# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.2 Quartic\1.2.2.4 (f x)^m (d+e x^2)^q (a+b x^2+c x^4)^p.txt"

lst:=[

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

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

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

# p>0
[x^3*(d+e*x^2)*(a+c*x^4)^5,x,3,1/4*a^5*d*x^4+1/6*a^5*e*x^6+5/8*a^4*c*d*x^8+1/2*a^4*c*e*x^10+5/6*a^3*c^2*d*x^12+5/7*a^3*c^2*e*x^14+5/8*a^2*c^3*d*x^16+5/9*a^2*c^3*e*x^18+1/4*a*c^4*d*x^20+5/22*a*c^4*e*x^22+1/24*c^5*d*x^24+1/26*c^5*e*x^26],
[x^2*(d+e*x^2)*(a+c*x^4)^5,x,2,1/3*a^5*d*x^3+1/5*a^5*e*x^5+5/7*a^4*c*d*x^7+5/9*a^4*c*e*x^9+10/11*a^3*c^2*d*x^11+10/13*a^3*c^2*e*x^13+2/3*a^2*c^3*d*x^15+10/17*a^2*c^3*e*x^17+5/19*a*c^4*d*x^19+5/21*a*c^4*e*x^21+1/23*c^5*d*x^23+1/25*c^5*e*x^25],
[x*(d+e*x^2)*(a+c*x^4)^5,x,4,1/2*a^5*d*x^2+5/6*a^4*c*d*x^6+a^3*c^2*d*x^10+5/7*a^2*c^3*d*x^14+5/18*a*c^4*d*x^18+1/22*c^5*d*x^22+1/24*e*(a+c*x^4)^6/c],
[(d+e*x^2)*(a+c*x^4)^5,x,2,a^5*d*x+1/3*a^5*e*x^3+a^4*c*d*x^5+5/7*a^4*c*e*x^7+10/9*a^3*c^2*d*x^9+10/11*a^3*c^2*e*x^11+10/13*a^2*c^3*d*x^13+2/3*a^2*c^3*e*x^15+5/17*a*c^4*d*x^17+5/19*a*c^4*e*x^19+1/21*c^5*d*x^21+1/23*c^5*e*x^23],
[(d+e*x^2)*(a+c*x^4)^5/x,x,3,1/2*a^5*e*x^2+5/4*a^4*c*d*x^4+5/6*a^4*c*e*x^6+5/4*a^3*c^2*d*x^8+a^3*c^2*e*x^10+5/6*a^2*c^3*d*x^12+5/7*a^2*c^3*e*x^14+5/16*a*c^4*d*x^16+5/18*a*c^4*e*x^18+1/20*c^5*d*x^20+1/22*c^5*e*x^22+a^5*d*log(x)],
[(d+e*x^2)*(a+c*x^4)^5/x^2,x,2,-a^5*d/x+a^5*e*x+5/3*a^4*c*d*x^3+a^4*c*e*x^5+10/7*a^3*c^2*d*x^7+10/9*a^3*c^2*e*x^9+10/11*a^2*c^3*d*x^11+10/13*a^2*c^3*e*x^13+1/3*a*c^4*d*x^15+5/17*a*c^4*e*x^17+1/19*c^5*d*x^19+1/21*c^5*e*x^21],
[(d+e*x^2)*(a+c*x^4)^5/x^3,x,3,-1/2*a^5*d/x^2+5/2*a^4*c*d*x^2+5/4*a^4*c*e*x^4+5/3*a^3*c^2*d*x^6+5/4*a^3*c^2*e*x^8+a^2*c^3*d*x^10+5/6*a^2*c^3*e*x^12+5/14*a*c^4*d*x^14+5/16*a*c^4*e*x^16+1/18*c^5*d*x^18+1/20*c^5*e*x^20+a^5*e*log(x)],

# p<0

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

# p>0
[x^5*(2+3*x^2)*sqrt(5+x^4),x,5,3/10*x^4*(5+x^4)^(3/2)-1/4*(4-x^2)*(5+x^4)^(3/2)-25/8*arcsinh(x^2/sqrt(5))-5/8*x^2*sqrt(5+x^4)],
[x^3*(2+3*x^2)*sqrt(5+x^4),x,4,1/24*(8+9*x^2)*(5+x^4)^(3/2)-75/16*arcsinh(x^2/sqrt(5))-15/16*x^2*sqrt(5+x^4)],
[x*(2+3*x^2)*sqrt(5+x^4),x,4,1/2*(5+x^4)^(3/2)+5/2*arcsinh(x^2/sqrt(5))+1/2*x^2*sqrt(5+x^4)],
[(2+3*x^2)*sqrt(5+x^4)/x,x,7,15/4*arcsinh(x^2/sqrt(5))-arctanh(sqrt(5+x^4)/sqrt(5))*sqrt(5)+1/4*(4+3*x^2)*sqrt(5+x^4)],
[(2+3*x^2)*sqrt(5+x^4)/x^3,x,7,arcsinh(x^2/sqrt(5))-3/2*arctanh(sqrt(5+x^4)/sqrt(5))*sqrt(5)-1/2*(2-3*x^2)*sqrt(5+x^4)/x^2],
[(2+3*x^2)*sqrt(5+x^4)/x^5,x,7,3/2*arcsinh(x^2/sqrt(5))-1/2*arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)-1/2*(1+3*x^2)*sqrt(5+x^4)/x^4],
[(2+3*x^2)*sqrt(5+x^4)/x^7,x,6,-1/15*(5+x^4)^(3/2)/x^6-3/4*arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)-3/4*sqrt(5+x^4)/x^4],
[x^4*(2+3*x^2)*sqrt(5+x^4),x,6,20/21*x*sqrt(5+x^4)+2/3*x^3*sqrt(5+x^4)+1/21*x^5*(6+7*x^2)*sqrt(5+x^4)-10*x*sqrt(5+x^4)/(x^2+sqrt(5))+10*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)-5/21*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(21+2*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[x^2*(2+3*x^2)*sqrt(5+x^4),x,5,10/7*x*sqrt(5+x^4)+1/35*x^3*(14+15*x^2)*sqrt(5+x^4)+4*x*sqrt(5+x^4)/(x^2+sqrt(5))-4*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+1/7*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(14-5*sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)*sqrt(5+x^4),x,4,1/15*x*(10+9*x^2)*sqrt(5+x^4)+6*x*sqrt(5+x^4)/(x^2+sqrt(5))-6*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+1/3*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(9+2*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)*sqrt(5+x^4)/x^2,x,4,-(2-x^2)*sqrt(5+x^4)/x+4*x*sqrt(5+x^4)/(x^2+sqrt(5))-4*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(2+sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)*sqrt(5+x^4)/x^4,x,5,-6*sqrt(5+x^4)/x-1/3*(2-9*x^2)*sqrt(5+x^4)/x^3+6*x*sqrt(5+x^4)/(x^2+sqrt(5))-6*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+1/3*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(2+9*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(1/4)*sqrt(5+x^4))],
[x^5*(2+3*x^2)*(5+x^4)^(3/2),x,6,-5/24*x^2*(5+x^4)^(3/2)+3/14*x^4*(5+x^4)^(5/2)-1/42*(18-7*x^2)*(5+x^4)^(5/2)-125/16*arcsinh(x^2/sqrt(5))-25/16*x^2*sqrt(5+x^4)],
[x^3*(2+3*x^2)*(5+x^4)^(3/2),x,5,-5/16*x^2*(5+x^4)^(3/2)+1/20*(4+5*x^2)*(5+x^4)^(5/2)-375/32*arcsinh(x^2/sqrt(5))-75/32*x^2*sqrt(5+x^4)],
[x*(2+3*x^2)*(5+x^4)^(3/2),x,5,1/4*x^2*(5+x^4)^(3/2)+3/10*(5+x^4)^(5/2)+75/8*arcsinh(x^2/sqrt(5))+15/8*x^2*sqrt(5+x^4)],
[(2+3*x^2)*(5+x^4)^(3/2)/x,x,8,1/24*(8+9*x^2)*(5+x^4)^(3/2)+225/16*arcsinh(x^2/sqrt(5))-5*arctanh(sqrt(5+x^4)/sqrt(5))*sqrt(5)+5/16*(16+9*x^2)*sqrt(5+x^4)],
[(2+3*x^2)*(5+x^4)^(3/2)/x^3,x,8,-1/2*(2-x^2)*(5+x^4)^(3/2)/x^2+15/2*arcsinh(x^2/sqrt(5))-15/2*arctanh(sqrt(5+x^4)/sqrt(5))*sqrt(5)+3/2*(5+x^2)*sqrt(5+x^4)],
[(2+3*x^2)*(5+x^4)^(3/2)/x^5,x,8,-1/4*(2-3*x^2)*(5+x^4)^(3/2)/x^4+45/4*arcsinh(x^2/sqrt(5))-3/2*arctanh(sqrt(5+x^4)/sqrt(5))*sqrt(5)-3/4*(15-2*x^2)*sqrt(5+x^4)/x^2],
[(2+3*x^2)*(5+x^4)^(3/2)/x^7,x,8,-1/12*(4+9*x^2)*(5+x^4)^(3/2)/x^6+arcsinh(x^2/sqrt(5))-9/4*arctanh(sqrt(5+x^4)/sqrt(5))*sqrt(5)-1/4*(4-9*x^2)*sqrt(5+x^4)/x^2],
[x^4*(2+3*x^2)*(5+x^4)^(3/2),x,7,1/143*x^5*(26+33*x^2)*(5+x^4)^(3/2)+200/77*x*sqrt(5+x^4)+20/13*x^3*sqrt(5+x^4)+10/1001*x^5*(78+77*x^2)*sqrt(5+x^4)-300/13*x*sqrt(5+x^4)/(x^2+sqrt(5))+300/13*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)-50/1001*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(231+26*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[x^2*(2+3*x^2)*(5+x^4)^(3/2),x,6,1/99*x^3*(22+27*x^2)*(5+x^4)^(3/2)+300/77*x*sqrt(5+x^4)+2/231*x^3*(154+135*x^2)*sqrt(5+x^4)+40/3*x*sqrt(5+x^4)/(x^2+sqrt(5))-40/3*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+10/231*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(154-45*sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)*(5+x^4)^(3/2),x,5,1/21*x*(6+7*x^2)*(5+x^4)^(3/2)+2/7*x*(10+7*x^2)*sqrt(5+x^4)+20*x*sqrt(5+x^4)/(x^2+sqrt(5))-20*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+10/7*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(7+2*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)*(5+x^4)^(3/2)/x^2,x,5,-1/7*(14-3*x^2)*(5+x^4)^(3/2)/x+6/35*x*(25+14*x^2)*sqrt(5+x^4)+24*x*sqrt(5+x^4)/(x^2+sqrt(5))-24*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+6/7*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(14+5*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)*(5+x^4)^(3/2)/x^4,x,5,-1/15*(10-9*x^2)*(5+x^4)^(3/2)/x^3-2/3*(27-2*x^2)*sqrt(5+x^4)/x+36*x*sqrt(5+x^4)/(x^2+sqrt(5))-36*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+2/3*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(27+2*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],

# p<0
[x^7*(2+3*x^2)/sqrt(5+x^4),x,5,225/16*arcsinh(x^2/sqrt(5))+1/3*x^4*sqrt(5+x^4)+3/8*x^6*sqrt(5+x^4)-5/48*(32+27*x^2)*sqrt(5+x^4)],
[x^5*(2+3*x^2)/sqrt(5+x^4),x,4,-5/2*arcsinh(x^2/sqrt(5))+1/2*x^4*sqrt(5+x^4)-1/2*(10-x^2)*sqrt(5+x^4)],
[x^3*(2+3*x^2)/sqrt(5+x^4),x,3,-15/4*arcsinh(x^2/sqrt(5))+1/4*(4+3*x^2)*sqrt(5+x^4)],
[x*(2+3*x^2)/sqrt(5+x^4),x,3,arcsinh(x^2/sqrt(5))+3/2*sqrt(5+x^4)],
[(2+3*x^2)/(x*sqrt(5+x^4)),x,6,3/2*arcsinh(x^2/sqrt(5))-arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)],
[(2+3*x^2)/(x^3*sqrt(5+x^4)),x,5,-3/2*arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)-1/5*sqrt(5+x^4)/x^2],
[(2+3*x^2)/(x^5*sqrt(5+x^4)),x,6,1/10*arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)-1/10*sqrt(5+x^4)/x^4-3/10*sqrt(5+x^4)/x^2],
[x^4*(2+3*x^2)/sqrt(5+x^4),x,5,2/3*x*sqrt(5+x^4)+3/5*x^3*sqrt(5+x^4)-9*x*sqrt(5+x^4)/(x^2+sqrt(5))+9*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)-1/6*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(27+2*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[x^2*(2+3*x^2)/sqrt(5+x^4),x,4,x*sqrt(5+x^4)+2*x*sqrt(5+x^4)/(x^2+sqrt(5))-2*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+1/2*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(2-sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)],
[(2+3*x^2)/sqrt(5+x^4),x,3,3*x*sqrt(5+x^4)/(x^2+sqrt(5))-3*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+1/2*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(2+3*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(1/4)*sqrt(5+x^4))],
[(2+3*x^2)/(x^2*sqrt(5+x^4)),x,4,-2/5*sqrt(5+x^4)/x+2/5*x*sqrt(5+x^4)/(x^2+sqrt(5))-2*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))+1/2*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(2+3*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))],
[(2+3*x^2)/(x^4*sqrt(5+x^4)),x,5,-2/15*sqrt(5+x^4)/x^3-3/5*sqrt(5+x^4)/x+3/5*x*sqrt(5+x^4)/(x^2+sqrt(5))-3*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))-1/30*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(2-9*sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(1/4)*sqrt(5+x^4))],
[x^7*(2+3*x^2)/(5+x^4)^(3/2),x,4,-45/4*arcsinh(x^2/sqrt(5))-1/2*x^4*(2+3*x^2)/sqrt(5+x^4)+1/4*(8+9*x^2)*sqrt(5+x^4)],
[x^5*(2+3*x^2)/(5+x^4)^(3/2),x,4,arcsinh(x^2/sqrt(5))-1/2*x^2*(2+3*x^2)/sqrt(5+x^4)+3*sqrt(5+x^4)],
[x^3*(2+3*x^2)/(5+x^4)^(3/2),x,3,3/2*arcsinh(x^2/sqrt(5))+1/2*(-2-3*x^2)/sqrt(5+x^4)],
[x*(2+3*x^2)/(5+x^4)^(3/2),x,2,1/10*(-15+2*x^2)/sqrt(5+x^4)],
[(2+3*x^2)/(x*(5+x^4)^(3/2)),x,6,-1/5*arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)+1/10*(2+3*x^2)/sqrt(5+x^4)],
[(2+3*x^2)/(x^3*(5+x^4)^(3/2)),x,6,-3/10*arctanh(sqrt(5+x^4)/sqrt(5))/sqrt(5)+1/10*(2+3*x^2)/(x^2*sqrt(5+x^4))-2/25*sqrt(5+x^4)/x^2],
[x^4*(2+3*x^2)/(5+x^4)^(3/2),x,5,-1/10*x^3*(15-2*x^2)/sqrt(5+x^4)-1/5*x*sqrt(5+x^4)+9/2*x*sqrt(5+x^4)/(x^2+sqrt(5))-9/2*5^(1/4)*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/sqrt(5+x^4)+1/4*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*(2+9*sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(1/4)*sqrt(5+x^4))],
[x^2*(2+3*x^2)/(5+x^4)^(3/2),x,4,-1/10*x*(15-2*x^2)/sqrt(5+x^4)-1/5*x*sqrt(5+x^4)/(x^2+sqrt(5))+sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))-1/4*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(2-3*sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))],
[(2+3*x^2)/(5+x^4)^(3/2),x,4,1/10*x*(2+3*x^2)/sqrt(5+x^4)-3/10*x*sqrt(5+x^4)/(x^2+sqrt(5))+3/2*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))+1/20*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(2-3*sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(1/4)*sqrt(5+x^4))],
[(2+3*x^2)/(x^2*(5+x^4)^(3/2)),x,5,1/10*(2+3*x^2)/(x*sqrt(5+x^4))-3/25*sqrt(5+x^4)/x+3/25*x*sqrt(5+x^4)/(x^2+sqrt(5))-3/5*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))+3/20*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(2+sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))],
[(2+3*x^2)/(x^4*(5+x^4)^(3/2)),x,6,1/10*(2+3*x^2)/(x^3*sqrt(5+x^4))-1/15*sqrt(5+x^4)/x^3-9/50*sqrt(5+x^4)/x+9/50*x*sqrt(5+x^4)/(x^2+sqrt(5))-9/10*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticE(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))+1/60*sqrt(cos(2*arctan(x/5^(1/4)))^2)/cos(2*arctan(x/5^(1/4)))*EllipticF(sin(2*arctan(x/5^(1/4))),sqrt(1/2))*(27-2*sqrt(5))*(x^2+sqrt(5))*sqrt((5+x^4)/(x^2+sqrt(5))^2)/(5^(3/4)*sqrt(5+x^4))],

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

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

# p>0
[(f*x)^m*(d+e*x^2)*(1+2*x^2+x^4)^5,x,3,d*(f*x)^(1+m)/(f*(1+m))+(10*d+e)*(f*x)^(3+m)/(f^3*(3+m))+5*(9*d+2*e)*(f*x)^(5+m)/(f^5*(5+m))+15*(8*d+3*e)*(f*x)^(7+m)/(f^7*(7+m))+30*(7*d+4*e)*(f*x)^(9+m)/(f^9*(9+m))+42*(6*d+5*e)*(f*x)^(11+m)/(f^11*(11+m))+42*(5*d+6*e)*(f*x)^(13+m)/(f^13*(13+m))+30*(4*d+7*e)*(f*x)^(15+m)/(f^15*(15+m))+15*(3*d+8*e)*(f*x)^(17+m)/(f^17*(17+m))+5*(2*d+9*e)*(f*x)^(19+m)/(f^19*(19+m))+(d+10*e)*(f*x)^(21+m)/(f^21*(21+m))+e*(f*x)^(23+m)/(f^23*(23+m))],
[x^5*(d+e*x^2)*(1+2*x^2+x^4)^5,x,4,1/22*(d-e)*(1+x^2)^11-1/24*(2*d-3*e)*(1+x^2)^12+1/26*(d-3*e)*(1+x^2)^13+1/28*e*(1+x^2)^14],
[x^4*(d+e*x^2)*(1+2*x^2+x^4)^5,x,3,1/5*d*x^5+1/7*(10*d+e)*x^7+5/9*(9*d+2*e)*x^9+15/11*(8*d+3*e)*x^11+30/13*(7*d+4*e)*x^13+14/5*(6*d+5*e)*x^15+42/17*(5*d+6*e)*x^17+30/19*(4*d+7*e)*x^19+5/7*(3*d+8*e)*x^21+5/23*(2*d+9*e)*x^23+1/25*(d+10*e)*x^25+1/27*e*x^27],
[x^3*(d+e*x^2)*(1+2*x^2+x^4)^5,x,4,-1/22*(d-e)*(1+x^2)^11+1/24*(d-2*e)*(1+x^2)^12+1/26*e*(1+x^2)^13],
[x^2*(d+e*x^2)*(1+2*x^2+x^4)^5,x,3,1/3*d*x^3+1/5*(10*d+e)*x^5+5/7*(9*d+2*e)*x^7+5/3*(8*d+3*e)*x^9+30/11*(7*d+4*e)*x^11+42/13*(6*d+5*e)*x^13+14/5*(5*d+6*e)*x^15+30/17*(4*d+7*e)*x^17+15/19*(3*d+8*e)*x^19+5/21*(2*d+9*e)*x^21+1/23*(d+10*e)*x^23+1/25*e*x^25],
[x*(d+e*x^2)*(1+2*x^2+x^4)^5,x,4,1/22*(d-e)*(1+x^2)^11+1/24*e*(1+x^2)^12],
[(d+e*x^2)*(1+2*x^2+x^4)^5,x,3,d*x+1/3*(10*d+e)*x^3+(9*d+2*e)*x^5+15/7*(8*d+3*e)*x^7+10/3*(7*d+4*e)*x^9+42/11*(6*d+5*e)*x^11+42/13*(5*d+6*e)*x^13+2*(4*d+7*e)*x^15+15/17*(3*d+8*e)*x^17+5/19*(2*d+9*e)*x^19+1/21*(d+10*e)*x^21+1/23*e*x^23],
[(d+e*x^2)*(1+2*x^2+x^4)^5/x,x,5,5*d*x^2+45/4*d*x^4+20*d*x^6+105/4*d*x^8+126/5*d*x^10+35/2*d*x^12+60/7*d*x^14+45/16*d*x^16+5/9*d*x^18+1/20*d*x^20+1/22*e*(1+x^2)^11+d*log(x)],
[(d+e*x^2)*(1+2*x^2+x^4)^5/x^2,x,3,-d/x+(10*d+e)*x+5/3*(9*d+2*e)*x^3+3*(8*d+3*e)*x^5+30/7*(7*d+4*e)*x^7+14/3*(6*d+5*e)*x^9+42/11*(5*d+6*e)*x^11+30/13*(4*d+7*e)*x^13+(3*d+8*e)*x^15+5/17*(2*d+9*e)*x^17+1/19*(d+10*e)*x^19+1/21*e*x^21],
[(d+e*x^2)*(1+2*x^2+x^4)^5/x^3,x,4,-1/2*d/x^2+5/2*(9*d+2*e)*x^2+15/4*(8*d+3*e)*x^4+5*(7*d+4*e)*x^6+21/4*(6*d+5*e)*x^8+21/5*(5*d+6*e)*x^10+5/2*(4*d+7*e)*x^12+15/14*(3*d+8*e)*x^14+5/16*(2*d+9*e)*x^16+1/18*(d+10*e)*x^18+1/20*e*x^20+(10*d+e)*log(x)],
[(f*x)^m*(1+x^2)*(1+2*x^2+x^4)^5,x,3,(f*x)^(1+m)/(f*(1+m))+11*(f*x)^(3+m)/(f^3*(3+m))+55*(f*x)^(5+m)/(f^5*(5+m))+165*(f*x)^(7+m)/(f^7*(7+m))+330*(f*x)^(9+m)/(f^9*(9+m))+462*(f*x)^(11+m)/(f^11*(11+m))+462*(f*x)^(13+m)/(f^13*(13+m))+330*(f*x)^(15+m)/(f^15*(15+m))+165*(f*x)^(17+m)/(f^17*(17+m))+55*(f*x)^(19+m)/(f^19*(19+m))+11*(f*x)^(21+m)/(f^21*(21+m))+(f*x)^(23+m)/(f^23*(23+m))],
[x^5*(1+x^2)*(1+2*x^2+x^4)^5,x,4,1/24*(1+x^2)^12-1/13*(1+x^2)^13+1/28*(1+x^2)^14],
[x^4*(1+x^2)*(1+2*x^2+x^4)^5,x,3,1/5*x^5+11/7*x^7+55/9*x^9+15*x^11+330/13*x^13+154/5*x^15+462/17*x^17+330/19*x^19+55/7*x^21+55/23*x^23+11/25*x^25+1/27*x^27],
[x^3*(1+x^2)*(1+2*x^2+x^4)^5,x,4,-1/24*(1+x^2)^12+1/26*(1+x^2)^13],
[x^2*(1+x^2)*(1+2*x^2+x^4)^5,x,3,1/3*x^3+11/5*x^5+55/7*x^7+55/3*x^9+30*x^11+462/13*x^13+154/5*x^15+330/17*x^17+165/19*x^19+55/21*x^21+11/23*x^23+1/25*x^25],
[x*(1+x^2)*(1+2*x^2+x^4)^5,x,2,1/24*(1+x^2)^12],
[(1+x^2)*(1+2*x^2+x^4)^5,x,3,x+11/3*x^3+11*x^5+165/7*x^7+110/3*x^9+42*x^11+462/13*x^13+22*x^15+165/17*x^17+55/19*x^19+11/21*x^21+1/23*x^23],
[(1+x^2)*(1+2*x^2+x^4)^5/x,x,4,11/2*x^2+55/4*x^4+55/2*x^6+165/4*x^8+231/5*x^10+77/2*x^12+165/7*x^14+165/16*x^16+55/18*x^18+11/20*x^20+1/22*x^22+log(x)],
[(1+x^2)*(1+2*x^2+x^4)^5/x^2,x,3,(-1)/x+11*x+55/3*x^3+33*x^5+330/7*x^7+154/3*x^9+42*x^11+330/13*x^13+11*x^15+55/17*x^17+11/19*x^19+1/21*x^21],
[(1+x^2)*(1+2*x^2+x^4)^5/x^3,x,4,(-1/2)/x^2+55/2*x^2+165/4*x^4+55*x^6+231/4*x^8+231/5*x^10+55/2*x^12+165/14*x^14+55/16*x^16+11/18*x^18+1/20*x^20+11*log(x)],

# p<0

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

# p>0

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

# Integrands of the form (f x)^m (d+e x^2) (a^2+2 a b x^2+b^2 x^4)^p with m symbolic
[(f*x)^m*(d+e*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,a^5*d*(f*x)^(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f*(1+m)*(a+b*x^2))+a^4*(5*b*d+a*e)*(f*x)^(3+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^3*(3+m)*(a+b*x^2))+5*a^3*b*(2*b*d+a*e)*(f*x)^(5+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^5*(5+m)*(a+b*x^2))+10*a^2*b^2*(b*d+a*e)*(f*x)^(7+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^7*(7+m)*(a+b*x^2))+5*a*b^3*(b*d+2*a*e)*(f*x)^(9+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^9*(9+m)*(a+b*x^2))+b^4*(b*d+5*a*e)*(f*x)^(11+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^11*(11+m)*(a+b*x^2))+b^5*e*(f*x)^(13+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^13*(13+m)*(a+b*x^2))],
[(f*x)^m*(d+e*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,a^3*d*(f*x)^(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f*(1+m)*(a+b*x^2))+a^2*(3*b*d+a*e)*(f*x)^(3+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^3*(3+m)*(a+b*x^2))+3*a*b*(b*d+a*e)*(f*x)^(5+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^5*(5+m)*(a+b*x^2))+b^2*(b*d+3*a*e)*(f*x)^(7+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^7*(7+m)*(a+b*x^2))+b^3*e*(f*x)^(9+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^9*(9+m)*(a+b*x^2))],
[(f*x)^m*(d+e*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(1/2),x,3,a*d*(f*x)^(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f*(1+m)*(a+b*x^2))+(b*d+a*e)*(f*x)^(3+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^3*(3+m)*(a+b*x^2))+b*e*(f*x)^(5+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(f^5*(5+m)*(a+b*x^2))],
[(f*x)^m*(d+e*x^2)/(a^2+2*a*b*x^2+b^2*x^4)^(1/2),x,3,e*(f*x)^(1+m)*(a+b*x^2)/(b*f*(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+(b*d-a*e)*(f*x)^(1+m)*(a+b*x^2)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*b*f*(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(f*x)^m*(d+e*x^2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/4*(b*d-a*e)*(f*x)^(1+m)/(a*b*f*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4*(b*d*(3-m)+a*e*(1+m))*(f*x)^(1+m)*(a+b*x^2)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^3*b*f*(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],

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

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

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

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

# p<0
[x^5*(A+B*x^2)/(a+b*x^2+c*x^4),x,7,-1/2*(b*B-A*c)*x^2/c^2+1/4*B*x^4/c+1/4*(b^2*B-A*b*c-a*B*c)*log(a+b*x^2+c*x^4)/c^3+1/2*(b^3*B-A*b^2*c-3*a*b*B*c+2*a*A*c^2)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[x^3*(A+B*x^2)/(a+b*x^2+c*x^4),x,6,1/2*B*x^2/c-1/4*(b*B-A*c)*log(a+b*x^2+c*x^4)/c^2-1/2*(b^2*B-A*b*c-2*a*B*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[x*(A+B*x^2)/(a+b*x^2+c*x^4),x,5,1/4*B*log(a+b*x^2+c*x^4)/c+1/2*(b*B-2*A*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[(A+B*x^2)/(x*(a+b*x^2+c*x^4)),x,7,A*log(x)/a-1/4*A*log(a+b*x^2+c*x^4)/a+1/2*(A*b-2*a*B)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[(A+B*x^2)/(x^3*(a+b*x^2+c*x^4)),x,7,-1/2*A/(a*x^2)-(A*b-a*B)*log(x)/a^2+1/4*(A*b-a*B)*log(a+b*x^2+c*x^4)/a^2-1/2*(A*b^2-a*b*B-2*a*A*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[x^4*(A+B*x^2)/(a+b*x^2+c*x^4),x,5,-(b*B-A*c)*x/c^2+1/3*B*x^3/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2*B-A*b*c-a*B*c+(-b^3*B+A*b^2*c+3*a*b*B*c-2*a*A*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2*B-A*b*c-a*B*c+(b^3*B-A*b^2*c-3*a*b*B*c+2*a*A*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2*(A+B*x^2)/(a+b*x^2+c*x^4),x,4,B*x/c-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b*B-A*c+(-b^2*B+A*b*c+2*a*B*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b*B-A*c+(b^2*B-A*b*c-2*a*B*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(a+b*x^2+c*x^4),x,3,arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(B+(-b*B+2*A*c)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(B+(b*B-2*A*c)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(x^2*(a+b*x^2+c*x^4)),x,4,-A/(a*x)-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(A+(A*b-2*a*B)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(A+(-A*b+2*a*B)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(x^4*(a+b*x^2+c*x^4)),x,5,-1/3*A/(a*x^3)+(A*b-a*B)/(a^2*x)-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(b+sqrt(b^2-4*a*c))-A*(b^2-2*a*c+b*sqrt(b^2-4*a*c)))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(b-sqrt(b^2-4*a*c))-A*(b^2-2*a*c-b*sqrt(b^2-4*a*c)))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^7*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,7,1/2*(2*b^2*B-A*b*c-6*a*B*c)*x^2/(c^2*(b^2-4*a*c))-1/2*x^4*(a*(b*B-2*A*c)+(b^2*B-A*b*c-2*a*B*c)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*(2*b^4*B-A*b^3*c-12*a*b^2*B*c+6*a*A*b*c^2+12*a^2*B*c^2)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(3/2))-1/4*(2*b*B-A*c)*log(a+b*x^2+c*x^4)/c^3],
[x^5*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,6,-1/2*x^2*(a*(b*B-2*A*c)+(b^2*B-A*b*c-2*a*B*c)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(b^3*B-6*a*b*B*c+4*a*A*c^2)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/2))+1/4*B*log(a+b*x^2+c*x^4)/c^2],
[x^3*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,4,1/2*(-a*(b*B-2*A*c)-(b^2*B-A*b*c-2*a*B*c)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))-(A*b-2*a*B)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[x*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,4,1/2*(-A*b+2*a*B+(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-(b*B-2*A*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[(A+B*x^2)/(x*(a+b*x^2+c*x^4)^2),x,8,1/2*(-a*b*B+A*(b^2-2*a*c)+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(4*a^2*B*c+A*(b^3-6*a*b*c))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2))+A*log(x)/a^2-1/4*A*log(a+b*x^2+c*x^4)/a^2],
[(A+B*x^2)/(x^3*(a+b*x^2+c*x^4)^2),x,8,1/2*(-2*A*b^2+a*b*B+6*a*A*c)/(a^2*(b^2-4*a*c)*x^2)+1/2*(-a*b*B+A*(b^2-2*a*c)+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*x^2*(a+b*x^2+c*x^4))+1/2*(a*b*B*(b^2-6*a*c)-2*A*(b^4-6*a*b^2*c+6*a^2*c^2))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2))-(2*A*b-a*B)*log(x)/a^3+1/4*(2*A*b-a*B)*log(a+b*x^2+c*x^4)/a^3],
[x^6*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,6,1/2*(3*b^2*B-A*b*c-10*a*B*c)*x/(c^2*(b^2-4*a*c))-1/2*(b*B-2*A*c)*x^3/(c*(b^2-4*a*c))-1/2*x^5*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(3*b^3*B-A*b^2*c-13*a*b*B*c+6*a*A*c^2+(-3*b^4*B+A*b^3*c+19*a*b^2*B*c-8*a*A*b*c^2-20*a^2*B*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(3*b^3*B-A*b^2*c-13*a*b*B*c+6*a*A*c^2+(3*b^4*B-A*b^3*c-19*a*b^2*B*c+8*a*A*b*c^2+20*a^2*B*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^4*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,5,-1/2*(b*B-2*A*c)*x/(c*(b^2-4*a*c))-1/2*x^3*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2*B+A*b*c-6*a*B*c+(-b^3*B-A*b^2*c+8*a*b*B*c-4*a*A*c^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2*B+A*b*c-6*a*B*c+(b^3*B+A*b^2*c-8*a*b*B*c+4*a*A*c^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2*(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,4,-1/2*x*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b*B-2*A*c+(-b^2*B+4*A*b*c-4*a*B*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b*B-2*A*c+(b^2*B-4*A*b*c+4*a*B*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(A*b^2-a*b*B-2*a*A*c+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(A*b-2*a*B+(4*a*b*B+A*(b^2-12*a*c))/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(A*b-2*a*B+(-A*b^2-4*a*b*B+12*a*A*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(x^2*(a+b*x^2+c*x^4)^2),x,5,1/2*(-3*A*b^2+a*b*B+10*a*A*c)/(a^2*(b^2-4*a*c)*x)+1/2*(-a*b*B+A*(b^2-2*a*c)+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*x*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(b^2-12*a*c+b*sqrt(b^2-4*a*c))-A*(3*b^3-16*a*b*c+3*b^2*sqrt(b^2-4*a*c)-10*a*c*sqrt(b^2-4*a*c)))/(a^2*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*A*b^2-a*b*B-10*a*A*c+(a*B*(b^2-12*a*c)-A*(3*b^3-16*a*b*c))/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(x^4*(a+b*x^2+c*x^4)^2),x,6,1/6*(-5*A*b^2+3*a*b*B+14*a*A*c)/(a^2*(b^2-4*a*c)*x^3)+1/2*(-a*B*(3*b^2-10*a*c)+A*(5*b^3-19*a*b*c))/(a^3*(b^2-4*a*c)*x)+1/2*(-a*b*B+A*(b^2-2*a*c)+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*x^3*(a+b*x^2+c*x^4))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(3*b^3-16*a*b*c+3*b^2*sqrt(b^2-4*a*c)-10*a*c*sqrt(b^2-4*a*c))-A*(5*b^4-29*a*b^2*c+28*a^2*c^2+5*b^3*sqrt(b^2-4*a*c)-19*a*b*c*sqrt(b^2-4*a*c)))/(a^3*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(3*b^3-16*a*b*c-3*b^2*sqrt(b^2-4*a*c)+10*a*c*sqrt(b^2-4*a*c))-A*(5*b^4-29*a*b^2*c+28*a^2*c^2-5*b^3*sqrt(b^2-4*a*c)+19*a*b*c*sqrt(b^2-4*a*c)))/(a^3*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^11*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,8,1/2*(3*b^4*B-A*b^3*c-21*a*b^2*B*c+7*a*A*b*c^2+30*a^2*B*c^2)*x^2/(c^3*(b^2-4*a*c)^2)-1/4*x^8*(a*(b*B-2*A*c)+(b^2*B-A*b*c-2*a*B*c)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/4*x^4*(a*(3*b^3*B-A*b^2*c-18*a*b*B*c+16*a*A*c^2)+(3*b^4*B-A*b^3*c-20*a*b^2*B*c+10*a*A*b*c^2+20*a^2*B*c^2)*x^2)/(c^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))-1/2*(3*b^6*B-A*b^5*c-30*a*b^4*B*c+10*a*A*b^3*c^2+90*a^2*b^2*B*c^2-30*a^2*A*b*c^3-60*a^3*B*c^3)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(5/2))-1/4*(3*b*B-A*c)*log(a+b*x^2+c*x^4)/c^4],
[x^9*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,7,-1/4*x^6*(a*(b*B-2*A*c)+(b^2*B-A*b*c-2*a*B*c)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/4*x^2*(2*a*(b^3*B-7*a*b*B*c+6*a*A*c^2)+(2*b^4*B-15*a*b^2*B*c+6*a*A*b*c^2+16*a^2*B*c^2)*x^2)/(c^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/2*(b^5*B-10*a*b^3*B*c+30*a^2*b*B*c^2-12*a^2*A*c^3)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(5/2))+1/4*B*log(a+b*x^2+c*x^4)/c^3],
[x^7*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,-1/4*x^6*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/4*(A*b-2*a*B)*x^2*(2*a+b*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3*a*(A*b-2*a*B)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x^5*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,-1/4*x^4*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/4*(-a*(b^2*B-6*A*b*c+8*a*B*c)-(b^3*B-4*A*b^2*c+2*a*b*B*c+4*a*A*c^2)*x^2)/(c*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+(3*a*b*B-A*(b^2+2*a*c))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x^3*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,1/4*(-a*(b*B-2*A*c)-(b^2*B-A*b*c-2*a*B*c)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/4*(b^2*B-3*A*b*c+2*a*B*c)*(b+2*c*x^2)/(c*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))-(b^2*B-3*A*b*c+2*a*B*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,1/4*(-A*b+2*a*B+(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-3/4*(b*B-2*A*c)*(b+2*c*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3*c*(b*B-2*A*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[(A+B*x^2)/(x*(a+b*x^2+c*x^4)^3),x,9,1/4*(-a*b*B+A*(b^2-2*a*c)+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/4*(6*a^2*b*B*c+A*(2*b^4-15*a*b^2*c+16*a^2*c^2)+2*c*(6*a^2*B*c+A*(b^3-7*a*b*c))*x^2)/(a^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))-1/2*(12*a^3*B*c^2-A*(b^5-10*a*b^3*c+30*a^2*b*c^2))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2))+A*log(x)/a^3-1/4*A*log(a+b*x^2+c*x^4)/a^3],
[(A+B*x^2)/(x^3*(a+b*x^2+c*x^4)^3),x,9,1/2*(a*b*B*(b^2-7*a*c)-3*A*(b^4-7*a*b^2*c+10*a^2*c^2))/(a^3*(b^2-4*a*c)^2*x^2)+1/4*(-a*b*B+A*(b^2-2*a*c)+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*x^2*(a+b*x^2+c*x^4)^2)+1/4*(-a*b*B*(b^2-10*a*c)+A*(3*b^4-20*a*b^2*c+20*a^2*c^2)-c*(a*B*(b^2-16*a*c)-3*A*(b^3-6*a*b*c))*x^2)/(a^2*(b^2-4*a*c)^2*x^2*(a+b*x^2+c*x^4))+1/2*(a*b*B*(b^4-10*a*b^2*c+30*a^2*c^2)-3*A*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(5/2))-(3*A*b-a*B)*log(x)/a^4+1/4*(3*A*b-a*B)*log(a+b*x^2+c*x^4)/a^4],
[x^8*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,7,-1/8*(3*b^3*B+A*b^2*c-24*a*b*B*c+20*a*A*c^2)*x/(c^2*(b^2-4*a*c)^2)+1/8*(b^2*B+12*A*b*c-28*a*B*c)*x^3/(c*(b^2-4*a*c)^2)-1/4*x^7*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/8*x^5*(7*A*b^2-12*a*b*B-4*a*A*c+(b^2*B+12*A*b*c-28*a*B*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(3*b^4*B+A*b^3*c-27*a*b^2*B*c-16*a*A*b*c^2+84*a^2*B*c^2+(-3*b^5*B-A*b^4*c+33*a*b^3*B*c+18*a*A*b^2*c^2-132*a^2*b*B*c^2+40*a^2*A*c^3)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(3*b^4*B+A*b^3*c-27*a*b^2*B*c-16*a*A*b*c^2+84*a^2*B*c^2+(3*b^5*B+A*b^4*c-33*a*b^3*B*c-18*a*A*b^2*c^2+132*a^2*b*B*c^2-40*a^2*A*c^3)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^6*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,6,-1/8*(b^2*B-12*A*b*c+20*a*B*c)*x/(c*(b^2-4*a*c)^2)-1/4*x^5*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/8*x^3*(5*A*b^2-12*a*b*B+4*a*A*c-(b^2*B-12*A*b*c+20*a*B*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^3*B+3*A*b^2*c-16*a*b*B*c+12*a*A*c^2+(-b^4*B-3*A*b^3*c+18*a*b^2*B*c-36*a*A*b*c^2+40*a^2*B*c^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^3*B+3*A*b^2*c-16*a*b*B*c+12*a*A*c^2+(b^4*B+3*A*b^3*c-18*a*b^2*B*c+36*a*A*b*c^2-40*a^2*B*c^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^4*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,-1/4*x^3*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/8*x*(4*a*b*B-A*(b^2+4*a*c)+(b^2*B-4*A*b*c+4*a*B*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2*B-4*A*b*c+4*a*B*c+(-b^3*B+6*A*b^2*c-12*a*b*B*c+8*a*A*c^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^2*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2*B-4*A*b*c+4*a*B*c+(b^3*B-6*A*b^2*c+12*a*b*B*c-8*a*A*c^2)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^2*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2*(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,-1/4*x*(A*b-2*a*B-(b*B-2*A*c)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/8*x*(a*B*(7*b^2-4*a*c)-A*(b^3+8*a*b*c)+c*(12*a*b*B-A*(b^2+20*a*c))*x^2)/(a*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(6*a*B*(3*b^2+4*a*c-2*b*sqrt(b^2-4*a*c))+A*(b^3-52*a*b*c+b^2*sqrt(b^2-4*a*c)+20*a*c*sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(6*a*B*(3*b^2+4*a*c+2*b*sqrt(b^2-4*a*c))+A*(b^3-52*a*b*c-b^2*sqrt(b^2-4*a*c)-20*a*c*sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x^2)/(a+b*x^2+c*x^4)^3,x,5,1/4*x*(A*b^2-a*b*B-2*a*A*c+(A*b-2*a*B)*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/8*x*(a*b*B*(b^2+8*a*c)+A*(3*b^4-25*a*b^2*c+28*a^2*c^2)+c*(a*B*(b^2+20*a*c)+3*A*(b^3-8*a*b*c))*x^2)/(a^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(b^2+20*a*c)+3*A*(b^3-8*a*b*c)+(a*b*B*(b^2-52*a*c)+3*A*(b^4-10*a*b^2*c+56*a^2*c^2))/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(a*B*(b^2+20*a*c)+3*A*(b^3-8*a*b*c)+(-a*b*B*(b^2-52*a*c)-3*A*(b^4-10*a*b^2*c+56*a^2*c^2))/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],

#  {x^(-2)*(A + B*x^2)/(a + b*x^2 + c*x^4)^3, x, 6, -((3*(5*A*b^4 - a*b^3*B - 37*a*A*b^2*c + 4*a^2*c*(2*b*B + 15*A*c)))/(8*a^3*(b^2 - 4*a*c)^2*x)) - ((-A)*b^2 + a*b*B + 2*a*A*c + ((-A)*b*c + 2*a*B*c)*x^2)/(4*a*(b^2 - 4*a*c)*x*(a + b*x^2 + c*x^4)^2) + (5*A*b^4 - a*b^3*B - 35*a*A*b^2*c + 4*a^2*c*(4*b*B + 9*A*c) + c*(5*A*b^3 - a*b^2*B - 32*a*A*b*c + 28*a^2*B*c)*x^2)/(8*a^2*(b^2 - 4*a*c)^2*x*(a + b*x^2 + c*x^4)) - (3*Sqrt[c]*(5*A*b^5 - 56*a^3*B*c^2 - 37*a*A*b^2*c*Sqrt[b^2 - 4*a*c] - b^4*(a*B - 5*A*Sqrt[b^2 - 4*a*c]) - a*b^3*(47*A*c + B*Sqrt[b^2 - 4*a*c]) + 2*a^2*c*(5*b^2*B + 4*b*B*Sqrt[b^2 - 4*a*c] + 2*A*c*(31*b + 15*Sqrt[b^2 - 4*a*c])))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(5*A*b^5 - 56*a^3*B*c^2 + 37*a*A*b^2*c*Sqrt[b^2 - 4*a*c] - b^4*(a*B + 5*A*Sqrt[b^2 - 4*a*c]) - a*b^3*(47*A*c - B*Sqrt[b^2 - 4*a*c]) + 2*a^2*c*(5*b^2*B - 4*b*B*Sqrt[b^2 - 4*a*c] + 2*A*c*(31*b - 15*Sqrt[b^2 - 4*a*c])))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])} 
[x*(-7+4*x^2)/(4-5*x^2+x^4),x,4,1/2*log(1-x^2)+3/2*log(4-x^2)],
[(-7*x+4*x^3)/(4-5*x^2+x^4),x,5,1/2*log(1-x^2)+3/2*log(4-x^2)],
[x*(2+x^2)/(1+x^2+x^4),x,5,1/4*log(1+x^2+x^4)+1/2*arctan((1+2*x^2)/sqrt(3))*sqrt(3)],
[(2*x+x^3)/(1+x^2+x^4),x,6,1/4*log(1+x^2+x^4)+1/2*arctan((1+2*x^2)/sqrt(3))*sqrt(3)],
[(11*x+2*x^3)/(3+2*x^2+x^4)^2,x,5,1/8*(5+9*x^2)/(3+2*x^2+x^4)+9/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],

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

# p>0
[x^5*(2+3*x^2)*sqrt(3+5*x^2+x^4),x,6,3/10*x^4*(3+5*x^2+x^4)^(3/2)+1/480*(1837-510*x^2)*(3+5*x^2+x^4)^(3/2)+21229/512*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-1633/256*(5+2*x^2)*sqrt(3+5*x^2+x^4)],
[x^3*(2+3*x^2)*sqrt(3+5*x^2+x^4),x,5,-1/48*(59-18*x^2)*(3+5*x^2+x^4)^(3/2)-3367/256*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))+259/128*(5+2*x^2)*sqrt(3+5*x^2+x^4)],
[x*(2+3*x^2)*sqrt(3+5*x^2+x^4),x,5,1/2*(3+5*x^2+x^4)^(3/2)+143/32*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-11/16*(5+2*x^2)*sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x,x,7,1/16*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))*sqrt(3)+1/8*(23+6*x^2)*sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^3,x,7,19/4*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-7*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-1/2*(2-3*x^2)*sqrt(3+5*x^2+x^4)/x^2],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^5,x,7,3/2*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-77/24*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-1/12*(6+23*x^2)*sqrt(3+5*x^2+x^4)/x^4],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^7,x,5,-1/9*(3+5*x^2+x^4)^(3/2)/x^6+13/36*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-1/18*(6+5*x^2)*sqrt(3+5*x^2+x^4)/x^4],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^9,x,6,-1/12*(3+5*x^2+x^4)^(3/2)/x^8-11/216*(3+5*x^2+x^4)^(3/2)/x^6-871/3456*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)+67/1728*(6+5*x^2)*sqrt(3+5*x^2+x^4)/x^4],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^11,x,7,-1/15*(3+5*x^2+x^4)^(3/2)/x^10-1/36*(3+5*x^2+x^4)^(3/2)/x^8+173/3240*(3+5*x^2+x^4)^(3/2)/x^6+2093/10368*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-161/5184*(6+5*x^2)*sqrt(3+5*x^2+x^4)/x^4],
[x^4*(2+3*x^2)*sqrt(3+5*x^2+x^4),x,6,-1924/105*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)+13/3*x*sqrt(3+5*x^2+x^4)-26/35*x^3*sqrt(3+5*x^2+x^4)+1/21*x^5*(11+7*x^2)*sqrt(3+5*x^2+x^4)+962/105*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-13*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],
[x^2*(2+3*x^2)*sqrt(3+5*x^2+x^4),x,5,1247/210*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-4/3*x*sqrt(3+5*x^2+x^4)+1/35*x^3*(29+15*x^2)*sqrt(3+5*x^2+x^4)+2*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-1247/210*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*sqrt(3+5*x^2+x^4),x,4,-23/15*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)+1/15*x*(25+9*x^2)*sqrt(3+5*x^2+x^4)+23/15*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)+sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^2,x,4,9/2*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-(2-x^2)*sqrt(3+5*x^2+x^4)/x+8*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-3/2*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(3/2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*sqrt(3+5*x^2+x^4)/x^4,x,5,32/9*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-64/9*sqrt(3+5*x^2+x^4)/x-1/3*(2-9*x^2)*sqrt(3+5*x^2+x^4)/x^3-16/9*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)+49/3*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],
[x^5*(2+3*x^2)*(3+5*x^2+x^4)^(3/2),x,7,-2183/768*(5+2*x^2)*(3+5*x^2+x^4)^(3/2)+3/14*x^4*(3+5*x^2+x^4)^(5/2)+1/1680*(3313-1070*x^2)*(3+5*x^2+x^4)^(5/2)-368927/4096*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))+28379/2048*(5+2*x^2)*sqrt(3+5*x^2+x^4)],
[x^3*(2+3*x^2)*(3+5*x^2+x^4)^(3/2),x,6,123/128*(5+2*x^2)*(3+5*x^2+x^4)^(3/2)-1/40*(27-10*x^2)*(3+5*x^2+x^4)^(5/2)+62361/2048*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-4797/1024*(5+2*x^2)*sqrt(3+5*x^2+x^4)],
[x*(2+3*x^2)*(3+5*x^2+x^4)^(3/2),x,6,-11/32*(5+2*x^2)*(3+5*x^2+x^4)^(3/2)+3/10*(3+5*x^2+x^4)^(5/2)-5577/512*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))+429/256*(5+2*x^2)*sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x,x,8,1/48*(61+18*x^2)*(3+5*x^2+x^4)^(3/2)+2401/256*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-3*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))*sqrt(3)+1/128*(199-74*x^2)*sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x^3,x,8,-1/2*(2-x^2)*(3+5*x^2+x^4)^(3/2)/x^2+609/32*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-12*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))*sqrt(3)+3/16*(109+18*x^2)*sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x^5,x,8,-1/4*(2-3*x^2)*(3+5*x^2+x^4)^(3/2)/x^4+453/16*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-127/8*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))*sqrt(3)-3/8*(28-19*x^2)*sqrt(3+5*x^2+x^4)/x^2],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x^7,x,8,-1/6*(2+7*x^2)*(3+5*x^2+x^4)^(3/2)/x^6+49/4*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-527/24*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-1/12*(67-32*x^2)*sqrt(3+5*x^2+x^4)/x^2],
[x^4*(2+3*x^2)*(3+5*x^2+x^4)^(3/2),x,7,1/143*x^5*(71+33*x^2)*(3+5*x^2+x^4)^(3/2)+176723/4290*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-4210/429*x*sqrt(3+5*x^2+x^4)+1251/715*x^3*sqrt(3+5*x^2+x^4)-1/429*x^5*(283+272*x^2)*sqrt(3+5*x^2+x^4)+2105/143*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-176723/4290*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[x^2*(2+3*x^2)*(3+5*x^2+x^4)^(3/2),x,6,1/99*x^3*(67+27*x^2)*(3+5*x^2+x^4)^(3/2)-49949/3465*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)+353/99*x*sqrt(3+5*x^2+x^4)-1/1155*x^3*(911+890*x^2)*sqrt(3+5*x^2+x^4)+49949/3465*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-353/33*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2),x,5,1/3*x*(3+x^2)*(3+5*x^2+x^4)^(3/2)+203/30*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-1/15*x*(5+12*x^2)*sqrt(3+5*x^2+x^4)+5*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-203/30*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x^2,x,5,-1/7*(14-3*x^2)*(3+5*x^2+x^4)^(3/2)/x+412/35*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)+1/35*x*(655+129*x^2)*sqrt(3+5*x^2+x^4)+19*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(3/2/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-206/35*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x^4,x,5,-1/15*(10-9*x^2)*(3+5*x^2+x^4)^(3/2)/x^3+949/30*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-13/15*(24-5*x^2)*sqrt(3+5*x^2+x^4)/x+65*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-949/30*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)*(3+5*x^2+x^4)^(3/2)/x^6,x,6,-1/5*(2-5*x^2)*(3+5*x^2+x^4)^(3/2)/x^5+361/15*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-722/15*sqrt(3+5*x^2+x^4)/x-1/5*(40-87*x^2)*sqrt(3+5*x^2+x^4)/x^3-361/15*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)+103*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],

# p<0
[x^5*(A+B*x^2)/sqrt(a+b*x^2+c*x^4),x,5,-1/32*(5*b^3*B-6*A*b^2*c-12*a*b*B*c+8*a*A*c^2)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(7/2)+1/6*B*x^4*sqrt(a+b*x^2+c*x^4)/c+1/48*(15*b^2*B-18*A*b*c-16*a*B*c-2*c*(5*b*B-6*A*c)*x^2)*sqrt(a+b*x^2+c*x^4)/c^3],
[x^3*(A+B*x^2)/sqrt(a+b*x^2+c*x^4),x,4,1/16*(3*b^2*B-4*A*b*c-4*a*B*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(5/2)-1/8*(3*b*B-4*A*c-2*B*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^2],
[x*(A+B*x^2)/sqrt(a+b*x^2+c*x^4),x,4,-1/4*(b*B-2*A*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(3/2)+1/2*B*sqrt(a+b*x^2+c*x^4)/c],
[(A+B*x^2)/(x*sqrt(a+b*x^2+c*x^4)),x,6,-1/2*A*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/sqrt(a)+1/2*B*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/sqrt(c)],
[(A+B*x^2)/(x^3*sqrt(a+b*x^2+c*x^4)),x,4,1/4*(A*b-2*a*B)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(3/2)-1/2*A*sqrt(a+b*x^2+c*x^4)/(a*x^2)],
[(A+B*x^2)/(x^5*sqrt(a+b*x^2+c*x^4)),x,5,-1/16*(3*A*b^2-4*a*b*B-4*a*A*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(5/2)-1/4*A*sqrt(a+b*x^2+c*x^4)/(a*x^4)+1/8*(3*A*b-4*a*B)*sqrt(a+b*x^2+c*x^4)/(a^2*x^2)],
[(A+B*x^2)/(x^7*sqrt(a+b*x^2+c*x^4)),x,6,1/32*(5*A*b^3-6*a*b^2*B-12*a*A*b*c+8*a^2*B*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(7/2)-1/6*A*sqrt(a+b*x^2+c*x^4)/(a*x^6)+1/24*(5*A*b-6*a*B)*sqrt(a+b*x^2+c*x^4)/(a^2*x^4)-1/48*(15*A*b^2-18*a*b*B-16*a*A*c)*sqrt(a+b*x^2+c*x^4)/(a^3*x^2)],
[x^4*(A+B*x^2)/sqrt(a+b*x^2+c*x^4),x,5,-1/15*(4*b*B-5*A*c)*x*sqrt(a+b*x^2+c*x^4)/c^2+1/5*B*x^3*sqrt(a+b*x^2+c*x^4)/c+1/15*(8*b^2*B-10*A*b*c-9*a*B*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(5/2)*(sqrt(a)+x^2*sqrt(c)))-1/15*a^(1/4)*(8*b^2*B-10*A*b*c-9*a*B*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(11/4)*sqrt(a+b*x^2+c*x^4))+1/30*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(8*b^2*B-10*A*b*c-9*a*B*c+(4*b*B-5*A*c)*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(11/4)*sqrt(a+b*x^2+c*x^4))],
[x^2*(A+B*x^2)/sqrt(a+b*x^2+c*x^4),x,4,1/3*B*x*sqrt(a+b*x^2+c*x^4)/c-1/3*(2*b*B-3*A*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))+1/3*a^(1/4)*(2*b*B-3*A*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))-1/6*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*b*B-3*A*c+B*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))],
[(A+B*x^2)/sqrt(a+b*x^2+c*x^4),x,3,B*x*sqrt(a+b*x^2+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*B*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+b*x^2+c*x^4))+1/2*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(B+A*sqrt(c)/sqrt(a))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+b*x^2+c*x^4))],
[(A+B*x^2)/(x^2*sqrt(a+b*x^2+c*x^4)),x,4,-A*sqrt(a+b*x^2+c*x^4)/(a*x)+A*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a*(sqrt(a)+x^2*sqrt(c)))-A*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+b*x^2+c*x^4))+1/2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(B*sqrt(a)+A*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*c^(1/4)*sqrt(a+b*x^2+c*x^4))],
[(A+B*x^2)/(x^4*sqrt(a+b*x^2+c*x^4)),x,5,-1/3*A*sqrt(a+b*x^2+c*x^4)/(a*x^3)+1/3*(2*A*b-3*a*B)*sqrt(a+b*x^2+c*x^4)/(a^2*x)-1/3*(2*A*b-3*a*B)*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a^2*(sqrt(a)+x^2*sqrt(c)))+1/3*(2*A*b-3*a*B)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))-1/6*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*A*b-3*a*B+A*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))],
[x^7*(2+3*x^2)/sqrt(3+5*x^2+x^4),x,6,32801/256*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-89/48*x^4*sqrt(3+5*x^2+x^4)+3/8*x^6*sqrt(3+5*x^2+x^4)-1/384*(24243-3802*x^2)*sqrt(3+5*x^2+x^4)],
[x^5*(2+3*x^2)/sqrt(3+5*x^2+x^4),x,5,-1083/32*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))+1/2*x^4*sqrt(3+5*x^2+x^4)+3/16*(89-14*x^2)*sqrt(3+5*x^2+x^4)],
[x^3*(2+3*x^2)/sqrt(3+5*x^2+x^4),x,4,149/16*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-1/8*(37-6*x^2)*sqrt(3+5*x^2+x^4)],
[x*(2+3*x^2)/sqrt(3+5*x^2+x^4),x,4,-11/4*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))+3/2*sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(x*sqrt(3+5*x^2+x^4)),x,6,3/2*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)],
[(2+3*x^2)/(x^3*sqrt(3+5*x^2+x^4)),x,4,-2/3*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-1/3*sqrt(3+5*x^2+x^4)/x^2],
[(2+3*x^2)/(x^5*sqrt(3+5*x^2+x^4)),x,5,1/8*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))*sqrt(3)-1/6*sqrt(3+5*x^2+x^4)/x^4-1/12*sqrt(3+5*x^2+x^4)/x^2],
[(2+3*x^2)/(x^7*sqrt(3+5*x^2+x^4)),x,6,-61/216*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)-1/9*sqrt(3+5*x^2+x^4)/x^6-1/54*sqrt(3+5*x^2+x^4)/x^4+13/108*sqrt(3+5*x^2+x^4)/x^2],
[x^4*(2+3*x^2)/sqrt(3+5*x^2+x^4),x,5,419/30*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-10/3*x*sqrt(3+5*x^2+x^4)+3/5*x^3*sqrt(3+5*x^2+x^4)+5*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-419/30*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[x^2*(2+3*x^2)/sqrt(3+5*x^2+x^4),x,4,-4*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)+x*sqrt(3+5*x^2+x^4)-sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(3/2/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)+2*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/sqrt(3+5*x^2+x^4),x,3,3/2*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)+sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-1/2*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(3/2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(x^2*sqrt(3+5*x^2+x^4)),x,4,1/3*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-2/3*sqrt(3+5*x^2+x^4)/x+sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(3/2/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-1/3*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(x^4*sqrt(3+5*x^2+x^4)),x,5,7/54*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-2/9*sqrt(3+5*x^2+x^4)/x^3-7/27*sqrt(3+5*x^2+x^4)/x-1/9*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-7/54*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[x^5*(2+3*x^2)/(3+5*x^2+x^4)^(3/2),x,5,-41/4*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))-1/13*x^2*(33+47*x^2)/sqrt(3+5*x^2+x^4)+133/26*sqrt(3+5*x^2+x^4)],
[x^3*(2+3*x^2)/(3+5*x^2+x^4)^(3/2),x,4,3/2*arctanh(1/2*(5+2*x^2)/sqrt(3+5*x^2+x^4))+1/13*(-33-47*x^2)/sqrt(3+5*x^2+x^4)],
[x*(2+3*x^2)/(3+5*x^2+x^4)^(3/2),x,2,1/13*(8+11*x^2)/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(x*(3+5*x^2+x^4)^(3/2)),x,5,-1/3*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)+1/39*(-7-8*x^2)/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(x^3*(3+5*x^2+x^4)^(3/2)),x,5,1/3*arctanh(1/2*(6+5*x^2)/(sqrt(3)*sqrt(3+5*x^2+x^4)))/sqrt(3)+1/39*(-7-8*x^2)/(x^2*sqrt(3+5*x^2+x^4))-2/39*sqrt(3+5*x^2+x^4)/x^2],
[x^4*(2+3*x^2)/(3+5*x^2+x^4)^(3/2),x,5,1/13*x^3*(8+11*x^2)/sqrt(3+5*x^2+x^4)+43/13*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-11/13*x*sqrt(3+5*x^2+x^4)+11/13*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(3/2/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-43/13*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[x^2*(2+3*x^2)/(3+5*x^2+x^4)^(3/2),x,4,1/13*x*(8+11*x^2)/sqrt(3+5*x^2+x^4)-11/26*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-4/13*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)+11/26*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(3+5*x^2+x^4)^(3/2),x,4,-1/39*x*(7+8*x^2)/sqrt(3+5*x^2+x^4)+4/39*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-2/39*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)+11/13*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],
[(2+3*x^2)/(x^2*(3+5*x^2+x^4)^(3/2)),x,5,1/39*(-7-8*x^2)/(x*sqrt(3+5*x^2+x^4))+19/234*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-19/117*sqrt(3+5*x^2+x^4)/x-4/39*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(2/3/(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-19/234*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)],
[(2+3*x^2)/(x^4*(3+5*x^2+x^4)^(3/2)),x,6,1/39*(-7-8*x^2)/(x^3*sqrt(3+5*x^2+x^4))-133/1053*x*(5+2*x^2+sqrt(13))/sqrt(3+5*x^2+x^4)-5/351*sqrt(3+5*x^2+x^4)/x^3+266/1053*sqrt(3+5*x^2+x^4)/x+133/1053*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticE(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt(1/6*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/sqrt(3+5*x^2+x^4)-5/351*sqrt(cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))^2)/cos(arctan(x*sqrt(1/6*(5+sqrt(13)))))*EllipticF(sin(arctan(x*sqrt(1/6*(5+sqrt(13))))),sqrt(1/6*(-13+5*sqrt(13))))*(6+x^2*(5+sqrt(13)))*sqrt((6+x^2*(5-sqrt(13)))/(6+x^2*(5+sqrt(13))))/(sqrt(3+5*x^2+x^4)*sqrt(6*(5+sqrt(13))))],

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

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

# p>0
[(f*x)^(3/2)*(d+e*x^2)*sqrt(a+b*x^2+c*x^4),x,6,2/5*d*(f*x)^(5/2)*AppellF1(5/4,-1/2,-1/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+2/9*e*(f*x)^(9/2)*AppellF1(9/4,-1/2,-1/2,13/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(f*x)^(1/2)*(d+e*x^2)*sqrt(a+b*x^2+c*x^4),x,6,2/3*d*(f*x)^(3/2)*AppellF1(3/4,-1/2,-1/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+2/7*e*(f*x)^(7/2)*AppellF1(7/4,-1/2,-1/2,11/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)*sqrt(a+b*x^2+c*x^4)/(f*x)^(1/2),x,6,2/5*e*(f*x)^(5/2)*AppellF1(5/4,-1/2,-1/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+2*d*AppellF1(1/4,-1/2,-1/2,5/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(f*x)*sqrt(a+b*x^2+c*x^4)/(f*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)*sqrt(a+b*x^2+c*x^4)/(f*x)^(3/2),x,6,2/3*e*(f*x)^(3/2)*AppellF1(3/4,-1/2,-1/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))-2*d*AppellF1(-1/4,-1/2,-1/2,3/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*sqrt(f*x)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(f*x)^(3/2)*(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2),x,6,2/5*a*d*(f*x)^(5/2)*AppellF1(5/4,-3/2,-3/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+2/9*a*e*(f*x)^(9/2)*AppellF1(9/4,-3/2,-3/2,13/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(f*x)^(1/2)*(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2),x,6,2/3*a*d*(f*x)^(3/2)*AppellF1(3/4,-3/2,-3/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+2/7*a*e*(f*x)^(7/2)*AppellF1(7/4,-3/2,-3/2,11/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)/(f*x)^(1/2),x,6,2/5*a*e*(f*x)^(5/2)*AppellF1(5/4,-3/2,-3/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+2*a*d*AppellF1(1/4,-3/2,-3/2,5/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(f*x)*sqrt(a+b*x^2+c*x^4)/(f*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)/(f*x)^(3/2),x,6,2/3*a*e*(f*x)^(3/2)*AppellF1(3/4,-3/2,-3/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))-2*a*d*AppellF1(-1/4,-3/2,-3/2,3/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*sqrt(f*x)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],

# p<0
[(f*x)^(3/2)*(d+e*x^2)/sqrt(a+b*x^2+c*x^4),x,6,2/5*d*(f*x)^(5/2)*AppellF1(5/4,1/2,1/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f*sqrt(a+b*x^2+c*x^4))+2/9*e*(f*x)^(9/2)*AppellF1(9/4,1/2,1/2,13/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f^3*sqrt(a+b*x^2+c*x^4))],
[(f*x)^(1/2)*(d+e*x^2)/sqrt(a+b*x^2+c*x^4),x,6,2/3*d*(f*x)^(3/2)*AppellF1(3/4,1/2,1/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f*sqrt(a+b*x^2+c*x^4))+2/7*e*(f*x)^(7/2)*AppellF1(7/4,1/2,1/2,11/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f^3*sqrt(a+b*x^2+c*x^4))],
[(d+e*x^2)/((f*x)^(1/2)*sqrt(a+b*x^2+c*x^4)),x,6,2/5*e*(f*x)^(5/2)*AppellF1(5/4,1/2,1/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f^3*sqrt(a+b*x^2+c*x^4))+2*d*AppellF1(1/4,1/2,1/2,5/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(f*x)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f*sqrt(a+b*x^2+c*x^4))],
[(d+e*x^2)/((f*x)^(3/2)*sqrt(a+b*x^2+c*x^4)),x,6,2/3*e*(f*x)^(3/2)*AppellF1(3/4,1/2,1/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f^3*sqrt(a+b*x^2+c*x^4))-2*d*AppellF1(-1/4,1/2,1/2,3/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f*sqrt(f*x)*sqrt(a+b*x^2+c*x^4))],
[(f*x)^(3/2)*(d+e*x^2)/(a+b*x^2+c*x^4)^(3/2),x,6,2/5*d*(f*x)^(5/2)*AppellF1(5/4,3/2,3/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f*sqrt(a+b*x^2+c*x^4))+2/9*e*(f*x)^(9/2)*AppellF1(9/4,3/2,3/2,13/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f^3*sqrt(a+b*x^2+c*x^4))],
[(f*x)^(1/2)*(d+e*x^2)/(a+b*x^2+c*x^4)^(3/2),x,6,2/3*d*(f*x)^(3/2)*AppellF1(3/4,3/2,3/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f*sqrt(a+b*x^2+c*x^4))+2/7*e*(f*x)^(7/2)*AppellF1(7/4,3/2,3/2,11/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f^3*sqrt(a+b*x^2+c*x^4))],
[(d+e*x^2)/((f*x)^(1/2)*(a+b*x^2+c*x^4)^(3/2)),x,6,2/5*e*(f*x)^(5/2)*AppellF1(5/4,3/2,3/2,9/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f^3*sqrt(a+b*x^2+c*x^4))+2*d*AppellF1(1/4,3/2,3/2,5/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(f*x)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f*sqrt(a+b*x^2+c*x^4))],
[(d+e*x^2)/((f*x)^(3/2)*(a+b*x^2+c*x^4)^(3/2)),x,6,2/3*e*(f*x)^(3/2)*AppellF1(3/4,3/2,3/2,7/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f^3*sqrt(a+b*x^2+c*x^4))-2*d*AppellF1(-1/4,3/2,3/2,3/4,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f*sqrt(f*x)*sqrt(a+b*x^2+c*x^4))],

# Integrands of the form (f x)^m (d+e x^2) (a+b x^2+c x^4)^p with m symbolic
[(f*x)^m*(d+e*x^2)*(a+b*x^2+c*x^4)^3,x,2,a^3*d*(f*x)^(1+m)/(f*(1+m))+a^2*(3*b*d+a*e)*(f*x)^(3+m)/(f^3*(3+m))+3*a*(b^2*d+a*c*d+a*b*e)*(f*x)^(5+m)/(f^5*(5+m))+(b^3*d+6*a*b*c*d+3*a*b^2*e+3*a^2*c*e)*(f*x)^(7+m)/(f^7*(7+m))+(3*b^2*c*d+3*a*c^2*d+b^3*e+6*a*b*c*e)*(f*x)^(9+m)/(f^9*(9+m))+3*c*(b*c*d+b^2*e+a*c*e)*(f*x)^(11+m)/(f^11*(11+m))+c^2*(c*d+3*b*e)*(f*x)^(13+m)/(f^13*(13+m))+c^3*e*(f*x)^(15+m)/(f^15*(15+m))],
[(f*x)^m*(d+e*x^2)*(a+b*x^2+c*x^4)^2,x,2,a^2*d*(f*x)^(1+m)/(f*(1+m))+a*(2*b*d+a*e)*(f*x)^(3+m)/(f^3*(3+m))+(b^2*d+2*a*c*d+2*a*b*e)*(f*x)^(5+m)/(f^5*(5+m))+(2*b*c*d+b^2*e+2*a*c*e)*(f*x)^(7+m)/(f^7*(7+m))+c*(c*d+2*b*e)*(f*x)^(9+m)/(f^9*(9+m))+c^2*e*(f*x)^(11+m)/(f^11*(11+m))],
[(f*x)^m*(d+e*x^2)*(a+b*x^2+c*x^4),x,2,a*d*(f*x)^(1+m)/(f*(1+m))+(b*d+a*e)*(f*x)^(3+m)/(f^3*(3+m))+(c*d+b*e)*(f*x)^(5+m)/(f^5*(5+m))+c*e*(f*x)^(7+m)/(f^7*(7+m))],
[(f*x)^m*(d+e*x^2)/(a+b*x^2+c*x^4),x,3,(f*x)^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-2*c*x^2/(b-sqrt(b^2-4*a*c)))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/(f*(1+m)*(b-sqrt(b^2-4*a*c)))+(f*x)^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-2*c*x^2/(b+sqrt(b^2-4*a*c)))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(f*(1+m)*(b+sqrt(b^2-4*a*c)))],
[(f*x)^m*(d+e*x^2)/(a+b*x^2+c*x^4)^2,x,4,1/2*(f*x)^(1+m)*(b^2*d-2*a*c*d-a*b*e+c*(b*d-2*a*e)*x^2)/(a*(b^2-4*a*c)*f*(a+b*x^2+c*x^4))-1/2*c*(f*x)^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-2*c*x^2/(b+sqrt(b^2-4*a*c)))*(b^2*d*(1-m)+b*(4*a*e-d*(1-m)*sqrt(b^2-4*a*c))+2*a*(-2*c*d*(3-m)+e*(1-m)*sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*f*(1+m)*(b+sqrt(b^2-4*a*c)))+1/2*c*(f*x)^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-2*c*x^2/(b-sqrt(b^2-4*a*c)))*(b^2*(d-d*m)+b*(4*a*e+d*(1-m)*sqrt(b^2-4*a*c))-2*a*(2*c*d*(3-m)+e*(1-m)*sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*f*(1+m)*(b-sqrt(b^2-4*a*c)))],
[(f*x)^m*(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2),x,6,a*d*(f*x)^(1+m)*AppellF1(1/2*(1+m),-3/2,-3/2,1/2*(3+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*(1+m)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+a*e*(f*x)^(3+m)*AppellF1(1/2*(3+m),-3/2,-3/2,1/2*(5+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*(3+m)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(f*x)^m*(d+e*x^2)*(a+b*x^2+c*x^4)^(1/2),x,6,d*(f*x)^(1+m)*AppellF1(1/2*(1+m),-1/2,-1/2,1/2*(3+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f*(1+m)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+e*(f*x)^(3+m)*AppellF1(1/2*(3+m),-1/2,-1/2,1/2*(5+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(a+b*x^2+c*x^4)/(f^3*(3+m)*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[(f*x)^m*(d+e*x^2)/(a+b*x^2+c*x^4)^(1/2),x,6,d*(f*x)^(1+m)*AppellF1(1/2*(1+m),1/2,1/2,1/2*(3+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f*(1+m)*sqrt(a+b*x^2+c*x^4))+e*(f*x)^(3+m)*AppellF1(1/2*(3+m),1/2,1/2,1/2*(5+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(f^3*(3+m)*sqrt(a+b*x^2+c*x^4))],
[(f*x)^m*(d+e*x^2)/(a+b*x^2+c*x^4)^(3/2),x,6,d*(f*x)^(1+m)*AppellF1(1/2*(1+m),3/2,3/2,1/2*(3+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f*(1+m)*sqrt(a+b*x^2+c*x^4))+e*(f*x)^(3+m)*AppellF1(1/2*(3+m),3/2,3/2,1/2*(5+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-2*c*x^2/(b+sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))/(a*f^3*(3+m)*sqrt(a+b*x^2+c*x^4))],

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

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

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

# p>0

# p<0
[x^9/((d+e*x^2)*(a+c*x^4)),x,6,-1/2*d*x^2/(c*e^2)+1/4*x^4/(c*e)+1/2*a^(3/2)*d*arctan(x^2*sqrt(c)/sqrt(a))/(c^(3/2)*(c*d^2+a*e^2))+1/2*d^4*log(d+e*x^2)/(e^3*(c*d^2+a*e^2))-1/4*a^2*e*log(a+c*x^4)/(c^2*(c*d^2+a*e^2))],
[x^7/((d+e*x^2)*(a+c*x^4)),x,6,1/2*x^2/(c*e)-1/2*a^(3/2)*e*arctan(x^2*sqrt(c)/sqrt(a))/(c^(3/2)*(c*d^2+a*e^2))-1/2*d^3*log(d+e*x^2)/(e^2*(c*d^2+a*e^2))-1/4*a*d*log(a+c*x^4)/(c*(c*d^2+a*e^2))],
[x^5/((d+e*x^2)*(a+c*x^4)),x,6,1/2*d^2*log(d+e*x^2)/(e*(c*d^2+a*e^2))+1/4*a*e*log(a+c*x^4)/(c*(c*d^2+a*e^2))-1/2*d*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(a)/((c*d^2+a*e^2)*sqrt(c))],
[x^3/((d+e*x^2)*(a+c*x^4)),x,6,-1/2*d*log(d+e*x^2)/(c*d^2+a*e^2)+1/4*d*log(a+c*x^4)/(c*d^2+a*e^2)+1/2*e*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(a)/((c*d^2+a*e^2)*sqrt(c))],
[x/((d+e*x^2)*(a+c*x^4)),x,6,1/2*e*log(d+e*x^2)/(c*d^2+a*e^2)-1/4*e*log(a+c*x^4)/(c*d^2+a*e^2)+1/2*d*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^2+a*e^2)*sqrt(a))],
[1/(x*(d+e*x^2)*(a+c*x^4)),x,6,log(x)/(a*d)-1/2*e^2*log(d+e*x^2)/(d*(c*d^2+a*e^2))-1/4*c*d*log(a+c*x^4)/(a*(c*d^2+a*e^2))-1/2*e*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^2+a*e^2)*sqrt(a))],
[1/(x^3*(d+e*x^2)*(a+c*x^4)),x,6,(-1/2)/(a*d*x^2)-1/2*c^(3/2)*d*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*(c*d^2+a*e^2))-e*log(x)/(a*d^2)+1/2*e^3*log(d+e*x^2)/(d^2*(c*d^2+a*e^2))+1/4*c*e*log(a+c*x^4)/(a*(c*d^2+a*e^2))],
[1/(x^5*(d+e*x^2)*(a+c*x^4)),x,6,(-1/4)/(a*d*x^4)+1/2*e/(a*d^2*x^2)+1/2*c^(3/2)*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*(c*d^2+a*e^2))-(c*d^2-a*e^2)*log(x)/(a^2*d^3)-1/2*e^4*log(d+e*x^2)/(d^3*(c*d^2+a*e^2))+1/4*c^2*d*log(a+c*x^4)/(a^2*(c*d^2+a*e^2))],
[x^8/((d+e*x^2)*(a+c*x^4)),x,12,-d*x/(c*e^2)+1/3*x^3/(c*e)+d^(7/2)*arctan(x*sqrt(e)/sqrt(d))/(e^(5/2)*(c*d^2+a*e^2))-1/2*a^(5/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*a^(5/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))-1/4*a^(5/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*a^(5/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))],
[x^6/((d+e*x^2)*(a+c*x^4)),x,12,x/(c*e)-d^(5/2)*arctan(x*sqrt(e)/sqrt(d))/(e^(3/2)*(c*d^2+a*e^2))-1/4*a^(3/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(c^(5/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*a^(3/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(c^(5/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*a^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(c^(5/4)*(c*d^2+a*e^2)*sqrt(2))-1/2*a^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(c^(5/4)*(c*d^2+a*e^2)*sqrt(2))],
[x^4/((d+e*x^2)*(a+c*x^4)),x,12,1/2*a^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)*sqrt(2))-1/2*a^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*a^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)*sqrt(2))-1/4*a^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)*sqrt(2))+d^(3/2)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2+a*e^2)*sqrt(e))],
[x^2/((d+e*x^2)*(a+c*x^4)),x,12,1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))-1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))-arctan(x*sqrt(e)/sqrt(d))*sqrt(d)*sqrt(e)/(c*d^2+a*e^2)],
[1/((d+e*x^2)*(a+c*x^4)),x,12,-1/2*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)*sqrt(2))-1/4*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)*sqrt(2))+e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2+a*e^2)*sqrt(d))],
[1/(x^2*(d+e*x^2)*(a+c*x^4)),x,12,(-1)/(a*d*x)-e^(5/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c*d^2+a*e^2))-1/4*c^(3/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(a^(5/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*c^(3/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(a^(5/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(5/4)*(c*d^2+a*e^2)*sqrt(2))-1/2*c^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(5/4)*(c*d^2+a*e^2)*sqrt(2))],
[1/(x^4*(d+e*x^2)*(a+c*x^4)),x,12,(-1/3)/(a*d*x^3)+e/(a*d^2*x)+e^(7/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*(c*d^2+a*e^2))+1/2*c^(5/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))-1/2*c^(5/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*c^(5/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))-1/4*c^(5/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))],
[x^9/((d+e*x^2)*(a+c*x^4)^2),x,7,1/4*a*(a*e+c*d*x^2)/(c^2*(c*d^2+a*e^2)*(a+c*x^4))+1/2*d^4*log(d+e*x^2)/(e*(c*d^2+a*e^2)^2)+1/4*a*e*(2*c*d^2+a*e^2)*log(a+c*x^4)/(c^2*(c*d^2+a*e^2)^2)-1/4*d*(3*c*d^2+a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(a)/(c^(3/2)*(c*d^2+a*e^2)^2)],
[x^7/((d+e*x^2)*(a+c*x^4)^2),x,7,1/4*a*(d-e*x^2)/(c*(c*d^2+a*e^2)*(a+c*x^4))-1/2*d^3*log(d+e*x^2)/(c*d^2+a*e^2)^2+1/4*d^3*log(a+c*x^4)/(c*d^2+a*e^2)^2+1/4*e*(3*c*d^2+a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(a)/(c^(3/2)*(c*d^2+a*e^2)^2)],
[x^5/((d+e*x^2)*(a+c*x^4)^2),x,7,1/4*(-a*e-c*d*x^2)/(c*(c*d^2+a*e^2)*(a+c*x^4))+1/2*d^2*e*log(d+e*x^2)/(c*d^2+a*e^2)^2-1/4*d^2*e*log(a+c*x^4)/(c*d^2+a*e^2)^2+1/4*d*(c*d^2-a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))/((c*d^2+a*e^2)^2*sqrt(a)*sqrt(c))],
[x^3/((d+e*x^2)*(a+c*x^4)^2),x,7,1/4*(-d+e*x^2)/((c*d^2+a*e^2)*(a+c*x^4))-1/2*d*e^2*log(d+e*x^2)/(c*d^2+a*e^2)^2+1/4*d*e^2*log(a+c*x^4)/(c*d^2+a*e^2)^2-1/4*e*(c*d^2-a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))/((c*d^2+a*e^2)^2*sqrt(a)*sqrt(c))],
[x/((d+e*x^2)*(a+c*x^4)^2),x,7,1/4*(a*e+c*d*x^2)/(a*(c*d^2+a*e^2)*(a+c*x^4))+1/2*e^3*log(d+e*x^2)/(c*d^2+a*e^2)^2-1/4*e^3*log(a+c*x^4)/(c*d^2+a*e^2)^2+1/4*d*(c*d^2+3*a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^2+a*e^2)^2)],
[1/(x*(d+e*x^2)*(a+c*x^4)^2),x,8,1/4*c*(d-e*x^2)/(a*(c*d^2+a*e^2)*(a+c*x^4))+log(x)/(a^2*d)-1/2*e^4*log(d+e*x^2)/(d*(c*d^2+a*e^2)^2)-1/4*c*d*(c*d^2+2*a*e^2)*log(a+c*x^4)/(a^2*(c*d^2+a*e^2)^2)-1/4*e*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^2+a*e^2))-1/2*e^3*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^2+a*e^2)^2*sqrt(a))],
[1/(x^3*(d+e*x^2)*(a+c*x^4)^2),x,8,(-1/2)/(a^2*d*x^2)-1/4*c*(a*e+c*d*x^2)/(a^2*(c*d^2+a*e^2)*(a+c*x^4))-1/4*c^(3/2)*d*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*(c*d^2+a*e^2))-1/2*c^(3/2)*d*(c*d^2+2*a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*(c*d^2+a*e^2)^2)-e*log(x)/(a^2*d^2)+1/2*e^5*log(d+e*x^2)/(d^2*(c*d^2+a*e^2)^2)+1/4*c*e*(c*d^2+2*a*e^2)*log(a+c*x^4)/(a^2*(c*d^2+a*e^2)^2)],
[1/(x^5*(d+e*x^2)*(a+c*x^4)^2),x,8,(-1/4)/(a^2*d*x^4)+1/2*e/(a^2*d^2*x^2)-1/4*c^2*(d-e*x^2)/(a^2*(c*d^2+a*e^2)*(a+c*x^4))+1/4*c^(3/2)*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*(c*d^2+a*e^2))+1/2*c^(3/2)*e*(c*d^2+2*a*e^2)*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*(c*d^2+a*e^2)^2)-(2*c*d^2-a*e^2)*log(x)/(a^3*d^3)-1/2*e^6*log(d+e*x^2)/(d^3*(c*d^2+a*e^2)^2)+1/4*c^2*d*(2*c*d^2+3*a*e^2)*log(a+c*x^4)/(a^3*(c*d^2+a*e^2)^2)],
[x^8/((d+e*x^2)*(a+c*x^4)^2),x,24,1/4*d*x/(c*(c*d^2+a*e^2))-1/4*x^3*(a*e+c*d*x^2)/(c*(c*d^2+a*e^2)*(a+c*x^4))+1/8*a^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-3*e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))-1/8*a^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-3*e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*a^(1/4)*d^2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/2*a^(1/4)*d^2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/4*a^(1/4)*d^2*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*a^(1/4)*d^2*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(c^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/16*a^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))-1/16*a^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*e*sqrt(a)+d*sqrt(c))/(c^(7/4)*(c*d^2+a*e^2)*sqrt(2))+d^(7/2)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2+a*e^2)^2*sqrt(e))],
[x^6/((d+e*x^2)*(a+c*x^4)^2),x,23,-1/4*x*(a*e+c*d*x^2)/(c*(c*d^2+a*e^2)*(a+c*x^4))+1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(5/4)*(c*d^2+a*e^2)*sqrt(2))-1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(5/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*d^2*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*d^2*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/2*d^2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/2*d^2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(1/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(5/4)*(c*d^2+a*e^2)*sqrt(2))+1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(1/4)*c^(5/4)*(c*d^2+a*e^2)*sqrt(2))-d^(5/2)*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/(c*d^2+a*e^2)^2],
[x^4/((d+e*x^2)*(a+c*x^4)^2),x,23,-1/4*x*(d-e*x^2)/((c*d^2+a*e^2)*(a+c*x^4))+d^(3/2)*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/(c*d^2+a*e^2)^2-1/2*c^(1/4)*d^2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/2*c^(1/4)*d^2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*c^(1/4)*d^2*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/4*c^(1/4)*d^2*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+3*d*sqrt(c))/(a^(3/4)*c^(3/4)*(c*d^2+a*e^2)*sqrt(2))-1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+3*d*sqrt(c))/(a^(3/4)*c^(3/4)*(c*d^2+a*e^2)*sqrt(2))+1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+3*d*sqrt(c))/(a^(3/4)*c^(3/4)*(c*d^2+a*e^2)*sqrt(2))-1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+3*d*sqrt(c))/(a^(3/4)*c^(3/4)*(c*d^2+a*e^2)*sqrt(2))],
[x^2/((d+e*x^2)*(a+c*x^4)^2),x,23,1/4*x*(a*e+c*d*x^2)/(a*(c*d^2+a*e^2)*(a+c*x^4))+1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*e*sqrt(a)+d*sqrt(c))/(a^(5/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))-1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*e*sqrt(a)+d*sqrt(c))/(a^(5/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))+1/2*c^(1/4)*d*e*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/2*c^(1/4)*d*e*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/4*c^(1/4)*d*e*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*c^(1/4)*d*e*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*e*sqrt(a)+d*sqrt(c))/(a^(5/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))+1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*e*sqrt(a)+d*sqrt(c))/(a^(5/4)*c^(1/4)*(c*d^2+a*e^2)*sqrt(2))-e^(5/2)*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/(c*d^2+a*e^2)^2],
[1/((d+e*x^2)*(a+c*x^4)^2),x,22,1/4*c*x*(d-e*x^2)/(a*(c*d^2+a*e^2)*(a+c*x^4))-1/2*c^(1/4)*e^2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/2*c^(1/4)*e^2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*c^(1/4)*e^2*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/4*c^(1/4)*e^2*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/8*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+3*d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))+1/8*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+3*d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))-1/16*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+3*d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))+1/16*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+3*d*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)*sqrt(2))+e^(7/2)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2+a*e^2)^2*sqrt(d))],
[1/(x^2*(d+e*x^2)*(a+c*x^4)^2),x,22,(-1)/(a^2*d*x)-1/4*c*x*(a*e+c*d*x^2)/(a^2*(c*d^2+a*e^2)*(a+c*x^4))-e^(9/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c*d^2+a*e^2)^2)-1/16*c^(3/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*e*sqrt(a)+d*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)*sqrt(2))+1/16*c^(3/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*e*sqrt(a)+d*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)*sqrt(2))+1/8*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*e*sqrt(a)+d*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)*sqrt(2))-1/8*c^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*e*sqrt(a)+d*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)*sqrt(2))+1/4*c^(3/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(a^(3/2)*e^3-d*(c*d^2+2*a*e^2)*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*c^(3/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(a^(3/2)*e^3-d*(c*d^2+2*a*e^2)*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/2*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(a^(3/2)*e^3+d*(c*d^2+2*a*e^2)*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/2*c^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(a^(3/2)*e^3+d*(c*d^2+2*a*e^2)*sqrt(c))/(a^(9/4)*(c*d^2+a*e^2)^2*sqrt(2))],
[1/(x^4*(d+e*x^2)*(a+c*x^4)^2),x,22,(-1/3)/(a^2*d*x^3)+e/(a^2*d^2*x)-1/4*c^2*x*(d-e*x^2)/(a^2*(c*d^2+a*e^2)*(a+c*x^4))+e^(11/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*(c*d^2+a*e^2)^2)+1/2*c^(5/4)*(c*d^2+2*a*e^2)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/2*c^(5/4)*(c*d^2+2*a*e^2)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/4*c^(5/4)*(c*d^2+2*a*e^2)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*c^(5/4)*(c*d^2+2*a*e^2)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/8*c^(5/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+3*d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)*sqrt(2))-1/8*c^(5/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(-e*sqrt(a)+3*d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)*sqrt(2))+1/16*c^(5/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+3*d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)*sqrt(2))-1/16*c^(5/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(e*sqrt(a)+3*d*sqrt(c))/(a^(11/4)*(c*d^2+a*e^2)*sqrt(2))],
[x^2/((1+x^2)*sqrt(1+x^4)),x,4,-1/2*arctan(x*sqrt(2)/sqrt(1+x^4))/sqrt(2)+1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^2/((1-x^2)*sqrt(1+x^4)),x,4,1/2*arctanh(x*sqrt(2)/sqrt(1+x^4))/sqrt(2)-1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(1+x^4)],
[x^2/((1+x^2)*sqrt(1-x^4)),x,6,-1/2*x*(1-x^2)/sqrt(1-x^4)-1/2*EllipticE(x,I)*sqrt(1-x^2)*sqrt(1+x^2)/sqrt(1-x^4)+EllipticF(x,I)*sqrt(1-x^2)*sqrt(1+x^2)/sqrt(1-x^4)],
[x^2/((1-x^2)*sqrt(1-x^4)),x,3,1/2*x*(1+x^2)/sqrt(1-x^4)-1/2*EllipticE(x,I)*sqrt(1-x^2)*sqrt(1+x^2)/sqrt(1-x^4)],
[x^2/((1+x^2)*sqrt(-1+x^4)),x,7,-1/2*x*(1-x^2)/sqrt(-1+x^4)-1/2*EllipticE(x,I)*sqrt(1-x^2)*sqrt(1+x^2)/sqrt(-1+x^4)+EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1+x^4))],
[x^2/((1-x^2)*sqrt(-1+x^4)),x,4,1/2*x*(1+x^2)/sqrt(-1+x^4)-1/2*EllipticE(x,I)*sqrt(1-x^2)*sqrt(1+x^2)/sqrt(-1+x^4)],
[x^2/((1+x^2)*sqrt(-1-x^4)),x,4,-1/2*arctanh(x*sqrt(2)/sqrt(-1-x^4))/sqrt(2)+1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(-1-x^4)],
[x^2/((1-x^2)*sqrt(-1-x^4)),x,4,1/2*arctan(x*sqrt(2)/sqrt(-1-x^4))/sqrt(2)-1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/2))*sqrt((1+x^4)/(1+x^2)^2)/sqrt(-1-x^4)],

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

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

# Integrands of the form (f x)^m (d+e x^2)^(q/2) (a^2+2 a b x^2+b^2 x^4)^(p/2)
[x^2*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,6,1/6*b*x^3*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+1/16*c^2*(b*c-2*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^(5/2)*(a+b*x^2))-1/16*c*(b*c-2*a*d)*x*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^2*(a+b*x^2))-1/8*(b*c-2*a*d)*x^3*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))],
[x*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,4,-1/3*(b*c-a*d)*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^2*(a+b*x^2))+1/5*b*(c+d*x^2)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^2*(a+b*x^2))],
[sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,5,1/4*b*x*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))-1/8*c*(b*c-4*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^(3/2)*(a+b*x^2))-1/8*(b*c-4*a*d)*x*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))],
[sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/x,x,6,1/3*b*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))-a*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(c)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^2,x,5,-a*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(c*x*(a+b*x^2))+1/2*(b*c+2*a*d)*arctanh(x*sqrt(d)/sqrt(c+d*x^2))*sqrt(a^2+2*a*b*x^2+b^2*x^4)/((a+b*x^2)*sqrt(d))+1/2*(b*c+2*a*d)*x*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(c*(a+b*x^2))],
[sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^3,x,6,-1/2*a*(c+d*x^2)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(c*x^2*(a+b*x^2))-1/2*(2*b*c+a*d)*arctanh(sqrt(c+d*x^2)/sqrt(c))*sqrt(a^2+2*a*b*x^2+b^2*x^4)/((a+b*x^2)*sqrt(c))+1/2*(2*b*c+a*d)*sqrt(c+d*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(c*(a+b*x^2))],

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

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

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

# q<0
[x^6*(a+b*x^2+c*x^4)/(d+e*x^2)^2,x,4,-d*(4*c*d^2-e*(3*b*d-2*a*e))*x/e^5+1/3*(3*c*d^2-e*(2*b*d-a*e))*x^3/e^4-1/5*(2*c*d-b*e)*x^5/e^3+1/7*c*x^7/e^2-1/2*d^2*(c*d^2-b*d*e+a*e^2)*x/(e^5*(d+e*x^2))+1/2*d^(3/2)*(9*c*d^2-e*(7*b*d-5*a*e))*arctan(x*sqrt(e)/sqrt(d))/e^(11/2)],
[x^4*(a+b*x^2+c*x^4)/(d+e*x^2)^2,x,4,(3*c*d^2-e*(2*b*d-a*e))*x/e^4-1/3*(2*c*d-b*e)*x^3/e^3+1/5*c*x^5/e^2+1/2*d*(c*d^2-b*d*e+a*e^2)*x/(e^4*(d+e*x^2))-1/2*(7*c*d^2-e*(5*b*d-3*a*e))*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/e^(9/2)],
[x^2*(a+b*x^2+c*x^4)/(d+e*x^2)^2,x,4,-(2*c*d-b*e)*x/e^3+1/3*c*x^3/e^2-1/2*(c*d^2-b*d*e+a*e^2)*x/(e^3*(d+e*x^2))+1/2*(5*c*d^2-e*(3*b*d-a*e))*arctan(x*sqrt(e)/sqrt(d))/(e^(7/2)*sqrt(d))],
[(a+b*x^2+c*x^4)/(d+e*x^2)^2,x,3,c*x/e^2+1/2*(a+d*(c*d-b*e)/e^2)*x/(d*(d+e*x^2))-1/2*(3*c*d^2-e*(b*d+a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(5/2))],
[(a+b*x^2+c*x^4)/(x^2*(d+e*x^2)^2),x,3,-a/(d^2*x)-1/2*(c*d^2-b*d*e+a*e^2)*x/(d^2*e*(d+e*x^2))+1/2*(c*d^2+e*(b*d-3*a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*e^(3/2))],
[(a+b*x^2+c*x^4)/(x^4*(d+e*x^2)^2),x,4,-1/3*a/(d^2*x^3)+(-b*d+2*a*e)/(d^3*x)+1/2*(c*d^2-b*d*e+a*e^2)*x/(d^3*(d+e*x^2))+1/2*(c*d^2-e*(3*b*d-5*a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*sqrt(e))],
[(a+b*x^2+c*x^4)/(x^6*(d+e*x^2)^2),x,4,-1/5*a/(d^2*x^5)+1/3*(-b*d+2*a*e)/(d^3*x^3)+(-c*d^2+e*(2*b*d-3*a*e))/(d^4*x)-1/2*e*(c*d^2-b*d*e+a*e^2)*x/(d^4*(d+e*x^2))-1/2*(3*c*d^2-e*(5*b*d-7*a*e))*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(9/2)],
[(a+b*x^2+c*x^4)/(x^8*(d+e*x^2)^2),x,4,-1/7*a/(d^2*x^7)+1/5*(-b*d+2*a*e)/(d^3*x^5)+1/3*(-c*d^2+e*(2*b*d-3*a*e))/(d^4*x^3)+e*(2*c*d^2-e*(3*b*d-4*a*e))/(d^5*x)+1/2*e^2*(c*d^2-b*d*e+a*e^2)*x/(d^5*(d+e*x^2))+1/2*e^(3/2)*(5*c*d^2-e*(7*b*d-9*a*e))*arctan(x*sqrt(e)/sqrt(d))/d^(11/2)],
[x^6*(a+b*x^2+c*x^4)/(d+e*x^2)^3,x,5,(6*c*d^2-e*(3*b*d-a*e))*x/e^5-1/3*(3*c*d-b*e)*x^3/e^4+1/5*c*x^5/e^3-1/4*d^2*(c*d^2-b*d*e+a*e^2)*x/(e^5*(d+e*x^2)^2)+1/8*d*(17*c*d^2-e*(13*b*d-9*a*e))*x/(e^5*(d+e*x^2))-1/8*(63*c*d^2-35*b*d*e+15*a*e^2)*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/e^(11/2)],
[x^4*(a+b*x^2+c*x^4)/(d+e*x^2)^3,x,5,-(3*c*d-b*e)*x/e^4+1/3*c*x^3/e^3+1/4*d*(c*d^2-b*d*e+a*e^2)*x/(e^4*(d+e*x^2)^2)-1/8*(13*c*d^2-e*(9*b*d-5*a*e))*x/(e^4*(d+e*x^2))+1/8*(35*c*d^2-3*e*(5*b*d-a*e))*arctan(x*sqrt(e)/sqrt(d))/(e^(9/2)*sqrt(d))],
[x^2*(a+b*x^2+c*x^4)/(d+e*x^2)^3,x,4,c*x/e^3-1/4*(c*d^2-b*d*e+a*e^2)*x/(e^3*(d+e*x^2)^2)+1/8*(9*c*d^2-e*(5*b*d-a*e))*x/(d*e^3*(d+e*x^2))-1/8*(15*c*d^2-e*(3*b*d+a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(7/2))],
[(a+b*x^2+c*x^4)/(d+e*x^2)^3,x,3,1/4*(a+d*(c*d-b*e)/e^2)*x/(d*(d+e*x^2)^2)-1/8*(5*c*d^2-e*(b*d+3*a*e))*x/(d^2*e^2*(d+e*x^2))+1/8*(3*c*d^2+e*(b*d+3*a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*e^(5/2))],
[(a+b*x^2+c*x^4)/(x^2*(d+e*x^2)^3),x,4,-a/(d^3*x)-1/4*(c*d^2-b*d*e+a*e^2)*x/(d^2*e*(d+e*x^2)^2)+1/8*(c*d^2+e*(3*b*d-7*a*e))*x/(d^3*e*(d+e*x^2))+1/8*(c*d^2+3*e*(b*d-5*a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*e^(3/2))],
[(a+b*x^2+c*x^4)/(x^4*(d+e*x^2)^3),x,5,-1/3*a/(d^3*x^3)+(-b*d+3*a*e)/(d^4*x)+1/4*(c*d^2-b*d*e+a*e^2)*x/(d^3*(d+e*x^2)^2)+1/8*(3*c*d^2-e*(7*b*d-11*a*e))*x/(d^4*(d+e*x^2))+1/8*(3*c*d^2-15*b*d*e+35*a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(d^(9/2)*sqrt(e))],
[(a+b*x^2+c*x^4)/(x^6*(d+e*x^2)^3),x,5,-1/5*a/(d^3*x^5)+1/3*(-b*d+3*a*e)/(d^4*x^3)+(-c*d^2+3*b*d*e-6*a*e^2)/(d^5*x)-1/4*e*(c*d^2-b*d*e+a*e^2)*x/(d^4*(d+e*x^2)^2)-1/8*e*(7*c*d^2-e*(11*b*d-15*a*e))*x/(d^5*(d+e*x^2))-1/8*(15*c*d^2-35*b*d*e+63*a*e^2)*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(11/2)],

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

# p>0

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

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

# p>0

# p<0
[1/((d+e*x^2)*(a+b*x^2+c*x^4)*sqrt(f*x)),x,19,-e^(7/4)*arctan(1-e^(1/4)*sqrt(2)*sqrt(f*x)/(d^(1/4)*sqrt(f)))/(d^(3/4)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(f))+e^(7/4)*arctan(1+e^(1/4)*sqrt(2)*sqrt(f*x)/(d^(1/4)*sqrt(f)))/(d^(3/4)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(f))-1/2*e^(7/4)*log(sqrt(d)*sqrt(f)+x*sqrt(e)*sqrt(f)-d^(1/4)*e^(1/4)*sqrt(2)*sqrt(f*x))/(d^(3/4)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(f))+1/2*e^(7/4)*log(sqrt(d)*sqrt(f)+x*sqrt(e)*sqrt(f)+d^(1/4)*e^(1/4)*sqrt(2)*sqrt(f*x))/(d^(3/4)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(f))+c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(f*x)/((-b-sqrt(b^2-4*a*c))^(1/4)*sqrt(f)))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2^(1/4)*(c*d^2-b*d*e+a*e^2)*(-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c)*sqrt(f))+c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(f*x)/((-b-sqrt(b^2-4*a*c))^(1/4)*sqrt(f)))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2^(1/4)*(c*d^2-b*d*e+a*e^2)*(-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c)*sqrt(f))-c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(f*x)/((-b+sqrt(b^2-4*a*c))^(1/4)*sqrt(f)))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2^(1/4)*(c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4)*sqrt(f))-c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(f*x)/((-b+sqrt(b^2-4*a*c))^(1/4)*sqrt(f)))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2^(1/4)*(c*d^2-b*d*e+a*e^2)*sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4)*sqrt(f))],

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

# p>0
[x^5*sqrt(a+b*x^2+c*x^4)/(d+e*x^2),x,8,1/6*(a+b*x^2+c*x^4)^(3/2)/(c*e)-1/32*(16*c^3*d^3-b^3*e^3-2*b*c*e^2*(b*d-2*a*e)-8*c^2*d*e*(b*d-a*e))*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(5/2)*e^4)+1/2*d^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))*sqrt(c*d^2-b*d*e+a*e^2)/e^4+1/16*((2*c*d-b*e)*(4*c*d+b*e)-2*c*e*(2*c*d+b*e)*x^2)*sqrt(a+b*x^2+c*x^4)/(c^2*e^3)],
[x^3*sqrt(a+b*x^2+c*x^4)/(d+e*x^2),x,7,1/16*(8*c^2*d^2-b^2*e^2-4*c*e*(b*d-a*e))*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(3/2)*e^3)-1/2*d*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))*sqrt(c*d^2-b*d*e+a*e^2)/e^3-1/8*(4*c*d-b*e-2*c*e*x^2)*sqrt(a+b*x^2+c*x^4)/(c*e^2)],
[x*sqrt(a+b*x^2+c*x^4)/(d+e*x^2),x,7,-1/4*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(e^2*sqrt(c))+1/2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))*sqrt(c*d^2-b*d*e+a*e^2)/e^2+1/2*sqrt(a+b*x^2+c*x^4)/e],
[sqrt(a+b*x^2+c*x^4)/(x*(d+e*x^2)),x,9,-1/2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))*sqrt(a)/d+1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))*sqrt(c)/e-1/2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))*sqrt(c*d^2-b*d*e+a*e^2)/(d*e)],
[sqrt(a+b*x^2+c*x^4)/(x^3*(d+e*x^2)),x,21,-1/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(d*sqrt(a))+1/2*e*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))*sqrt(a)/d^2-1/4*b*e*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(d^2*sqrt(c))-1/4*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(d^2*sqrt(c))+1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))*sqrt(c)/d+1/2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))*sqrt(c*d^2-b*d*e+a*e^2)/d^2-1/2*sqrt(a+b*x^2+c*x^4)/(d*x^2)],
[x^4*sqrt(1+2*x^2+2*x^4)/(3+2*x^2),x,17,3/16*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(15)-1/60*x*(13-6*x^2)*sqrt(1+2*x^2+2*x^4)+109/60*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-109/60*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+15/16*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+1/60*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(-70+263*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),3/16*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(15)-1/4*x*sqrt(1+2*x^2+2*x^4)+1/30*x*(1+3*x^2)*sqrt(1+2*x^2+2*x^4)+109/60*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-109/60*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-139/240*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/4*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+45/112*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-15/224*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[x^2*sqrt(1+2*x^2+2*x^4)/(3+2*x^2),x,13,-1/8*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(15)+1/6*x*sqrt(1+2*x^2+2*x^4)-7/6*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))+7/6*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-5/8*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))-1/6*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(-4+17*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),-1/8*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(15)+1/6*x*sqrt(1+2*x^2+2*x^4)-7/6*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))+7/6*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+3/8*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+1/6*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-15/56*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+5/112*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[sqrt(1+2*x^2+2*x^4)/(3+2*x^2),x,7,1/4*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(5/3)+x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+5/12*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+2^(3/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/((-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),1/4*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(5/3)+x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-1/4*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+5/28*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-5/168*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[sqrt(1+2*x^2+2*x^4)/(x^2*(3+2*x^2)),x,8,-1/6*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(5/3)-1/3*sqrt(1+2*x^2+2*x^4)/x+1/3*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-1/3*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+1/21*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+5/252*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[sqrt(1+2*x^2+2*x^4)/(x^4*(3+2*x^2)),x,7,1/9*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(5/3)-1/9*sqrt(1+2*x^2+2*x^4)/x^3-1/9*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+5/63*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-5/378*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[sqrt(1+2*x^2+2*x^4)/(x^6*(3+2*x^2)),x,13,-2/27*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(5/3)-1/15*sqrt(1+2*x^2+2*x^4)/x^5+4/135*sqrt(1+2*x^2+2*x^4)/x^3-4/45*sqrt(1+2*x^2+2*x^4)/x+4/45*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-4/45*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+5/189*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(5-3*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-1/135*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(19-2*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+5/567*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[x^5*(a+b*x^2+c*x^4)^(3/2)/(d+e*x^2),x,9,1/96*(16*c^2*d^2-6*b*c*d*e-3*b^2*e^2-6*c*e*(2*c*d+b*e)*x^2)*(a+b*x^2+c*x^4)^(3/2)/(c^2*e^3)+1/10*(a+b*x^2+c*x^4)^(5/2)/(c*e)-1/512*(256*c^5*d^5+3*b^5*e^5+6*b^3*c*e^4*(b*d-4*a*e)-384*c^4*d^3*e*(b*d-a*e)+96*c^3*d*e^2*(b*d-a*e)^2+16*b*c^2*e^3*(b^2*d^2-3*a*b*d*e+3*a^2*e^2))*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(7/2)*e^6)+1/2*d^2*(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/e^6+1/256*(128*c^4*d^4+3*b^4*e^4-32*c^3*d^2*e*(5*b*d-4*a*e)+8*b*c^2*d*e^2*(2*b*d-3*a*e)+6*b^2*c*e^3*(b*d-2*a*e)-2*c*e*(32*c^3*d^3-3*b^3*e^3-8*c^2*d*e*(2*b*d-3*a*e)-6*b*c*e^2*(b*d-2*a*e))*x^2)*sqrt(a+b*x^2+c*x^4)/(c^3*e^5)],
[x^3*(a+b*x^2+c*x^4)^(3/2)/(d+e*x^2),x,8,-1/48*(8*c*d-3*b*e-6*c*e*x^2)*(a+b*x^2+c*x^4)^(3/2)/(c*e^2)+1/256*(128*c^4*d^4+3*b^4*e^4+8*b^2*c*e^3*(b*d-3*a*e)-192*c^3*d^2*e*(b*d-a*e)+48*c^2*e^2*(b*d-a*e)^2)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(5/2)*e^5)-1/2*d*(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/e^5-1/128*(64*c^3*d^3+3*b^3*e^3-16*c^2*d*e*(5*b*d-4*a*e)+4*b*c*e^2*(2*b*d-3*a*e)-2*c*e*(16*c^2*d^2-3*b^2*e^2-4*c*e*(2*b*d-3*a*e))*x^2)*sqrt(a+b*x^2+c*x^4)/(c^2*e^4)],
[x*(a+b*x^2+c*x^4)^(3/2)/(d+e*x^2),x,8,1/6*(a+b*x^2+c*x^4)^(3/2)/e-1/32*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(3/2)*e^4)+1/2*(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/e^4+1/16*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x^2)*sqrt(a+b*x^2+c*x^4)/(c*e^3)],
[(a+b*x^2+c*x^4)^(3/2)/(x*(d+e*x^2)),x,14,-1/2*a^(3/2)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/d-1/2*(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(d*e^3)+1/4*a*b*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(d*sqrt(c))+1/16*(8*c^2*d^3+b*e^2*(3*b*d-4*a*e)-12*c*d*e*(b*d-a*e))*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(d*e^3*sqrt(c))+1/2*a*sqrt(a+b*x^2+c*x^4)/d-1/8*(4*c*d^2-e*(5*b*d-4*a*e)-2*c*d*e*x^2)*sqrt(a+b*x^2+c*x^4)/(d*e^2)],
[(a+b*x^2+c*x^4)^(3/2)/(x^3*(d+e*x^2)),x,24,-1/2*(a+b*x^2+c*x^4)^(3/2)/(d*x^2)+1/2*a^(3/2)*e*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/d^2+1/32*b*(b^2-12*a*c)*e*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(3/2)*d^2)-1/32*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(3/2)*d^2*e^2)+1/2*(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(d^2*e^2)-3/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))*sqrt(a)/d+3/16*(b^2+4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(d*sqrt(c))+3/8*(3*b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/d-1/16*e*(b^2+8*a*c+2*b*c*x^2)*sqrt(a+b*x^2+c*x^4)/(c*d^2)+1/16*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x^2)*sqrt(a+b*x^2+c*x^4)/(c*d^2*e)],
[x^2*(1+2*x^2+2*x^4)^(3/2)/(3-2*x^2),x,19,-1/14*x*(1+2*x^2+2*x^4)^(3/2)+17/16*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(51)-213/140*x*sqrt(1+2*x^2+2*x^4)-27/70*x^3*sqrt(1+2*x^2+2*x^4)-2211/140*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))+2211/140*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-289/16*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4))-3/140*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(514+2717*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),-1/14*x*(1+2*x^2+2*x^4)^(3/2)+17/16*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(51)-3/35*x*(2+x^2)*sqrt(1+2*x^2+2*x^4)-3/20*x*(9+2*x^2)*sqrt(1+2*x^2+2*x^4)-309/20*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-6/35*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))+309/20*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+6/35*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-289/224*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(11-6*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+867/112*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-51/16*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(5+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-3/70*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+2*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-3/20*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(9+8*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))],
[(1+2*x^2+2*x^4)^(3/2)/(3-2*x^2),x,12,17/8*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(17/3)-1/10*x*(9+2*x^2)*sqrt(1+2*x^2+2*x^4)-103/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))+103/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-289/24*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(66+383*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),17/8*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(17/3)-1/10*x*(9+2*x^2)*sqrt(1+2*x^2+2*x^4)-103/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))+103/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-289/336*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(11-6*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+289/56*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-17/8*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(5+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(9+8*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))],
[(1+2*x^2+2*x^4)^(3/2)/(x^2*(3-2*x^2)),x,13,17/12*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(17/3)-1/3*(1+x^2)*sqrt(1+2*x^2+2*x^4)/x-17/3*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))+1/3*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))+17/3*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-1/3*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+1/3*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-289/504*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(11-6*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+289/84*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-17/12*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(5+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[(1+2*x^2+2*x^4)^(3/2)/(x^4*(3-2*x^2)),x,13,17/18*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(17/3)-2*sqrt(1+2*x^2+2*x^4)/x-1/9*(1-8*x^2)*sqrt(1+2*x^2+2*x^4)/x^3+1/9*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-1/9*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-289/756*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(11-6*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+289/126*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-17/18*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(5+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+1/9*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(9+5*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)],
[(1+2*x^2+2*x^4)^(3/2)/(x^6*(3-2*x^2)),x,15,17/27*arctanh(x*sqrt(17/3)/sqrt(1+2*x^2+2*x^4))*sqrt(17/3)+74/135*sqrt(1+2*x^2+2*x^4)/x^3-262/135*sqrt(1+2*x^2+2*x^4)/x-1/45*(3+40*x^2)*sqrt(1+2*x^2+2*x^4)/x^5+262/135*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-262/135*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-289/1134*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12+11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(11-6*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+85/189*2^(3/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+1/135*2^(3/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(37+23*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)],

# p<0
[x^5/((d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,7,-1/4*(2*c*d+b*e)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(3/2)*e^2)+1/2*d^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(e^2*sqrt(c*d^2-b*d*e+a*e^2))+1/2*sqrt(a+b*x^2+c*x^4)/(c*e)],
[x^3/((d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,6,1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(e*sqrt(c))-1/2*d*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(e*sqrt(c*d^2-b*d*e+a*e^2))],
[x/((d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,3,1/2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/sqrt(c*d^2-b*d*e+a*e^2)],
[1/(x*(d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,7,-1/2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(d*sqrt(a))-1/2*e*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(d*sqrt(c*d^2-b*d*e+a*e^2))],
[1/(x^3*(d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,10,1/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(a^(3/2)*d)+1/2*e*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(d^2*sqrt(a))+1/2*e^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(d^2*sqrt(c*d^2-b*d*e+a*e^2))-1/2*sqrt(a+b*x^2+c*x^4)/(a*d*x^2)],
[x^4/((3+2*x^2)*sqrt(1+2*x^2+2*x^4)),x,4,-3/4*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/10)*(3-sqrt(2))/(2-3*sqrt(2))+1/2*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/2*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+1/2*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-3*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+3/8*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))],
[x^2/((3+2*x^2)*sqrt(1+2*x^2+2*x^4)),x,3,-1/4*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)-1/14*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+1/56*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[1/((3+2*x^2)*sqrt(1+2*x^2+2*x^4)),x,3,1/2*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)+1/14*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/84*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[1/(x^2*(3+2*x^2)*sqrt(1+2*x^2+2*x^4)),x,6,-1/3*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)-1/3*sqrt(1+2*x^2+2*x^4)/x+1/3*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-1/3*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+1/21*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(5-3*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+1/126*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[1/(x^4*(3+2*x^2)*sqrt(1+2*x^2+2*x^4)),x,7,2/9*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)-1/9*sqrt(1+2*x^2+2*x^4)/x^3+2/3*sqrt(1+2*x^2+2*x^4)/x-2/3*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))+2/3*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-1/189*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/63*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+19*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[x^7/((d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)),x,7,1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/(c^(3/2)*e)-1/2*d^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(e*(c*d^2-b*d*e+a*e^2)^(3/2))+(a*(b^2*d-2*a*c*d-a*b*e)+(b^3*d-3*a*b*c*d-a*b^2*e+2*a^2*c*e)*x^2)/(c*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))],
[x^5/((d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)),x,5,1/2*d^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(c*d^2-b*d*e+a*e^2)^(3/2)+(-a*(b*d-2*a*e)-(b^2*d-2*a*c*d-a*b*e)*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))],
[x^3/((d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)),x,5,-1/2*d*e*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(c*d^2-b*d*e+a*e^2)^(3/2)+(a*(2*c*d-b*e)+c*(b*d-2*a*e)*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))],
[x/((d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)),x,5,1/2*e^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(c*d^2-b*d*e+a*e^2)^(3/2)+(-b*c*d+b^2*e-2*a*c*e-c*(2*c*d-b*e)*x^2)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))],
[1/(x*(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)),x,11,-1/2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(a^(3/2)*d)-1/2*e^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(d*(c*d^2-b*d*e+a*e^2)^(3/2))+(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*d*sqrt(a+b*x^2+c*x^4))+e*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x^2)/((b^2-4*a*c)*d*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))],
[1/(x^3*(d+e*x^2)*(a+b*x^2+c*x^4)^(3/2)),x,15,3/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(a^(5/2)*d)+1/2*e*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/(a^(3/2)*d^2)+1/2*e^4*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x^2)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4)))/(d^2*(c*d^2-b*d*e+a*e^2)^(3/2))-e*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*d^2*sqrt(a+b*x^2+c*x^4))+(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*d*x^2*sqrt(a+b*x^2+c*x^4))-e^2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x^2)/((b^2-4*a*c)*d^2*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))-1/2*(3*b^2-8*a*c)*sqrt(a+b*x^2+c*x^4)/(a^2*(b^2-4*a*c)*d*x^2)],
[x^8/((3+2*x^2)*(1+2*x^2+2*x^4)^(3/2)),x,10,27/80*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)+1/20*x^3*(1-2*x^2)/sqrt(1+2*x^2+2*x^4)+1/20*x*sqrt(1+2*x^2+2*x^4)+1/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+27/80*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+1/8*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(-2+7*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),-27/40*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/10)*(3-sqrt(2))/(2-3*sqrt(2))+1/20*x^3*(1-2*x^2)/sqrt(1+2*x^2+2*x^4)+1/20*x*sqrt(1+2*x^2+2*x^4)+1/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+9/20*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-3*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+27/80*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))-1/40*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(7+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))],
[x^6/((3+2*x^2)*(1+2*x^2+2*x^4)^(3/2)),x,8,-9/40*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)+1/20*x*(1-2*x^2)/sqrt(1+2*x^2+2*x^4)+1/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-9/40*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))-1/8*(2^(1/4)+2^(3/4))*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/((-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),-9/40*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)+1/20*x*(1-2*x^2)/sqrt(1+2*x^2+2*x^4)+1/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))-1/40*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-9/140*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+9/560*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[x^4/((3+2*x^2)*(1+2*x^2+2*x^4)^(3/2)),x,8,3/20*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)-1/10*x*(2+x^2)/sqrt(1+2*x^2+2*x^4)+1/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+3/20*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+1/4*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(2+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),3/20*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)-1/10*x*(2+x^2)/sqrt(1+2*x^2+2*x^4)+1/10*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+1/20*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+9/140*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-3/280*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[x^2/((3+2*x^2)*(1+2*x^2+2*x^4)^(3/2)),x,8,-1/10*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)+1/10*x*(3+4*x^2)/sqrt(1+2*x^2+2*x^4)-1/5*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))+1/5*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))-1/4*(2^(1/4)+2^(3/4))*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/((-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),-1/10*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))*sqrt(3/5)+1/10*x*(3+4*x^2)/sqrt(1+2*x^2+2*x^4)-1/5*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))+1/5*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-3/70*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+1/140*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/20*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+2*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))],
[1/((3+2*x^2)*(1+2*x^2+2*x^4)^(3/2)),x,8,1/5*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)-1/5*x*(1+3*x^2)/sqrt(1+2*x^2+2*x^4)+3/5*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-3/5*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+1/15*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+1/2*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(2+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),1/5*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)-1/5*x*(1+3*x^2)/sqrt(1+2*x^2+2*x^4)+3/5*x*sqrt(1+2*x^2+2*x^4)/(sqrt(2)*(1+x^2*sqrt(2)))-3/5*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))+1/35*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/210*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))+1/10*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+2*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))],
[1/(x^2*(3+2*x^2)*(1+2*x^2+2*x^4)^(3/2)),x,15,-2/15*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)-1/3*x/sqrt(1+2*x^2+2*x^4)+2/15*x*(1+3*x^2)/sqrt(1+2*x^2+2*x^4)-1/3*sqrt(1+2*x^2+2*x^4)/x+2/15*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-2/15*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-1/45*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/((2-3*sqrt(2))*sqrt(1+2*x^2+2*x^4))+1/3*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(-7+3*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*(-2+3*sqrt(2))*sqrt(1+2*x^2+2*x^4)),-2/15*arctan(x*sqrt(5/3)/sqrt(1+2*x^2+2*x^4))/sqrt(15)-1/3*x/sqrt(1+2*x^2+2*x^4)+2/15*x*(1+3*x^2)/sqrt(1+2*x^2+2*x^4)-1/3*sqrt(1+2*x^2+2*x^4)/x+2/15*x*sqrt(2)*sqrt(1+2*x^2+2*x^4)/(1+x^2*sqrt(2))-2/15*2^(1/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticE(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)-1/6*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(1-sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/105*2^(3/4)*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/sqrt(1+2*x^2+2*x^4)+1/315*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticPi(sin(2*arctan(2^(1/4)*x)),1/24*(12-11*sqrt(2)),sqrt(1/4*(2-sqrt(2))))*(3+sqrt(2))^2*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(1/4)*sqrt(1+2*x^2+2*x^4))-1/15*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/4*(2-sqrt(2))))*(3+2*sqrt(2))*(1+x^2*sqrt(2))*sqrt((1+2*x^2+2*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(1+2*x^2+2*x^4))],

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

# q>0
[x^7*sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,7,-1/3*(c*d+b*e)*(d+e*x^2)^(3/2)/(c^2*e^2)+1/5*(d+e*x^2)^(5/2)/(c*e^2)+(b^2-a*c)*sqrt(d+e*x^2)/c^3-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e+(-b^3*c*d+3*a*b*c^2*d+b^4*e-4*a*b^2*c*e+2*a^2*c^2*e)/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e+(b^3*c*d-3*a*b*c^2*d-b^4*e+4*a*b^2*c*e-2*a^2*c^2*e)/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^5*sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,7,1/3*(d+e*x^2)^(3/2)/(c*e)-b*sqrt(d+e*x^2)/c^2+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(-b^2*c*d+2*a*c^2*d+b^3*e-3*a*b*c*e)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^3*sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,6,sqrt(d+e*x^2)/c+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+2*a*c*e-(c*d-b*e)*sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+2*a*c*e+(c*d-b*e)*sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x*sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,5,-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[sqrt(d+e*x^2)/(x*(a+b*x^2+c*x^4)),x,8,-arctanh(sqrt(d+e*x^2)/sqrt(d))*sqrt(d)/a+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(b*d-2*a*e+d*sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(b*d-2*a*e-d*sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x^2)/(x^3*(a+b*x^2+c*x^4)),x,10,1/2*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/(a*sqrt(d))+(b*d-a*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/(a^2*sqrt(d))-1/2*sqrt(d+e*x^2)/(a*x^2)-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(b^2*d-2*a*c*d-a*b*e+(b*d-a*e)*sqrt(b^2-4*a*c))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(b^2*d-2*a*c*d-a*b*e-(b*d-a*e)*sqrt(b^2-4*a*c))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x^2)/(x^5*(a+b*x^2+c*x^4)),x,13,-3/8*e^2*arctanh(sqrt(d+e*x^2)/sqrt(d))/(a*d^(3/2))-1/2*e*(b*d-a*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/(a^2*d^(3/2))-(b^2*d-a*c*d-a*b*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/(a^3*sqrt(d))-1/4*sqrt(d+e*x^2)/(a*x^4)+3/8*e*sqrt(d+e*x^2)/(a*d*x^2)+1/2*(b*d-a*e)*sqrt(d+e*x^2)/(a^2*d*x^2)+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(b^3*d-a*c*(-2*a*e+d*sqrt(b^2-4*a*c))+b^2*(-a*e+d*sqrt(b^2-4*a*c))-a*b*(3*c*d+e*sqrt(b^2-4*a*c)))/(a^3*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(b^3*d-b^2*(a*e+d*sqrt(b^2-4*a*c))+a*c*(2*a*e+d*sqrt(b^2-4*a*c))-a*b*(3*c*d-e*sqrt(b^2-4*a*c)))/(a^3*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^4*sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,10,1/2*(c*d-2*b*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^2*sqrt(e))+1/2*x*sqrt(d+e*x^2)/c-arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(-b^2*c*d+2*a*c^2*d+b^3*e-3*a*b*c*e)/sqrt(b^2-4*a*c))/(c^2*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)/sqrt(b^2-4*a*c))/(c^2*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^2*sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,9,arctanh(x*sqrt(e)/sqrt(d+e*x^2))*sqrt(e)/c+arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(c*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(c*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x^2)/(a+b*x^2+c*x^4),x,11,arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(b^2-4*a*c)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[sqrt(d+e*x^2)/(x^2*(a+b*x^2+c*x^4)),x,8,-sqrt(d+e*x^2)/(a*x)-c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/(a*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/(a*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x^2)/(x^4*(a+b*x^2+c*x^4)),x,12,-1/3*sqrt(d+e*x^2)/(a*x^3)+2/3*e*sqrt(d+e*x^2)/(a*d*x)+(b*d-a*e)*sqrt(d+e*x^2)/(a^2*d*x)+c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b*d-a*e+(b^2*d-2*a*c*d-a*b*e)/sqrt(b^2-4*a*c))/(a^2*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b*d-a*e+(-b^2*d+2*a*c*d+a*b*e)/sqrt(b^2-4*a*c))/(a^2*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x^2)/(x^6*(a+b*x^2+c*x^4)),x,15,-1/5*sqrt(d+e*x^2)/(a*x^5)+4/15*e*sqrt(d+e*x^2)/(a*d*x^3)+1/3*(b*d-a*e)*sqrt(d+e*x^2)/(a^2*d*x^3)-8/15*e^2*sqrt(d+e*x^2)/(a*d^2*x)-2/3*e*(b*d-a*e)*sqrt(d+e*x^2)/(a^2*d^2*x)-(b^2*d-a*c*d-a*b*e)*sqrt(d+e*x^2)/(a^3*d*x)-c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^2*d-a*c*d-a*b*e+(b^3*d-3*a*b*c*d-a*b^2*e+2*a^2*c*e)/sqrt(b^2-4*a*c))/(a^3*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^2*d-a*c*d-a*b*e+(-b^3*d+3*a*b*c*d+a*b^2*e-2*a^2*c*e)/sqrt(b^2-4*a*c))/(a^3*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^3*(d+e*x^2)^(3/2)/(a+b*x^2+c*x^4),x,7,1/3*(d+e*x^2)^(3/2)/c+(c*d-b*e)*sqrt(d+e*x^2)/c^2+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b^3*e^2-b^2*e*(2*c*d+e*sqrt(b^2-4*a*c))+c*(a*e^2*sqrt(b^2-4*a*c)-c*d*(-4*a*e+d*sqrt(b^2-4*a*c)))+b*c*(c*d^2+e*(-3*a*e+2*d*sqrt(b^2-4*a*c))))/(c^(5/2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b^3*e^2-b^2*e*(2*c*d-e*sqrt(b^2-4*a*c))-c*(a*e^2*sqrt(b^2-4*a*c)-c*d*(4*a*e+d*sqrt(b^2-4*a*c)))+b*c*(c*d^2-e*(3*a*e+2*d*sqrt(b^2-4*a*c))))/(c^(5/2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x*(d+e*x^2)^(3/2)/(a+b*x^2+c*x^4),x,6,e*sqrt(d+e*x^2)/c-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e-d*sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(2*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e+d*sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)^(3/2)/(x*(a+b*x^2+c*x^4)),x,8,-d^(3/2)*arctanh(sqrt(d+e*x^2)/sqrt(d))/a-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(-b*(c*d^2+a*e^2)+a*e^2*sqrt(b^2-4*a*c)-c*d*(-4*a*e+d*sqrt(b^2-4*a*c)))/(a*sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b*(c*d^2+a*e^2)+a*e^2*sqrt(b^2-4*a*c)-c*d*(4*a*e+d*sqrt(b^2-4*a*c)))/(a*sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)^(3/2)/(x^3*(a+b*x^2+c*x^4)),x,10,1/2*e*arctanh(sqrt(d+e*x^2)/sqrt(d))*sqrt(d)/a+(b*d-2*a*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))*sqrt(d)/a^2-1/2*d*sqrt(d+e*x^2)/(a*x^2)-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(c)*(b^2*d^2+b*d*(-2*a*e+d*sqrt(b^2-4*a*c))-2*a*(c*d^2+e*(-a*e+d*sqrt(b^2-4*a*c))))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x^2)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(c)*(b^2*d^2-b*d*(2*a*e+d*sqrt(b^2-4*a*c))-2*a*(c*d^2-e*(a*e+d*sqrt(b^2-4*a*c))))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^4*(d+e*x^2)^(3/2)/(a+b*x^2+c*x^4),x,17,1/4*x*(d+e*x^2)^(3/2)/c+1/8*d*(3*c*d-4*b*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^2*sqrt(e))-1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(b*c*d-b^2*e+a*c*e+(-b^2*c*d+2*a*c^2*d+b^3*e-3*a*b*c*e)/sqrt(b^2-4*a*c))*sqrt(e)/c^3-1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(b*c*d-b^2*e+a*c*e+(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)/sqrt(b^2-4*a*c))*sqrt(e)/c^3+1/8*(3*c*d-4*b*e)*x*sqrt(d+e*x^2)/c^2-1/2*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(-b^2*c*d+2*a*c^2*d+b^3*e-3*a*b*c*e)/sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c^3*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)/sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^3*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2*(d+e*x^2)^(3/2)/(a+b*x^2+c*x^4),x,16,1/2*d*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*sqrt(e)/c+1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))*sqrt(e)/c^2+1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))*sqrt(e)/c^2+1/2*e*x*sqrt(d+e*x^2)/c+1/2*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c^2*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^2*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)^(3/2)/(a+b*x^2+c*x^4),x,13,1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(3*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(e)/(c*sqrt(b^2-4*a*c))-1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(3*c*d-e*(b+sqrt(b^2-4*a*c)))*sqrt(e)/(c*sqrt(b^2-4*a*c))+arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(2*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e-d*sqrt(b^2-4*a*c)))/(c*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(2*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e+d*sqrt(b^2-4*a*c)))/(c*sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)^(3/2)/(x^2*(a+b*x^2+c*x^4)),x,-16,-arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))^(3/2)/((b-sqrt(b^2-4*a*c))^(3/2)*sqrt(b^2-4*a*c))+arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(2*c*d-e*(b+sqrt(b^2-4*a*c)))^(3/2)/(sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c))^(3/2))-d*sqrt(d+e*x^2)/(a*x)],
[(d+e*x^2)^(3/2)/(x^4*(a+b*x^2+c*x^4)),x,19,-1/3*(d+e*x^2)^(3/2)/(a*x^3)-(b*d-a*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*sqrt(e)/a^2+1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(b*d-a*e+(-b^2*d+2*a*c*d+a*b*e)/sqrt(b^2-4*a*c))*sqrt(e)/a^2+1/2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*(b*d-a*e+(b^2*d-2*a*c*d-a*b*e)/sqrt(b^2-4*a*c))*sqrt(e)/a^2+(b*d-a*e)*sqrt(d+e*x^2)/(a^2*x)+1/2*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b*d-a*e+(b^2*d-2*a*c*d-a*b*e)/sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(a^2*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b*d-a*e+(-b^2*d+2*a*c*d+a*b*e)/sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(a^2*sqrt(b+sqrt(b^2-4*a*c)))],
[x^5*sqrt(1-x^2)/(a+b*x^2+c*x^4),x,7,-1/3*(1-x^2)^(3/2)/c-b*sqrt(1-x^2)/c^2+arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b^2-a*c+b*c+(-b^3+3*a*b*c-b^2*c+2*a*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+2*c-sqrt(b^2-4*a*c)))+arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b^2-a*c+b*c+(b^3-3*a*b*c+b^2*c-2*a*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[x^3*sqrt(1-x^2)/(a+b*x^2+c*x^4),x,6,sqrt(1-x^2)/c-arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b+c+(-b^2+2*a*c-b*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+2*c-sqrt(b^2-4*a*c)))-arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b+c+(b^2-2*a*c+b*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[x*sqrt(1-x^2)/(a+b*x^2+c*x^4),x,5,-arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*sqrt(b+2*c-sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))+arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*sqrt(b+2*c+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[sqrt(1-x^2)/(x*(a+b*x^2+c*x^4)),x,8,-arctanh(sqrt(1-x^2))/a+arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*sqrt(c)*(2*a+b+sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b+2*c-sqrt(b^2-4*a*c)))-arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*sqrt(c)*(2*a+b-sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[sqrt(1-x^2)/(x^3*(a+b*x^2+c*x^4)),x,8,1/2*(a+2*b)*arctanh(sqrt(1-x^2))/a^2+(-1/4)/(a*(1-sqrt(1-x^2)))+1/4/(a*(1+sqrt(1-x^2)))-arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*sqrt(c)*(a+b+(b^2+a*(b-2*c))/sqrt(b^2-4*a*c))/(a^2*sqrt(2)*sqrt(b+2*c-sqrt(b^2-4*a*c)))-arctanh(sqrt(2)*sqrt(c)*sqrt(1-x^2)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*sqrt(c)*(a+b+(-b^2-a*(b-2*c))/sqrt(b^2-4*a*c))/(a^2*sqrt(2)*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[x^4*sqrt(1-x^2)/(a+b*x^2+c*x^4),x,9,1/2*(2*b+c)*arcsin(x)/c^2+1/2*x*sqrt(1-x^2)/c-arctan(x*sqrt(b+2*c-sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^2-a*c+b*c+(-b^3+3*a*b*c-b^2*c+2*a*c^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-arctan(x*sqrt(b+2*c+sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^2-a*c+b*c+(b^3-3*a*b*c+b^2*c-2*a*c^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[x^2*sqrt(1-x^2)/(a+b*x^2+c*x^4),x,8,-arcsin(x)/c+arctan(x*sqrt(b+2*c-sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b+c+(-b^2+2*a*c-b*c)/sqrt(b^2-4*a*c))/(c*sqrt(b-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))+arctan(x*sqrt(b+2*c+sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b+c+(b^2-2*a*c+b*c)/sqrt(b^2-4*a*c))/(c*sqrt(b+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[sqrt(1-x^2)/(a+b*x^2+c*x^4),x,9,arctan(x*sqrt(b+2*c-sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b-sqrt(b^2-4*a*c))))*sqrt(b+2*c-sqrt(b^2-4*a*c))/(sqrt(b^2-4*a*c)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(b+2*c+sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b+sqrt(b^2-4*a*c))))*sqrt(b+2*c+sqrt(b^2-4*a*c))/(sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[sqrt(1-x^2)/(x^2*(a+b*x^2+c*x^4)),x,8,-sqrt(1-x^2)/(a*x)-c*arctan(x*sqrt(b+2*c-sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(1+(2*a+b)/sqrt(b^2-4*a*c))/(a*sqrt(b-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(b+2*c+sqrt(b^2-4*a*c))/(sqrt(1-x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(1+(-2*a-b)/sqrt(b^2-4*a*c))/(a*sqrt(b+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[x^2*sqrt(1-x^2)/(-1+x^2+x^4),x,8,-arcsin(x)-arctanh(x*sqrt(1/2*(-1+sqrt(5)))/sqrt(1-x^2))*sqrt(1/5*(-2+sqrt(5)))+arctan(x*sqrt(1/2*(1+sqrt(5)))/sqrt(1-x^2))*sqrt(1/5*(2+sqrt(5)))],

# q<0
[x^8/((a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,17,3/8*d^2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*e^(5/2))+1/2*b*d*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^2*e^(3/2))+(b^2-a*c)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^3*sqrt(e))-3/8*d*x*sqrt(d+e*x^2)/(c*e^2)-1/2*b*x*sqrt(d+e*x^2)/(c^2*e)+1/4*x^3*sqrt(d+e*x^2)/(c*e)-arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^3-2*a*b*c+(-b^4+4*a*b^2*c-2*a^2*c^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^3-2*a*b*c+(b^4-4*a*b^2*c+2*a^2*c^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^6/((a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,13,-1/2*d*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*e^(3/2))-b*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^2*sqrt(e))+1/2*x*sqrt(d+e*x^2)/(c*e)+arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^2-a*c-b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c^2*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^2-a*c+b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c^2*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^4/((a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,10,arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*sqrt(e))-arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^2/((a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,6,-arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*sqrt(b-sqrt(b^2-4*a*c))/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,5,2*c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-2*c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))/(sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/(x^2*(a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,9,-sqrt(d+e*x^2)/(a*d*x)-c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/(a*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/(a*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/(x^4*(a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,11,-1/3*sqrt(d+e*x^2)/(a*d*x^3)+b*sqrt(d+e*x^2)/(a^2*d*x)+2/3*e*sqrt(d+e*x^2)/(a*d^2*x)+c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/(x^6*(a+b*x^2+c*x^4)*sqrt(d+e*x^2)),x,14,-1/5*sqrt(d+e*x^2)/(a*d*x^5)+1/3*b*sqrt(d+e*x^2)/(a^2*d*x^3)+4/15*e*sqrt(d+e*x^2)/(a*d^2*x^3)-(b^2-a*c)*sqrt(d+e*x^2)/(a^3*d*x)-2/3*b*e*sqrt(d+e*x^2)/(a^2*d^2*x)-8/15*e^2*sqrt(d+e*x^2)/(a*d^3*x)-c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^2-a*c+b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(a^3*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^2-a*c-b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(a^3*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^6/((d+e*x^2)^(3/2)*(a+b*x^2+c*x^4)),x,14,arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*e^(3/2))-d^2*x/(e*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))+2*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^2-a*c-b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c*(2*c*d-e*(b-sqrt(b^2-4*a*c)))^(3/2)*sqrt(b-sqrt(b^2-4*a*c)))+2*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^2-a*c+b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c*(2*c*d-e*(b+sqrt(b^2-4*a*c)))^(3/2)*sqrt(b+sqrt(b^2-4*a*c))),d^2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(e^(3/2)*(c*d^2-b*d*e+a*e^2))-(b*d-a*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*(c*d^2-b*d*e+a*e^2)*sqrt(e))-d^2*x/(e*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))+arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b^2*d-a*c*d-a*b*e+(-b^3*d+3*a*b*c*d+a*b^2*e-2*a^2*c*e)/sqrt(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b^2*d-a*c*d-a*b*e+(b^3*d-3*a*b*c*d-a*b^2*e+2*a^2*c*e)/sqrt(b^2-4*a*c))/(c*(c*d^2-b*d*e+a*e^2)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^4/((d+e*x^2)^(3/2)*(a+b*x^2+c*x^4)),x,8,d*x/((c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))-arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b*d-a*e+(-b^2*d+2*a*c*d+a*b*e)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b*d-a*e+(b^2*d-2*a*c*d-a*b*e)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^2/((d+e*x^2)^(3/2)*(a+b*x^2+c*x^4)),x,8,-e*x/((c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))+c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(d+(-b*d+2*a*e)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(d+(b*d-2*a*e)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x^2)^(3/2)*(a+b*x^2+c*x^4)),x,8,e^2*x/(d*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))-c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(e+(2*c*d-b*e)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/(x^2*(d+e*x^2)^(3/2)*(a+b*x^2+c*x^4)),x,12,e*(c*d-b*e)*x/(a*d*(c*d^2+e*(-b*d+a*e))*sqrt(d+e*x^2))+(-d-2*e*x^2)/(a*d^2*x*sqrt(d+e*x^2))-2*c^2*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/(a*(2*c*d-e*(b-sqrt(b^2-4*a*c)))^(3/2)*sqrt(b-sqrt(b^2-4*a*c)))-2*c^2*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/(a*(2*c*d-e*(b+sqrt(b^2-4*a*c)))^(3/2)*sqrt(b+sqrt(b^2-4*a*c))),-e^2/(d*(c*d^2-b*d*e+a*e^2)*x*sqrt(d+e*x^2))-2*e^3*x/(d^2*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))-(c*d-b*e)*sqrt(d+e*x^2)/(a*d*(c*d^2-b*d*e+a*e^2)*x)-c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(c*d-b*e+(b*c*d-b^2*e+2*a*c*e)/sqrt(b^2-4*a*c))/(a*(c*d^2-b*d*e+a*e^2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))-c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(c*d-b*e+(-b*c*d+b^2*e-2*a*c*e)/sqrt(b^2-4*a*c))/(a*(c*d^2-b*d*e+a*e^2)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/(x^4*(d+e*x^2)^(3/2)*(a+b*x^2+c*x^4)),x,15,(-1/3)/(a*d*x^3*sqrt(d+e*x^2))+1/3*(3*b*d+4*a*e)/(a^2*d^2*x*sqrt(d+e*x^2))+2/3*e*(3*b*d+4*a*e)*x/(a^2*d^3*sqrt(d+e*x^2))-e*(b*c*d-b^2*e+a*c*e)*x/(a^2*d*(c*d^2+e*(-b*d+a*e))*sqrt(d+e*x^2))+2*c^2*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*(2*c*d-e*(b-sqrt(b^2-4*a*c)))^(3/2)*sqrt(b-sqrt(b^2-4*a*c)))+2*c^2*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*(2*c*d-e*(b+sqrt(b^2-4*a*c)))^(3/2)*sqrt(b+sqrt(b^2-4*a*c))),-1/3*e^2/(d*(c*d^2-b*d*e+a*e^2)*x^3*sqrt(d+e*x^2))+4/3*e^3/(d^2*(c*d^2-b*d*e+a*e^2)*x*sqrt(d+e*x^2))+8/3*e^4*x/(d^3*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x^2))-1/3*(c*d-b*e)*sqrt(d+e*x^2)/(a*d*(c*d^2-b*d*e+a*e^2)*x^3)+2/3*e*(c*d-b*e)*sqrt(d+e*x^2)/(a*d^2*(c*d^2-b*d*e+a*e^2)*x)+(b*c*d-b^2*e+a*c*e)*sqrt(d+e*x^2)/(a^2*d*(c*d^2-b*d*e+a*e^2)*x)+c*arctan(x*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b-sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)/sqrt(b^2-4*a*c))/(a^2*(c*d^2-b*d*e+a*e^2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c)))+c*arctan(x*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x^2)*sqrt(b+sqrt(b^2-4*a*c))))*(b*c*d-b^2*e+a*c*e+(-b^2*c*d+2*a*c^2*d+b^3*e-3*a*b*c*e)/sqrt(b^2-4*a*c))/(a^2*(c*d^2-b*d*e+a*e^2)*sqrt(b+sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],

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

# p>0

# p<0
[(f*x)^m*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,6,2*c*(f*x)^(1+m)*(d+e*x^2)^q*AppellF1(1/2*(1+m),1,-q,1/2*(3+m),-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)/(f*(1+m)*(1+e*x^2/d)^q*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c))-2*c*(f*x)^(1+m)*(d+e*x^2)^q*AppellF1(1/2*(1+m),1,-q,1/2*(3+m),-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)/(f*(1+m)*(1+e*x^2/d)^q*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[x^7*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,5,-1/2*(c*d+b*e)*(d+e*x^2)^(1+q)/(c^2*e^2*(1+q))+1/2*(d+e*x^2)^(2+q)/(c*e^2*(2+q))+1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(a-b^2/c+b*(b^2-3*a*c)/(c*sqrt(b^2-4*a*c)))/(c*(1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))+1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(a-b^2/c-b*(b^2-3*a*c)/(c*sqrt(b^2-4*a*c)))/(c*(1+q)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^5*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,5,1/2*(d+e*x^2)^(1+q)/(c*e*(1+q))+1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c*(1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))+1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c*(1+q)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^3*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,5,-1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/((1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))-1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/((1+q)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,5,-c*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))/((1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b^2-4*a*c))+c*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/((1+q)*sqrt(b^2-4*a*c)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)^q/(x*(a+b*x^2+c*x^4)),x,8,-1/2*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],1+e*x^2/d)/(a*d*(1+q))+1/2*c*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/(a*(1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))+1/2*c*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/(a*(1+q)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x^2)^q/(x^3*(a+b*x^2+c*x^4)),x,9,1/2*b*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],1+e*x^2/d)/(a^2*d*(1+q))+1/2*e*(d+e*x^2)^(1+q)*hypergeom([2,1+q],[2+q],1+e*x^2/d)/(a*d^2*(1+q))-1/2*c*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*(1+q)*(2*c*d-e*(b-sqrt(b^2-4*a*c))))-1/2*c*(d+e*x^2)^(1+q)*hypergeom([1,1+q],[2+q],2*c*(d+e*x^2)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*(1+q)*(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^6*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,12,-b*x*(d+e*x^2)^q*hypergeom([1/2,-q],[3/2],-e*x^2/d)/(c^2*(1+e*x^2/d)^q)+1/3*x^3*(d+e*x^2)^q*hypergeom([3/2,-q],[5/2],-e*x^2/d)/(c*(1+e*x^2/d)^q)+x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)*(b^2-a*c-b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c^2*(1+e*x^2/d)^q*(b-sqrt(b^2-4*a*c)))+x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)*(b^2-a*c+b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c^2*(1+e*x^2/d)^q*(b+sqrt(b^2-4*a*c)))],
[x^4*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,10,x*(d+e*x^2)^q*hypergeom([1/2,-q],[3/2],-e*x^2/d)/(c*(1+e*x^2/d)^q)-x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c*(1+e*x^2/d)^q*(b-sqrt(b^2-4*a*c)))-x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c*(1+e*x^2/d)^q*(b+sqrt(b^2-4*a*c)))],
[x^2*(d+e*x^2)^q/(a+b*x^2+c*x^4),x,6,-x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)/((1+e*x^2/d)^q*sqrt(b^2-4*a*c))+x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)/((1+e*x^2/d)^q*sqrt(b^2-4*a*c))],
[(d+e*x^2)^q/(a+b*x^2+c*x^4),x,5,-2*c*x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)/((1+e*x^2/d)^q*(b^2-4*a*c-b*sqrt(b^2-4*a*c)))-2*c*x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)/((1+e*x^2/d)^q*(b^2-4*a*c+b*sqrt(b^2-4*a*c)))],
[(d+e*x^2)^q/(x^2*(a+b*x^2+c*x^4)),x,10,-(d+e*x^2)^q*hypergeom([-1/2,-q],[1/2],-e*x^2/d)/(a*x*(1+e*x^2/d)^q)-c*x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)*(1+b/sqrt(b^2-4*a*c))/(a*(1+e*x^2/d)^q*(b-sqrt(b^2-4*a*c)))-c*x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)*(1-b/sqrt(b^2-4*a*c))/(a*(1+e*x^2/d)^q*(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)^q/(x^4*(a+b*x^2+c*x^4)),x,12,-1/3*(d+e*x^2)^q*hypergeom([-3/2,-q],[-1/2],-e*x^2/d)/(a*x^3*(1+e*x^2/d)^q)+b*(d+e*x^2)^q*hypergeom([-1/2,-q],[1/2],-e*x^2/d)/(a^2*x*(1+e*x^2/d)^q)+c*x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-sqrt(b^2-4*a*c)),-e*x^2/d)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*(1+e*x^2/d)^q*(b-sqrt(b^2-4*a*c)))+c*x*(d+e*x^2)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b+sqrt(b^2-4*a*c)),-e*x^2/d)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*(1+e*x^2/d)^q*(b+sqrt(b^2-4*a*c)))],

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

# Integrands of the form (f x)^m (d+e/x^2)^q (a+b x^2+c x^4)^p with b=0

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

# p<0 & q>0
[sqrt(1+1/(c^2*x^2))/sqrt(1-c^4*x^4),x,5,-arctanh(sqrt(1-c^4*x^4)/(c*x*sqrt(1+1/(c^2*x^2))))/c,-x*arctanh(sqrt(1-c^2*x^2))*sqrt(1+1/(c^2*x^2))/sqrt(1+c^2*x^2)]]:
