2)/(d*e^3)-2/195*c*(117*a^2*d^2+7*b*c*(11*b*c-26*a*d))*e^2*sqrt(e*x)*sqrt(c+d*x^2)/(d^(7/2)*(sqrt(c)+x*sqrt(d)))+2/195*c^(5/4)*(117*a^2*d^2+7*b*c*(11*b*c-26*a*d))*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(15/4)*sqrt(c+d*x^2))-1/195*c^(5/4)*(117*a^2*d^2+7*b*c*(11*b*c-26*a*d))*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(15/4)*sqrt(c+d*x^2))],
[(e*x)^(3/2)*(a+b*x^2)^2/sqrt(c+d*x^2),x,5,-2/77*b*(9*b*c-22*a*d)*(e*x)^(5/2)*sqrt(c+d*x^2)/(d^2*e)+2/11*b^2*(e*x)^(9/2)*sqrt(c+d*x^2)/(d*e^3)+2/231*(77*a^2*d^2+5*b*c*(9*b*c-22*a*d))*e*sqrt(e*x)*sqrt(c+d*x^2)/d^3-1/231*c^(3/4)*(77*a^2*d^2+5*b*c*(9*b*c-22*a*d))*e^(3/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(13/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(e*x)/sqrt(c+d*x^2),x,6,-2/45*b*(7*b*c-18*a*d)*(e*x)^(3/2)*sqrt(c+d*x^2)/(d^2*e)+2/9*b^2*(e*x)^(7/2)*sqrt(c+d*x^2)/(d*e^3)+2/15*(15*a^2*d^2+b*c*(7*b*c-18*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(d^(5/2)*(sqrt(c)+x*sqrt(d)))-2/15*c^(1/4)*(15*a^2*d^2+b*c*(7*b*c-18*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(11/4)*sqrt(c+d*x^2))+1/15*c^(1/4)*(15*a^2*d^2+b*c*(7*b*c-18*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(11/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(sqrt(e*x)*sqrt(c+d*x^2)),x,4,2/7*b^2*(e*x)^(5/2)*sqrt(c+d*x^2)/(d*e^3)-2/21*b*(5*b*c-14*a*d)*sqrt(e*x)*sqrt(c+d*x^2)/(d^2*e)+1/21*(5*b^2*c^2-14*a*b*c*d+21*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(1/4)*d^(9/4)*sqrt(e)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(3/2)*sqrt(c+d*x^2)),x,6,2/5*b^2*(e*x)^(3/2)*sqrt(c+d*x^2)/(d*e^3)-2*a^2*sqrt(c+d*x^2)/(c*e*sqrt(e*x))-2/5*(3*b^2*c^2-5*a*d*(2*b*c+a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(c*d^(3/2)*e^2*(sqrt(c)+x*sqrt(d)))+2/5*(3*b^2*c^2-5*a*d*(2*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(7/4)*e^(3/2)*sqrt(c+d*x^2))-1/5*(3*b^2*c^2-5*a*d*(2*b*c+a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(7/4)*e^(3/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(5/2)*sqrt(c+d*x^2)),x,4,-2/3*a^2*sqrt(c+d*x^2)/(c*e*(e*x)^(3/2))+2/3*b^2*sqrt(e*x)*sqrt(c+d*x^2)/(d*e^3)-1/3*(b^2*c^2-6*a*b*c*d+a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(5/4)*d^(5/4)*e^(5/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(7/2)*sqrt(c+d*x^2)),x,6,-2/5*a^2*sqrt(c+d*x^2)/(c*e*(e*x)^(5/2))-2/5*a*(10*b*c-3*a*d)*sqrt(c+d*x^2)/(c^2*e^3*sqrt(e*x))+2/5*(5*b^2*c^2+10*a*b*c*d-3*a^2*d^2)*sqrt(e*x)*sqrt(c+d*x^2)/(c^2*e^4*sqrt(d)*(sqrt(c)+x*sqrt(d)))-2/5*(5*b^2*c^2+10*a*b*c*d-3*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))+1/5*(5*b^2*c^2+10*a*b*c*d-3*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(9/2)*sqrt(c+d*x^2)),x,4,-2/7*a^2*sqrt(c+d*x^2)/(c*e*(e*x)^(7/2))-2/21*a*(14*b*c-5*a*d)*sqrt(c+d*x^2)/(c^2*e^3*(e*x)^(3/2))+1/21*(21*b^2*c^2-14*a*b*c*d+5*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(9/4)*d^(1/4)*e^(9/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(11/2)*sqrt(c+d*x^2)),x,7,-2/9*a^2*sqrt(c+d*x^2)/(c*e*(e*x)^(9/2))-2/45*a*(18*b*c-7*a*d)*sqrt(c+d*x^2)/(c^2*e^3*(e*x)^(5/2))-2/15*(15*b^2*c^2-18*a*b*c*d+7*a^2*d^2)*sqrt(c+d*x^2)/(c^3*e^5*sqrt(e*x))+2/15*(15*b^2*c^2-18*a*b*c*d+7*a^2*d^2)*sqrt(d)*sqrt(e*x)*sqrt(c+d*x^2)/(c^3*e^6*(sqrt(c)+x*sqrt(d)))-2/15*d^(1/4)*(15*b^2*c^2-18*a*b*c*d+7*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*e^(11/2)*sqrt(c+d*x^2))+1/15*d^(1/4)*(15*b^2*c^2-18*a*b*c*d+7*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*e^(11/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(13/2)*sqrt(c+d*x^2)),x,5,-2/11*a^2*sqrt(c+d*x^2)/(c*e*(e*x)^(11/2))-2/77*a*(22*b*c-9*a*d)*sqrt(c+d*x^2)/(c^2*e^3*(e*x)^(7/2))-2/231*(77*b^2*c^2-5*a*d*(22*b*c-9*a*d))*sqrt(c+d*x^2)/(c^3*e^5*(e*x)^(3/2))-1/231*d^(3/4)*(77*b^2*c^2-5*a*d*(22*b*c-9*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(13/4)*e^(13/2)*sqrt(c+d*x^2))],
[(e*x)^(7/2)*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,6,(b*c-a*d)^2*(e*x)^(9/2)/(c*d^2*e*sqrt(c+d*x^2))-1/77*(117*b^2*c^2-198*a*b*c*d+77*a^2*d^2)*e*(e*x)^(5/2)*sqrt(c+d*x^2)/(c*d^3)+2/11*b^2*(e*x)^(9/2)*sqrt(c+d*x^2)/(d^2*e)+5/231*(117*b^2*c^2-198*a*b*c*d+77*a^2*d^2)*e^3*sqrt(e*x)*sqrt(c+d*x^2)/d^4-5/462*c^(3/4)*(117*b^2*c^2-198*a*b*c*d+77*a^2*d^2)*e^(7/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(17/4)*sqrt(c+d*x^2))],
[(e*x)^(5/2)*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,7,(b*c-a*d)^2*(e*x)^(7/2)/(c*d^2*e*sqrt(c+d*x^2))-1/45*(77*b^2*c^2-126*a*b*c*d+45*a^2*d^2)*e*(e*x)^(3/2)*sqrt(c+d*x^2)/(c*d^3)+2/9*b^2*(e*x)^(7/2)*sqrt(c+d*x^2)/(d^2*e)+1/15*(77*b^2*c^2-126*a*b*c*d+45*a^2*d^2)*e^2*sqrt(e*x)*sqrt(c+d*x^2)/(d^(7/2)*(sqrt(c)+x*sqrt(d)))-1/15*c^(1/4)*(77*b^2*c^2-126*a*b*c*d+45*a^2*d^2)*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(15/4)*sqrt(c+d*x^2))+1/30*c^(1/4)*(77*b^2*c^2-126*a*b*c*d+45*a^2*d^2)*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(d^(15/4)*sqrt(c+d*x^2))],
[(e*x)^(3/2)*(a+b*x^2)^2/(c+d*x^2)^(3/2),x,5,(b*c-a*d)^2*(e*x)^(5/2)/(c*d^2*e*sqrt(c+d*x^2))+2/7*b^2*(e*x)^(5/2)*sqrt(c+d*x^2)/(d^2*e)-1/21*(45*b^2*c^2-70*a*b*c*d+21*a^2*d^2)*e*sqrt(e*x)*sqrt(c+d*x^2)/(c*d^3)+1/42*(45*b^2*c^2-70*a*b*c*d+21*a^2*d^2)*e^(3/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(1/4)*d^(13/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(e*x)/(c+d*x^2)^(3/2),x,6,(b*c-a*d)^2*(e*x)^(3/2)/(c*d^2*e*sqrt(c+d*x^2))+2/5*b^2*(e*x)^(3/2)*sqrt(c+d*x^2)/(d^2*e)-1/5*(21*b^2*c^2-30*a*b*c*d+5*a^2*d^2)*sqrt(e*x)*sqrt(c+d*x^2)/(c*d^(5/2)*(sqrt(c)+x*sqrt(d)))+1/5*(21*b^2*c^2-30*a*b*c*d+5*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(11/4)*sqrt(c+d*x^2))-1/10*(21*b^2*c^2-30*a*b*c*d+5*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(11/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((c+d*x^2)^(3/2)*sqrt(e*x)),x,4,(b*c-a*d)^2*sqrt(e*x)/(c*d^2*e*sqrt(c+d*x^2))+2/3*b^2*sqrt(e*x)*sqrt(c+d*x^2)/(d^2*e)-1/6*(5*b^2*c^2-6*a*b*c*d-3*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(5/4)*d^(9/4)*sqrt(e)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(3/2)*(c+d*x^2)^(3/2)),x,6,-(b^2*c^2-2*a*b*c*d+3*a^2*d^2)*(e*x)^(3/2)/(c^2*d*e^3*sqrt(c+d*x^2))-2*a^2/(c*e*sqrt(e*x)*sqrt(c+d*x^2))+(3*b^2*c^2-2*a*b*c*d+3*a^2*d^2)*sqrt(e*x)*sqrt(c+d*x^2)/(c^2*d^(3/2)*e^2*(sqrt(c)+x*sqrt(d)))-(3*b^2*c^2-2*a*b*c*d+3*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*d^(7/4)*e^(3/2)*sqrt(c+d*x^2))+1/2*(3*b^2*c^2-2*a*b*c*d+3*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*d^(7/4)*e^(3/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(5/2)*(c+d*x^2)^(3/2)),x,4,-2/3*a^2/(c*e*(e*x)^(3/2)*sqrt(c+d*x^2))-1/3*(3*b^2*c^2-6*a*b*c*d+5*a^2*d^2)*sqrt(e*x)/(c^2*d*e^3*sqrt(c+d*x^2))+1/6*(3*b^2*c^2+a*d*(6*b*c-5*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(9/4)*d^(5/4)*e^(5/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(7/2)*(c+d*x^2)^(3/2)),x,7,-2/5*a^2/(c*e*(e*x)^(5/2)*sqrt(c+d*x^2))+1/5*(5*b^2*c^2-3*a*d*(10*b*c-7*a*d))*(e*x)^(3/2)/(c^3*e^5*sqrt(c+d*x^2))-2/5*a*(10*b*c-7*a*d)/(c^2*e^3*sqrt(e*x)*sqrt(c+d*x^2))-1/5*(5*b^2*c^2-3*a*d*(10*b*c-7*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(c^3*e^4*sqrt(d)*(sqrt(c)+x*sqrt(d)))+1/5*(5*b^2*c^2-3*a*d*(10*b*c-7*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))-1/10*(5*b^2*c^2-3*a*d*(10*b*c-7*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))],
[(e*x)^(7/2)*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,6,1/3*(b*c-a*d)^2*(e*x)^(9/2)/(c*d^2*e*(c+d*x^2)^(3/2))+1/14*(39*b^2*c^2-42*a*b*c*d+7*a^2*d^2)*e*(e*x)^(5/2)/(c*d^3*sqrt(c+d*x^2))+2/7*b^2*(e*x)^(9/2)/(d^2*e*sqrt(c+d*x^2))-5/42*(39*b^2*c^2-42*a*b*c*d+7*a^2*d^2)*e^3*sqrt(e*x)*sqrt(c+d*x^2)/(c*d^4)+5/84*(39*b^2*c^2-42*a*b*c*d+7*a^2*d^2)*e^(7/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(1/4)*d^(17/4)*sqrt(c+d*x^2))],
[(e*x)^(5/2)*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,7,1/3*(b*c-a*d)^2*(e*x)^(7/2)/(c*d^2*e*(c+d*x^2)^(3/2))+1/30*(77*b^2*c^2-70*a*b*c*d+5*a^2*d^2)*e*(e*x)^(3/2)/(c*d^3*sqrt(c+d*x^2))+2/5*b^2*(e*x)^(7/2)/(d^2*e*sqrt(c+d*x^2))-1/10*(77*b^2*c^2-70*a*b*c*d+5*a^2*d^2)*e^2*sqrt(e*x)*sqrt(c+d*x^2)/(c*d^(7/2)*(sqrt(c)+x*sqrt(d)))+1/10*(77*b^2*c^2-70*a*b*c*d+5*a^2*d^2)*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(15/4)*sqrt(c+d*x^2))-1/20*(77*b^2*c^2-70*a*b*c*d+5*a^2*d^2)*e^(5/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(3/4)*d^(15/4)*sqrt(c+d*x^2))],
[(e*x)^(3/2)*(a+b*x^2)^2/(c+d*x^2)^(5/2),x,5,1/3*(b*c-a*d)^2*(e*x)^(5/2)/(c*d^2*e*(c+d*x^2)^(3/2))+2/3*b^2*(e*x)^(5/2)/(d^2*e*sqrt(c+d*x^2))+1/6*(15*b^2*c^2-10*a*b*c*d-a^2*d^2)*e*sqrt(e*x)/(c*d^3*sqrt(c+d*x^2))-1/12*(15*b^2*c^2-10*a*b*c*d-a^2*d^2)*e^(3/2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(5/4)*d^(13/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2*sqrt(e*x)/(c+d*x^2)^(5/2),x,6,1/3*(b*c-a*d)^2*(e*x)^(3/2)/(c*d^2*e*(c+d*x^2)^(3/2))-1/2*(b*c-a*d)*(3*b*c+a*d)*(e*x)^(3/2)/(c^2*d^2*e*sqrt(c+d*x^2))+1/2*(7*b^2*c^2-2*a*b*c*d-a^2*d^2)*sqrt(e*x)*sqrt(c+d*x^2)/(c^2*d^(5/2)*(sqrt(c)+x*sqrt(d)))-1/2*(7*b^2*c^2-2*a*b*c*d-a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*d^(11/4)*sqrt(c+d*x^2))+1/4*(7*b^2*c^2-2*a*b*c*d-a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt(e)*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(7/4)*d^(11/4)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((c+d*x^2)^(5/2)*sqrt(e*x)),x,4,1/3*(b*c-a*d)^2*sqrt(e*x)/(c*d^2*e*(c+d*x^2)^(3/2))-1/6*(b*c-a*d)*(7*b*c+5*a*d)*sqrt(e*x)/(c^2*d^2*e*sqrt(c+d*x^2))+1/12*(5*b^2*c^2+2*a*b*c*d+5*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(9/4)*d^(9/4)*sqrt(e)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(3/2)*(c+d*x^2)^(5/2)),x,7,-1/3*(b^2*c^2-2*a*b*c*d+7*a^2*d^2)*(e*x)^(3/2)/(c^2*d*e^3*(c+d*x^2)^(3/2))-2*a^2/(c*e*(c+d*x^2)^(3/2)*sqrt(e*x))+1/2*(b^2*c^2+a*d*(2*b*c-7*a*d))*(e*x)^(3/2)/(c^3*d*e^3*sqrt(c+d*x^2))-1/2*(b^2*c^2+a*d*(2*b*c-7*a*d))*sqrt(e*x)*sqrt(c+d*x^2)/(c^3*d^(3/2)*e^2*(sqrt(c)+x*sqrt(d)))+1/2*(b^2*c^2+a*d*(2*b*c-7*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*d^(7/4)*e^(3/2)*sqrt(c+d*x^2))-1/4*(b^2*c^2+a*d*(2*b*c-7*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(11/4)*d^(7/4)*e^(3/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(5/2)*(c+d*x^2)^(5/2)),x,5,-2/3*a^2/(c*e*(e*x)^(3/2)*(c+d*x^2)^(3/2))-1/3*(b^2*c^2-2*a*b*c*d+3*a^2*d^2)*sqrt(e*x)/(c^2*d*e^3*(c+d*x^2)^(3/2))+1/6*(b^2*c^2+5*a*d*(2*b*c-3*a*d))*sqrt(e*x)/(c^3*d*e^3*sqrt(c+d*x^2))+1/12*(b^2*c^2+5*a*d*(2*b*c-3*a*d))*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(13/4)*d^(5/4)*e^(5/2)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/((e*x)^(7/2)*(c+d*x^2)^(5/2)),x,8,-2/5*a^2/(c*e*(e*x)^(5/2)*(c+d*x^2)^(3/2))+1/15*(5*b^2*c^2-70*a*b*c*d+77*a^2*d^2)*(e*x)^(3/2)/(c^3*e^5*(c+d*x^2)^(3/2))-2/5*a*(10*b*c-11*a*d)/(c^2*e^3*(c+d*x^2)^(3/2)*sqrt(e*x))+1/10*(5*b^2*c^2-70*a*b*c*d+77*a^2*d^2)*(e*x)^(3/2)/(c^4*e^5*sqrt(c+d*x^2))-1/10*(5*b^2*c^2-70*a*b*c*d+77*a^2*d^2)*sqrt(e*x)*sqrt(c+d*x^2)/(c^4*e^4*sqrt(d)*(sqrt(c)+x*sqrt(d)))+1/10*(5*b^2*c^2-70*a*b*c*d+77*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticE(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(15/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))-1/20*(5*b^2*c^2-70*a*b*c*d+77*a^2*d^2)*sqrt(cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))^2)/cos(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e))))*EllipticF(sin(2*arctan(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)))),sqrt(1/2))*(sqrt(c)+x*sqrt(d))*sqrt((c+d*x^2)/(sqrt(c)+x*sqrt(d))^2)/(c^(15/4)*d^(3/4)*e^(7/2)*sqrt(c+d*x^2))],

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

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

# p>0
[(e*x)^(7/2)*sqrt(c-d*x^2)/(a-b*x^2),x,11,-2/7*e*(e*x)^(5/2)*sqrt(c-d*x^2)/b+2/21*(2*b*c-7*a*d)*e^3*sqrt(e*x)*sqrt(c-d*x^2)/(b^2*d)-2/21*c^(1/4)*(2*b^2*c^2+14*a*b*c*d-21*a^2*d^2)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*d^(5/4)*sqrt(c-d*x^2))+a*c^(1/4)*(b*c-a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))+a*c^(1/4)*(b*c-a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(5/2)*sqrt(c-d*x^2)/(a-b*x^2),x,15,-2/5*e*(e*x)^(3/2)*sqrt(c-d*x^2)/b-2/5*c^(3/4)*(2*b*c-5*a*d)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(3/4)*sqrt(c-d*x^2))+2/5*c^(3/4)*(2*b*c-5*a*d)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(3/4)*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(3/2)*sqrt(c-d*x^2)/(a-b*x^2),x,10,-2/3*e*sqrt(e*x)*sqrt(c-d*x^2)/b-2/3*c^(1/4)*(2*b*c-3*a*d)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*sqrt(c-d*x^2))],
[sqrt(e*x)*sqrt(c-d*x^2)/(a-b*x^2),x,13,2*c^(3/4)*d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b*sqrt(c-d*x^2))-2*c^(3/4)*d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((a-b*x^2)*sqrt(e*x)),x,9,2*c^(1/4)*d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*sqrt(e)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((e*x)^(3/2)*(a-b*x^2)),x,15,-2*sqrt(c-d*x^2)/(a*e*sqrt(e*x))-2*c^(3/4)*d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*e^(3/2)*sqrt(c-d*x^2))+2*c^(3/4)*d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*e^(3/2)*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*e^(3/2)*sqrt(b)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*e^(3/2)*sqrt(b)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((e*x)^(5/2)*(a-b*x^2)),x,10,-2/3*sqrt(c-d*x^2)/(a*e*(e*x)^(3/2))+2/3*c^(1/4)*d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*e^(5/2)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((e*x)^(7/2)*(a-b*x^2)),x,16,-2/5*sqrt(c-d*x^2)/(a*e*(e*x)^(5/2))-2/5*(5*b*c-2*a*d)*sqrt(c-d*x^2)/(a^2*c*e^3*sqrt(e*x))-2/5*d^(1/4)*(5*b*c-2*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(1/4)*e^(7/2)*sqrt(c-d*x^2))+2/5*d^(1/4)*(5*b*c-2*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(1/4)*e^(7/2)*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c-d*x^2))],
[(e*x)^(5/2)*(c-d*x^2)^(3/2)/(a-b*x^2),x,16,-2/45*(11*b*c-9*a*d)*e*(e*x)^(3/2)*sqrt(c-d*x^2)/b^2+2/9*d*(e*x)^(7/2)*sqrt(c-d*x^2)/(b*e)-2/15*c^(3/4)*(4*b^2*c^2-21*a*b*c*d+15*a^2*d^2)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*d^(3/4)*sqrt(c-d*x^2))+2/15*c^(3/4)*(4*b^2*c^2-21*a*b*c*d+15*a^2*d^2)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*d^(3/4)*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)^2*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(7/2)*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(7/2)*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(3/2)*(c-d*x^2)^(3/2)/(a-b*x^2),x,11,2/7*d*(e*x)^(5/2)*sqrt(c-d*x^2)/(b*e)-2/21*(9*b*c-7*a*d)*e*sqrt(e*x)*sqrt(c-d*x^2)/b^2-2/21*c^(1/4)*(12*b^2*c^2-35*a*b*c*d+21*a^2*d^2)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)*sqrt(e*x)/(a-b*x^2),x,15,2/5*d*(e*x)^(3/2)*sqrt(c-d*x^2)/(b*e)+2/5*c^(3/4)*d^(1/4)*(7*b*c-5*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b^2*sqrt(c-d*x^2))-2/5*c^(3/4)*d^(1/4)*(7*b*c-5*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b^2*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((a-b*x^2)*sqrt(e*x)),x,10,2/3*d*sqrt(e*x)*sqrt(c-d*x^2)/(b*e)+2/3*c^(1/4)*d^(3/4)*(5*b*c-3*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*sqrt(e)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b^2*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b^2*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((e*x)^(3/2)*(a-b*x^2)),x,15,-2*c*sqrt(c-d*x^2)/(a*e*sqrt(e*x))-2*c^(3/4)*d^(1/4)*(b*c+a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*b*e^(3/2)*sqrt(c-d*x^2))+2*c^(3/4)*d^(1/4)*(b*c+a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*b*e^(3/2)*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(3/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(3/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((e*x)^(5/2)*(a-b*x^2)),x,10,-2/3*c*sqrt(c-d*x^2)/(a*e*(e*x)^(3/2))+2/3*c^(1/4)*d^(3/4)*(b*c-3*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*b*e^(5/2)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*b*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*b*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((e*x)^(7/2)*(a-b*x^2)),x,16,-2/5*c*sqrt(c-d*x^2)/(a*e*(e*x)^(5/2))-2/5*(5*b*c-7*a*d)*sqrt(c-d*x^2)/(a^2*e^3*sqrt(e*x))-2/5*c^(3/4)*d^(1/4)*(5*b*c-7*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*e^(7/2)*sqrt(c-d*x^2))+2/5*c^(3/4)*d^(1/4)*(5*b*c-7*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*e^(7/2)*sqrt(c-d*x^2))-c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(b)*sqrt(c-d*x^2))+c^(1/4)*(b*c-a*d)^2*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(b)*sqrt(c-d*x^2))],

# p<0
[(e*x)^(7/2)/((a-b*x^2)*sqrt(c-d*x^2)),x,10,2/3*e^3*sqrt(e*x)*sqrt(c-d*x^2)/(b*d)-2/3*c^(1/4)*(b*c+3*a*d)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(5/4)*sqrt(c-d*x^2))+a*c^(1/4)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*sqrt(c-d*x^2))+a*c^(1/4)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(5/2)/((a-b*x^2)*sqrt(c-d*x^2)),x,13,-2*c^(3/4)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(3/4)*sqrt(c-d*x^2))+2*c^(3/4)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(3/4)*sqrt(c-d*x^2))-c^(1/4)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(3/2)/((a-b*x^2)*sqrt(c-d*x^2)),x,9,-2*c^(1/4)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*sqrt(c-d*x^2))+c^(1/4)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*sqrt(c-d*x^2))],
[sqrt(e*x)/((a-b*x^2)*sqrt(c-d*x^2)),x,6,-c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(d^(1/4)*sqrt(a)*sqrt(b)*sqrt(c-d*x^2))+c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(d^(1/4)*sqrt(a)*sqrt(b)*sqrt(c-d*x^2))],
[1/((a-b*x^2)*sqrt(e*x)*sqrt(c-d*x^2)),x,6,c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))+c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))],
[1/((e*x)^(3/2)*(a-b*x^2)*sqrt(c-d*x^2)),x,15,-2*sqrt(c-d*x^2)/(a*c*e*sqrt(e*x))-2*d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(1/4)*e^(3/2)*sqrt(c-d*x^2))+2*d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(1/4)*e^(3/2)*sqrt(c-d*x^2))-c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*e^(3/2)*sqrt(c-d*x^2))+c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*e^(3/2)*sqrt(c-d*x^2))],
[1/((e*x)^(5/2)*(a-b*x^2)*sqrt(c-d*x^2)),x,10,-2/3*sqrt(c-d*x^2)/(a*c*e*(e*x)^(3/2))+2/3*d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(3/4)*e^(5/2)*sqrt(c-d*x^2))+b*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))+b*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))],
[1/((e*x)^(7/2)*(a-b*x^2)*sqrt(c-d*x^2)),x,16,-2/5*sqrt(c-d*x^2)/(a*c*e*(e*x)^(5/2))-2/5*(5*b*c+3*a*d)*sqrt(c-d*x^2)/(a^2*c^2*e^3*sqrt(e*x))-2/5*d^(1/4)*(5*b*c+3*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(5/4)*e^(7/2)*sqrt(c-d*x^2))+2/5*d^(1/4)*(5*b*c+3*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(5/4)*e^(7/2)*sqrt(c-d*x^2))-b^(3/2)*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c-d*x^2))+b^(3/2)*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(7/2)*sqrt(c-d*x^2))],
[(e*x)^(9/2)/((a-b*x^2)*(c-d*x^2)^(3/2)),x,15,-c*e^3*(e*x)^(3/2)/(d*(b*c-a*d)*sqrt(c-d*x^2))+c^(3/4)*(3*b*c-2*a*d)*e^(9/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(7/4)*(b*c-a*d)*sqrt(c-d*x^2))-c^(3/4)*(3*b*c-2*a*d)*e^(9/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(7/4)*(b*c-a*d)*sqrt(c-d*x^2))-a^(3/2)*c^(1/4)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))+a^(3/2)*c^(1/4)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))],
[(e*x)^(7/2)/((a-b*x^2)*(c-d*x^2)^(3/2)),x,10,-c*e^3*sqrt(e*x)/(d*(b*c-a*d)*sqrt(c-d*x^2))+c^(1/4)*(b*c-2*a*d)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(5/4)*(b*c-a*d)*sqrt(c-d*x^2))+a*c^(1/4)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))+a*c^(1/4)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))],
[(e*x)^(5/2)/((a-b*x^2)*(c-d*x^2)^(3/2)),x,15,-e*(e*x)^(3/2)/((b*c-a*d)*sqrt(c-d*x^2))+c^(3/4)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(d^(3/4)*(b*c-a*d)*sqrt(c-d*x^2))-c^(3/4)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(d^(3/4)*(b*c-a*d)*sqrt(c-d*x^2))-c^(1/4)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(b)*sqrt(c-d*x^2))+c^(1/4)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(b)*sqrt(c-d*x^2))],
[(e*x)^(3/2)/((a-b*x^2)*(c-d*x^2)^(3/2)),x,10,-e*sqrt(e*x)/((b*c-a*d)*sqrt(c-d*x^2))-c^(1/4)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))+c^(1/4)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))+c^(1/4)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))],
[sqrt(e*x)/((a-b*x^2)*(c-d*x^2)^(3/2)),x,15,-d*(e*x)^(3/2)/(c*(b*c-a*d)*e*sqrt(c-d*x^2))+d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(c^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))-d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(c^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))-c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(e)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(a)*sqrt(c-d*x^2))+c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(e)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)*sqrt(a)*sqrt(c-d*x^2))],
[1/((a-b*x^2)*(c-d*x^2)^(3/2)*sqrt(e*x)),x,10,-d*sqrt(e*x)/(c*(b*c-a*d)*e*sqrt(c-d*x^2))-d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(c^(3/4)*(b*c-a*d)*sqrt(e)*sqrt(c-d*x^2))+b*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*(b*c-a*d)*sqrt(e)*sqrt(c-d*x^2))+b*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*(b*c-a*d)*sqrt(e)*sqrt(c-d*x^2))],
[1/((e*x)^(3/2)*(a-b*x^2)*(c-d*x^2)^(3/2)),x,16,-d/(c*(b*c-a*d)*e*sqrt(e*x)*sqrt(c-d*x^2))-(2*b*c-3*a*d)*sqrt(c-d*x^2)/(a*c^2*(b*c-a*d)*e*sqrt(e*x))-d^(1/4)*(2*b*c-3*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(5/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))+d^(1/4)*(2*b*c-3*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(5/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))-b^(3/2)*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))+b^(3/2)*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))],
[1/((e*x)^(5/2)*(a-b*x^2)*(c-d*x^2)^(3/2)),x,11,-d/(c*(b*c-a*d)*e*(e*x)^(3/2)*sqrt(c-d*x^2))-1/3*(2*b*c-5*a*d)*sqrt(c-d*x^2)/(a*c^2*(b*c-a*d)*e*(e*x)^(3/2))+1/3*d^(3/4)*(2*b*c-5*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(7/4)*(b*c-a*d)*e^(5/2)*sqrt(c-d*x^2))+b^2*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)*e^(5/2)*sqrt(c-d*x^2))+b^2*c^(1/4)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)*e^(5/2)*sqrt(c-d*x^2))],

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

# p>0
[(e*x)^(7/2)*sqrt(c-d*x^2)/(a-b*x^2)^2,x,11,1/2*e*(e*x)^(5/2)*sqrt(c-d*x^2)/(b*(a-b*x^2))+7/6*e^3*sqrt(e*x)*sqrt(c-d*x^2)/b^2+1/6*c^(1/4)*(8*b*c-21*a*d)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-7*a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-7*a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^3*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(5/2)*sqrt(c-d*x^2)/(a-b*x^2)^2,x,15,1/2*e*(e*x)^(3/2)*sqrt(c-d*x^2)/(b*(a-b*x^2))-5/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*sqrt(c-d*x^2))+5/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c-5*a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))-1/4*c^(1/4)*(3*b*c-5*a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))],
[(e*x)^(3/2)*sqrt(c-d*x^2)/(a-b*x^2)^2,x,10,1/2*e*sqrt(e*x)*sqrt(c-d*x^2)/(b*(a-b*x^2))-3/2*c^(1/4)*d^(3/4)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c-3*a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b^2*d^(1/4)*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c-3*a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b^2*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(1/2)*sqrt(c-d*x^2)/(a-b*x^2)^2,x,15,1/2*(e*x)^(3/2)*sqrt(c-d*x^2)/(a*e*(a-b*x^2))-1/2*c^(3/4)*d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*b*sqrt(c-d*x^2))+1/2*c^(3/4)*d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*b*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c+a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*b^(3/2)*d^(1/4)*sqrt(c-d*x^2))+1/4*c^(1/4)*(b*c+a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*b^(3/2)*d^(1/4)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((e*x)^(1/2)*(a-b*x^2)^2),x,10,1/2*sqrt(e*x)*sqrt(c-d*x^2)/(a*e*(a-b*x^2))+1/2*c^(1/4)*d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*b*sqrt(e)*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*b*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*b*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((e*x)^(3/2)*(a-b*x^2)^2),x,16,-5/2*sqrt(c-d*x^2)/(a^2*e*sqrt(e*x))+1/2*sqrt(c-d*x^2)/(a*e*(a-b*x^2)*sqrt(e*x))-5/2*c^(3/4)*d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*e^(3/2)*sqrt(c-d*x^2))+5/2*c^(3/4)*d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*e^(3/2)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(3/2)*sqrt(b)*sqrt(c-d*x^2))+1/4*c^(1/4)*(5*b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*e^(3/2)*sqrt(b)*sqrt(c-d*x^2))],
[sqrt(c-d*x^2)/((e*x)^(5/2)*(a-b*x^2)^2),x,11,-7/6*sqrt(c-d*x^2)/(a^2*e*(e*x)^(3/2))+1/2*sqrt(c-d*x^2)/(a*e*(e*x)^(3/2)*(a-b*x^2))+7/6*c^(1/4)*d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*e^(5/2)*sqrt(c-d*x^2))+1/4*c^(1/4)*(7*b*c-5*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))+1/4*c^(1/4)*(7*b*c-5*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))],
[(e*x)^(7/2)*(c-d*x^2)^(3/2)/(a-b*x^2)^2,x,12,1/2*e*(e*x)^(5/2)*(c-d*x^2)^(3/2)/(b*(a-b*x^2))-11/14*d*e*(e*x)^(5/2)*sqrt(c-d*x^2)/b^2+1/42*(57*b*c-77*a*d)*e^3*sqrt(e*x)*sqrt(c-d*x^2)/b^3+1/42*c^(1/4)*(48*b^2*c^2-259*a*b*c*d+231*a^2*d^2)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^4*d^(1/4)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-11*a*d)*(b*c-a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^4*d^(1/4)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-11*a*d)*(b*c-a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^4*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(5/2)*(c-d*x^2)^(3/2)/(a-b*x^2)^2,x,16,1/2*e*(e*x)^(3/2)*(c-d*x^2)^(3/2)/(b*(a-b*x^2))-9/10*d*e*(e*x)^(3/2)*sqrt(c-d*x^2)/b^2-3/10*c^(3/4)*d^(1/4)*(11*b*c-15*a*d)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*sqrt(c-d*x^2))+3/10*c^(3/4)*d^(1/4)*(11*b*c-15*a*d)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*sqrt(c-d*x^2))+3/4*c^(1/4)*(b^2*c^2-4*a*b*c*d+3*a^2*d^2)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(7/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))-3/4*c^(1/4)*(b^2*c^2-4*a*b*c*d+3*a^2*d^2)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(7/2)*d^(1/4)*sqrt(a)*sqrt(c-d*x^2))],
[(e*x)^(3/2)*(c-d*x^2)^(3/2)/(a-b*x^2)^2,x,11,1/2*e*(c-d*x^2)^(3/2)*sqrt(e*x)/(b*(a-b*x^2))-7/6*d*e*sqrt(e*x)*sqrt(c-d*x^2)/b^2-1/6*c^(1/4)*d^(3/4)*(17*b*c-21*a*d)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^3*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c-7*a*d)*(b*c-a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b^3*d^(1/4)*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c-7*a*d)*(b*c-a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b^3*d^(1/4)*sqrt(c-d*x^2))],
[(e*x)^(1/2)*(c-d*x^2)^(3/2)/(a-b*x^2)^2,x,15,1/2*(b*c-a*d)*(e*x)^(3/2)*sqrt(c-d*x^2)/(a*b*e*(a-b*x^2))-1/2*c^(3/4)*d^(1/4)*(b*c-5*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*b^2*sqrt(c-d*x^2))+1/2*c^(3/4)*d^(1/4)*(b*c-5*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*b^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(b^2*c^2+4*a*b*c*d-5*a^2*d^2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*b^(5/2)*d^(1/4)*sqrt(c-d*x^2))+1/4*c^(1/4)*(b^2*c^2+4*a*b*c*d-5*a^2*d^2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*b^(5/2)*d^(1/4)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((e*x)^(1/2)*(a-b*x^2)^2),x,10,1/2*(b*c-a*d)*sqrt(e*x)*sqrt(c-d*x^2)/(a*b*e*(a-b*x^2))+1/2*c^(1/4)*d^(3/4)*(b*c+3*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*b^2*sqrt(e)*sqrt(c-d*x^2))+3/4*c^(1/4)*(b*c-a*d)*(b*c+a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*b^2*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))+3/4*c^(1/4)*(b*c-a*d)*(b*c+a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*b^2*d^(1/4)*sqrt(e)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((e*x)^(3/2)*(a-b*x^2)^2),x,16,-1/2*(5*b*c-a*d)*sqrt(c-d*x^2)/(a^2*b*e*sqrt(e*x))+1/2*(b*c-a*d)*sqrt(c-d*x^2)/(a*b*e*(a-b*x^2)*sqrt(e*x))-1/2*c^(3/4)*d^(1/4)*(5*b*c-a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*b*e^(3/2)*sqrt(c-d*x^2))+1/2*c^(3/4)*d^(1/4)*(5*b*c-a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*b*e^(3/2)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b^2*c^2-4*a*b*c*d-a^2*d^2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c-d*x^2))+1/4*c^(1/4)*(5*b^2*c^2-4*a*b*c*d-a^2*d^2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*b^(3/2)*d^(1/4)*e^(3/2)*sqrt(c-d*x^2))],
[(c-d*x^2)^(3/2)/((e*x)^(5/2)*(a-b*x^2)^2),x,11,-1/6*(7*b*c-3*a*d)*sqrt(c-d*x^2)/(a^2*b*e*(e*x)^(3/2))+1/2*(b*c-a*d)*sqrt(c-d*x^2)/(a*b*e*(e*x)^(3/2)*(a-b*x^2))+1/6*c^(1/4)*d^(3/4)*(7*b*c-3*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*b*e^(5/2)*sqrt(c-d*x^2))+1/4*c^(1/4)*(b*c-a*d)*(7*b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*b*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))+1/4*c^(1/4)*(b*c-a*d)*(7*b*c-a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*b*d^(1/4)*e^(5/2)*sqrt(c-d*x^2))],

# p<0
[(e*x)^(9/2)/((a-b*x^2)^2*sqrt(c-d*x^2)),x,15,1/2*a*e^3*(e*x)^(3/2)*sqrt(c-d*x^2)/(b*(b*c-a*d)*(a-b*x^2))+1/2*c^(3/4)*(4*b*c-5*a*d)*e^(9/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(3/4)*(b*c-a*d)*sqrt(c-d*x^2))-1/2*c^(3/4)*(4*b*c-5*a*d)*e^(9/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(3/4)*(b*c-a*d)*sqrt(c-d*x^2))+1/4*c^(1/4)*(7*b*c-5*a*d)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))-1/4*c^(1/4)*(7*b*c-5*a*d)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(5/2)*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))],
[(e*x)^(7/2)/((a-b*x^2)^2*sqrt(c-d*x^2)),x,10,1/2*a*e^3*sqrt(e*x)*sqrt(c-d*x^2)/(b*(b*c-a*d)*(a-b*x^2))+1/2*c^(1/4)*(4*b*c-3*a*d)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-3*a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-3*a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^2*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))],
[(e*x)^(5/2)/((a-b*x^2)^2*sqrt(c-d*x^2)),x,15,1/2*e*(e*x)^(3/2)*sqrt(c-d*x^2)/((b*c-a*d)*(a-b*x^2))-1/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*(b*c-a*d)*sqrt(c-d*x^2))+1/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*(b*c-a*d)*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c-a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*(b*c-a*d)*sqrt(a)*sqrt(c-d*x^2))-1/4*c^(1/4)*(3*b*c-a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*(b*c-a*d)*sqrt(a)*sqrt(c-d*x^2))],
[(e*x)^(3/2)/((a-b*x^2)^2*sqrt(c-d*x^2)),x,10,1/2*e*sqrt(e*x)*sqrt(c-d*x^2)/((b*c-a*d)*(a-b*x^2))+1/2*c^(1/4)*d^(3/4)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*(b*c-a*d)*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c+a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c+a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*b*d^(1/4)*(b*c-a*d)*sqrt(c-d*x^2))],
[(e*x)^(1/2)/((a-b*x^2)^2*sqrt(c-d*x^2)),x,15,1/2*b*(e*x)^(3/2)*sqrt(c-d*x^2)/(a*(b*c-a*d)*e*(a-b*x^2))-1/2*c^(3/4)*d^(1/4)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*(b*c-a*d)*sqrt(c-d*x^2))+1/2*c^(3/4)*d^(1/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*(b*c-a*d)*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)*sqrt(b)*sqrt(c-d*x^2))+1/4*c^(1/4)*(b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)*sqrt(b)*sqrt(c-d*x^2))],
[1/((e*x)^(1/2)*(a-b*x^2)^2*sqrt(c-d*x^2)),x,10,1/2*b*sqrt(e*x)*sqrt(c-d*x^2)/(a*(b*c-a*d)*e*(a-b*x^2))+1/2*c^(1/4)*d^(3/4)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*(b*c-a*d)*sqrt(e)*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c-5*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)*sqrt(e)*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c-5*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)*sqrt(e)*sqrt(c-d*x^2))],
[1/((e*x)^(3/2)*(a-b*x^2)^2*sqrt(c-d*x^2)),x,16,-1/2*(5*b*c-4*a*d)*sqrt(c-d*x^2)/(a^2*c*(b*c-a*d)*e*sqrt(e*x))+1/2*b*sqrt(c-d*x^2)/(a*(b*c-a*d)*e*(a-b*x^2)*sqrt(e*x))-1/2*d^(1/4)*(5*b*c-4*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(1/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))+1/2*d^(1/4)*(5*b*c-4*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(1/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c-7*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))+1/4*c^(1/4)*(5*b*c-7*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*(b*c-a*d)*e^(3/2)*sqrt(c-d*x^2))],
[1/((e*x)^(5/2)*(a-b*x^2)^2*sqrt(c-d*x^2)),x,11,-1/6*(7*b*c-4*a*d)*sqrt(c-d*x^2)/(a^2*c*(b*c-a*d)*e*(e*x)^(3/2))+1/2*b*sqrt(c-d*x^2)/(a*(b*c-a*d)*e*(e*x)^(3/2)*(a-b*x^2))+1/6*d^(3/4)*(7*b*c-4*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(3/4)*(b*c-a*d)*e^(5/2)*sqrt(c-d*x^2))+1/4*b*c^(1/4)*(7*b*c-9*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*(b*c-a*d)*e^(5/2)*sqrt(c-d*x^2))+1/4*b*c^(1/4)*(7*b*c-9*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*(b*c-a*d)*e^(5/2)*sqrt(c-d*x^2))],
[(e*x)^(9/2)/((a-b*x^2)^2*(c-d*x^2)^(3/2)),x,16,1/2*(2*b*c+a*d)*e^3*(e*x)^(3/2)/(b*(b*c-a*d)^2*sqrt(c-d*x^2))+1/2*a*e^3*(e*x)^(3/2)/(b*(b*c-a*d)*(a-b*x^2)*sqrt(c-d*x^2))-1/2*c^(3/4)*(2*b*c+a*d)*e^(9/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(3/4)*(b*c-a*d)^2*sqrt(c-d*x^2))+1/2*c^(3/4)*(2*b*c+a*d)*e^(9/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(3/4)*(b*c-a*d)^2*sqrt(c-d*x^2))+1/4*c^(1/4)*(7*b*c-a*d)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(7*b*c-a*d)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(b^(3/2)*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))],
[(e*x)^(7/2)/((a-b*x^2)^2*(c-d*x^2)^(3/2)),x,11,1/2*(2*b*c+a*d)*e^3*sqrt(e*x)/(b*(b*c-a*d)^2*sqrt(c-d*x^2))+1/2*a*e^3*sqrt(e*x)/(b*(b*c-a*d)*(a-b*x^2)*sqrt(c-d*x^2))+1/2*c^(1/4)*(2*b*c+a*d)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c+a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(5*b*c+a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(b*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))],
[(e*x)^(5/2)/((a-b*x^2)^2*(c-d*x^2)^(3/2)),x,16,3/2*d*e*(e*x)^(3/2)/((b*c-a*d)^2*sqrt(c-d*x^2))+1/2*e*(e*x)^(3/2)/((b*c-a*d)*(a-b*x^2)*sqrt(c-d*x^2))-3/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/((b*c-a*d)^2*sqrt(c-d*x^2))+3/2*c^(3/4)*d^(1/4)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/((b*c-a*d)^2*sqrt(c-d*x^2))+3/4*c^(1/4)*(b*c+a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^2*sqrt(a)*sqrt(b)*sqrt(c-d*x^2))-3/4*c^(1/4)*(b*c+a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^2*sqrt(a)*sqrt(b)*sqrt(c-d*x^2))],
[(e*x)^(3/2)/((a-b*x^2)^2*(c-d*x^2)^(3/2)),x,11,3/2*d*e*sqrt(e*x)/((b*c-a*d)^2*sqrt(c-d*x^2))+1/2*e*sqrt(e*x)/((b*c-a*d)*(a-b*x^2)*sqrt(c-d*x^2))+3/2*c^(1/4)*d^(3/4)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/((b*c-a*d)^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c+5*a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c+5*a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))],
[(e*x)^(1/2)/((a-b*x^2)^2*(c-d*x^2)^(3/2)),x,16,1/2*d*(b*c+2*a*d)*(e*x)^(3/2)/(a*c*(b*c-a*d)^2*e*sqrt(c-d*x^2))+1/2*b*(e*x)^(3/2)/(a*(b*c-a*d)*e*(a-b*x^2)*sqrt(c-d*x^2))-1/2*d^(1/4)*(b*c+2*a*d)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*c^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))+1/2*d^(1/4)*(b*c+2*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*c^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))-1/4*c^(1/4)*(b*c-7*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))+1/4*c^(1/4)*(b*c-7*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)^2*sqrt(c-d*x^2))],
[1/((e*x)^(1/2)*(a-b*x^2)^2*(c-d*x^2)^(3/2)),x,11,1/2*d*(b*c+2*a*d)*sqrt(e*x)/(a*c*(b*c-a*d)^2*e*sqrt(c-d*x^2))+1/2*b*sqrt(e*x)/(a*(b*c-a*d)*e*(a-b*x^2)*sqrt(c-d*x^2))+1/2*d^(3/4)*(b*c+2*a*d)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(3/4)*(b*c-a*d)^2*sqrt(e)*sqrt(c-d*x^2))+3/4*b*c^(1/4)*(b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)^2*sqrt(e)*sqrt(c-d*x^2))+3/4*b*c^(1/4)*(b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)^2*sqrt(e)*sqrt(c-d*x^2))],
[1/((e*x)^(3/2)*(a-b*x^2)^2*(c-d*x^2)^(3/2)),x,17,1/2*d*(b*c+2*a*d)/(a*c*(b*c-a*d)^2*e*sqrt(e*x)*sqrt(c-d*x^2))+1/2*b/(a*(b*c-a*d)*e*(a-b*x^2)*sqrt(e*x)*sqrt(c-d*x^2))-1/2*(5*b^2*c^2-8*a*b*c*d+6*a^2*d^2)*sqrt(c-d*x^2)/(a^2*c^2*(b*c-a*d)^2*e*sqrt(e*x))-1/2*d^(1/4)*(5*b^2*c^2-8*a*b*c*d+6*a^2*d^2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(5/4)*(b*c-a*d)^2*e^(3/2)*sqrt(c-d*x^2))+1/2*d^(1/4)*(5*b^2*c^2-8*a*b*c*d+6*a^2*d^2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(5/4)*(b*c-a*d)^2*e^(3/2)*sqrt(c-d*x^2))-1/4*b^(3/2)*c^(1/4)*(5*b*c-11*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*(b*c-a*d)^2*e^(3/2)*sqrt(c-d*x^2))+1/4*b^(3/2)*c^(1/4)*(5*b*c-11*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*(b*c-a*d)^2*e^(3/2)*sqrt(c-d*x^2))],
[1/((e*x)^(5/2)*(a-b*x^2)^2*(c-d*x^2)^(3/2)),x,12,1/2*d*(b*c+2*a*d)/(a*c*(b*c-a*d)^2*e*(e*x)^(3/2)*sqrt(c-d*x^2))+1/2*b/(a*(b*c-a*d)*e*(e*x)^(3/2)*(a-b*x^2)*sqrt(c-d*x^2))-1/6*(7*b^2*c^2-8*a*b*c*d+10*a^2*d^2)*sqrt(c-d*x^2)/(a^2*c^2*(b*c-a*d)^2*e*(e*x)^(3/2))+1/6*d^(3/4)*(7*b^2*c^2-8*a*b*c*d+10*a^2*d^2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(7/4)*(b*c-a*d)^2*e^(5/2)*sqrt(c-d*x^2))+1/4*b^2*c^(1/4)*(7*b*c-13*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*(b*c-a*d)^2*e^(5/2)*sqrt(c-d*x^2))+1/4*b^2*c^(1/4)*(7*b*c-13*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*(b*c-a*d)^2*e^(5/2)*sqrt(c-d*x^2))],
[(e*x)^(9/2)/((a-b*x^2)^2*(c-d*x^2)^(5/2)),x,17,1/6*(2*b*c+3*a*d)*e^3*(e*x)^(3/2)/(b*(b*c-a*d)^2*(c-d*x^2)^(3/2))+1/2*a*e^3*(e*x)^(3/2)/(b*(b*c-a*d)*(a-b*x^2)*(c-d*x^2)^(3/2))+1/2*(b*c+4*a*d)*e^3*(e*x)^(3/2)/((b*c-a*d)^3*sqrt(c-d*x^2))-1/2*c^(3/4)*(b*c+4*a*d)*e^(9/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(d^(3/4)*(b*c-a*d)^3*sqrt(c-d*x^2))+1/2*c^(3/4)*(b*c+4*a*d)*e^(9/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(d^(3/4)*(b*c-a*d)^3*sqrt(c-d*x^2))+1/4*c^(1/4)*(7*b*c+3*a*d)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(b)*sqrt(c-d*x^2))-1/4*c^(1/4)*(7*b*c+3*a*d)*e^(9/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(a)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(b)*sqrt(c-d*x^2))],
[(e*x)^(7/2)/((a-b*x^2)^2*(c-d*x^2)^(5/2)),x,12,1/6*(2*b*c+3*a*d)*e^3*sqrt(e*x)/(b*(b*c-a*d)^2*(c-d*x^2)^(3/2))+1/2*a*e^3*sqrt(e*x)/(b*(b*c-a*d)*(a-b*x^2)*(c-d*x^2)^(3/2))+5/6*(b*c+2*a*d)*e^3*sqrt(e*x)/((b*c-a*d)^3*sqrt(c-d*x^2))+5/6*c^(1/4)*(b*c+2*a*d)*e^(7/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))-5/4*c^(1/4)*(b*c+a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))-5/4*c^(1/4)*(b*c+a*d)*e^(7/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))],
[(e*x)^(5/2)/((a-b*x^2)^2*(c-d*x^2)^(5/2)),x,17,5/6*d*e*(e*x)^(3/2)/((b*c-a*d)^2*(c-d*x^2)^(3/2))+1/2*e*(e*x)^(3/2)/((b*c-a*d)*(a-b*x^2)*(c-d*x^2)^(3/2))+1/2*d*(4*b*c+a*d)*e*(e*x)^(3/2)/(c*(b*c-a*d)^3*sqrt(c-d*x^2))-1/2*d^(1/4)*(4*b*c+a*d)*e^(5/2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(c^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))+1/2*d^(1/4)*(4*b*c+a*d)*e^(5/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(c^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))+1/4*c^(1/4)*(3*b*c+7*a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(a)*sqrt(c-d*x^2))-1/4*c^(1/4)*(3*b*c+7*a*d)*e^(5/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(b)*sqrt(1-d*x^2/c)/(d^(1/4)*(b*c-a*d)^3*sqrt(a)*sqrt(c-d*x^2))],
[(e*x)^(3/2)/((a-b*x^2)^2*(c-d*x^2)^(5/2)),x,12,5/6*d*e*sqrt(e*x)/((b*c-a*d)^2*(c-d*x^2)^(3/2))+1/2*e*sqrt(e*x)/((b*c-a*d)*(a-b*x^2)*(c-d*x^2)^(3/2))+1/6*d*(14*b*c+a*d)*e*sqrt(e*x)/(c*(b*c-a*d)^3*sqrt(c-d*x^2))+1/6*d^(3/4)*(14*b*c+a*d)*e^(3/2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(c^(3/4)*(b*c-a*d)^3*sqrt(c-d*x^2))-1/4*b*c^(1/4)*(b*c+9*a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))-1/4*b*c^(1/4)*(b*c+9*a*d)*e^(3/2)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a*d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))],
[(e*x)^(1/2)/((a-b*x^2)^2*(c-d*x^2)^(5/2)),x,17,1/6*d*(3*b*c+2*a*d)*(e*x)^(3/2)/(a*c*(b*c-a*d)^2*e*(c-d*x^2)^(3/2))+1/2*b*(e*x)^(3/2)/(a*(b*c-a*d)*e*(a-b*x^2)*(c-d*x^2)^(3/2))+1/2*d*(b^2*c^2+5*a*b*c*d-a^2*d^2)*(e*x)^(3/2)/(a*c^2*(b*c-a*d)^3*e*sqrt(c-d*x^2))-1/2*d^(1/4)*(b^2*c^2+5*a*b*c*d-a^2*d^2)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*c^(5/4)*(b*c-a*d)^3*sqrt(c-d*x^2))+1/2*d^(1/4)*(b^2*c^2+5*a*b*c*d-a^2*d^2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a*c^(5/4)*(b*c-a*d)^3*sqrt(c-d*x^2))-1/4*b^(3/2)*c^(1/4)*(b*c-11*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))+1/4*b^(3/2)*c^(1/4)*(b*c-11*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(e)*sqrt(1-d*x^2/c)/(a^(3/2)*d^(1/4)*(b*c-a*d)^3*sqrt(c-d*x^2))],
[1/((e*x)^(1/2)*(a-b*x^2)^2*(c-d*x^2)^(5/2)),x,12,1/6*d*(3*b*c+2*a*d)*sqrt(e*x)/(a*c*(b*c-a*d)^2*e*(c-d*x^2)^(3/2))+1/2*b*sqrt(e*x)/(a*(b*c-a*d)*e*(a-b*x^2)*(c-d*x^2)^(3/2))+1/6*d*(3*b^2*c^2+17*a*b*c*d-5*a^2*d^2)*sqrt(e*x)/(a*c^2*(b*c-a*d)^3*e*sqrt(c-d*x^2))+1/6*d^(3/4)*(3*b^2*c^2+17*a*b*c*d-5*a^2*d^2)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a*c^(7/4)*(b*c-a*d)^3*sqrt(e)*sqrt(c-d*x^2))+1/4*b^2*c^(1/4)*(3*b*c-13*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)^3*sqrt(e)*sqrt(c-d*x^2))+1/4*b^2*c^(1/4)*(3*b*c-13*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^2*d^(1/4)*(b*c-a*d)^3*sqrt(e)*sqrt(c-d*x^2))],
[1/((e*x)^(3/2)*(a-b*x^2)^2*(c-d*x^2)^(5/2)),x,18,1/6*d*(3*b*c+2*a*d)/(a*c*(b*c-a*d)^2*e*(c-d*x^2)^(3/2)*sqrt(e*x))+1/2*b/(a*(b*c-a*d)*e*(a-b*x^2)*(c-d*x^2)^(3/2)*sqrt(e*x))+1/6*d*(3*b^2*c^2+19*a*b*c*d-7*a^2*d^2)/(a*c^2*(b*c-a*d)^3*e*sqrt(e*x)*sqrt(c-d*x^2))-1/2*(5*b^3*c^3-12*a*b^2*c^2*d+19*a^2*b*c*d^2-7*a^3*d^3)*sqrt(c-d*x^2)/(a^2*c^3*(b*c-a*d)^3*e*sqrt(e*x))-1/2*d^(1/4)*(5*b^3*c^3-12*a*b^2*c^2*d+19*a^2*b*c*d^2-7*a^3*d^3)*EllipticE(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(9/4)*(b*c-a*d)^3*e^(3/2)*sqrt(c-d*x^2))+1/2*d^(1/4)*(5*b^3*c^3-12*a*b^2*c^2*d+19*a^2*b*c*d^2-7*a^3*d^3)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(9/4)*(b*c-a*d)^3*e^(3/2)*sqrt(c-d*x^2))-5/4*b^(5/2)*c^(1/4)*(b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*(b*c-a*d)^3*e^(3/2)*sqrt(c-d*x^2))+5/4*b^(5/2)*c^(1/4)*(b*c-3*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^(5/2)*d^(1/4)*(b*c-a*d)^3*e^(3/2)*sqrt(c-d*x^2))],
[1/((e*x)^(5/2)*(a-b*x^2)^2*(c-d*x^2)^(5/2)),x,13,1/6*d*(3*b*c+2*a*d)/(a*c*(b*c-a*d)^2*e*(e*x)^(3/2)*(c-d*x^2)^(3/2))+1/2*b/(a*(b*c-a*d)*e*(e*x)^(3/2)*(a-b*x^2)*(c-d*x^2)^(3/2))+1/2*d*(b^2*c^2+7*a*b*c*d-3*a^2*d^2)/(a*c^2*(b*c-a*d)^3*e*(e*x)^(3/2)*sqrt(c-d*x^2))-1/6*(7*b^3*c^3-12*a*b^2*c^2*d+35*a^2*b*c*d^2-15*a^3*d^3)*sqrt(c-d*x^2)/(a^2*c^3*(b*c-a*d)^3*e*(e*x)^(3/2))+1/6*d^(3/4)*(7*b^3*c^3-12*a*b^2*c^2*d+35*a^2*b*c*d^2-15*a^3*d^3)*EllipticF(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),I)*sqrt(1-d*x^2/c)/(a^2*c^(11/4)*(b*c-a*d)^3*e^(5/2)*sqrt(c-d*x^2))+1/4*b^3*c^(1/4)*(7*b*c-17*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*(b*c-a*d)^3*e^(5/2)*sqrt(c-d*x^2))+1/4*b^3*c^(1/4)*(7*b*c-17*a*d)*EllipticPi(d^(1/4)*sqrt(e*x)/(c^(1/4)*sqrt(e)),sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)),I)*sqrt(1-d*x^2/c)/(a^3*d^(1/4)*(b*c-a*d)^3*e^(5/2)*sqrt(c-d*x^2))],

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

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

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

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

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

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

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

# p>0
[x^5*(a+b*x^2)^(1/2)/sqrt(c+d*x^2),x,7,-1/16*(b*c-a*d)*(5*b^2*c^2+2*a*b*c*d+a^2*d^2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(5/2)*d^(7/2))-1/24*(5*b*c+3*a*d)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b^2*d^2)+1/6*x^2*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b*d)+1/16*(5*b^2*c^2+2*a*b*c*d+a^2*d^2)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b^2*d^3)],
[x^3*(a+b*x^2)^(1/2)/sqrt(c+d*x^2),x,6,1/8*(b*c-a*d)*(3*b*c+a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(3/2)*d^(5/2))+1/4*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b*d)-1/8*(3*b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d^2)],
[x*(a+b*x^2)^(1/2)/sqrt(c+d*x^2),x,5,-1/2*(b*c-a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(d^(3/2)*sqrt(b))+1/2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^(1/2)/(x*sqrt(c+d*x^2)),x,8,-arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/sqrt(c)+arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))*sqrt(b)/sqrt(d)],
[(a+b*x^2)^(1/2)/(x^3*sqrt(c+d*x^2)),x,4,-1/2*(b*c-a*d)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/(c^(3/2)*sqrt(a))-1/2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*x^2)],
[(a+b*x^2)^(1/2)/(x^5*sqrt(c+d*x^2)),x,5,1/8*(b*c-a*d)*(b*c+3*a*d)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*c^(5/2))-1/4*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(a*c*x^4)+1/8*(b*c+3*a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a*c^2*x^2)],
[x^4*(a+b*x^2)^(1/2)/sqrt(c+d*x^2),x,6,1/15*(8*b^2*c^2-3*a*b*c*d-2*a^2*d^2)*x*sqrt(a+b*x^2)/(b^2*d^2*sqrt(c+d*x^2))+1/15*c^(3/2)*(4*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(b*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/15*(8*b^2*c^2-3*a*b*c*d-2*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b^2*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/15*(4*b*c-a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d^2)+1/5*x^3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[x^2*(a+b*x^2)^(1/2)/sqrt(c+d*x^2),x,5,-1/3*(2*b*c-a*d)*x*sqrt(a+b*x^2)/(b*d*sqrt(c+d*x^2))-1/3*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*(2*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^(1/2)/(x^2*sqrt(c+d*x^2)),x,6,d*x*sqrt(a+b*x^2)/(c*sqrt(c+d*x^2))+b*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*x)],
[(a+b*x^2)^(1/2)/(x^4*sqrt(c+d*x^2)),x,6,1/3*d*(b*c-2*a*d)*x*sqrt(a+b*x^2)/(a*c^2*sqrt(c+d*x^2))-1/3*(b*c-2*a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(a*c^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*b*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(a*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*x^3)-1/3*(b*c-2*a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a*c^2*x)],
[x^5*(a+b*x^2)^(3/2)/sqrt(c+d*x^2),x,8,1/128*(b*c-a*d)^2*(35*b^2*c^2+10*a*b*c*d+3*a^2*d^2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(5/2)*d^(9/2))+1/192*(35*b^2*c^2+10*a*b*c*d+3*a^2*d^2)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b^2*d^3)-1/48*(7*b*c+3*a*d)*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/(b^2*d^2)+1/8*x^2*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/(b*d)-1/128*(b*c-a*d)*(35*b^2*c^2+10*a*b*c*d+3*a^2*d^2)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b^2*d^4)],
[x^3*(a+b*x^2)^(3/2)/sqrt(c+d*x^2),x,7,-1/16*(b*c-a*d)^2*(5*b*c+a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(3/2)*d^(7/2))-1/24*(5*b*c+a*d)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b*d^2)+1/6*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/(b*d)+1/16*(b*c-a*d)*(5*b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d^3)],
[x*(a+b*x^2)^(3/2)/sqrt(c+d*x^2),x,6,3/8*(b*c-a*d)^2*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(d^(5/2)*sqrt(b))+1/4*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d-3/8*(b*c-a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^(3/2)/(x*sqrt(c+d*x^2)),x,8,-1/2*(b*c-3*a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))*sqrt(b)/d^(3/2)-a^(3/2)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/sqrt(c)+1/2*b*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^(3/2)/(x^3*sqrt(c+d*x^2)),x,8,-1/2*(3*b*c-a*d)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/c^(3/2)+b^(3/2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/sqrt(d)-1/2*a*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*x^2)],
[(a+b*x^2)^(3/2)/(x^5*sqrt(c+d*x^2)),x,5,-3/8*(b*c-a*d)^2*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/(c^(5/2)*sqrt(a))-1/4*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(c*x^4)-3/8*(b*c-a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c^2*x^2)],
[x^4*(a+b*x^2)^(3/2)/sqrt(c+d*x^2),x,7,-2/35*(2*b*c-a*d)*(4*b^2*c^2-4*a*b*c*d-a^2*d^2)*x*sqrt(a+b*x^2)/(b^2*d^3*sqrt(c+d*x^2))-1/35*c^(3/2)*(8*b^2*c^2-11*a*b*c*d+a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(b*d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+2/35*(2*b*c-a*d)*(4*b^2*c^2-4*a*b*c*d-a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b^2*d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/35*(8*b^2*c^2-11*a*b*c*d+a^2*d^2)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d^3)-2/35*(3*b*c-4*a*d)*x^3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2+1/7*b*x^5*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[x^2*(a+b*x^2)^(3/2)/sqrt(c+d*x^2),x,6,-1/15*(13*a*c-8*b*c^2/d-3*a^2*d/b)*x*sqrt(a+b*x^2)/(d*sqrt(c+d*x^2))+2/15*c^(3/2)*(2*b*c-3*a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/15*(8*b^2*c^2-13*a*b*c*d+3*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-2/15*(2*b*c-3*a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2+1/5*b*x^3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^(3/2)/(x^2*sqrt(c+d*x^2)),x,5,(b*c+a*d)*x*sqrt(a+b*x^2)/(c*sqrt(c+d*x^2))-(b*c+a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(sqrt(c)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+2*b*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-a*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*x)],
[(a+b*x^2)^(3/2)/(x^4*sqrt(c+d*x^2)),x,6,2/3*d*(2*b*c-a*d)*x*sqrt(a+b*x^2)/(c^2*sqrt(c+d*x^2))+1/3*b*(3*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(a*sqrt(c)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-2/3*(2*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(c^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*a*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*x^3)-2/3*(2*b*c-a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c^2*x)],
[x^5*(a+b*x^2)^(5/2)/sqrt(c+d*x^2),x,9,-1/256*(b*c-a*d)^3*(63*b^2*c^2+14*a*b*c*d+3*a^2*d^2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(5/2)*d^(11/2))-1/384*(b*c-a*d)*(63*b^2*c^2+14*a*b*c*d+3*a^2*d^2)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b^2*d^4)+1/480*(63*b^2*c^2+14*a*b*c*d+3*a^2*d^2)*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/(b^2*d^3)-3/80*(3*b*c+a*d)*(a+b*x^2)^(7/2)*sqrt(c+d*x^2)/(b^2*d^2)+1/10*x^2*(a+b*x^2)^(7/2)*sqrt(c+d*x^2)/(b*d)+1/256*(b*c-a*d)^2*(63*b^2*c^2+14*a*b*c*d+3*a^2*d^2)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b^2*d^5)],
[x^3*(a+b*x^2)^(5/2)/sqrt(c+d*x^2),x,8,5/128*(b*c-a*d)^3*(7*b*c+a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(3/2)*d^(9/2))+5/192*(b*c-a*d)*(7*b*c+a*d)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(b*d^3)-1/48*(7*b*c+a*d)*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/(b*d^2)+1/8*(a+b*x^2)^(7/2)*sqrt(c+d*x^2)/(b*d)-5/128*(b*c-a*d)^2*(7*b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d^4)],
[x*(a+b*x^2)^(5/2)/sqrt(c+d*x^2),x,7,-5/16*(b*c-a*d)^3*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(d^(7/2)*sqrt(b))-5/24*(b*c-a*d)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d^2+1/6*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/d+5/16*(b*c-a*d)^2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^3],
[(a+b*x^2)^(5/2)/(x*sqrt(c+d*x^2)),x,9,1/8*(3*b^2*c^2-10*a*b*c*d+15*a^2*d^2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))*sqrt(b)/d^(5/2)-a^(5/2)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/sqrt(c)+1/4*b*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d-1/8*b*(3*b*c-7*a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^(5/2)/(x^3*sqrt(c+d*x^2)),x,9,-1/2*a^(3/2)*(5*b*c-a*d)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/c^(3/2)-1/2*b^(3/2)*(b*c-5*a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/d^(3/2)-1/2*a*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(c*x^2)+1/2*b*(b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*d)],
[(a+b*x^2)^(5/2)/(x^5*sqrt(c+d*x^2)),x,9,-1/8*(15*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(a)/c^(5/2)+b^(5/2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/sqrt(d)-1/4*a*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(c*x^4)-1/8*a*(7*b*c-3*a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c^2*x^2)],
[x^4*(a+b*x^2)^(5/2)/sqrt(c+d*x^2),x,8,1/315*(128*b^4*c^4-328*a*b^3*c^3*d+243*a^2*b^2*c^2*d^2-25*a^3*b*c*d^3-10*a^4*d^4)*x*sqrt(a+b*x^2)/(b^2*d^4*sqrt(c+d*x^2))+1/315*c^(3/2)*(64*b^3*c^3-156*a*b^2*c^2*d+105*a^2*b*c*d^2-5*a^3*d^3)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(b*d^(9/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/315*(128*b^4*c^4-328*a*b^3*c^3*d+243*a^2*b^2*c^2*d^2-25*a^3*b*c*d^3-10*a^4*d^4)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b^2*d^(9/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/9*b*x^5*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d-1/315*(64*b^3*c^3-156*a*b^2*c^2*d+105*a^2*b*c*d^2-5*a^3*d^3)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d^4)+1/315*(48*b^2*c^2-115*a*b*c*d+75*a^2*d^2)*x^3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^3-4/63*b*(2*b*c-3*a*d)*x^5*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2],
[x^2*(a+b*x^2)^(5/2)/sqrt(c+d*x^2),x,7,-1/105*(48*b^3*c^3-128*a*b^2*c^2*d+103*a^2*b*c*d^2-15*a^3*d^3)*x*sqrt(a+b*x^2)/(b*d^3*sqrt(c+d*x^2))-1/105*c^(3/2)*(24*b^2*c^2-61*a*b*c*d+45*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/105*(48*b^3*c^3-128*a*b^2*c^2*d+103*a^2*b*c*d^2-15*a^3*d^3)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b*d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/7*b*x^3*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d+1/105*(24*b^2*c^2-61*a*b*c*d+45*a^2*d^2)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^3-2/35*b*(3*b*c-5*a*d)*x^3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^(5/2)/(x^2*sqrt(c+d*x^2)),x,6,1/3*(7*a*b-2*b^2*c/d+3*a^2*d/c)*x*sqrt(a+b*x^2)/sqrt(c+d*x^2)+1/3*(2*b^2*c^2-7*a*b*c*d-3*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(d^(3/2)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*b*(b*c-9*a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-a*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(c*x)+1/3*b*(b*c+3*a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*d)],
[(a+b*x^2)^(5/2)/(x^4*sqrt(c+d*x^2)),x,6,1/3*(3*b^2*c^2+7*a*b*c*d-2*a^2*d^2)*x*sqrt(a+b*x^2)/(c^2*sqrt(c+d*x^2))-1/3*(3*b^2*c^2+7*a*b*c*d-2*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(c^(3/2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*b*(9*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(sqrt(c)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*a*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(c*x^3)-2/3*a*(3*b*c-a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c^2*x)],
[x^4*sqrt(-1+3*x^2)/sqrt(2-3*x^2),x,5,-8/45*sqrt(cos(arccos(x*sqrt(3/2)))^2)/cos(arccos(x*sqrt(3/2)))*EllipticE(sin(arccos(x*sqrt(3/2))),sqrt(2))/sqrt(3)-2/27*sqrt(cos(arccos(x*sqrt(3/2)))^2)/cos(arccos(x*sqrt(3/2)))*EllipticF(sin(arccos(x*sqrt(3/2))),sqrt(2))/sqrt(3)-7/135*x*sqrt(2-3*x^2)*sqrt(-1+3*x^2)-1/15*x^3*sqrt(2-3*x^2)*sqrt(-1+3*x^2)],
[x^3*sqrt(-1+3*x^2)/sqrt(2-3*x^2),x,6,-7/144*arcsin(3-6*x^2)-1/36*(-1+3*x^2)^(3/2)*sqrt(2-3*x^2)-7/72*sqrt(2-3*x^2)*sqrt(-1+3*x^2)],
[x^2*sqrt(-1+3*x^2)/sqrt(2-3*x^2),x,4,-1/3*sqrt(cos(arccos(x*sqrt(3/2)))^2)/cos(arccos(x*sqrt(3/2)))*EllipticE(sin(arccos(x*sqrt(3/2))),sqrt(2))/sqrt(3)-1/9*sqrt(cos(arccos(x*sqrt(3/2)))^2)/cos(arccos(x*sqrt(3/2)))*EllipticF(sin(arccos(x*sqrt(3/2))),sqrt(2))/sqrt(3)-1/9*x*sqrt(2-3*x^2)*sqrt(-1+3*x^2)],
[x*sqrt(-1+3*x^2)/sqrt(2-3*x^2),x,5,-1/12*arcsin(3-6*x^2)-1/6*sqrt(2-3*x^2)*sqrt(-1+3*x^2)],
[x^2*sqrt(2+b*x^2)/sqrt(3+d*x^2),x,5,-2/3*(3*b-d)*x*sqrt(2+b*x^2)/(b*d*sqrt(3+d*x^2))+2/3*(3*b-d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(b*d^(3/2)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(d^(3/2)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))+1/3*x*sqrt(2+b*x^2)*sqrt(3+d*x^2)/d],

# p<0
[x^5/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,6,-1/8*(4*a*b*c*d-3*(b*c+a*d)^2)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(5/2)*d^(5/2))-3/8*(b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b^2*d^2)+1/4*x^2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d)],
[x^3/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,5,-1/2*(b*c+a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(3/2)*d^(3/2))+1/2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d)],
[x/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,4,arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(sqrt(b)*sqrt(d))],
[1/(x*(a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,3,-arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/(sqrt(a)*sqrt(c))],
[1/(x^3*(a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,4,1/2*(b*c+a*d)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*c^(3/2))-1/2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a*c*x^2)],
[1/(x^5*(a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,6,-1/8*(3*b^2*c^2+2*a*b*c*d+3*a^2*d^2)*arctanh(sqrt(c)*sqrt(a+b*x^2)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*c^(5/2))-1/4*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a*c*x^4)+3/8*(b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a^2*c^2*x^2)],
[x^6/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,6,1/15*(8*b^2*c^2+7*a*b*c*d+8*a^2*d^2)*x*sqrt(a+b*x^2)/(b^3*d^2*sqrt(c+d*x^2))+4/15*c^(3/2)*(b*c+a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(b^2*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/15*(8*b^2*c^2+7*a*b*c*d+8*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b^3*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-4/15*(b*c+a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b^2*d^2)+1/5*x^3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d)],
[x^4/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,5,-2/3*(b*c+a*d)*x*sqrt(a+b*x^2)/(b^2*d*sqrt(c+d*x^2))-1/3*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(a+b*x^2)/(b*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+2/3*(b*c+a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b^2*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b*d)],
[x^2/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,2,x*sqrt(a+b*x^2)/(b*sqrt(c+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)/(b*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/(x^2*(a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,4,d*x*sqrt(a+b*x^2)/(a*c*sqrt(c+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(a*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a*c*x)],
[1/(x^4*(a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,6,-2/3*d*(b*c+a*d)*x*sqrt(a+b*x^2)/(a^2*c^2*sqrt(c+d*x^2))+2/3*(b*c+a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(a^2*c^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*b*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(a+b*x^2)/(a^2*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a*c*x^3)+2/3*(b*c+a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(a^2*c^2*x)],
[x^5/((a+b*x^2)^(3/2)*sqrt(c+d*x^2)),x,6,-1/2*(b*c+3*a*d)*arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(5/2)*d^(3/2))-a^2*sqrt(c+d*x^2)/(b^2*(b*c-a*d)*sqrt(a+b*x^2))+1/2*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(b^2*d)],
[x^3/((a+b*x^2)^(3/2)*sqrt(c+d*x^2)),x,5,arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(3/2)*sqrt(d))+a*sqrt(c+d*x^2)/(b*(b*c-a*d)*sqrt(a+b*x^2))],
[x/((a+b*x^2)^(3/2)*sqrt(c+d*x^2)),x,2,-sqrt(c+d*x^2)/((b*c-a*d)*sqrt(a+b*x^2))],
[x^5/((a+b*x^2)^(5/2)*sqrt(c+d*x^2)),x,6,arctanh(sqrt(d)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(b^(5/2)*sqrt(d))-1/3*a^2*sqrt(c+d*x^2)/(b^2*(b*c-a*d)*(a+b*x^2)^(3/2))+2/3*a*(3*b*c-2*a*d)*sqrt(c+d*x^2)/(b^2*(b*c-a*d)^2*sqrt(a+b*x^2))],
[x^3/((a+b*x^2)^(5/2)*sqrt(c+d*x^2)),x,3,1/3*a*sqrt(c+d*x^2)/(b*(b*c-a*d)*(a+b*x^2)^(3/2))-1/3*(3*b*c-a*d)*sqrt(c+d*x^2)/(b*(b*c-a*d)^2*sqrt(a+b*x^2))],
[x/((a+b*x^2)^(5/2)*sqrt(c+d*x^2)),x,3,-1/3*sqrt(c+d*x^2)/((b*c-a*d)*(a+b*x^2)^(3/2))+2/3*d*sqrt(c+d*x^2)/((b*c-a*d)^2*sqrt(a+b*x^2))],
[x^5/((a+b*x^2)^(7/2)*sqrt(c+d*x^2)),x,4,-1/5*a^2*sqrt(c+d*x^2)/(b^2*(b*c-a*d)*(a+b*x^2)^(5/2))+2/15*a*(5*b*c-3*a*d)*sqrt(c+d*x^2)/(b^2*(b*c-a*d)^2*(a+b*x^2)^(3/2))-1/15*(15*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*sqrt(c+d*x^2)/(b^2*(b*c-a*d)^3*sqrt(a+b*x^2))],
[x^3/((a+b*x^2)^(7/2)*sqrt(c+d*x^2)),x,4,1/5*a*sqrt(c+d*x^2)/(b*(b*c-a*d)*(a+b*x^2)^(5/2))-1/15*(5*b*c-a*d)*sqrt(c+d*x^2)/(b*(b*c-a*d)^2*(a+b*x^2)^(3/2))+2/15*d*(5*b*c-a*d)*sqrt(c+d*x^2)/(b*(b*c-a*d)^3*sqrt(a+b*x^2))],
[x/((a+b*x^2)^(7/2)*sqrt(c+d*x^2)),x,4,-1/5*sqrt(c+d*x^2)/((b*c-a*d)*(a+b*x^2)^(5/2))+4/15*d*sqrt(c+d*x^2)/((b*c-a*d)^2*(a+b*x^2)^(3/2))-8/15*d^2*sqrt(c+d*x^2)/((b*c-a*d)^3*sqrt(a+b*x^2))],
[x^5/((a+b*x^2)^(9/2)*sqrt(c+d*x^2)),x,5,-1/7*a^2*sqrt(c+d*x^2)/(b^2*(b*c-a*d)*(a+b*x^2)^(7/2))+2/35*a*(7*b*c-4*a*d)*sqrt(c+d*x^2)/(b^2*(b*c-a*d)^2*(a+b*x^2)^(5/2))-1/105*(35*b^2*c^2-14*a*b*c*d+3*a^2*d^2)*sqrt(c+d*x^2)/(b^2*(b*c-a*d)^3*(a+b*x^2)^(3/2))+2/105*d*(35*b^2*c^2-14*a*b*c*d+3*a^2*d^2)*sqrt(c+d*x^2)/(b^2*(b*c-a*d)^4*sqrt(a+b*x^2))],
[x/(sqrt(a-b*x^2)*sqrt(c+d*x^2)),x,4,-arctan(sqrt(d)*sqrt(a-b*x^2)/(sqrt(b)*sqrt(c+d*x^2)))/(sqrt(b)*sqrt(d))],
[x/(sqrt(a-b*x^2)*sqrt(c-d*x^2)),x,4,-arctanh(sqrt(d)*sqrt(a-b*x^2)/(sqrt(b)*sqrt(c-d*x^2)))/(sqrt(b)*sqrt(d))],
[x^2/(sqrt(2+b*x^2)*sqrt(3+d*x^2)),x,2,x*sqrt(2+b*x^2)/(b*sqrt(3+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(b*sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))],
[x^2/(sqrt(4-x^2)*sqrt(c+d*x^2)),x,5,EllipticE(1/2*x,sqrt(-4*d/c))*sqrt(c+d*x^2)/(d*sqrt(1+d*x^2/c))-c*EllipticF(1/2*x,sqrt(-4*d/c))*sqrt(1+d*x^2/c)/(d*sqrt(c+d*x^2))],
[x^2/(sqrt(4+x^2)*sqrt(c+d*x^2)),x,2,x*sqrt(c+d*x^2)/(d*sqrt(4+x^2))-sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticE(sin(arctan(1/2*x)),sqrt(1-4*d/c))*sqrt(c+d*x^2)/(d*sqrt(4+x^2)*sqrt((c+d*x^2)/(c*(4+x^2))))],
[x^2/(sqrt(1-x^2)*sqrt(2+3*x^2)),x,3,1/3*EllipticE(x,sqrt(-3/2))*sqrt(2)-1/3*EllipticF(x,sqrt(-3/2))*sqrt(2)],
[x^2/(sqrt(2-3*x^2)*sqrt(1-x^2)),x,3,-1/3*EllipticE(x,sqrt(3/2))*sqrt(2)+1/3*EllipticF(x,sqrt(3/2))*sqrt(2)],
[x^2/(sqrt(4-x^2)*sqrt(2+3*x^2)),x,3,1/3*EllipticE(1/2*x,sqrt(-6))*sqrt(2)-1/3*EllipticF(1/2*x,sqrt(-6))*sqrt(2)],
[x^2/(sqrt(2-3*x^2)*sqrt(4-x^2)),x,3,-1/3*EllipticE(1/2*x,sqrt(6))*sqrt(2)+1/3*EllipticF(1/2*x,sqrt(6))*sqrt(2)],
[x^2/(sqrt(1-4*x^2)*sqrt(2+3*x^2)),x,3,1/3*EllipticE(2*x,sqrt(-3/8))/sqrt(2)-1/3*EllipticF(2*x,sqrt(-3/8))/sqrt(2)],
[x^2/(sqrt(1-4*x^2)*sqrt(2-3*x^2)),x,3,-1/3*EllipticE(2*x,sqrt(3/8))/sqrt(2)+1/3*EllipticF(2*x,sqrt(3/8))/sqrt(2)],
[x^2/(sqrt(2-3*x^2)*sqrt(1+x^2)),x,3,EllipticE(x*sqrt(3/2),sqrt(-2/3))/sqrt(3)-EllipticF(x*sqrt(3/2),sqrt(-2/3))/sqrt(3)],
[x^2/(sqrt(2-3*x^2)*sqrt(4+x^2)),x,3,2*EllipticE(x*sqrt(3/2),sqrt(-1/6))/sqrt(3)-2*EllipticF(x*sqrt(3/2),sqrt(-1/6))/sqrt(3)],
[x^2/(sqrt(2-3*x^2)*sqrt(1+4*x^2)),x,3,1/4*EllipticE(x*sqrt(3/2),sqrt(-8/3))/sqrt(3)-1/4*EllipticF(x*sqrt(3/2),sqrt(-8/3))/sqrt(3)],
[x^2/(sqrt(1+x^2)*sqrt(2+3*x^2)),x,2,1/3*x*sqrt(2+3*x^2)/sqrt(1+x^2)-1/3*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(-1/2))*sqrt(2)*sqrt(2+3*x^2)/(sqrt(1+x^2)*sqrt((2+3*x^2)/(1+x^2)))],
[x^2/(sqrt(4+x^2)*sqrt(2+3*x^2)),x,2,1/3*x*sqrt(2+3*x^2)/sqrt(4+x^2)-1/3*sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticE(sin(arctan(1/2*x)),sqrt(-5))*sqrt(2)*sqrt(2+3*x^2)/(sqrt(4+x^2)*sqrt((2+3*x^2)/(4+x^2)))],
[x^2/(sqrt(2+3*x^2)*sqrt(1+4*x^2)),x,2,1/3*x*sqrt(2+3*x^2)/sqrt(1+4*x^2)-1/3*sqrt(cos(arctan(2*x))^2)/cos(arctan(2*x))*EllipticE(sin(arctan(2*x)),sqrt(5/8))*sqrt(2+3*x^2)/(sqrt(2)*sqrt((2+3*x^2)/(1+4*x^2))*sqrt(1+4*x^2))],
[x^2/(sqrt(1-x^2)*sqrt(-1+2*x^2)),x,3,-1/2*sqrt(cos(arccos(x))^2)/cos(arccos(x))*EllipticE(sin(arccos(x)),sqrt(2))-1/2*sqrt(cos(arccos(x))^2)/cos(arccos(x))*EllipticF(sin(arccos(x)),sqrt(2))],

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

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

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

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

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

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

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

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

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

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

# Integrands of the form (e x)^m / (a+b x^2)^(1/3) (c+d x^2)^q when b c+3 a d=0

# q>0

# q<0
[x^5/((1-x^2)^(1/3)*(3+x^2)),x,7,3/2*(1-x^2)^(2/3)+3/10*(1-x^2)^(5/3)-9/4*log(3+x^2)/2^(2/3)+27/4*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+9/2*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[x^3/((1-x^2)^(1/3)*(3+x^2)),x,6,-3/4*(1-x^2)^(2/3)+3/4*log(3+x^2)/2^(2/3)-9/4*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)-3/2*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[x/((1-x^2)^(1/3)*(3+x^2)),x,5,-1/4*log(3+x^2)/2^(2/3)+3/4*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+1/2*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[1/(x*(1-x^2)^(1/3)*(3+x^2)),x,10,-1/6*log(x)+1/12*log(3+x^2)/2^(2/3)+1/4*log(1-(1-x^2)^(1/3))-1/4*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)-1/2*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))+1/2*arctan((1+2*(1-x^2)^(1/3))/sqrt(3))/sqrt(3)],
[1/(x^3*(1-x^2)^(1/3)*(3+x^2)),x,7,-1/6*(1-x^2)^(2/3)/x^2-1/36*log(3+x^2)/2^(2/3)+1/12*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+1/6*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))],
[1/(x^5*(1-x^2)^(1/3)*(3+x^2)),x,12,-1/12*(1-x^2)^(2/3)/x^4-1/18*(1-x^2)^(2/3)/x^2-1/27*log(x)+1/108*log(3+x^2)/2^(2/3)+1/18*log(1-(1-x^2)^(1/3))-1/36*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)-1/18*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))+1/9*arctan((1+2*(1-x^2)^(1/3))/sqrt(3))/sqrt(3)],
[x^4/((1-x^2)^(1/3)*(3+x^2)),x,7,-3/7*x*(1-x^2)^(2/3)-3/2*arctanh(x)/2^(2/3)+9/2*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+54/7*x/(1-(1-x^2)^(1/3)-sqrt(3))+3/2*arctan(sqrt(3)/x)*sqrt(3)/2^(2/3)+3/2*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)*sqrt(3)/2^(2/3)-18/7*3^(3/4)*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))+27/7*3^(1/4)*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[x^2/((1-x^2)^(1/3)*(3+x^2)),x,6,1/2*arctanh(x)/2^(2/3)-3/2*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)-3*x/(1-(1-x^2)^(1/3)-sqrt(3))-1/2*arctan(sqrt(3)/x)*sqrt(3)/2^(2/3)-1/2*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)*sqrt(3)/2^(2/3)+3^(3/4)*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))-3/2*3^(1/4)*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[1/((1-x^2)^(1/3)*(3+x^2)),x,1,-1/6*arctanh(x)/2^(2/3)+1/2*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+1/2*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))+1/2*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))],
[1/(x^2*(1-x^2)^(1/3)*(3+x^2)),x,7,-1/3*(1-x^2)^(2/3)/x+1/18*arctanh(x)/2^(2/3)-1/6*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+1/3*x/(1-(1-x^2)^(1/3)-sqrt(3))-1/6*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))-1/6*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))-1/3*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(3^(1/4)*x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))+1/2*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[1/(x^4*(1-x^2)^(1/3)*(3+x^2)),x,8,-1/9*(1-x^2)^(2/3)/x^3-2/27*(1-x^2)^(2/3)/x-1/54*arctanh(x)/2^(2/3)+1/18*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+2/27*x/(1-(1-x^2)^(1/3)-sqrt(3))+1/18*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))+1/18*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))-2/27*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(3^(1/4)*x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))+1/9*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[x^7/((1-x^2)^(1/3)*(3+x^2)^2),x,7,-3/10*x^4*(1-x^2)^(2/3)/(3+x^2)+9/40*(1-x^2)^(2/3)*(69+14*x^2)/(3+x^2)-99/16*log(3+x^2)/2^(2/3)+297/16*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+99/8*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[x^5/((1-x^2)^(1/3)*(3+x^2)^2),x,7,-3/4*(1-x^2)^(2/3)-9/8*(1-x^2)^(2/3)/(3+x^2)+21/16*log(3+x^2)/2^(2/3)-63/16*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)-21/8*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[x^3/((1-x^2)^(1/3)*(3+x^2)^2),x,6,3/8*(1-x^2)^(2/3)/(3+x^2)-3/16*log(3+x^2)/2^(2/3)+9/16*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+3/8*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[x/((1-x^2)^(1/3)*(3+x^2)^2),x,6,-1/8*(1-x^2)^(2/3)/(3+x^2)-1/48*log(3+x^2)/2^(2/3)+1/16*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+1/8*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))],
[1/(x*(1-x^2)^(1/3)*(3+x^2)^2),x,11,1/24*(1-x^2)^(2/3)/(3+x^2)-1/18*log(x)+5/144*log(3+x^2)/2^(2/3)+1/12*log(1-(1-x^2)^(1/3))-5/48*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)-5/24*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))+1/6*arctan((1+2*(1-x^2)^(1/3))/sqrt(3))/sqrt(3)],
[1/(x^3*(1-x^2)^(1/3)*(3+x^2)^2),x,12,-5/72*(1-x^2)^(2/3)/(3+x^2)-1/6*(1-x^2)^(2/3)/(x^2*(3+x^2))+1/54*log(x)-1/48*log(3+x^2)/2^(2/3)-1/36*log(1-(1-x^2)^(1/3))+1/16*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)+1/8*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))-1/18*arctan((1+2*(1-x^2)^(1/3))/sqrt(3))/sqrt(3)],
[1/(x^5*(1-x^2)^(1/3)*(3+x^2)^2),x,13,1/216*(1-x^2)^(2/3)/(3+x^2)-1/12*(1-x^2)^(2/3)/(x^4*(3+x^2))-1/36*(1-x^2)^(2/3)/(x^2*(3+x^2))-1/54*log(x)+13/1296*log(3+x^2)/2^(2/3)+1/36*log(1-(1-x^2)^(1/3))-13/432*log(2^(2/3)-(1-x^2)^(1/3))/2^(2/3)-13/216*arctan((1+(2-2*x^2)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))+1/18*arctan((1+2*(1-x^2)^(1/3))/sqrt(3))/sqrt(3)],
[x^4/((1-x^2)^(1/3)*(3+x^2)^2),x,7,3/8*x*(1-x^2)^(2/3)/(3+x^2)+5/8*arctanh(x)/2^(2/3)-15/8*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)-27/8*x/(1-(1-x^2)^(1/3)-sqrt(3))-5/8*arctan(sqrt(3)/x)*sqrt(3)/2^(2/3)-5/8*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)*sqrt(3)/2^(2/3)+9/4*3^(3/4)*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(x*sqrt(2)*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))-27/16*3^(1/4)*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[x^2/((1-x^2)^(1/3)*(3+x^2)^2),x,7,-1/8*x*(1-x^2)^(2/3)/(3+x^2)-1/24*arctanh(x)/2^(2/3)+1/8*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+1/8*x/(1-(1-x^2)^(1/3)-sqrt(3))+1/8*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))+1/8*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))-1/4*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(3^(1/4)*x*sqrt(2)*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))+1/16*3^(1/4)*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[1/((1-x^2)^(1/3)*(3+x^2)^2),x,7,1/24*x*(1-x^2)^(2/3)/(3+x^2)-1/24*arctanh(x)/2^(2/3)+1/8*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)-1/24*x/(1-(1-x^2)^(1/3)-sqrt(3))+1/8*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))+1/8*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))+1/12*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(3^(1/4)*x*sqrt(2)*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))-1/16*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[1/(x^2*(1-x^2)^(1/3)*(3+x^2)^2),x,8,-1/8*(1-x^2)^(2/3)/x+1/24*(1-x^2)^(2/3)/(x*(3+x^2))+7/216*arctanh(x)/2^(2/3)-7/72*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+1/8*x/(1-(1-x^2)^(1/3)-sqrt(3))-7/72*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))-7/72*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))-1/4*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(3^(1/4)*x*sqrt(2)*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))+1/16*3^(1/4)*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],
[1/(x^4*(1-x^2)^(1/3)*(3+x^2)^2),x,9,-11/216*(1-x^2)^(2/3)/x^3+11/648*(1-x^2)^(2/3)/x+1/24*(1-x^2)^(2/3)/(x^3*(3+x^2))-11/648*arctanh(x)/2^(2/3)+11/216*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)-11/648*x/(1-(1-x^2)^(1/3)-sqrt(3))+11/216*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))+11/216*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))+11/324*(1-(1-x^2)^(1/3))*EllipticF((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)/(3^(1/4)*x*sqrt(2)*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))-11/432*(1-(1-x^2)^(1/3))*EllipticE((1-(1-x^2)^(1/3)+sqrt(3))/(1-(1-x^2)^(1/3)-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1+(1-x^2)^(1/3)+(1-x^2)^(2/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2)*sqrt(2+sqrt(3))/(3^(3/4)*x*sqrt((-1+(1-x^2)^(1/3))/(1-(1-x^2)^(1/3)-sqrt(3))^2))],

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

# Integrands of the form e^m (a+b x^2)^(1/4) (c+d x^2)^q when b c-2 a d=0

# a>0
[x^7/((2-3*x^2)^(1/4)*(4-3*x^2)),x,10,56/243*(2-3*x^2)^(3/4)-16/567*(2-3*x^2)^(7/4)+2/891*(2-3*x^2)^(11/4)+32/81*2^(1/4)*arctan((sqrt(2)-sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))+32/81*2^(1/4)*arctanh((sqrt(2)+sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))],
[x^5/((2-3*x^2)^(1/4)*(4-3*x^2)),x,7,4/27*(2-3*x^2)^(3/4)-2/189*(2-3*x^2)^(7/4)+8/27*2^(1/4)*arctan((sqrt(2)-sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))+8/27*2^(1/4)*arctanh((sqrt(2)+sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))],
[x^3/((2-3*x^2)^(1/4)*(4-3*x^2)),x,4,2/27*(2-3*x^2)^(3/4)+2/9*2^(1/4)*arctan((sqrt(2)-sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))+2/9*2^(1/4)*arctanh((sqrt(2)+sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))],
[x/((2-3*x^2)^(1/4)*(4-3*x^2)),x,1,1/3*arctan((sqrt(2)-sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))/2^(3/4)+1/3*arctanh((sqrt(2)+sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))/2^(3/4)],
[1/(x*(2-3*x^2)^(1/4)*(4-3*x^2)),x,8,1/4*arctan((2-3*x^2)^(1/4)/2^(1/4))/2^(1/4)+1/4*arctan((sqrt(2)-sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))/2^(3/4)-1/4*arctanh((2-3*x^2)^(1/4)/2^(1/4))/2^(1/4)+1/4*arctanh((sqrt(2)+sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))/2^(3/4)],
[1/(x^3*(2-3*x^2)^(1/4)*(4-3*x^2)),x,14,-1/16*(2-3*x^2)^(3/4)/x^2+9/32*arctan((2-3*x^2)^(1/4)/2^(1/4))/2^(1/4)+3/16*arctan((sqrt(2)-sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))/2^(3/4)-9/32*arctanh((2-3*x^2)^(1/4)/2^(1/4))/2^(1/4)+3/16*arctanh((sqrt(2)+sqrt(2-3*x^2))/(2^(3/4)*(2-3*x^2)^(1/4)))/2^(3/4)],
[x^4/((2-3*x^2)^(1/4)*(4-3*x^2)),x,6,2/45*x*(2-3*x^2)^(3/4)+4/9*2^(1/4)*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)+4/9*2^(1/4)*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)-16/15*2^(1/4)*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticE(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))/sqrt(3)],
[x^2/((2-3*x^2)^(1/4)*(4-3*x^2)),x,4,1/3*2^(1/4)*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)+1/3*2^(1/4)*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)-2/3*2^(1/4)*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticE(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))/sqrt(3)],
[1/((2-3*x^2)^(1/4)*(4-3*x^2)),x,1,1/2*arctan((2-sqrt(2)*sqrt(2-3*x^2))/(2^(1/4)*x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(3/4)*sqrt(3))+1/2*arctanh((2+sqrt(2)*sqrt(2-3*x^2))/(2^(1/4)*x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(3/4)*sqrt(3))],
[1/(x^2*(2-3*x^2)^(1/4)*(4-3*x^2)),x,5,-1/8*(2-3*x^2)^(3/4)/x+1/8*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(3/4)+1/8*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(3/4)-1/4*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticE(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))*sqrt(3)/2^(3/4)],
[1/(x^4*(2-3*x^2)^(1/4)*(4-3*x^2)),x,8,-1/24*(2-3*x^2)^(3/4)/x^3-3/16*(2-3*x^2)^(3/4)/x+3/32*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(3/4)+3/32*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(3/4)-3/8*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticE(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))*sqrt(3)/2^(3/4)],

# a<0
[x^7/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,7,14/243*(-1+3*x^2)^(3/4)+8/567*(-1+3*x^2)^(7/4)+2/891*(-1+3*x^2)^(11/4)+8/81*arctan((-1+3*x^2)^(1/4))-8/81*arctanh((-1+3*x^2)^(1/4))],
[x^5/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,7,2/27*(-1+3*x^2)^(3/4)+2/189*(-1+3*x^2)^(7/4)+4/27*arctan((-1+3*x^2)^(1/4))-4/27*arctanh((-1+3*x^2)^(1/4))],
[x^3/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,6,2/27*(-1+3*x^2)^(3/4)+2/9*arctan((-1+3*x^2)^(1/4))-2/9*arctanh((-1+3*x^2)^(1/4))],
[x/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,5,1/3*arctan((-1+3*x^2)^(1/4))-1/3*arctanh((-1+3*x^2)^(1/4))],
[1/(x*(-2+3*x^2)*(-1+3*x^2)^(1/4)),x,16,1/2*arctan((-1+3*x^2)^(1/4))-1/2*arctanh((-1+3*x^2)^(1/4))+1/2*arctan(1-(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)-1/2*arctan(1+(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)-1/4*log(1-(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)+1/4*log(1+(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)],
[1/(x^3*(-2+3*x^2)*(-1+3*x^2)^(1/4)),x,17,-1/4*(-1+3*x^2)^(3/4)/x^2+3/4*arctan((-1+3*x^2)^(1/4))-3/4*arctanh((-1+3*x^2)^(1/4))+9/8*arctan(1-(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)-9/8*arctan(1+(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)-9/16*log(1-(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)+9/16*log(1+(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)],
[x^4/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,12,2/45*x*(-1+3*x^2)^(3/4)-1/9*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(2/3)-1/9*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(2/3)+8/15*x*(-1+3*x^2)^(1/4)/(1+sqrt(-1+3*x^2))-8/15*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticE(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))+4/15*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))],
[x^2/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,7,-1/3*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)-1/3*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)+2/3*x*(-1+3*x^2)^(1/4)/(1+sqrt(-1+3*x^2))-2/3*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticE(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))+1/3*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))],
[1/((-2+3*x^2)*(-1+3*x^2)^(1/4)),x,1,-1/2*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)-1/2*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)],
[1/(x^2*(-2+3*x^2)*(-1+3*x^2)^(1/4)),x,8,-1/2*(-1+3*x^2)^(3/4)/x-1/4*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)-1/4*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)+3/2*x*(-1+3*x^2)^(1/4)/(1+sqrt(-1+3*x^2))-1/2*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticE(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*sqrt(3)*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/x+1/4*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*sqrt(3)*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/x],
[1/(x^4*(-2+3*x^2)*(-1+3*x^2)^(1/4)),x,14,-1/6*(-1+3*x^2)^(3/4)/x^3-3/2*(-1+3*x^2)^(3/4)/x-3/8*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)-3/8*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)+9/2*x*(-1+3*x^2)^(1/4)/(1+sqrt(-1+3*x^2))-3/2*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticE(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*sqrt(3)*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/x+3/4*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*sqrt(3)*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/x],

# Integrands of the form e^m (a+b x^2)^(3/4) (c+d x^2)^q when b c-2 a d=0

# a>0
[x^2/((2+3*x^2)^(3/4)*(4+3*x^2)),x,1,-1/3*arctan(1/2*(2*2^(3/4)+2*2^(1/4)*sqrt(2+3*x^2))/(x*(2+3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))+1/3*arctanh(1/2*(2*2^(3/4)-2*2^(1/4)*sqrt(2+3*x^2))/(x*(2+3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))],
[x^2/((2-3*x^2)^(3/4)*(4-3*x^2)),x,1,1/3*arctan((2-sqrt(2)*sqrt(2-3*x^2))/(2^(1/4)*x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))-1/3*arctanh((2+sqrt(2)*sqrt(2-3*x^2))/(2^(1/4)*x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))],
[x^2/((2+b*x^2)^(3/4)*(4+b*x^2)),x,1,-arctan(1/2*(2*2^(3/4)+2*2^(1/4)*sqrt(2+b*x^2))/(x*(2+b*x^2)^(1/4)*sqrt(b)))/(2^(1/4)*b^(3/2))+arctanh(1/2*(2*2^(3/4)-2*2^(1/4)*sqrt(2+b*x^2))/(x*(2+b*x^2)^(1/4)*sqrt(b)))/(2^(1/4)*b^(3/2))],
[x^2/((2-b*x^2)^(3/4)*(4-b*x^2)),x,1,arctan((2-sqrt(2)*sqrt(2-b*x^2))/(2^(1/4)*x*(2-b*x^2)^(1/4)*sqrt(b)))/(2^(1/4)*b^(3/2))-arctanh((2+sqrt(2)*sqrt(2-b*x^2))/(2^(1/4)*x*(2-b*x^2)^(1/4)*sqrt(b)))/(2^(1/4)*b^(3/2))],
[x^2/((a+3*x^2)^(3/4)*(2*a+3*x^2)),x,1,-1/3*arctan(a^(3/4)*(1+sqrt(a+3*x^2)/sqrt(a))/(x*(a+3*x^2)^(1/4)*sqrt(3)))/(a^(1/4)*sqrt(3))+1/3*arctanh(a^(3/4)*(1-sqrt(a+3*x^2)/sqrt(a))/(x*(a+3*x^2)^(1/4)*sqrt(3)))/(a^(1/4)*sqrt(3))],
[x^2/((a-3*x^2)^(3/4)*(2*a-3*x^2)),x,1,1/3*arctan(a^(3/4)*(1-sqrt(a-3*x^2)/sqrt(a))/(x*(a-3*x^2)^(1/4)*sqrt(3)))/(a^(1/4)*sqrt(3))-1/3*arctanh(a^(3/4)*(1+sqrt(a-3*x^2)/sqrt(a))/(x*(a-3*x^2)^(1/4)*sqrt(3)))/(a^(1/4)*sqrt(3))],
[x^2/((a+b*x^2)^(3/4)*(2*a+b*x^2)),x,1,-arctan(a^(3/4)*(1+sqrt(a+b*x^2)/sqrt(a))/(x*(a+b*x^2)^(1/4)*sqrt(b)))/(a^(1/4)*b^(3/2))+arctanh(a^(3/4)*(1-sqrt(a+b*x^2)/sqrt(a))/(x*(a+b*x^2)^(1/4)*sqrt(b)))/(a^(1/4)*b^(3/2))],
[x^2/((a-b*x^2)^(3/4)*(2*a-b*x^2)),x,1,arctan(a^(3/4)*(1-sqrt(a-b*x^2)/sqrt(a))/(x*(a-b*x^2)^(1/4)*sqrt(b)))/(a^(1/4)*b^(3/2))-arctanh(a^(3/4)*(1+sqrt(a-b*x^2)/sqrt(a))/(x*(a-b*x^2)^(1/4)*sqrt(b)))/(a^(1/4)*b^(3/2))],
[x^7/((2-3*x^2)^(3/4)*(4-3*x^2)),x,20,56/81*(2-3*x^2)^(1/4)-16/405*(2-3*x^2)^(5/4)+2/729*(2-3*x^2)^(9/4)-16/81*2^(3/4)*arctan(1+(4-6*x^2)^(1/4))+16/81*2^(3/4)*arctan(1-2^(1/4)*(2-3*x^2)^(1/4))+8/81*2^(3/4)*log(-2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))-8/81*2^(3/4)*log(2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))],
[x^5/((2-3*x^2)^(3/4)*(4-3*x^2)),x,17,4/9*(2-3*x^2)^(1/4)-2/135*(2-3*x^2)^(5/4)-4/27*2^(3/4)*arctan(1+(4-6*x^2)^(1/4))+4/27*2^(3/4)*arctan(1-2^(1/4)*(2-3*x^2)^(1/4))+2/27*2^(3/4)*log(-2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))-2/27*2^(3/4)*log(2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))],
[x^3/((2-3*x^2)^(3/4)*(4-3*x^2)),x,14,2/9*(2-3*x^2)^(1/4)-1/9*2^(3/4)*arctan(1+(4-6*x^2)^(1/4))+1/9*2^(3/4)*arctan(1-2^(1/4)*(2-3*x^2)^(1/4))+1/9*log(-2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)-1/9*log(2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)],
[x/((2-3*x^2)^(3/4)*(4-3*x^2)),x,11,-1/6*arctan(1+(4-6*x^2)^(1/4))/2^(1/4)+1/6*arctan(1-2^(1/4)*(2-3*x^2)^(1/4))/2^(1/4)+1/12*log(-2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)-1/12*log(2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)],
[1/(x*(2-3*x^2)^(3/4)*(4-3*x^2)),x,18,-1/4*arctan((2-3*x^2)^(1/4)/2^(1/4))/2^(3/4)-1/8*arctan(1+(4-6*x^2)^(1/4))/2^(1/4)+1/8*arctan(1-2^(1/4)*(2-3*x^2)^(1/4))/2^(1/4)-1/4*arctanh((2-3*x^2)^(1/4)/2^(1/4))/2^(3/4)+1/16*log(-2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)-1/16*log(2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)],
[1/(x^3*(2-3*x^2)^(3/4)*(4-3*x^2)),x,24,-1/16*(2-3*x^2)^(1/4)/x^2-15/32*arctan((2-3*x^2)^(1/4)/2^(1/4))/2^(3/4)-3/32*arctan(1+(4-6*x^2)^(1/4))/2^(1/4)+3/32*arctan(1-2^(1/4)*(2-3*x^2)^(1/4))/2^(1/4)-15/32*arctanh((2-3*x^2)^(1/4)/2^(1/4))/2^(3/4)+3/64*log(-2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)-3/64*log(2^(3/4)*(2-3*x^2)^(1/4)+sqrt(2)+sqrt(2-3*x^2))/2^(1/4)],
[x^6/((2-3*x^2)^(3/4)*(4-3*x^2)),x,11,80/567*x*(2-3*x^2)^(1/4)+2/63*x^3*(2-3*x^2)^(1/4)+8/27*2^(3/4)*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)-8/27*2^(3/4)*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)-160/567*2^(3/4)*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticF(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))/sqrt(3)],
[x^4/((2-3*x^2)^(3/4)*(4-3*x^2)),x,8,2/27*x*(2-3*x^2)^(1/4)+2/9*2^(3/4)*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)-2/9*2^(3/4)*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/sqrt(3)-4/27*2^(3/4)*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticF(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))/sqrt(3)],
[x^2/((2-3*x^2)^(3/4)*(4-3*x^2)),x,1,1/3*arctan((2-sqrt(2)*sqrt(2-3*x^2))/(2^(1/4)*x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))-1/3*arctanh((2+sqrt(2)*sqrt(2-3*x^2))/(2^(1/4)*x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))],
[1/((2-3*x^2)^(3/4)*(4-3*x^2)),x,3,1/4*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))-1/4*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))/(2^(1/4)*sqrt(3))+1/2*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticF(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))/(2^(1/4)*sqrt(3))],
[1/(x^2*(2-3*x^2)^(3/4)*(4-3*x^2)),x,7,-1/8*(2-3*x^2)^(1/4)/x+1/16*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(1/4)-1/16*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(1/4)+1/4*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticF(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))*sqrt(3)/2^(1/4)],
[1/(x^4*(2-3*x^2)^(3/4)*(4-3*x^2)),x,10,-1/24*(2-3*x^2)^(1/4)/x^3-1/4*(2-3*x^2)^(1/4)/x+3/64*arctan((2^(3/4)-2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(1/4)-3/64*arctanh((2^(3/4)+2^(1/4)*sqrt(2-3*x^2))/(x*(2-3*x^2)^(1/4)*sqrt(3)))*sqrt(3)/2^(1/4)+11/32*sqrt(cos(1/2*arcsin(x*sqrt(3/2)))^2)/cos(1/2*arcsin(x*sqrt(3/2)))*EllipticF(sin(1/2*arcsin(x*sqrt(3/2))),sqrt(2))*sqrt(3)/2^(1/4)],

# a<0
[x^2/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,1,1/3*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)-1/3*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)],
[x^2/((-2-3*x^2)*(-1-3*x^2)^(3/4)),x,1,1/3*arctan(x*sqrt(3/2)/(-1-3*x^2)^(1/4))/sqrt(6)-1/3*arctanh(x*sqrt(3/2)/(-1-3*x^2)^(1/4))/sqrt(6)],
[x^2/((-2+b*x^2)*(-1+b*x^2)^(3/4)),x,1,arctan(x*sqrt(b)/((-1+b*x^2)^(1/4)*sqrt(2)))/(b^(3/2)*sqrt(2))-arctanh(x*sqrt(b)/((-1+b*x^2)^(1/4)*sqrt(2)))/(b^(3/2)*sqrt(2))],
[x^2/((-2-b*x^2)*(-1-b*x^2)^(3/4)),x,1,arctan(x*sqrt(b)/((-1-b*x^2)^(1/4)*sqrt(2)))/(b^(3/2)*sqrt(2))-arctanh(x*sqrt(b)/((-1-b*x^2)^(1/4)*sqrt(2)))/(b^(3/2)*sqrt(2))],
[x^2/((-2*a+3*x^2)*(-a+3*x^2)^(3/4)),x,1,1/3*arctan(x*sqrt(3/2)/(a^(1/4)*(-a+3*x^2)^(1/4)))/(a^(1/4)*sqrt(6))-1/3*arctanh(x*sqrt(3/2)/(a^(1/4)*(-a+3*x^2)^(1/4)))/(a^(1/4)*sqrt(6))],
[x^2/((-2*a-3*x^2)*(-a-3*x^2)^(3/4)),x,1,1/3*arctan(x*sqrt(3/2)/(a^(1/4)*(-a-3*x^2)^(1/4)))/(a^(1/4)*sqrt(6))-1/3*arctanh(x*sqrt(3/2)/(a^(1/4)*(-a-3*x^2)^(1/4)))/(a^(1/4)*sqrt(6))],
[x^2/((-2*a+b*x^2)*(-a+b*x^2)^(3/4)),x,1,arctan(x*sqrt(b)/(a^(1/4)*(-a+b*x^2)^(1/4)*sqrt(2)))/(a^(1/4)*b^(3/2)*sqrt(2))-arctanh(x*sqrt(b)/(a^(1/4)*(-a+b*x^2)^(1/4)*sqrt(2)))/(a^(1/4)*b^(3/2)*sqrt(2))],
[x^2/((-2*a-b*x^2)*(-a-b*x^2)^(3/4)),x,1,arctan(x*sqrt(b)/(a^(1/4)*(-a-b*x^2)^(1/4)*sqrt(2)))/(a^(1/4)*b^(3/2)*sqrt(2))-arctanh(x*sqrt(b)/(a^(1/4)*(-a-b*x^2)^(1/4)*sqrt(2)))/(a^(1/4)*b^(3/2)*sqrt(2))],
[x^7/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,7,14/81*(-1+3*x^2)^(1/4)+8/405*(-1+3*x^2)^(5/4)+2/729*(-1+3*x^2)^(9/4)-8/81*arctan((-1+3*x^2)^(1/4))-8/81*arctanh((-1+3*x^2)^(1/4))],
[x^5/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,7,2/9*(-1+3*x^2)^(1/4)+2/135*(-1+3*x^2)^(5/4)-4/27*arctan((-1+3*x^2)^(1/4))-4/27*arctanh((-1+3*x^2)^(1/4))],
[x^3/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,6,2/9*(-1+3*x^2)^(1/4)-2/9*arctan((-1+3*x^2)^(1/4))-2/9*arctanh((-1+3*x^2)^(1/4))],
[x/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,5,-1/3*arctan((-1+3*x^2)^(1/4))-1/3*arctanh((-1+3*x^2)^(1/4))],
[1/(x*(-2+3*x^2)*(-1+3*x^2)^(3/4)),x,16,-1/2*arctan((-1+3*x^2)^(1/4))-1/2*arctanh((-1+3*x^2)^(1/4))+1/2*arctan(1-(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)-1/2*arctan(1+(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)+1/4*log(1-(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)-1/4*log(1+(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)],
[1/(x^3*(-2+3*x^2)*(-1+3*x^2)^(3/4)),x,17,-1/4*(-1+3*x^2)^(1/4)/x^2-3/4*arctan((-1+3*x^2)^(1/4))-3/4*arctanh((-1+3*x^2)^(1/4))+15/8*arctan(1-(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)-15/8*arctan(1+(-1+3*x^2)^(1/4)*sqrt(2))/sqrt(2)+15/16*log(1-(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)-15/16*log(1+(-1+3*x^2)^(1/4)*sqrt(2)+sqrt(-1+3*x^2))/sqrt(2)],
[x^6/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,15,40/567*x*(-1+3*x^2)^(1/4)+2/63*x^3*(-1+3*x^2)^(1/4)+2/27*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(2/3)-2/27*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(2/3)+40/567*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))],
[x^4/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,11,2/27*x*(-1+3*x^2)^(1/4)+1/9*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(2/3)-1/9*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(2/3)+2/27*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))],
[x^2/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,1,1/3*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)-1/3*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)],
[1/((-2+3*x^2)*(-1+3*x^2)^(3/4)),x,4,1/2*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)-1/2*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))/sqrt(6)-1/2*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/(x*sqrt(3))],
[1/(x^2*(-2+3*x^2)*(-1+3*x^2)^(3/4)),x,9,-1/2*(-1+3*x^2)^(1/4)/x+1/4*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)-1/4*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)-1/2*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*sqrt(3)*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/x],
[1/(x^4*(-2+3*x^2)*(-1+3*x^2)^(3/4)),x,13,-1/6*(-1+3*x^2)^(1/4)/x^3-2*(-1+3*x^2)^(1/4)/x+3/8*arctan(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)-3/8*arctanh(x*sqrt(3/2)/(-1+3*x^2)^(1/4))*sqrt(3/2)-11/8*sqrt(cos(2*arctan((-1+3*x^2)^(1/4)))^2)/cos(2*arctan((-1+3*x^2)^(1/4)))*EllipticF(sin(2*arctan((-1+3*x^2)^(1/4))),sqrt(1/2))*sqrt(3)*(1+sqrt(-1+3*x^2))*sqrt(x^2/(1+sqrt(-1+3*x^2))^2)/x],

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

# p>0

# p<0
[(e*x)^(5/2)*(c+d*x^2)/(a+b*x^2)^(3/4),x,7,1/16*(8*b*c-7*a*d)*e*(e*x)^(3/2)*(a+b*x^2)^(1/4)/b^2+1/4*d*(e*x)^(7/2)*(a+b*x^2)^(1/4)/(b*e)+3/32*a*(8*b*c-7*a*d)*e^(5/2)*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(11/4)-3/32*a*(8*b*c-7*a*d)*e^(5/2)*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(11/4)],
[(e*x)^(1/2)*(c+d*x^2)/(a+b*x^2)^(3/4),x,6,1/2*d*(e*x)^(3/2)*(a+b*x^2)^(1/4)/(b*e)-1/4*(4*b*c-3*a*d)*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))*sqrt(e)/b^(7/4)+1/4*(4*b*c-3*a*d)*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))*sqrt(e)/b^(7/4)],
[(c+d*x^2)/((e*x)^(3/2)*(a+b*x^2)^(3/4)),x,6,-d*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/(b^(3/4)*e^(3/2))+d*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/(b^(3/4)*e^(3/2))-2*c*(a+b*x^2)^(1/4)/(a*e*sqrt(e*x))],
[(c+d*x^2)/((e*x)^(7/2)*(a+b*x^2)^(3/4)),x,2,-2/5*c*(a+b*x^2)^(1/4)/(a*e*(e*x)^(5/2))+2/5*(4*b*c-5*a*d)*(a+b*x^2)^(1/4)/(a^2*e^3*sqrt(e*x))],
[(c+d*x^2)/((e*x)^(11/2)*(a+b*x^2)^(3/4)),x,3,-2/9*c*(a+b*x^2)^(1/4)/(a*e*(e*x)^(9/2))+2/9*(8*b*c-9*a*d)*(a+b*x^2)^(1/4)/(a^2*e^3*(e*x)^(5/2))-8/45*(8*b*c-9*a*d)*(a+b*x^2)^(5/4)/(a^3*e^3*(e*x)^(5/2))],
[(c+d*x^2)/((e*x)^(15/2)*(a+b*x^2)^(3/4)),x,4,-2/13*c*(a+b*x^2)^(1/4)/(a*e*(e*x)^(13/2))+2/13*(12*b*c-13*a*d)*(a+b*x^2)^(1/4)/(a^2*e^3*(e*x)^(9/2))-16/65*(12*b*c-13*a*d)*(a+b*x^2)^(5/4)/(a^3*e^3*(e*x)^(9/2))+64/585*(12*b*c-13*a*d)*(a+b*x^2)^(9/4)/(a^4*e^3*(e*x)^(9/2))],
[(e*x)^(7/2)*(c+d*x^2)/(a+b*x^2)^(3/4),x,8,1/30*(10*b*c-9*a*d)*e*(e*x)^(5/2)*(a+b*x^2)^(1/4)/b^2+1/5*d*(e*x)^(9/2)*(a+b*x^2)^(1/4)/(b*e)-1/12*a^(3/2)*(10*b*c-9*a*d)*e^2*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(b^(5/2)*(a+b*x^2)^(3/4))-1/12*a*(10*b*c-9*a*d)*e^3*(a+b*x^2)^(1/4)*sqrt(e*x)/b^3],
[(e*x)^(3/2)*(c+d*x^2)/(a+b*x^2)^(3/4),x,7,1/3*d*(e*x)^(5/2)*(a+b*x^2)^(1/4)/(b*e)+1/6*(6*b*c-5*a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/(b^(3/2)*(a+b*x^2)^(3/4))+1/6*(6*b*c-5*a*d)*e*(a+b*x^2)^(1/4)*sqrt(e*x)/b^2],
[(c+d*x^2)/((e*x)^(1/2)*(a+b*x^2)^(3/4)),x,6,-(2*b*c-a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(e^2*(a+b*x^2)^(3/4)*sqrt(a)*sqrt(b))+d*(a+b*x^2)^(1/4)*sqrt(e*x)/(b*e)],
[(c+d*x^2)/((e*x)^(5/2)*(a+b*x^2)^(3/4)),x,6,-2/3*c*(a+b*x^2)^(1/4)/(a*e*(e*x)^(3/2))+2/3*(2*b*c-3*a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(3/2)*e^4*(a+b*x^2)^(3/4))],
[(c+d*x^2)/((e*x)^(9/2)*(a+b*x^2)^(3/4)),x,7,-2/7*c*(a+b*x^2)^(1/4)/(a*e*(e*x)^(7/2))+2/21*(6*b*c-7*a*d)*(a+b*x^2)^(1/4)/(a^2*e^3*(e*x)^(3/2))-4/21*b^(3/2)*(6*b*c-7*a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(a^(5/2)*e^6*(a+b*x^2)^(3/4))],
[(c+d*x^2)/((e*x)^(13/2)*(a+b*x^2)^(3/4)),x,8,-2/11*c*(a+b*x^2)^(1/4)/(a*e*(e*x)^(11/2))+2/77*(10*b*c-11*a*d)*(a+b*x^2)^(1/4)/(a^2*e^3*(e*x)^(7/2))-4/77*b*(10*b*c-11*a*d)*(a+b*x^2)^(1/4)/(a^3*e^5*(e*x)^(3/2))+8/77*b^(5/2)*(10*b*c-11*a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*e^8*(a+b*x^2)^(3/4))],
[(e*x)^(3/2)*(c+d*x^2)/(a+b*x^2)^(5/4),x,7,1/2*d*(e*x)^(5/2)/(b*e*(a+b*x^2)^(1/4))+1/4*(4*b*c-5*a*d)*e^(3/2)*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(9/4)+1/4*(4*b*c-5*a*d)*e^(3/2)*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(9/4)-1/2*(4*b*c-5*a*d)*e*sqrt(e*x)/(b^2*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(1/2)*(a+b*x^2)^(5/4)),x,6,d*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/(b^(5/4)*sqrt(e))+d*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/(b^(5/4)*sqrt(e))+2*(b*c-a*d)*sqrt(e*x)/(a*b*e*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(5/2)*(a+b*x^2)^(5/4)),x,2,-2/3*c/(a*e*(e*x)^(3/2)*(a+b*x^2)^(1/4))-2/3*(4*b*c-3*a*d)*sqrt(e*x)/(a^2*e^3*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(9/2)*(a+b*x^2)^(5/4)),x,3,-2/7*c/(a*e*(e*x)^(7/2)*(a+b*x^2)^(1/4))-2/7*(8*b*c-7*a*d)/(a^2*e^3*(e*x)^(3/2)*(a+b*x^2)^(1/4))+8/21*(8*b*c-7*a*d)*(a+b*x^2)^(3/4)/(a^3*e^3*(e*x)^(3/2))],
[(c+d*x^2)/((e*x)^(13/2)*(a+b*x^2)^(5/4)),x,4,-2/11*c/(a*e*(e*x)^(11/2)*(a+b*x^2)^(1/4))-2/11*(12*b*c-11*a*d)/(a^2*e^3*(e*x)^(7/2)*(a+b*x^2)^(1/4))+16/33*(12*b*c-11*a*d)*(a+b*x^2)^(3/4)/(a^3*e^3*(e*x)^(7/2))-64/231*(12*b*c-11*a*d)*(a+b*x^2)^(7/4)/(a^4*e^3*(e*x)^(7/2))],
[(e*x)^(9/2)*(c+d*x^2)/(a+b*x^2)^(5/4),x,6,-7/60*a*(10*b*c-11*a*d)*e^3*(e*x)^(3/2)/(b^3*(a+b*x^2)^(1/4))+1/30*(10*b*c-11*a*d)*e*(e*x)^(7/2)/(b^2*(a+b*x^2)^(1/4))+1/5*d*(e*x)^(11/2)/(b*e*(a+b*x^2)^(1/4))-7/20*a^(3/2)*(10*b*c-11*a*d)*e^4*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(b^(7/2)*(a+b*x^2)^(1/4))],
[(e*x)^(5/2)*(c+d*x^2)/(a+b*x^2)^(5/4),x,5,1/6*(6*b*c-7*a*d)*e*(e*x)^(3/2)/(b^2*(a+b*x^2)^(1/4))+1/3*d*(e*x)^(7/2)/(b*e*(a+b*x^2)^(1/4))+1/2*(6*b*c-7*a*d)*e^2*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(e*x)/(b^(5/2)*(a+b*x^2)^(1/4))],
[(e*x)^(1/2)*(c+d*x^2)/(a+b*x^2)^(5/4),x,4,d*(e*x)^(3/2)/(b*e*(a+b*x^2)^(1/4))-(2*b*c-3*a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(b^(3/2)*(a+b*x^2)^(1/4)*sqrt(a))],
[(c+d*x^2)/((e*x)^(3/2)*(a+b*x^2)^(5/4)),x,4,-2*c/(a*e*(a+b*x^2)^(1/4)*sqrt(e*x))+2*(2*b*c-a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(a^(3/2)*e^2*(a+b*x^2)^(1/4)*sqrt(b))],
[(c+d*x^2)/((e*x)^(7/2)*(a+b*x^2)^(5/4)),x,5,-2/5*c/(a*e*(e*x)^(5/2)*(a+b*x^2)^(1/4))+2/5*(6*b*c-5*a*d)/(a^2*e^3*(a+b*x^2)^(1/4)*sqrt(e*x))-4/5*(6*b*c-5*a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)*sqrt(e*x)/(a^(5/2)*e^4*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(11/2)*(a+b*x^2)^(5/4)),x,6,-2/9*c/(a*e*(e*x)^(9/2)*(a+b*x^2)^(1/4))+2/45*(10*b*c-9*a*d)/(a^2*e^3*(e*x)^(5/2)*(a+b*x^2)^(1/4))-4/15*b*(10*b*c-9*a*d)/(a^3*e^5*(a+b*x^2)^(1/4)*sqrt(e*x))+8/15*b^(3/2)*(10*b*c-9*a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(a^(7/2)*e^6*(a+b*x^2)^(1/4))],
[(e*x)^(5/2)*(c+d*x^2)/(a+b*x^2)^(7/4),x,7,2/3*(b*c-a*d)*(e*x)^(7/2)/(a*b*e*(a+b*x^2)^(3/4))-1/6*(4*b*c-7*a*d)*e*(e*x)^(3/2)*(a+b*x^2)^(1/4)/(a*b^2)-1/4*(4*b*c-7*a*d)*e^(5/2)*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(11/4)+1/4*(4*b*c-7*a*d)*e^(5/2)*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(11/4)],
[(e*x)^(1/2)*(c+d*x^2)/(a+b*x^2)^(7/4),x,6,2/3*(b*c-a*d)*(e*x)^(3/2)/(a*b*e*(a+b*x^2)^(3/4))-d*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))*sqrt(e)/b^(7/4)+d*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))*sqrt(e)/b^(7/4)],
[(c+d*x^2)/((e*x)^(3/2)*(a+b*x^2)^(7/4)),x,2,-2/3*(4*b*c-a*d)*(e*x)^(3/2)/(a^2*e^3*(a+b*x^2)^(3/4))-2*c/(a*e*(a+b*x^2)^(3/4)*sqrt(e*x))],
[(c+d*x^2)/((e*x)^(7/2)*(a+b*x^2)^(7/4)),x,3,-2/5*c/(a*e*(e*x)^(5/2)*(a+b*x^2)^(3/4))-2/15*(8*b*c-5*a*d)/(a^2*e^3*(a+b*x^2)^(3/4)*sqrt(e*x))+8/15*(8*b*c-5*a*d)*(a+b*x^2)^(1/4)/(a^3*e^3*sqrt(e*x))],
[(c+d*x^2)/((e*x)^(11/2)*(a+b*x^2)^(7/4)),x,4,-2/9*c/(a*e*(e*x)^(9/2)*(a+b*x^2)^(3/4))-2/9*(4*b*c-3*a*d)/(a^2*e^3*(e*x)^(5/2)*(a+b*x^2)^(3/4))+16/9*(4*b*c-3*a*d)*(a+b*x^2)^(1/4)/(a^3*e^3*(e*x)^(5/2))-64/45*(4*b*c-3*a*d)*(a+b*x^2)^(5/4)/(a^4*e^3*(e*x)^(5/2))],
[(e*x)^(7/2)*(c+d*x^2)/(a+b*x^2)^(7/4),x,8,2/3*(b*c-a*d)*(e*x)^(9/2)/(a*b*e*(a+b*x^2)^(3/4))-1/3*(2*b*c-3*a*d)*e*(e*x)^(5/2)*(a+b*x^2)^(1/4)/(a*b^2)+5/6*(2*b*c-3*a*d)*e^2*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)/(b^(5/2)*(a+b*x^2)^(3/4))+5/6*(2*b*c-3*a*d)*e^3*(a+b*x^2)^(1/4)*sqrt(e*x)/b^3],
[(e*x)^(3/2)*(c+d*x^2)/(a+b*x^2)^(7/4),x,7,2/3*(b*c-a*d)*(e*x)^(5/2)/(a*b*e*(a+b*x^2)^(3/4))-1/3*(2*b*c-5*a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(b^(3/2)*(a+b*x^2)^(3/4)*sqrt(a))-1/3*(2*b*c-5*a*d)*e*(a+b*x^2)^(1/4)*sqrt(e*x)/(a*b^2)],
[(c+d*x^2)/((e*x)^(1/2)*(a+b*x^2)^(7/4)),x,6,-2/3*(2*b*c+a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(a^(3/2)*e^2*(a+b*x^2)^(3/4)*sqrt(b))+2/3*(b*c-a*d)*sqrt(e*x)/(a*b*e*(a+b*x^2)^(3/4))],
[(c+d*x^2)/((e*x)^(5/2)*(a+b*x^2)^(7/4)),x,7,-2/3*c/(a*e*(e*x)^(3/2)*(a+b*x^2)^(3/4))+4/3*(2*b*c-a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(5/2)*e^4*(a+b*x^2)^(3/4))-2/3*(2*b*c-a*d)*sqrt(e*x)/(a^2*e^3*(a+b*x^2)^(3/4))],
[(c+d*x^2)/((e*x)^(9/2)*(a+b*x^2)^(7/4)),x,8,-2/7*c/(a*e*(e*x)^(7/2)*(a+b*x^2)^(3/4))-2/21*(10*b*c-7*a*d)/(a^2*e^3*(e*x)^(3/2)*(a+b*x^2)^(3/4))+4/21*(10*b*c-7*a*d)*(a+b*x^2)^(1/4)/(a^3*e^3*(e*x)^(3/2))-8/21*b^(3/2)*(10*b*c-7*a*d)*(1+a/(b*x^2))^(3/4)*(e*x)^(3/2)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))/(a^(7/2)*e^6*(a+b*x^2)^(3/4))],
[(e*x)^(7/2)*(c+d*x^2)/(a+b*x^2)^(9/4),x,8,2/5*(b*c-a*d)*(e*x)^(9/2)/(a*b*e*(a+b*x^2)^(5/4))-1/10*(4*b*c-9*a*d)*e*(e*x)^(5/2)/(a*b^2*(a+b*x^2)^(1/4))+1/4*(4*b*c-9*a*d)*e^(7/2)*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(13/4)+1/4*(4*b*c-9*a*d)*e^(7/2)*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(13/4)-1/2*(4*b*c-9*a*d)*e^3*sqrt(e*x)/(b^3*(a+b*x^2)^(1/4))],
[(e*x)^(3/2)*(c+d*x^2)/(a+b*x^2)^(9/4),x,7,2/5*(b*c-a*d)*(e*x)^(5/2)/(a*b*e*(a+b*x^2)^(5/4))+d*e^(3/2)*arctan(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(9/4)+d*e^(3/2)*arctanh(b^(1/4)*sqrt(e*x)/((a+b*x^2)^(1/4)*sqrt(e)))/b^(9/4)-2*d*e*sqrt(e*x)/(b^2*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(1/2)*(a+b*x^2)^(9/4)),x,2,2/5*(b*c-a*d)*sqrt(e*x)/(a*b*e*(a+b*x^2)^(5/4))+2/5*(4*b*c+a*d)*sqrt(e*x)/(a^2*b*e*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(5/2)*(a+b*x^2)^(9/4)),x,3,-2/3*c/(a*e*(e*x)^(3/2)*(a+b*x^2)^(5/4))-2/15*(8*b*c-3*a*d)*sqrt(e*x)/(a^2*e^3*(a+b*x^2)^(5/4))-8/15*(8*b*c-3*a*d)*sqrt(e*x)/(a^3*e^3*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(9/2)*(a+b*x^2)^(9/4)),x,4,-2/7*c/(a*e*(e*x)^(7/2)*(a+b*x^2)^(5/4))-2/35*(12*b*c-7*a*d)/(a^2*e^3*(e*x)^(3/2)*(a+b*x^2)^(5/4))-16/35*(12*b*c-7*a*d)/(a^3*e^3*(e*x)^(3/2)*(a+b*x^2)^(1/4))+64/105*(12*b*c-7*a*d)*(a+b*x^2)^(3/4)/(a^4*e^3*(e*x)^(3/2))],
[(c+d*x^2)/((e*x)^(13/2)*(a+b*x^2)^(9/4)),x,5,-2/11*c/(a*e*(e*x)^(11/2)*(a+b*x^2)^(5/4))-2/55*(16*b*c-11*a*d)/(a^2*e^3*(e*x)^(7/2)*(a+b*x^2)^(5/4))-24/55*(16*b*c-11*a*d)/(a^3*e^3*(e*x)^(7/2)*(a+b*x^2)^(1/4))+64/55*(16*b*c-11*a*d)*(a+b*x^2)^(3/4)/(a^4*e^3*(e*x)^(7/2))-256/385*(16*b*c-11*a*d)*(a+b*x^2)^(7/4)/(a^5*e^3*(e*x)^(7/2))],
[(e*x)^(13/2)*(c+d*x^2)/(a+b*x^2)^(9/4),x,7,2/5*(b*c-a*d)*(e*x)^(15/2)/(a*b*e*(a+b*x^2)^(5/4))-77/60*a*(2*b*c-3*a*d)*e^5*(e*x)^(3/2)/(b^4*(a+b*x^2)^(1/4))+11/30*(2*b*c-3*a*d)*e^3*(e*x)^(7/2)/(b^3*(a+b*x^2)^(1/4))-1/5*(2*b*c-3*a*d)*e*(e*x)^(11/2)/(a*b^2*(a+b*x^2)^(1/4))-77/20*a^(3/2)*(2*b*c-3*a*d)*e^6*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(b^(9/2)*(a+b*x^2)^(1/4))],
[(e*x)^(9/2)*(c+d*x^2)/(a+b*x^2)^(9/4),x,6,2/5*(b*c-a*d)*(e*x)^(11/2)/(a*b*e*(a+b*x^2)^(5/4))+7/30*(6*b*c-11*a*d)*e^3*(e*x)^(3/2)/(b^3*(a+b*x^2)^(1/4))-1/15*(6*b*c-11*a*d)*e*(e*x)^(7/2)/(a*b^2*(a+b*x^2)^(1/4))+7/10*(6*b*c-11*a*d)*e^4*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(e*x)/(b^(7/2)*(a+b*x^2)^(1/4))],
[(e*x)^(5/2)*(c+d*x^2)/(a+b*x^2)^(9/4),x,5,2/5*(b*c-a*d)*(e*x)^(7/2)/(a*b*e*(a+b*x^2)^(5/4))-1/5*(2*b*c-7*a*d)*e*(e*x)^(3/2)/(a*b^2*(a+b*x^2)^(1/4))-3/5*(2*b*c-7*a*d)*e^2*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(b^(5/2)*(a+b*x^2)^(1/4)*sqrt(a))],
[(e*x)^(1/2)*(c+d*x^2)/(a+b*x^2)^(9/4),x,4,2/5*(b*c-a*d)*(e*x)^(3/2)/(a*b*e*(a+b*x^2)^(5/4))-2/5*(2*b*c+3*a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(a^(3/2)*b^(3/2)*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(3/2)*(a+b*x^2)^(9/4)),x,5,-2/5*(6*b*c-a*d)*(e*x)^(3/2)/(a^2*e^3*(a+b*x^2)^(5/4))-2*c/(a*e*(a+b*x^2)^(5/4)*sqrt(e*x))+4/5*(6*b*c-a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(a^(5/2)*e^2*(a+b*x^2)^(1/4)*sqrt(b))],
[(c+d*x^2)/((e*x)^(7/2)*(a+b*x^2)^(9/4)),x,6,-2/5*c/(a*e*(e*x)^(5/2)*(a+b*x^2)^(5/4))-2/5*(2*b*c-a*d)/(a^2*e^3*(a+b*x^2)^(5/4)*sqrt(e*x))+12/5*(2*b*c-a*d)/(a^3*e^3*(a+b*x^2)^(1/4)*sqrt(e*x))-24/5*(2*b*c-a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)*sqrt(e*x)/(a^(7/2)*e^4*(a+b*x^2)^(1/4))],
[(c+d*x^2)/((e*x)^(11/2)*(a+b*x^2)^(9/4)),x,7,-2/9*c/(a*e*(e*x)^(9/2)*(a+b*x^2)^(5/4))-2/45*(14*b*c-9*a*d)/(a^2*e^3*(e*x)^(5/2)*(a+b*x^2)^(5/4))+4/45*(14*b*c-9*a*d)/(a^3*e^3*(e*x)^(5/2)*(a+b*x^2)^(1/4))-8/15*b*(14*b*c-9*a*d)/(a^4*e^5*(a+b*x^2)^(1/4)*sqrt(e*x))+16/15*b^(3/2)*(14*b*c-9*a*d)*(1+a/(b*x^2))^(1/4)*sqrt(cos(1/2*arccot(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arccot(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arccot(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(e*x)/(a^(9/2)*e^6*(a+b*x^2)^(1/4))],

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

# Integrands of the form (e x)^(m/2) (a+b x^2)^p (c+d x^2)^q with p and/or q symbolic
[(e*x)^(5/2)*(a+b*x^2)^p*(c+d*x^2)^q,x,3,2/7*(e*x)^(7/2)*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(7/4,-p,-q,11/4,-b*x^2/a,-d*x^2/c)/(e*(1+b*x^2/a)^p*(1+d*x^2/c)^q)],
[(e*x)^(3/2)*(a+b*x^2)^p*(c+d*x^2)^q,x,3,2/5*(e*x)^(5/2)*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(5/4,-p,-q,9/4,-b*x^2/a,-d*x^2/c)/(e*(1+b*x^2/a)^p*(1+d*x^2/c)^q)],
[(e*x)^(1/2)*(a+b*x^2)^p*(c+d*x^2)^q,x,3,2/3*(e*x)^(3/2)*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(3/4,-p,-q,7/4,-b*x^2/a,-d*x^2/c)/(e*(1+b*x^2/a)^p*(1+d*x^2/c)^q)],
[(a+b*x^2)^p*(c+d*x^2)^q/(e*x)^(1/2),x,3,2*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(1/4,-p,-q,5/4,-b*x^2/a,-d*x^2/c)*sqrt(e*x)/(e*(1+b*x^2/a)^p*(1+d*x^2/c)^q)],
[(a+b*x^2)^p*(c+d*x^2)^q/(e*x)^(3/2),x,3,-2*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(-1/4,-p,-q,3/4,-b*x^2/a,-d*x^2/c)/(e*(1+b*x^2/a)^p*(1+d*x^2/c)^q*sqrt(e*x))],
[(a+b*x^2)^p*(c+d*x^2)^q/(e*x)^(5/2),x,3,-2/3*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(-3/4,-p,-q,1/4,-b*x^2/a,-d*x^2/c)/(e*(e*x)^(3/2)*(1+b*x^2/a)^p*(1+d*x^2/c)^q)]]:
