# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.2 Quartic\1.2.2.3 (d+e x^2)^m (a+b x^2+c x^4)^p.txt"

lst:=[

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

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

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

# p>0

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

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

# p>0

# p<0
[(1+b*x^2)/sqrt(1-b^2*x^4),x,2,EllipticE(x*sqrt(b),I)/sqrt(b)],
[(1-b*x^2)/sqrt(1-b^2*x^4),x,5,-EllipticE(x*sqrt(b),I)/sqrt(b)+2*EllipticF(x*sqrt(b),I)/sqrt(b)],
[(1+b*x^2)/sqrt(-1+b^2*x^4),x,3,EllipticE(x*sqrt(b),I)*sqrt(1-b^2*x^4)/(sqrt(b)*sqrt(-1+b^2*x^4))],
[(1-b*x^2)/sqrt(-1+b^2*x^4),x,6,-EllipticE(x*sqrt(b),I)*sqrt(1-b^2*x^4)/(sqrt(b)*sqrt(-1+b^2*x^4))+2*EllipticF(x*sqrt(b),I)*sqrt(1-b^2*x^4)/(sqrt(b)*sqrt(-1+b^2*x^4))],
[(1-b*x^2)/sqrt(1+b^2*x^4),x,1,-x*sqrt(1+b^2*x^4)/(1+b*x^2)+(1+b*x^2)*sqrt(cos(2*arctan(x*sqrt(b)))^2)/cos(2*arctan(x*sqrt(b)))*EllipticE(sin(2*arctan(x*sqrt(b))),sqrt(1/2))*sqrt((1+b^2*x^4)/(1+b*x^2)^2)/(sqrt(b)*sqrt(1+b^2*x^4))],
[(1+b*x^2)/sqrt(1+b^2*x^4),x,3,x*sqrt(1+b^2*x^4)/(1+b*x^2)-(1+b*x^2)*sqrt(cos(2*arctan(x*sqrt(b)))^2)/cos(2*arctan(x*sqrt(b)))*EllipticE(sin(2*arctan(x*sqrt(b))),sqrt(1/2))*sqrt((1+b^2*x^4)/(1+b*x^2)^2)/(sqrt(b)*sqrt(1+b^2*x^4))+(1+b*x^2)*sqrt(cos(2*arctan(x*sqrt(b)))^2)/cos(2*arctan(x*sqrt(b)))*EllipticF(sin(2*arctan(x*sqrt(b))),sqrt(1/2))*sqrt((1+b^2*x^4)/(1+b*x^2)^2)/(sqrt(b)*sqrt(1+b^2*x^4))],
[(1-b*x^2)/sqrt(-1-b^2*x^4),x,1,x*sqrt(-1-b^2*x^4)/(1+b*x^2)+(1+b*x^2)*sqrt(cos(2*arctan(x*sqrt(b)))^2)/cos(2*arctan(x*sqrt(b)))*EllipticE(sin(2*arctan(x*sqrt(b))),sqrt(1/2))*sqrt((1+b^2*x^4)/(1+b*x^2)^2)/(sqrt(b)*sqrt(-1-b^2*x^4))],
[(1+b*x^2)/sqrt(-1-b^2*x^4),x,3,-x*sqrt(-1-b^2*x^4)/(1+b*x^2)-(1+b*x^2)*sqrt(cos(2*arctan(x*sqrt(b)))^2)/cos(2*arctan(x*sqrt(b)))*EllipticE(sin(2*arctan(x*sqrt(b))),sqrt(1/2))*sqrt((1+b^2*x^4)/(1+b*x^2)^2)/(sqrt(b)*sqrt(-1-b^2*x^4))+(1+b*x^2)*sqrt(cos(2*arctan(x*sqrt(b)))^2)/cos(2*arctan(x*sqrt(b)))*EllipticF(sin(2*arctan(x*sqrt(b))),sqrt(1/2))*sqrt((1+b^2*x^4)/(1+b*x^2)^2)/(sqrt(b)*sqrt(-1-b^2*x^4))],

#  Following pairs of integrands are equal. 
[sqrt(1+c^2*x^2)/sqrt(1-c^2*x^2),x,1,EllipticE(c*x,I)/c],
[(1+c^2*x^2)/sqrt(1-c^4*x^4),x,2,EllipticE(c*x,I)/c],
[sqrt(1-c^2*x^2)/sqrt(1+c^2*x^2),x,4,-EllipticE(c*x,I)/c+2*EllipticF(c*x,I)/c],
[(1-c^2*x^2)/sqrt(1-c^4*x^4),x,5,-EllipticE(c*x,I)/c+2*EllipticF(c*x,I)/c],

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

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

# p>0

# p<0
[(d+e*x^2)/(d^2+b*x^2+e^2*x^4),x,5,-arctan((-2*e*x+sqrt(-b+2*d*e))/sqrt(b+2*d*e))/sqrt(b+2*d*e)+arctan((2*e*x+sqrt(-b+2*d*e))/sqrt(b+2*d*e))/sqrt(b+2*d*e)],
[(d+e*x^2)/(d^2+f*x^2+e^2*x^4),x,5,-arctan((-2*e*x+sqrt(2*d*e-f))/sqrt(2*d*e+f))/sqrt(2*d*e+f)+arctan((2*e*x+sqrt(2*d*e-f))/sqrt(2*d*e+f))/sqrt(2*d*e+f)],
[(d+e*x^2)/(d^2-b*x^2+e^2*x^4),x,5,arctanh((-2*e*x+sqrt(b+2*d*e))/sqrt(b-2*d*e))/sqrt(b-2*d*e)-arctanh((2*e*x+sqrt(b+2*d*e))/sqrt(b-2*d*e))/sqrt(b-2*d*e)],
[(d+e*x^2)/(d^2-f*x^2+e^2*x^4),x,5,-arctan((-2*e*x+sqrt(2*d*e+f))/sqrt(2*d*e-f))/sqrt(2*d*e-f)+arctan((2*e*x+sqrt(2*d*e+f))/sqrt(2*d*e-f))/sqrt(2*d*e-f)],
[(d-e*x^2)/(d^2+b*x^2+e^2*x^4),x,3,-1/2*log(d+e*x^2-x*sqrt(-b+2*d*e))/sqrt(-b+2*d*e)+1/2*log(d+e*x^2+x*sqrt(-b+2*d*e))/sqrt(-b+2*d*e)],
[(d-e*x^2)/(d^2+f*x^2+e^2*x^4),x,3,-1/2*log(d+e*x^2-x*sqrt(2*d*e-f))/sqrt(2*d*e-f)+1/2*log(d+e*x^2+x*sqrt(2*d*e-f))/sqrt(2*d*e-f)],
[(d-e*x^2)/(d^2-b*x^2+e^2*x^4),x,3,-1/2*log(d+e*x^2-x*sqrt(b+2*d*e))/sqrt(b+2*d*e)+1/2*log(d+e*x^2+x*sqrt(b+2*d*e))/sqrt(b+2*d*e)],
[(d-e*x^2)/(d^2-f*x^2+e^2*x^4),x,3,-1/2*log(d+e*x^2-x*sqrt(2*d*e+f))/sqrt(2*d*e+f)+1/2*log(d+e*x^2+x*sqrt(2*d*e+f))/sqrt(2*d*e+f)],
[(d-e*x^2)/(c*d^2/e^2+b*x^2+c*x^4),x,3,-1/2*e^(3/2)*log(d*sqrt(c)+e*x^2*sqrt(c)-x*sqrt(e)*sqrt(2*c*d-b*e))/(sqrt(c)*sqrt(2*c*d-b*e))+1/2*e^(3/2)*log(d*sqrt(c)+e*x^2*sqrt(c)+x*sqrt(e)*sqrt(2*c*d-b*e))/(sqrt(c)*sqrt(2*c*d-b*e))],
[(d+e*x^2)/(c*d^2/e^2+b*x^2+c*x^4),x,5,-e^(3/2)*arctan((-2*x*sqrt(c)*sqrt(e)+sqrt(2*c*d-b*e))/sqrt(2*c*d+b*e))/(sqrt(c)*sqrt(2*c*d+b*e))+e^(3/2)*arctan((2*x*sqrt(c)*sqrt(e)+sqrt(2*c*d-b*e))/sqrt(2*c*d+b*e))/(sqrt(c)*sqrt(2*c*d+b*e))],
[(d+e*x^2)/(b*x^2+c*(d^2/e^2+x^4)),x,6,-e^(3/2)*arctan((-2*x*sqrt(c)*sqrt(e)+sqrt(2*c*d-b*e))/sqrt(2*c*d+b*e))/(sqrt(c)*sqrt(2*c*d+b*e))+e^(3/2)*arctan((2*x*sqrt(c)*sqrt(e)+sqrt(2*c*d-b*e))/sqrt(2*c*d+b*e))/(sqrt(c)*sqrt(2*c*d+b*e))],
[(a-b*x^2)/(a^2+(-1+2*a*b)*x^2+b^2*x^4),x,3,-1/2*log(a-x+b*x^2)+1/2*log(a+x+b*x^2)],
[(a+b*x^2)/(a^2+(-1+2*a*b)*x^2+b^2*x^4),x,5,arctanh((1-2*b*x)/sqrt(1-4*a*b))/sqrt(1-4*a*b)-arctanh((1+2*b*x)/sqrt(1-4*a*b))/sqrt(1-4*a*b)],
[(1+2*x^2)/(1+b*x^2+4*x^4),x,5,-arctan((-4*x+sqrt(4-b))/sqrt(4+b))/sqrt(4+b)+arctan((4*x+sqrt(4-b))/sqrt(4+b))/sqrt(4+b)],
[(1+2*x^2)/(1-b*x^2+4*x^4),x,5,-arctan((-4*x+sqrt(4+b))/sqrt(4-b))/sqrt(4-b)+arctan((4*x+sqrt(4+b))/sqrt(4-b))/sqrt(4-b)],
[(1+2*x^2)/(1+6*x^2+4*x^4),x,3,arctan(2*x/sqrt(3-sqrt(5)))/sqrt(10)+arctan(2*x/sqrt(3+sqrt(5)))/sqrt(10)],
[(1+2*x^2)/(1+5*x^2+4*x^4),x,3,1/3*arctan(x)+1/3*arctan(2*x)],
[(1+2*x^2)/(1+4*x^2+4*x^4),x,3,arctan(x*sqrt(2))/sqrt(2)],
[(1+2*x^2)/(1+3*x^2+4*x^4),x,5,-arctan((1-4*x)/sqrt(7))/sqrt(7)+arctan((1+4*x)/sqrt(7))/sqrt(7)],
[(1+2*x^2)/(1+2*x^2+4*x^4),x,5,-arctan((1-2*x*sqrt(2))/sqrt(3))/sqrt(6)+arctan((1+2*x*sqrt(2))/sqrt(3))/sqrt(6)],
[(1+2*x^2)/(1+x^2+4*x^4),x,5,-arctan((-4*x+sqrt(3))/sqrt(5))/sqrt(5)+arctan((4*x+sqrt(3))/sqrt(5))/sqrt(5)],
[(1+2*x^2)/(1+4*x^4),x,5,-1/2*arctan(1-2*x)+1/2*arctan(1+2*x)],
[(1+2*x^2)/(1-x^2+4*x^4),x,5,-arctan((-4*x+sqrt(5))/sqrt(3))/sqrt(3)+arctan((4*x+sqrt(5))/sqrt(3))/sqrt(3)],
[(1+2*x^2)/(1-2*x^2+4*x^4),x,5,-arctan(-2*x*sqrt(2)+sqrt(3))/sqrt(2)+arctan(2*x*sqrt(2)+sqrt(3))/sqrt(2)],
[(1+2*x^2)/(1-3*x^2+4*x^4),x,5,-arctan(-4*x+sqrt(7))+arctan(4*x+sqrt(7))],
[(1+2*x^2)/(1-4*x^2+4*x^4),x,2,x/(1-2*x^2)],
[(1+2*x^2)/(1-5*x^2+4*x^4),x,7,-1/2*log(1-2*x)+1/2*log(1-x)-1/2*log(1+x)+1/2*log(1+2*x)],
[(1+2*x^2)/(1-6*x^2+4*x^4),x,5,arctanh(-2*x*sqrt(2)+sqrt(5))/sqrt(2)-arctanh(2*x*sqrt(2)+sqrt(5))/sqrt(2)],
[(1-2*x^2)/(1+b*x^2+4*x^4),x,3,-1/2*log(1+2*x^2-x*sqrt(4-b))/sqrt(4-b)+1/2*log(1+2*x^2+x*sqrt(4-b))/sqrt(4-b)],
[(1-2*x^2)/(1+6*x^2+4*x^4),x,3,arctan(2*x/sqrt(3-sqrt(5)))/sqrt(2)-arctan(2*x/sqrt(3+sqrt(5)))/sqrt(2)],
[(1-2*x^2)/(1+5*x^2+4*x^4),x,3,-arctan(x)+arctan(2*x)],
[(1-2*x^2)/(1+4*x^2+4*x^4),x,2,x/(1+2*x^2)],
[(1-2*x^2)/(1+3*x^2+4*x^4),x,3,-1/2*log(1-x+2*x^2)+1/2*log(1+x+2*x^2)],
[(1-2*x^2)/(1+2*x^2+4*x^4),x,3,-1/2*log(1+2*x^2-x*sqrt(2))/sqrt(2)+1/2*log(1+2*x^2+x*sqrt(2))/sqrt(2)],
[(1-2*x^2)/(1+x^2+4*x^4),x,3,-1/2*log(1+2*x^2-x*sqrt(3))/sqrt(3)+1/2*log(1+2*x^2+x*sqrt(3))/sqrt(3)],
[(1-2*x^2)/(1+4*x^4),x,3,-1/4*log(1-2*x+2*x^2)+1/4*log(1+2*x+2*x^2)],
[(1-2*x^2)/(1-x^2+4*x^4),x,3,-1/2*log(1+2*x^2-x*sqrt(5))/sqrt(5)+1/2*log(1+2*x^2+x*sqrt(5))/sqrt(5)],
[(1-2*x^2)/(1-2*x^2+4*x^4),x,3,-1/2*log(1+2*x^2-x*sqrt(6))/sqrt(6)+1/2*log(1+2*x^2+x*sqrt(6))/sqrt(6)],
[(1-2*x^2)/(1-3*x^2+4*x^4),x,3,-1/2*log(1+2*x^2-x*sqrt(7))/sqrt(7)+1/2*log(1+2*x^2+x*sqrt(7))/sqrt(7)],
[(1-2*x^2)/(1-4*x^2+4*x^4),x,3,arctanh(x*sqrt(2))/sqrt(2)],
[(1-2*x^2)/(1-5*x^2+4*x^4),x,7,-1/6*log(1-2*x)-1/6*log(1-x)+1/6*log(1+x)+1/6*log(1+2*x)],
[(1-2*x^2)/(1-6*x^2+4*x^4),x,5,-arctanh((1-2*x*sqrt(2))/sqrt(5))/sqrt(10)+arctanh((1+2*x*sqrt(2))/sqrt(5))/sqrt(10)],
[(1+x^2)/(1+b*x^2+x^4),x,5,-arctan((-2*x+sqrt(2-b))/sqrt(2+b))/sqrt(2+b)+arctan((2*x+sqrt(2-b))/sqrt(2+b))/sqrt(2+b)],
[(1+x^2)/(1+5*x^2+x^4),x,3,arctan(x*sqrt(2/(5+sqrt(21))))/sqrt(7)+arctan(x*sqrt(1/2*(5+sqrt(21))))/sqrt(7)],
[(1+x^2)/(1+4*x^2+x^4),x,3,arctan(x/sqrt(2-sqrt(3)))/sqrt(6)+arctan(x/sqrt(2+sqrt(3)))/sqrt(6)],
[(1+x^2)/(1+3*x^2+x^4),x,3,arctan(x*sqrt(2/(3+sqrt(5))))/sqrt(5)+arctan(x*sqrt(1/2*(3+sqrt(5))))/sqrt(5)],
[(1+x^2)/(1+2*x^2+x^4),x,2,arctan(x)],
[(1+x^2)/(1+x^2+x^4),x,5,-arctan((1-2*x)/sqrt(3))/sqrt(3)+arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(1+x^2)/(1+x^4),x,5,-arctan(1-x*sqrt(2))/sqrt(2)+arctan(1+x*sqrt(2))/sqrt(2)],
[(1+x^2)/(1-x^2+x^4),x,5,-arctan(-2*x+sqrt(3))+arctan(2*x+sqrt(3))],
[(1+x^2)/(1-2*x^2+x^4),x,2,x/(1-x^2)],
[(1+x^2)/(1-3*x^2+x^4),x,7,1/2*log(1-2*x-sqrt(5))-1/2*log(1+2*x-sqrt(5))+1/2*log(1-2*x+sqrt(5))-1/2*log(1+2*x+sqrt(5))],
[(1+x^2)/(1-4*x^2+x^4),x,5,arctanh(-x*sqrt(2)+sqrt(3))/sqrt(2)-arctanh(x*sqrt(2)+sqrt(3))/sqrt(2)],
[(1+x^2)/(1-5*x^2+x^4),x,5,arctanh((-2*x+sqrt(7))/sqrt(3))/sqrt(3)-arctanh((2*x+sqrt(7))/sqrt(3))/sqrt(3)],
[(1-x^2)/(1+b*x^2+x^4),x,3,-1/2*log(1+x^2-x*sqrt(2-b))/sqrt(2-b)+1/2*log(1+x^2+x*sqrt(2-b))/sqrt(2-b)],
[(1-x^2)/(1+5*x^2+x^4),x,3,-arctan(x*sqrt(2/(5+sqrt(21))))/sqrt(3)+arctan(x*sqrt(1/2*(5+sqrt(21))))/sqrt(3)],
[(1-x^2)/(1+4*x^2+x^4),x,3,arctan(x/sqrt(2-sqrt(3)))/sqrt(2)-arctan(x/sqrt(2+sqrt(3)))/sqrt(2)],
[(1-x^2)/(1+3*x^2+x^4),x,3,-arctan(x*sqrt(2/(3+sqrt(5))))+arctan(x*sqrt(1/2*(3+sqrt(5))))],
[(1-x^2)/(1+2*x^2+x^4),x,2,x/(1+x^2)],
[(1-x^2)/(1+x^2+x^4),x,3,-1/2*log(1-x+x^2)+1/2*log(1+x+x^2)],
[(1-x^2)/(1+x^4),x,3,-1/2*log(1+x^2-x*sqrt(2))/sqrt(2)+1/2*log(1+x^2+x*sqrt(2))/sqrt(2)],
[(1-x^2)/(1-x^2+x^4),x,3,-1/2*log(1+x^2-x*sqrt(3))/sqrt(3)+1/2*log(1+x^2+x*sqrt(3))/sqrt(3)],
[(1-x^2)/(1-2*x^2+x^4),x,3,arctanh(x)],
[(1-x^2)/(1-3*x^2+x^4),x,5,-arctanh((1-2*x)/sqrt(5))/sqrt(5)+arctanh((1+2*x)/sqrt(5))/sqrt(5)],
[(1-x^2)/(1-4*x^2+x^4),x,5,-arctanh((1-x*sqrt(2))/sqrt(3))/sqrt(6)+arctanh((1+x*sqrt(2))/sqrt(3))/sqrt(6)],
[(1-x^2)/(1-5*x^2+x^4),x,5,-arctanh((-2*x+sqrt(3))/sqrt(7))/sqrt(7)+arctanh((2*x+sqrt(3))/sqrt(7))/sqrt(7)],
[(-1-3*x^2)/(1+2*x^2+9*x^4),x,5,1/2*arctan((1-3*x)/sqrt(2))/sqrt(2)-1/2*arctan((1+3*x)/sqrt(2))/sqrt(2)],
[(1+3*x^2)/(-1-2*x^2-9*x^4),x,5,1/2*arctan((1-3*x)/sqrt(2))/sqrt(2)-1/2*arctan((1+3*x)/sqrt(2))/sqrt(2)],

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

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

# p>0

# p<0
[(3+2*x^2)/(1-2*x^2+x^4),x,3,5/2*x/(1-x^2)+1/2*arctanh(x)],
[(2+3*x^2)/(5-8*x^2+3*x^4),x,3,5/2*arctanh(x)-7/2*arctanh(x*sqrt(3/5))*sqrt(3/5)],
[(d+e*x^2)/(5-8*x^2+3*x^4),x,3,1/2*(d+e)*arctanh(x)-1/2*(3*d+5*e)*arctanh(x*sqrt(3/5))/sqrt(15)],
[(3+x^2)/(1+3*x^2+x^4),x,3,1/2*arctan(x*sqrt(1/2*(3+sqrt(5))))*(3+sqrt(5))^(3/2)/sqrt(10)-1/10*arctan(x*sqrt(2/(3+sqrt(5))))*sqrt(180-80*sqrt(5))],
[(a+b*x^2)/(1+x^2+x^4),x,9,-1/4*(a-b)*log(1-x+x^2)+1/4*(a-b)*log(1+x+x^2)-1/2*(a+b)*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/2*(a+b)*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(a+b*x^2)/(1+x^2+x^4)^2,x,10,1/6*x*(a+b-(a-2*b)*x^2)/(1+x^2+x^4)-1/8*(2*a-b)*log(1-x+x^2)+1/8*(2*a-b)*log(1+x+x^2)-1/12*(4*a+b)*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/12*(4*a+b)*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(a+b*x^2)/(2+x^2+x^4),x,9,-1/2*arctan((-2*x+sqrt(-1+2*sqrt(2)))/sqrt(1+2*sqrt(2)))*(a+b*sqrt(2))*sqrt(1/14*(-1+2*sqrt(2)))+1/2*arctan((2*x+sqrt(-1+2*sqrt(2)))/sqrt(1+2*sqrt(2)))*(a+b*sqrt(2))*sqrt(1/14*(-1+2*sqrt(2)))-1/4*log(x^2+sqrt(2)-x*sqrt(-1+2*sqrt(2)))*(a-b*sqrt(2))/sqrt(2*(-1+2*sqrt(2)))+1/4*log(x^2+sqrt(2)+x*sqrt(-1+2*sqrt(2)))*(a-b*sqrt(2))/sqrt(2*(-1+2*sqrt(2)))],
[(a+b*x^2)/(2+x^2+x^4)^2,x,10,1/28*x*(3*a+2*b-(a-4*b)*x^2)/(2+x^2+x^4)-1/56*arctan((-2*x+sqrt(-1+2*sqrt(2)))/sqrt(1+2*sqrt(2)))*(-b*(2-4*sqrt(2))+a*(11-sqrt(2)))*sqrt(1/14*(-1+2*sqrt(2)))+1/56*arctan((2*x+sqrt(-1+2*sqrt(2)))/sqrt(1+2*sqrt(2)))*(-b*(2-4*sqrt(2))+a*(11-sqrt(2)))*sqrt(1/14*(-1+2*sqrt(2)))-1/112*log(x^2+sqrt(2)-x*sqrt(-1+2*sqrt(2)))*(11*a-2*b+(a-4*b)*sqrt(2))/sqrt(2*(-1+2*sqrt(2)))+1/112*log(x^2+sqrt(2)+x*sqrt(-1+2*sqrt(2)))*(a*(11+sqrt(2))-2*(b+2*b*sqrt(2)))/sqrt(2*(-1+2*sqrt(2)))],
[(-x^2+sqrt(2))/(1+x^4-x^2*sqrt(2)),x,9,-1/4*log(1+x^2-x*sqrt(2+sqrt(2)))*sqrt(1+1/sqrt(2))+1/4*log(1+x^2+x*sqrt(2+sqrt(2)))*sqrt(1+1/sqrt(2))-1/2*arctan((-2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2+sqrt(2))+1/2*arctan((2*x+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))/sqrt(2+sqrt(2))],
[(x^2+sqrt(2))/(1+x^4+x^2*sqrt(2)),x,9,-1/4*log(1+x^2-x*sqrt(2-sqrt(2)))*sqrt(1+(-1)/sqrt(2))+1/4*log(1+x^2+x*sqrt(2-sqrt(2)))*sqrt(1+(-1)/sqrt(2))-1/2*arctan((-2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2-sqrt(2))+1/2*arctan((2*x+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))/sqrt(2-sqrt(2))],
[(-x^2+sqrt(2))/(1+b*x^2+x^4),x,9,-1/4*log(1+x^2-x*sqrt(2-b))*(1+sqrt(2))/sqrt(2-b)+1/4*log(1+x^2+x*sqrt(2-b))*(1+sqrt(2))/sqrt(2-b)+1/2*arctan((-2*x+sqrt(2-b))/sqrt(2+b))*(1-sqrt(2))/sqrt(2+b)-1/2*arctan((2*x+sqrt(2-b))/sqrt(2+b))*(1-sqrt(2))/sqrt(2+b)],
[(x^2+sqrt(2))/(1+b*x^2+x^4),x,9,1/4*log(1+x^2-x*sqrt(2-b))*(1-sqrt(2))/sqrt(2-b)-1/4*log(1+x^2+x*sqrt(2-b))*(1-sqrt(2))/sqrt(2-b)-1/2*arctan((-2*x+sqrt(2-b))/sqrt(2+b))*(1+sqrt(2))/sqrt(2+b)+1/2*arctan((2*x+sqrt(2-b))/sqrt(2+b))*(1+sqrt(2))/sqrt(2+b)],
[(2*a-x^2)/(a^2-a*x^2+x^4),x,9,-1/2*arctan(sqrt(3)-2*x/sqrt(a))/sqrt(a)+1/2*arctan(sqrt(3)+2*x/sqrt(a))/sqrt(a)-1/4*log(a+x^2-x*sqrt(3)*sqrt(a))*sqrt(3)/sqrt(a)+1/4*log(a+x^2+x*sqrt(3)*sqrt(a))*sqrt(3)/sqrt(a)],
[(-x^2+2*sqrt(a))/(a+x^4-x^2*sqrt(a)),x,9,-1/2*arctan(-2*x/a^(1/4)+sqrt(3))/a^(1/4)+1/2*arctan(2*x/a^(1/4)+sqrt(3))/a^(1/4)-1/4*log(x^2-a^(1/4)*x*sqrt(3)+sqrt(a))*sqrt(3)/a^(1/4)+1/4*log(x^2+a^(1/4)*x*sqrt(3)+sqrt(a))*sqrt(3)/a^(1/4)],
[(2*b^(2/3)+x^2)/(b^(4/3)+b^(2/3)*x^2+x^4),x,9,-1/4*log(b^(2/3)-b^(1/3)*x+x^2)/b^(1/3)+1/4*log(b^(2/3)+b^(1/3)*x+x^2)/b^(1/3)-1/2*arctan((b^(1/3)-2*x)/(b^(1/3)*sqrt(3)))*sqrt(3)/b^(1/3)+1/2*arctan((b^(1/3)+2*x)/(b^(1/3)*sqrt(3)))*sqrt(3)/b^(1/3)],
[(A+B*x^2)/(a^2-a*x^2+x^4),x,9,-1/2*(A+a*B)*arctan(sqrt(3)-2*x/sqrt(a))/a^(3/2)+1/2*(A+a*B)*arctan(sqrt(3)+2*x/sqrt(a))/a^(3/2)-1/4*(A-a*B)*log(a+x^2-x*sqrt(3)*sqrt(a))/(a^(3/2)*sqrt(3))+1/4*(A-a*B)*log(a+x^2+x*sqrt(3)*sqrt(a))/(a^(3/2)*sqrt(3))],
[(A+B*x^2)/(a+x^4-x^2*sqrt(a)),x,9,-1/4*log(x^2-a^(1/4)*x*sqrt(3)+sqrt(a))*(A-B*sqrt(a))/(a^(3/4)*sqrt(3))+1/4*log(x^2+a^(1/4)*x*sqrt(3)+sqrt(a))*(A-B*sqrt(a))/(a^(3/4)*sqrt(3))-1/2*arctan(-2*x/a^(1/4)+sqrt(3))*(A+B*sqrt(a))/a^(3/4)+1/2*arctan(2*x/a^(1/4)+sqrt(3))*(A+B*sqrt(a))/a^(3/4)],
[(A+B*x^2)/(a+c*x^4-x^2*sqrt(a*c)),x,9,-1/2*arctan((-2*x*sqrt(c)+sqrt(2*sqrt(a)*sqrt(c)+sqrt(a*c)))/sqrt(2*sqrt(a)*sqrt(c)-sqrt(a*c)))*(B*sqrt(a)+A*sqrt(c))/(sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c)-sqrt(a*c)))+1/2*arctan((2*x*sqrt(c)+sqrt(2*sqrt(a)*sqrt(c)+sqrt(a*c)))/sqrt(2*sqrt(a)*sqrt(c)-sqrt(a*c)))*(B*sqrt(a)+A*sqrt(c))/(sqrt(a)*sqrt(c)*sqrt(2*sqrt(a)*sqrt(c)-sqrt(a*c)))-1/4*log(sqrt(a)+x^2*sqrt(c)-x*sqrt(2*sqrt(a)*sqrt(c)+sqrt(a*c)))*(A-B*sqrt(a)/sqrt(c))/(sqrt(a)*sqrt(2*sqrt(a)*sqrt(c)+sqrt(a*c)))+1/4*log(sqrt(a)+x^2*sqrt(c)+x*sqrt(2*sqrt(a)*sqrt(c)+sqrt(a*c)))*(A-B*sqrt(a)/sqrt(c))/(sqrt(a)*sqrt(2*sqrt(a)*sqrt(c)+sqrt(a*c)))],
[(A+B*x^2)/(a+c*x^4-x^2*sqrt(a)*sqrt(c)),x,9,-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(3)+sqrt(a)+x^2*sqrt(c))*(A-B*sqrt(a)/sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(3))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(3)+sqrt(a)+x^2*sqrt(c))*(A-B*sqrt(a)/sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(3))-1/2*arctan(-2*c^(1/4)*x/a^(1/4)+sqrt(3))*(B*sqrt(a)+A*sqrt(c))/(a^(3/4)*c^(3/4))+1/2*arctan(2*c^(1/4)*x/a^(1/4)+sqrt(3))*(B*sqrt(a)+A*sqrt(c))/(a^(3/4)*c^(3/4))],

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

# p>0

# p<0
[(3-x^2)/sqrt(3+x^2-x^4),x,4,-EllipticE(x*sqrt(2/(1+sqrt(13))),sqrt(1/6*(-7-sqrt(13))))*sqrt(1/2*(-1+sqrt(13)))+EllipticF(x*sqrt(2/(1+sqrt(13))),sqrt(1/6*(-7-sqrt(13))))*sqrt(7+2*sqrt(13))],
[(3-x^2)/sqrt(3+2*x^2-x^4),x,5,-EllipticE(x/sqrt(3),sqrt(-3))+4*EllipticF(x/sqrt(3),sqrt(-3))],
[(3-x^2)/sqrt(3+3*x^2-x^4),x,4,-EllipticE(x*sqrt(2/(3+sqrt(21))),sqrt(1/2*(-5-sqrt(21))))*sqrt(1/2*(-3+sqrt(21)))+EllipticF(x*sqrt(2/(3+sqrt(21))),sqrt(1/2*(-5-sqrt(21))))*sqrt(9+2*sqrt(21))],
[(3-x^2)/sqrt(3-x^2-x^4),x,4,-EllipticE(x*sqrt(2/(-1+sqrt(13))),sqrt(1/6*(-7+sqrt(13))))*sqrt(1/2*(1+sqrt(13)))+EllipticF(x*sqrt(2/(-1+sqrt(13))),sqrt(1/6*(-7+sqrt(13))))*sqrt(5+2*sqrt(13))],
[(3-x^2)/sqrt(3-2*x^2-x^4),x,4,-EllipticE(x,sqrt(-1/3))*sqrt(3)+2*EllipticF(x,sqrt(-1/3))*sqrt(3)],
[(3-x^2)/sqrt(3-3*x^2-x^4),x,4,-EllipticE(x*sqrt(2/(-3+sqrt(21))),sqrt(1/2*(-5+sqrt(21))))*sqrt(1/2*(3+sqrt(21)))+EllipticF(x*sqrt(2/(-3+sqrt(21))),sqrt(1/2*(-5+sqrt(21))))*sqrt(3+2*sqrt(21))],
[(b+2*c*x^2-sqrt(b^2-4*a*c))/sqrt(a+b*x^2+c*x^4),x,3,2*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))-2*a^(1/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(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)/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)))))*(sqrt(a)+x^2*sqrt(c))*(b+2*sqrt(a)*sqrt(c)-sqrt(b^2-4*a*c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*sqrt(a+b*x^2+c*x^4))],

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

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

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

# p>0
[(d+e*x^2)^4*(a+c*x^4),x,2,a*d^4*x+4/3*a*d^3*e*x^3+1/5*d^2*(c*d^2+6*a*e^2)*x^5+4/7*d*e*(c*d^2+a*e^2)*x^7+1/9*e^2*(6*c*d^2+a*e^2)*x^9+4/11*c*d*e^3*x^11+1/13*c*e^4*x^13],
[(d+e*x^2)^3*(a+c*x^4),x,2,a*d^3*x+a*d^2*e*x^3+1/5*d*(c*d^2+3*a*e^2)*x^5+1/7*e*(3*c*d^2+a*e^2)*x^7+1/3*c*d*e^2*x^9+1/11*c*e^3*x^11],
[(d+e*x^2)^2*(a+c*x^4),x,2,a*d^2*x+2/3*a*d*e*x^3+1/5*(c*d^2+a*e^2)*x^5+2/7*c*d*e*x^7+1/9*c*e^2*x^9],
[(d+e*x^2)*(a+c*x^4),x,2,a*d*x+1/3*a*e*x^3+1/5*c*d*x^5+1/7*c*e*x^7],
[(a+c*x^4)/(d+e*x^2),x,3,-c*d*x/e^2+1/3*c*x^3/e+(c*d^2+a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(e^(5/2)*sqrt(d))],
[(a+c*x^4)/(d+e*x^2)^2,x,3,c*x/e^2+1/2*(a+c*d^2/e^2)*x/(d*(d+e*x^2))-1/2*(3*c*d^2-a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(5/2))],
[(a+c*x^4)/(d+e*x^2)^3,x,3,1/4*(a+c*d^2/e^2)*x/(d*(d+e*x^2)^2)+1/8*(3*a/d^2-5*c/e^2)*x/(d+e*x^2)+3/8*(c*d^2+a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*e^(5/2))],
[(a+c*x^4)/(d+e*x^2)^4,x,4,1/6*(a+c*d^2/e^2)*x/(d*(d+e*x^2)^3)+1/24*(5*a/d^2-7*c/e^2)*x/(d+e*x^2)^2+1/16*(5*a/d^2+c/e^2)*x/(d*(d+e*x^2))+1/16*(c*d^2+5*a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*e^(5/2))],
[(d+e*x^2)^3*(a+c*x^4)^2,x,2,a^2*d^3*x+a^2*d^2*e*x^3+1/5*a*d*(2*c*d^2+3*a*e^2)*x^5+1/7*a*e*(6*c*d^2+a*e^2)*x^7+1/9*c*d*(c*d^2+6*a*e^2)*x^9+1/11*c*e*(3*c*d^2+2*a*e^2)*x^11+3/13*c^2*d*e^2*x^13+1/15*c^2*e^3*x^15],
[(d+e*x^2)^2*(a+c*x^4)^2,x,2,a^2*d^2*x+2/3*a^2*d*e*x^3+1/5*a*(2*c*d^2+a*e^2)*x^5+4/7*a*c*d*e*x^7+1/9*c*(c*d^2+2*a*e^2)*x^9+2/11*c^2*d*e*x^11+1/13*c^2*e^2*x^13],
[(d+e*x^2)*(a+c*x^4)^2,x,2,a^2*d*x+1/3*a^2*e*x^3+2/5*a*c*d*x^5+2/7*a*c*e*x^7+1/9*c^2*d*x^9+1/11*c^2*e*x^11],
[(a+c*x^4)^2,x,2,a^2*x+2/5*a*c*x^5+1/9*c^2*x^9],
[(a+c*x^4)^2/(d+e*x^2),x,3,-c*d*(c*d^2+2*a*e^2)*x/e^4+1/3*c*(c*d^2+2*a*e^2)*x^3/e^3-1/5*c^2*d*x^5/e^2+1/7*c^2*x^7/e+(c*d^2+a*e^2)^2*arctan(x*sqrt(e)/sqrt(d))/(e^(9/2)*sqrt(d))],
[(a+c*x^4)^2/(d+e*x^2)^2,x,4,c*(3*c*d^2+2*a*e^2)*x/e^4-2/3*c^2*d*x^3/e^3+1/5*c^2*x^5/e^2+1/2*(c*d^2+a*e^2)^2*x/(d*e^4*(d+e*x^2))-1/2*(7*c*d^2-a*e^2)*(c*d^2+a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(9/2))],
[(a+c*x^4)^2/(d+e*x^2)^3,x,5,-3*c^2*d*x/e^4+1/3*c^2*x^3/e^3+1/4*(c*d^2+a*e^2)^2*x/(d*e^4*(d+e*x^2)^2)+1/8*(3*a^2-13*c^2*d^4/e^4-10*a*c*d^2/e^2)*x/(d^2*(d+e*x^2))+1/8*(35*c^2*d^4+6*a*c*d^2*e^2+3*a^2*e^4)*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*e^(9/2))],
[(a+c*x^4)^2/(d+e*x^2)^4,x,5,c^2*x/e^4+1/6*(c*d^2+a*e^2)^2*x/(d*e^4*(d+e*x^2)^3)+1/24*(5*a^2-19*c^2*d^4/e^4-14*a*c*d^2/e^2)*x/(d^2*(d+e*x^2)^2)+1/16*(5*a^2+29*c^2*d^4/e^4+2*a*c*d^2/e^2)*x/(d^3*(d+e*x^2))-1/16*(35*c^2*d^4-2*a*c*d^2*e^2-5*a^2*e^4)*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*e^(9/2))],
[(a+c*x^4)^2/(d+e*x^2)^5,x,5,1/8*(c*d^2+a*e^2)^2*x/(d*e^4*(d+e*x^2)^4)+1/48*(7*a^2-25*c^2*d^4/e^4-18*a*c*d^2/e^2)*x/(d^2*(d+e*x^2)^3)+1/192*(35*a^2+163*c^2*d^4/e^4+6*a*c*d^2/e^2)*x/(d^3*(d+e*x^2)^2)-1/128*(93*c^2*d^4-6*a*c*d^2*e^2-35*a^2*e^4)*x/(d^4*e^4*(d+e*x^2))+1/128*(35*c^2*d^4+6*a*c*d^2*e^2+35*a^2*e^4)*arctan(x*sqrt(e)/sqrt(d))/(d^(9/2)*e^(9/2))],

# p<0
[(d+e*x^2)^4/(a+c*x^4),x,11,e^2*(6*c*d^2-a*e^2)*x/c^2+4/3*d*e^3*x^3/c+1/5*e^4*x^5/c-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4-4*d*e*(c*d^2-a*e^2)*sqrt(a)*sqrt(c))/(a^(3/4)*c^(9/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4-4*d*e*(c*d^2-a*e^2)*sqrt(a)*sqrt(c))/(a^(3/4)*c^(9/4)*sqrt(2))-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4+4*d*e*(c*d^2-a*e^2)*sqrt(a)*sqrt(c))/(a^(3/4)*c^(9/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(c^2*d^4-6*a*c*d^2*e^2+a^2*e^4+4*d*e*(c*d^2-a*e^2)*sqrt(a)*sqrt(c))/(a^(3/4)*c^(9/4)*sqrt(2))],
[(d+e*x^2)^3/(a+c*x^4),x,11,3*d*e^2*x/c+1/3*e^3*x^3/c-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*(3*c*d^2-a*e^2)*sqrt(a)+d*(c*d^2-3*a*e^2)*sqrt(c))/(a^(3/4)*c^(7/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e*(3*c*d^2-a*e^2)*sqrt(a)+d*(c*d^2-3*a*e^2)*sqrt(c))/(a^(3/4)*c^(7/4)*sqrt(2))-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*(3*c*d^2-a*e^2)*sqrt(a)+d*(c*d^2-3*a*e^2)*sqrt(c))/(a^(3/4)*c^(7/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*(3*c*d^2-a*e^2)*sqrt(a)+d*(c*d^2-3*a*e^2)*sqrt(c))/(a^(3/4)*c^(7/4)*sqrt(2))],
[(d+e*x^2)^2/(a+c*x^4),x,11,e^2*x/c-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(c*d^2-a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*c^(5/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(c*d^2-a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*c^(5/4)*sqrt(2))-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(c*d^2-a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*c^(5/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(c*d^2-a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*c^(5/4)*sqrt(2))],
[(d+e*x^2)/(a+c*x^4),x,9,-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^(3/4)*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^(3/4)*c^(3/4)*sqrt(2))-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))],
[1/(a+c*x^4),x,9,-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))],
[1/((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/((d+e*x^2)^2*(a+c*x^4)),x,14,1/2*e^2*x/(d*(c*d^2+a*e^2)*(d+e*x^2))+1/2*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c*d^2+a*e^2))-1/2*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(c*d^2-a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(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))*(c*d^2-a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(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))*(c*d^2-a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(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))*(c*d^2-a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^2*sqrt(2))+2*c*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/(c*d^2+a*e^2)^2],
[(d+e*x^2)^3/(a+c*x^4)^2,x,11,-e^3*x^3/(c*(a+c*x^4))+1/4*x*(d*(c*d^2-3*a*e^2)+3*e*(c*d^2+a*e^2)*x^2)/(a*c*(a+c*x^4))-3/16*(c*d^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^(7/4)*c^(7/4)*sqrt(2))+3/16*(c*d^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^(7/4)*c^(7/4)*sqrt(2))-3/8*(c*d^2+a*e^2)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(7/4)*c^(7/4)*sqrt(2))+3/8*(c*d^2+a*e^2)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+d*sqrt(c))/(a^(7/4)*c^(7/4)*sqrt(2))],
[(d+e*x^2)^2/(a+c*x^4)^2,x,11,-1/3*e^2*x/(c*(a+c*x^4))+1/12*x*(3*c*d^2+a*e^2+6*c*d*e*x^2)/(a*c*(a+c*x^4))-1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*c*d^2+a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*c^(5/4)*sqrt(2))+1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*c*d^2+a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*c^(5/4)*sqrt(2))-1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c*d^2+a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*c^(5/4)*sqrt(2))+1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c*d^2+a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*c^(5/4)*sqrt(2))],
[(d+e*x^2)/(a+c*x^4)^2,x,10,1/4*x*(d+e*x^2)/(a*(a+c*x^4))-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^(7/4)*c^(3/4)*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^(7/4)*c^(3/4)*sqrt(2))-1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+3*d*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))+1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e*sqrt(a)+3*d*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))],
[1/(a+c*x^4)^2,x,10,1/4*x/(a*(a+c*x^4))-3/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))+3/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))-3/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))+3/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))],
[1/((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/((d+e*x^2)^2*(a+c*x^4)^2),x,24,1/2*e^4*x/(d*(c*d^2+a*e^2)^2*(d+e*x^2))+1/4*c*x*(c*d^2-a*e^2-2*c*d*e*x^2)/(a*(c*d^2+a*e^2)^2*(a+c*x^4))+1/2*e^(7/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c*d^2+a*e^2)^2)-1/2*c^(3/4)*e^2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c*d^2-a*e^2-4*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^3*sqrt(2))+1/2*c^(3/4)*e^2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c*d^2-a*e^2-4*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^3*sqrt(2))-1/8*c^(3/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c*d^2-3*a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)^2*sqrt(2))+1/8*c^(3/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c*d^2-3*a*e^2-2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)^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*c*d^2-3*a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)^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*c*d^2-3*a*e^2+2*d*e*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^2+a*e^2)^2*sqrt(2))-1/4*c^(3/4)*e^2*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*c*d^2-a*e^2+4*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^3*sqrt(2))+1/4*c^(3/4)*e^2*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*c*d^2-a*e^2+4*d*e*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^2+a*e^2)^3*sqrt(2))+4*c*e^(7/2)*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/(c*d^2+a*e^2)^3],

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

# p>0

# p<0
[(d+e*x^2)^4/sqrt(a+c*x^4),x,6,1/21*e^2*(42*c*d^2-5*a*e^2)*x*sqrt(a+c*x^4)/c^2+4/5*d*e^3*x^3*sqrt(a+c*x^4)/c+1/7*e^4*x^5*sqrt(a+c*x^4)/c+4/5*d*e*(5*c*d^2-3*a*e^2)*x*sqrt(a+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))-4/5*a^(1/4)*d*e*(5*c*d^2-3*a*e^2)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+c*x^4))+1/210*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(105*c^2*d^4-210*a*c*d^2*e^2+25*a^2*e^4+420*c^(3/2)*d^3*e*sqrt(a)-252*a^(3/2)*d*e^3*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(9/4)*sqrt(a+c*x^4))],
[(d+e*x^2)^3/sqrt(a+c*x^4),x,5,d*e^2*x*sqrt(a+c*x^4)/c+1/5*e^3*x^3*sqrt(a+c*x^4)/c+3/5*e*(5*c*d^2-a*e^2)*x*sqrt(a+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))-3/5*a^(1/4)*e*(5*c*d^2-a*e^2)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+c*x^4))+1/10*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(1/2))*(sqrt(a)+x^2*sqrt(c))*(15*c*d^2*e-3*a*e^3+5*d*(c*d^2-a*e^2)*sqrt(c)/sqrt(a))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+c*x^4))],
[(d+e*x^2)^2/sqrt(a+c*x^4),x,4,1/3*e^2*x*sqrt(a+c*x^4)/c+2*d*e*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-2*a^(1/4)*d*e*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/6*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(3*c*d^2-a*e^2+6*d*e*sqrt(a)*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(5/4)*sqrt(a+c*x^4))],
[(d+e*x^2)/sqrt(a+c*x^4),x,3,e*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*e*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/2*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(1/2))*(sqrt(a)+x^2*sqrt(c))*(e+d*sqrt(c)/sqrt(a))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[1/((d+e*x^2)*sqrt(a+c*x^4)),x,3,1/2*arctan(x*sqrt(c*d^2+a*e^2)/(sqrt(d)*sqrt(e)*sqrt(a+c*x^4)))*sqrt(e)/(sqrt(d)*sqrt(c*d^2+a*e^2))+1/2*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+c*x^4))-1/4*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(e+d*sqrt(c)/sqrt(a))^2*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*d*(c*d^2-a*e^2)*sqrt(a+c*x^4))],
[1/((d+e*x^2)^2*sqrt(a+c*x^4)),x,6,1/4*(3*c*d^2+a*e^2)*arctan(x*sqrt(c*d^2+a*e^2)/(sqrt(d)*sqrt(e)*sqrt(a+c*x^4)))*sqrt(e)/(d^(3/2)*(c*d^2+a*e^2)^(3/2))+1/2*e^2*x*sqrt(a+c*x^4)/(d*(c*d^2+a*e^2)*(d+e*x^2))-1/2*e*x*sqrt(c)*sqrt(a+c*x^4)/(d*(c*d^2+a*e^2)*(sqrt(a)+x^2*sqrt(c)))+1/2*a^(1/4)*c^(1/4)*e*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(d*(c*d^2+a*e^2)*sqrt(a+c*x^4))+1/2*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*d*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+c*x^4))-1/8*(3*c*d^2+a*e^2)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/2))*(e*sqrt(a)+d*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d^2*(c*d^2+a*e^2)*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+c*x^4))],
[1/((d+e*x^2)^3*sqrt(a+c*x^4)),x,7,3/16*(5*c^2*d^4+2*a*c*d^2*e^2+a^2*e^4)*arctan(x*sqrt(c*d^2+a*e^2)/(sqrt(d)*sqrt(e)*sqrt(a+c*x^4)))*sqrt(e)/(d^(5/2)*(c*d^2+a*e^2)^(5/2))+1/4*e^2*x*sqrt(a+c*x^4)/(d*(c*d^2+a*e^2)*(d+e*x^2)^2)+3/8*e^2*(3*c*d^2+a*e^2)*x*sqrt(a+c*x^4)/(d^2*(c*d^2+a*e^2)^2*(d+e*x^2))-3/8*e*(3*c*d^2+a*e^2)*x*sqrt(c)*sqrt(a+c*x^4)/(d^2*(c*d^2+a*e^2)^2*(sqrt(a)+x^2*sqrt(c)))+3/8*a^(1/4)*c^(1/4)*e*(3*c*d^2+a*e^2)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(d^2*(c*d^2+a*e^2)^2*sqrt(a+c*x^4))-3/32*(5*c^2*d^4+2*a*c*d^2*e^2+a^2*e^4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/2))*(e*sqrt(a)+d*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d^3*(c*d^2+a*e^2)^2*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+c*x^4))+1/8*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(4*c*d^2+3*a*e^2-d*e*sqrt(a)*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*d^2*(c*d^2+a*e^2)*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+c*x^4))],
[(d+e*x^2)^3/sqrt(a-c*x^4),x,8,-d*e^2*x*sqrt(a-c*x^4)/c-1/5*e^3*x^3*sqrt(a-c*x^4)/c+3/5*a^(3/4)*e*(5*c*d^2+a*e^2)*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(c^(7/4)*sqrt(a-c*x^4))+1/5*a^(3/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(-3*e*(5*c*d^2+a*e^2)+5*d*(c*d^2+a*e^2)*sqrt(c)/sqrt(a))*sqrt(1-c*x^4/a)/(c^(7/4)*sqrt(a-c*x^4))],
[(d+e*x^2)^2/sqrt(a-c*x^4),x,7,-1/3*e^2*x*sqrt(a-c*x^4)/c+2*a^(3/4)*d*e*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(c^(3/4)*sqrt(a-c*x^4))+1/3*a^(1/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(3*c*d^2+a*e^2-6*d*e*sqrt(a)*sqrt(c))*sqrt(1-c*x^4/a)/(c^(5/4)*sqrt(a-c*x^4))],
[(d+e*x^2)/sqrt(a-c*x^4),x,6,a^(3/4)*e*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(c^(3/4)*sqrt(a-c*x^4))+a^(3/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(-e+d*sqrt(c)/sqrt(a))*sqrt(1-c*x^4/a)/(c^(3/4)*sqrt(a-c*x^4))],
[1/((d+e*x^2)*sqrt(a-c*x^4)),x,2,a^(1/4)*EllipticPi(c^(1/4)*x/a^(1/4),-e*sqrt(a)/(d*sqrt(c)),I)*sqrt(1-c*x^4/a)/(c^(1/4)*d*sqrt(a-c*x^4))],
[1/((d+e*x^2)^2*sqrt(a-c*x^4)),x,10,-1/2*e^2*x*sqrt(a-c*x^4)/(d*(c*d^2-a*e^2)*(d+e*x^2))-1/2*a^(3/4)*c^(1/4)*e*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(d*(c*d^2-a*e^2)*sqrt(a-c*x^4))+1/2*a^(1/4)*(3*c*d^2-a*e^2)*EllipticPi(c^(1/4)*x/a^(1/4),-e*sqrt(a)/(d*sqrt(c)),I)*sqrt(1-c*x^4/a)/(c^(1/4)*d^2*(c*d^2-a*e^2)*sqrt(a-c*x^4))-1/2*a^(1/4)*c^(1/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(d*(e*sqrt(a)+d*sqrt(c))*sqrt(a-c*x^4))],
[1/((d+e*x^2)^3*sqrt(a-c*x^4)),x,11,-1/4*e^2*x*sqrt(a-c*x^4)/(d*(c*d^2-a*e^2)*(d+e*x^2)^2)-3/8*e^2*(3*c*d^2-a*e^2)*x*sqrt(a-c*x^4)/(d^2*(c*d^2-a*e^2)^2*(d+e*x^2))-3/8*a^(3/4)*c^(1/4)*e*(3*c*d^2-a*e^2)*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(d^2*(c*d^2-a*e^2)^2*sqrt(a-c*x^4))+3/8*a^(1/4)*(5*c^2*d^4-2*a*c*d^2*e^2+a^2*e^4)*EllipticPi(c^(1/4)*x/a^(1/4),-e*sqrt(a)/(d*sqrt(c)),I)*sqrt(1-c*x^4/a)/(c^(1/4)*d^3*(c*d^2-a*e^2)^2*sqrt(a-c*x^4))-1/8*a^(1/4)*c^(1/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(7*c*d^2-3*a*e^2-2*d*e*sqrt(a)*sqrt(c))*sqrt(1-c*x^4/a)/(d^2*(c*d^2-a*e^2)*(e*sqrt(a)+d*sqrt(c))*sqrt(a-c*x^4))],
[1/((d+e*x^2)^4*sqrt(a-c*x^4)),x,12,-1/6*e^2*x*sqrt(a-c*x^4)/(d*(c*d^2-a*e^2)*(d+e*x^2)^3)-5/24*e^2*(3*c*d^2-a*e^2)*x*sqrt(a-c*x^4)/(d^2*(c*d^2-a*e^2)^2*(d+e*x^2)^2)-1/16*e^2*(29*c^2*d^4-14*a*c*d^2*e^2+5*a^2*e^4)*x*sqrt(a-c*x^4)/(d^3*(c*d^2-a*e^2)^3*(d+e*x^2))-1/16*a^(3/4)*c^(1/4)*e*(29*c^2*d^4-14*a*c*d^2*e^2+5*a^2*e^4)*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(d^3*(c*d^2-a*e^2)^3*sqrt(a-c*x^4))+1/16*a^(1/4)*(35*c^3*d^6-7*a*c^2*d^4*e^2+17*a^2*c*d^2*e^4-5*a^3*e^6)*EllipticPi(c^(1/4)*x/a^(1/4),-e*sqrt(a)/(d*sqrt(c)),I)*sqrt(1-c*x^4/a)/(c^(1/4)*d^4*(c*d^2-a*e^2)^3*sqrt(a-c*x^4))-1/48*a^(1/4)*c^(1/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(57*c^2*d^4-32*a*c*d^2*e^2+15*a^2*e^4-30*c^(3/2)*d^3*e*sqrt(a)+10*a^(3/2)*d*e^3*sqrt(c))*sqrt(1-c*x^4/a)/(d^3*(-e*sqrt(a)+d*sqrt(c))^2*(e*sqrt(a)+d*sqrt(c))^3*sqrt(a-c*x^4))],
[(d+e*x^2)/sqrt(-a+c*x^4),x,6,a^(3/4)*e*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(c^(3/4)*sqrt(-a+c*x^4))+a^(3/4)*EllipticF(c^(1/4)*x/a^(1/4),I)*(-e+d*sqrt(c)/sqrt(a))*sqrt(1-c*x^4/a)/(c^(3/4)*sqrt(-a+c*x^4))],
[1/((d+e*x^2)*sqrt(-a+c*x^4)),x,2,a^(1/4)*EllipticPi(c^(1/4)*x/a^(1/4),-e*sqrt(a)/(d*sqrt(c)),I)*sqrt(1-c*x^4/a)/(c^(1/4)*d*sqrt(-a+c*x^4))],
[(sqrt(a)+x^2*sqrt(c))/sqrt(-a+c*x^4),x,3,a^(3/4)*EllipticE(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(c^(1/4)*sqrt(-a+c*x^4))],
[(1+x^2*sqrt(c/a))/sqrt(-a+c*x^4),x,3,EllipticE((c/a)^(1/4)*x,I)*sqrt(1-c*x^4/a)/((c/a)^(1/4)*sqrt(-a+c*x^4))],
[(d+e*x^2)/sqrt(-a-c*x^4),x,3,-e*x*sqrt(-a-c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*e*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(-a-c*x^4))+1/2*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(1/2))*(sqrt(a)+x^2*sqrt(c))*(e+d*sqrt(c)/sqrt(a))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(-a-c*x^4))],
[1/((d+e*x^2)*sqrt(-a-c*x^4)),x,3,1/2*arctan(x*sqrt(-c*d^2-a*e^2)/(sqrt(d)*sqrt(e)*sqrt(-a-c*x^4)))*sqrt(e)/(sqrt(d)*sqrt(-c*d^2-a*e^2))+1/2*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(-e*sqrt(a)+d*sqrt(c))*sqrt(-a-c*x^4))-1/4*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(e+d*sqrt(c)/sqrt(a))^2*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*d*(c*d^2-a*e^2)*sqrt(-a-c*x^4))],
[1/((a+b*x^2)*sqrt(4-5*x^4)),x,2,EllipticPi(5^(1/4)*x/sqrt(2),-2*b/(a*sqrt(5)),I)/(5^(1/4)*a*sqrt(2))],
[1/((a+b*x^2)*sqrt(4+5*x^4)),x,3,1/2*arctan(x*sqrt(5*a^2+4*b^2)/(sqrt(a)*sqrt(b)*sqrt(4+5*x^4)))*sqrt(b)/(sqrt(a)*sqrt(5*a^2+4*b^2))+1/2*5^(1/4)*sqrt(cos(2*arctan(5^(1/4)*x/sqrt(2)))^2)/cos(2*arctan(5^(1/4)*x/sqrt(2)))*EllipticF(sin(2*arctan(5^(1/4)*x/sqrt(2))),sqrt(1/2))*(2*b+a*sqrt(5))*(2+x^2*sqrt(5))*sqrt((4+5*x^4)/(2+x^2*sqrt(5))^2)/((5*a^2-4*b^2)*sqrt(2)*sqrt(4+5*x^4))-1/4*sqrt(cos(2*arctan(5^(1/4)*x/sqrt(2)))^2)/cos(2*arctan(5^(1/4)*x/sqrt(2)))*EllipticPi(sin(2*arctan(5^(1/4)*x/sqrt(2))),-1/8*(-2*b+a*sqrt(5))^2/(a*b*sqrt(5)),sqrt(1/2))*(2*b+a*sqrt(5))^2*(2+x^2*sqrt(5))*sqrt((4+5*x^4)/(2+x^2*sqrt(5))^2)/(5^(1/4)*a*(5*a^2-4*b^2)*sqrt(2)*sqrt(4+5*x^4))],
[1/((a+b*x^2)*sqrt(4-d*x^4)),x,1,EllipticPi(d^(1/4)*x/sqrt(2),-2*b/(a*sqrt(d)),I)/(a*d^(1/4)*sqrt(2))],
[1/((a+b*x^2)*sqrt(4+d*x^4)),x,3,1/2*arctan(x*sqrt(4*b^2+a^2*d)/(sqrt(a)*sqrt(b)*sqrt(4+d*x^4)))*sqrt(b)/(sqrt(a)*sqrt(4*b^2+a^2*d))-1/2*d^(1/4)*sqrt(cos(2*arctan(d^(1/4)*x/sqrt(2)))^2)/cos(2*arctan(d^(1/4)*x/sqrt(2)))*EllipticF(sin(2*arctan(d^(1/4)*x/sqrt(2))),sqrt(1/2))*(2+x^2*sqrt(d))*sqrt((4+d*x^4)/(2+x^2*sqrt(d))^2)/(sqrt(2)*(2*b-a*sqrt(d))*sqrt(4+d*x^4))+1/4*sqrt(cos(2*arctan(d^(1/4)*x/sqrt(2)))^2)/cos(2*arctan(d^(1/4)*x/sqrt(2)))*EllipticPi(sin(2*arctan(d^(1/4)*x/sqrt(2))),-1/8*(2*b-a*sqrt(d))^2/(a*b*sqrt(d)),sqrt(1/2))*(2*b+a*sqrt(d))*(2+x^2*sqrt(d))*sqrt((4+d*x^4)/(2+x^2*sqrt(d))^2)/(a*d^(1/4)*sqrt(2)*(2*b-a*sqrt(d))*sqrt(4+d*x^4))],

# Integrands of the form (d+e x^2)^(q/2) (a+c x^4)^(p/2)
[sqrt(a+b*x^2)/sqrt(1-x^4),x,-1,a*EllipticPi(x*sqrt(a+b)/sqrt(a+b*x^2),b/(a+b),sqrt((-a+b)/(a+b)))*sqrt(1-x^2)*sqrt(a*(1+x^2)/(a+b*x^2))/(sqrt(a+b)*sqrt(1+x^2)*sqrt(a*(1-x^2)/(a+b*x^2)))],

# Integrands of the form (d+e x^2)^q (a+c x^4)^p with p symbolic
[(c+e*x^2)^q*(a+b*x^4)^p,x,0,Unintegrable((c+e*x^2)^q*(a+b*x^4)^p,x)],
[(c+e*x^2)^3*(a+b*x^4)^p,x,9,e^3*x^3*(a+b*x^4)^(1+p)/(b*(7+4*p))+c^3*x*(a+b*x^4)^p*hypergeom([1/4,-p],[5/4],-b*x^4/a)/(1+b*x^4/a)^p-e*(a*e^2-b*c^2*(7+4*p))*x^3*(a+b*x^4)^p*hypergeom([3/4,-p],[7/4],-b*x^4/a)/(b*(7+4*p)*(1+b*x^4/a)^p)+3/5*c*e^2*x^5*(a+b*x^4)^p*hypergeom([5/4,-p],[9/4],-b*x^4/a)/(1+b*x^4/a)^p],
[(c+e*x^2)^2*(a+b*x^4)^p,x,7,e^2*x*(a+b*x^4)^(1+p)/(b*(5+4*p))-(a*e^2-b*c^2*(5+4*p))*x*(a+b*x^4)^p*hypergeom([1/4,-p],[5/4],-b*x^4/a)/(b*(5+4*p)*(1+b*x^4/a)^p)+2/3*c*e*x^3*(a+b*x^4)^p*hypergeom([3/4,-p],[7/4],-b*x^4/a)/(1+b*x^4/a)^p],
[(c+e*x^2)*(a+b*x^4)^p,x,6,c*x*(a+b*x^4)^p*hypergeom([1/4,-p],[5/4],-b*x^4/a)/(1+b*x^4/a)^p+1/3*e*x^3*(a+b*x^4)^p*hypergeom([3/4,-p],[7/4],-b*x^4/a)/(1+b*x^4/a)^p],
[(a+b*x^4)^p,x,2,x*(a+b*x^4)^p*hypergeom([1/4,-p],[5/4],-b*x^4/a)/(1+b*x^4/a)^p],
[(a+b*x^4)^p/(c+e*x^2),x,6,x*(a+b*x^4)^p*AppellF1(1/4,-p,1,5/4,-b*x^4/a,e^2*x^4/c^2)/(c*(1+b*x^4/a)^p)-1/3*e*x^3*(a+b*x^4)^p*AppellF1(3/4,-p,1,7/4,-b*x^4/a,e^2*x^4/c^2)/(c^2*(1+b*x^4/a)^p)],
[(a+b*x^4)^p/(c+e*x^2)^2,x,8,x*(a+b*x^4)^p*AppellF1(1/4,-p,2,5/4,-b*x^4/a,e^2*x^4/c^2)/(c^2*(1+b*x^4/a)^p)-2/3*e*x^3*(a+b*x^4)^p*AppellF1(3/4,-p,2,7/4,-b*x^4/a,e^2*x^4/c^2)/(c^3*(1+b*x^4/a)^p)+1/5*e^2*x^5*(a+b*x^4)^p*AppellF1(5/4,-p,2,9/4,-b*x^4/a,e^2*x^4/c^2)/(c^4*(1+b*x^4/a)^p)],
[(1-x^2)^3*(1+b*x^4)^p,x,6,-x^3*(1+b*x^4)^(1+p)/(b*(7+4*p))+x*hypergeom([1/4,-p],[5/4],-b*x^4)+(1-b*(7+4*p))*x^3*hypergeom([3/4,-p],[7/4],-b*x^4)/(b*(7+4*p))+3/5*x^5*hypergeom([5/4,-p],[9/4],-b*x^4)],
[(1-x^2)^2*(1+b*x^4)^p,x,5,x*(1+b*x^4)^(1+p)/(b*(5+4*p))-(1-b*(5+4*p))*x*hypergeom([1/4,-p],[5/4],-b*x^4)/(b*(5+4*p))-2/3*x^3*hypergeom([3/4,-p],[7/4],-b*x^4)],
[(1-x^2)*(1+b*x^4)^p,x,4,x*hypergeom([1/4,-p],[5/4],-b*x^4)-1/3*x^3*hypergeom([3/4,-p],[7/4],-b*x^4)],
[(1+b*x^4)^p,x,1,x*hypergeom([1/4,-p],[5/4],-b*x^4)],
[(1+b*x^4)^p/(1-x^2),x,4,x*AppellF1(1/4,1,-p,5/4,x^4,-b*x^4)+1/3*x^3*AppellF1(3/4,1,-p,7/4,x^4,-b*x^4)],
[(1+b*x^4)^p/(1-x^2)^2,x,5,x*AppellF1(1/4,2,-p,5/4,x^4,-b*x^4)+2/3*x^3*AppellF1(3/4,2,-p,7/4,x^4,-b*x^4)+1/5*x^5*AppellF1(5/4,2,-p,9/4,x^4,-b*x^4)],
[(1+b*x^4)^p/(1-x^2)^3,x,6,x*AppellF1(1/4,3,-p,5/4,x^4,-b*x^4)+x^3*AppellF1(3/4,3,-p,7/4,x^4,-b*x^4)+3/5*x^5*AppellF1(5/4,3,-p,9/4,x^4,-b*x^4)+1/7*x^7*AppellF1(7/4,3,-p,11/4,x^4,-b*x^4)],

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

# Integrands of the form (d+e x^2)^q (d^2-e^2 x^4)^p

# p>0

# p<0
[(d+e*x^2)^4/(d^2-e^2*x^4),x,4,-7*d^2*x-4/3*d*e*x^3-1/5*e^2*x^5+8*d^(5/2)*arctanh(x*sqrt(e)/sqrt(d))/sqrt(e)],
[(d+e*x^2)^3/(d^2-e^2*x^4),x,4,-3*d*x-1/3*e*x^3+4*d^(3/2)*arctanh(x*sqrt(e)/sqrt(d))/sqrt(e)],
[(d+e*x^2)^2/(d^2-e^2*x^4),x,3,-x+2*arctanh(x*sqrt(e)/sqrt(d))*sqrt(d)/sqrt(e)],
[(d+e*x^2)/(d^2-e^2*x^4),x,2,arctanh(x*sqrt(e)/sqrt(d))/(sqrt(d)*sqrt(e))],
[1/((d+e*x^2)*(d^2-e^2*x^4)),x,5,1/4*x/(d^2*(d+e*x^2))+1/2*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(e))+1/4*arctanh(x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(e))],
[1/((d+e*x^2)^2*(d^2-e^2*x^4)),x,6,1/8*x/(d^2*(d+e*x^2)^2)+5/16*x/(d^3*(d+e*x^2))+7/16*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*sqrt(e))+1/8*arctanh(x*sqrt(e)/sqrt(d))/(d^(7/2)*sqrt(e))],

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

# p>0

# p<0
[(d+e*x^2)^(3/2)/(d^2-e^2*x^4),x,6,-arctanh(x*sqrt(e)/sqrt(d+e*x^2))/sqrt(e)+arctanh(x*sqrt(2)*sqrt(e)/sqrt(d+e*x^2))*sqrt(2)/sqrt(e)],
[(d+e*x^2)^(1/2)/(d^2-e^2*x^4),x,3,arctanh(x*sqrt(2)*sqrt(e)/sqrt(d+e*x^2))/(d*sqrt(2)*sqrt(e))],
[1/((d+e*x^2)^(1/2)*(d^2-e^2*x^4)),x,4,1/2*arctanh(x*sqrt(2)*sqrt(e)/sqrt(d+e*x^2))/(d^2*sqrt(2)*sqrt(e))+1/2*x/(d^2*sqrt(d+e*x^2))],
[1/((d+e*x^2)^(3/2)*(d^2-e^2*x^4)),x,6,1/6*x/(d^2*(d+e*x^2)^(3/2))+1/4*arctanh(x*sqrt(2)*sqrt(e)/sqrt(d+e*x^2))/(d^3*sqrt(2)*sqrt(e))+7/12*x/(d^3*sqrt(d+e*x^2))],

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

# p>0

# p<0
[(a+b*x^2)^(5/2)/sqrt(a^2-b^2*x^4),x,5,-1/4*x*(a-b*x^2)*(a+b*x^2)^(3/2)/sqrt(a^2-b^2*x^4)-9/8*a*x*(a-b*x^2)*sqrt(a+b*x^2)/sqrt(a^2-b^2*x^4)+19/8*a^2*arctan(x*sqrt(b)/sqrt(a-b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(a^2-b^2*x^4))],
[(a+b*x^2)^(3/2)/sqrt(a^2-b^2*x^4),x,4,-1/2*x*(a-b*x^2)*sqrt(a+b*x^2)/sqrt(a^2-b^2*x^4)+3/2*a*arctan(x*sqrt(b)/sqrt(a-b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(a^2-b^2*x^4))],
[(a+b*x^2)^(1/2)/sqrt(a^2-b^2*x^4),x,3,arctan(x*sqrt(b)/sqrt(a-b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(a^2-b^2*x^4))],
[1/((a+b*x^2)^(1/2)*sqrt(a^2-b^2*x^4)),x,3,arctan(x*sqrt(2)*sqrt(b)/sqrt(a-b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(a*sqrt(2)*sqrt(b)*sqrt(a^2-b^2*x^4))],
[1/((a+b*x^2)^(3/2)*sqrt(a^2-b^2*x^4)),x,4,1/4*x*(a-b*x^2)/(a^2*sqrt(a+b*x^2)*sqrt(a^2-b^2*x^4))+3/4*arctan(x*sqrt(2)*sqrt(b)/sqrt(a-b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(a^2*sqrt(2)*sqrt(b)*sqrt(a^2-b^2*x^4))],
[1/((a+b*x^2)^(5/2)*sqrt(a^2-b^2*x^4)),x,6,1/8*x*(a-b*x^2)/(a^2*(a+b*x^2)^(3/2)*sqrt(a^2-b^2*x^4))+9/32*x*(a-b*x^2)/(a^3*sqrt(a+b*x^2)*sqrt(a^2-b^2*x^4))+19/32*arctan(x*sqrt(2)*sqrt(b)/sqrt(a-b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(a^3*sqrt(2)*sqrt(b)*sqrt(a^2-b^2*x^4))],
[(a-b*x^2)^(5/2)/sqrt(a^2-b^2*x^4),x,5,-1/4*x*(a-b*x^2)^(3/2)*(a+b*x^2)/sqrt(a^2-b^2*x^4)-9/8*a*x*(a+b*x^2)*sqrt(a-b*x^2)/sqrt(a^2-b^2*x^4)+19/8*a^2*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(a^2-b^2*x^4))],
[(a-b*x^2)^(3/2)/sqrt(a^2-b^2*x^4),x,4,-1/2*x*(a+b*x^2)*sqrt(a-b*x^2)/sqrt(a^2-b^2*x^4)+3/2*a*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(a^2-b^2*x^4))],
[(a-b*x^2)^(1/2)/sqrt(a^2-b^2*x^4),x,3,arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(sqrt(b)*sqrt(a^2-b^2*x^4))],
[1/((a-b*x^2)^(1/2)*sqrt(a^2-b^2*x^4)),x,3,arctanh(x*sqrt(2)*sqrt(b)/sqrt(a+b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(a*sqrt(2)*sqrt(b)*sqrt(a^2-b^2*x^4))],
[1/((a-b*x^2)^(3/2)*sqrt(a^2-b^2*x^4)),x,4,1/4*x*(a+b*x^2)/(a^2*sqrt(a-b*x^2)*sqrt(a^2-b^2*x^4))+3/4*arctanh(x*sqrt(2)*sqrt(b)/sqrt(a+b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(a^2*sqrt(2)*sqrt(b)*sqrt(a^2-b^2*x^4))],
[1/((a-b*x^2)^(5/2)*sqrt(a^2-b^2*x^4)),x,6,1/8*x*(a+b*x^2)/(a^2*(a-b*x^2)^(3/2)*sqrt(a^2-b^2*x^4))+9/32*x*(a+b*x^2)/(a^3*sqrt(a-b*x^2)*sqrt(a^2-b^2*x^4))+19/32*arctanh(x*sqrt(2)*sqrt(b)/sqrt(a+b*x^2))*sqrt(a-b*x^2)*sqrt(a+b*x^2)/(a^3*sqrt(2)*sqrt(b)*sqrt(a^2-b^2*x^4))],
[sqrt(-1+x^2)/sqrt(-1+x^4),x,2,arcsinh(x)*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)],
[sqrt(1+x^2)/sqrt(-1+x^4),x,3,-arcsin(x)*sqrt(-1+x^4)/sqrt(1-x^4),arctanh(x/sqrt(-1+x^2))*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)],
[(-sqrt(-1+x^2)+sqrt(1+x^2))/sqrt(-1+x^4),x,7,-arcsin(x)*sqrt(-1+x^4)/(sqrt(1-x^2)*sqrt(1+x^2))+arcsinh(x)*sqrt(-1+x^2)*sqrt(-1+x^4)/((1-x^2)*sqrt(1+x^2)),-arcsinh(x)*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)+arctanh(x/sqrt(-1+x^2))*sqrt(-1+x^2)*sqrt(1+x^2)/sqrt(-1+x^4)],

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

# Integrands of the form (d+e x^2)^q (-c d^2+b d e+b e^2 x^2+c e^2 x^4)^p
[(d+e*x^2)^4/(-c*d^2+b*d*e+b*e^2*x^2+c*e^2*x^4),x,4,(7*c^2*d^2-5*b*c*d*e+b^2*e^2)*x/c^3+1/3*e*(4*c*d-b*e)*x^3/c^2+1/5*e^2*x^5/c-(2*c*d-b*e)^3*arctanh(x*sqrt(c)*sqrt(e)/sqrt(c*d-b*e))/(c^(7/2)*sqrt(e)*sqrt(c*d-b*e))],
[(d+e*x^2)^3/(-c*d^2+b*d*e+b*e^2*x^2+c*e^2*x^4),x,4,(3*c*d-b*e)*x/c^2+1/3*e*x^3/c-(2*c*d-b*e)^2*arctanh(x*sqrt(c)*sqrt(e)/sqrt(c*d-b*e))/(c^(5/2)*sqrt(e)*sqrt(c*d-b*e))],
[(d+e*x^2)^2/(-c*d^2+b*d*e+b*e^2*x^2+c*e^2*x^4),x,3,x/c-(2*c*d-b*e)*arctanh(x*sqrt(c)*sqrt(e)/sqrt(c*d-b*e))/(c^(3/2)*sqrt(e)*sqrt(c*d-b*e))],
[(d+e*x^2)/(-c*d^2+b*d*e+b*e^2*x^2+c*e^2*x^4),x,2,-arctanh(x*sqrt(c)*sqrt(e)/sqrt(c*d-b*e))/(sqrt(c)*sqrt(e)*sqrt(c*d-b*e))],
[1/((d+e*x^2)*(-c*d^2+b*d*e+b*e^2*x^2+c*e^2*x^4)),x,5,-1/2*x/(d*(2*c*d-b*e)*(d+e*x^2))-1/2*(4*c*d-b*e)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(2*c*d-b*e)^2*sqrt(e))-c^(3/2)*arctanh(x*sqrt(c)*sqrt(e)/sqrt(c*d-b*e))/((2*c*d-b*e)^2*sqrt(e)*sqrt(c*d-b*e))],
[1/((d+e*x^2)^2*(-c*d^2+b*d*e+b*e^2*x^2+c*e^2*x^4)),x,6,-1/4*x/(d*(2*c*d-b*e)*(d+e*x^2)^2)-1/8*(10*c*d-3*b*e)*x/(d^2*(2*c*d-b*e)^2*(d+e*x^2))-1/8*(28*c^2*d^2-16*b*c*d*e+3*b^2*e^2)*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*(2*c*d-b*e)^3*sqrt(e))-c^(5/2)*arctanh(x*sqrt(c)*sqrt(e)/sqrt(c*d-b*e))/((2*c*d-b*e)^3*sqrt(e)*sqrt(c*d-b*e))],

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

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

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

# p>0
[(1+x^2)^3*sqrt(1+x^2+x^4),x,6,1/3*x*(1+x^2+x^4)^(3/2)+1/9*x^3*(1+x^2+x^4)^(3/2)+26/45*x*sqrt(1+x^2+x^4)/(1+x^2)+2/45*x*(7+6*x^2)*sqrt(1+x^2+x^4)-26/45*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+7/15*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)^2*sqrt(1+x^2+x^4),x,5,1/7*x*(1+x^2+x^4)^(3/2)+2/3*x*sqrt(1+x^2+x^4)/(1+x^2)+2/21*x*(4+3*x^2)*sqrt(1+x^2+x^4)-2/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+4/7*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)*sqrt(1+x^2+x^4),x,4,3/5*x*sqrt(1+x^2+x^4)/(1+x^2)+1/5*x*(2+x^2)*sqrt(1+x^2+x^4)-3/5*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+3/5*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[sqrt(1+x^2+x^4)/(1+x^2),x,8,1/2*arctan(x/sqrt(1+x^2+x^4))+x*sqrt(1+x^2+x^4)/(1+x^2)-(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+3/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[sqrt(1+x^2+x^4)/(1+x^2)^2,x,1,1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[sqrt(1+x^2+x^4)/(1+x^2)^3,x,23,1/4*arctan(x/sqrt(1+x^2+x^4))+1/4*x*sqrt(1+x^2+x^4)/(1+x^2)^2+1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[sqrt(1+x^2+x^4)/(1+x^2)^4,x,26,1/4*arctan(x/sqrt(1+x^2+x^4))+1/6*x*sqrt(1+x^2+x^4)/(1+x^2)^3+1/6*x*sqrt(1+x^2+x^4)/(1+x^2)^2+1/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)-1/8*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],

# p<0
[(1+x^2)^3/sqrt(1+x^2+x^4),x,5,11/15*x*sqrt(1+x^2+x^4)+1/5*x^3*sqrt(1+x^2+x^4)+14/15*x*sqrt(1+x^2+x^4)/(1+x^2)-14/15*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+3/5*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)^2/sqrt(1+x^2+x^4),x,4,1/3*x*sqrt(1+x^2+x^4)+4/3*x*sqrt(1+x^2+x^4)/(1+x^2)-4/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)/sqrt(1+x^2+x^4),x,3,x*sqrt(1+x^2+x^4)/(1+x^2)-(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[1/((1+x^2)*sqrt(1+x^2+x^4)),x,4,1/2*arctan(x/sqrt(1+x^2+x^4))+1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[1/((1+x^2)^2*sqrt(1+x^2+x^4)),x,8,1/2*arctan(x/sqrt(1+x^2+x^4))+1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)-1/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[1/((1+x^2)^3*sqrt(1+x^2+x^4)),x,9,1/4*arctan(x/sqrt(1+x^2+x^4))+1/4*x*sqrt(1+x^2+x^4)/(1+x^2)^2+3/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)-1/2*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)^3/(1+x^2+x^4)^(3/2),x,4,-1/3*x*(1-x^2)/sqrt(1+x^2+x^4)+2/3*x*sqrt(1+x^2+x^4)/(1+x^2)-2/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)^2/(1+x^2+x^4)^(3/2),x,2,1/3*x*(1+2*x^2)/sqrt(1+x^2+x^4)-2/3*x*sqrt(1+x^2+x^4)/(1+x^2)+2/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[(1+x^2)/(1+x^2+x^4)^(3/2),x,2,1/3*x*(2+x^2)/sqrt(1+x^2+x^4)-1/3*x*sqrt(1+x^2+x^4)/(1+x^2)+1/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[1/((1+x^2)*(1+x^2+x^4)^(3/2)),x,9,1/2*arctan(x/sqrt(1+x^2+x^4))-1/3*x*(1+2*x^2)/sqrt(1+x^2+x^4)+2/3*x*sqrt(1+x^2+x^4)/(1+x^2)-2/3*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)+3/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[1/((1+x^2)^2*(1+x^2+x^4)^(3/2)),x,16,arctan(x/sqrt(1+x^2+x^4))-1/3*x*(2+x^2)/sqrt(1+x^2+x^4)+1/3*x*sqrt(1+x^2+x^4)/(1+x^2)+1/6*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],
[1/((1+x^2)^3*(1+x^2+x^4)^(3/2)),x,23,3/4*arctan(x/sqrt(1+x^2+x^4))-1/3*x*(1-x^2)/sqrt(1+x^2+x^4)+1/4*x*sqrt(1+x^2+x^4)/(1+x^2)^2-1/3*x*sqrt(1+x^2+x^4)/(1+x^2)+19/12*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)-5/4*(1+x^2)*sqrt(cos(2*arctan(x))^2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),sqrt(1/4))*sqrt((1+x^2+x^4)/(1+x^2)^2)/sqrt(1+x^2+x^4)],

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

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

# p>0
[(d+e*x^2)^4*(a+b*x^2+c*x^4),x,2,a*d^4*x+1/3*d^3*(b*d+4*a*e)*x^3+1/5*d^2*(c*d^2+4*b*d*e+6*a*e^2)*x^5+2/7*d*e*(2*c*d^2+e*(3*b*d+2*a*e))*x^7+1/9*e^2*(6*c*d^2+e*(4*b*d+a*e))*x^9+1/11*e^3*(4*c*d+b*e)*x^11+1/13*c*e^4*x^13],
[(d+e*x^2)^3*(a+b*x^2+c*x^4),x,2,a*d^3*x+1/3*d^2*(b*d+3*a*e)*x^3+1/5*d*(c*d^2+3*e*(b*d+a*e))*x^5+1/7*e*(3*c*d^2+e*(3*b*d+a*e))*x^7+1/9*e^2*(3*c*d+b*e)*x^9+1/11*c*e^3*x^11],
[(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)*(a+b*x^2+c*x^4),x,2,a*d*x+1/3*(b*d+a*e)*x^3+1/5*(c*d+b*e)*x^5+1/7*c*e*x^7],
[(a+b*x^2+c*x^4)/(d+e*x^2),x,3,-(c*d-b*e)*x/e^2+1/3*c*x^3/e+(c*d^2-b*d*e+a*e^2)*arctan(x*sqrt(e)/sqrt(d))/(e^(5/2)*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)/(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)/(d+e*x^2)^4,x,4,1/6*(a+d*(c*d-b*e)/e^2)*x/(d*(d+e*x^2)^3)-1/24*(7*c*d^2-e*(b*d+5*a*e))*x/(d^2*e^2*(d+e*x^2)^2)+1/16*(c*d^2+e*(b*d+5*a*e))*x/(d^3*e^2*(d+e*x^2))+1/16*(c*d^2+e*(b*d+5*a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*e^(5/2))],
[(d+e*x^2)^3*(a+b*x^2+c*x^4)^2,x,2,a^2*d^3*x+1/3*a*d^2*(2*b*d+3*a*e)*x^3+1/5*d*(b^2*d^2+6*a*b*d*e+a*(2*c*d^2+3*a*e^2))*x^5+1/7*(2*b*c*d^3+3*b^2*d^2*e+6*a*c*d^2*e+6*a*b*d*e^2+a^2*e^3)*x^7+1/9*(c^2*d^3+6*c*d*e*(b*d+a*e)+b*e^2*(3*b*d+2*a*e))*x^9+1/11*e*(3*c^2*d^2+b^2*e^2+2*c*e*(3*b*d+a*e))*x^11+1/13*c*e^2*(3*c*d+2*b*e)*x^13+1/15*c^2*e^3*x^15],
[(d+e*x^2)^2*(a+b*x^2+c*x^4)^2,x,2,a^2*d^2*x+2/3*a*d*(b*d+a*e)*x^3+1/5*(b^2*d^2+4*a*b*d*e+a*(2*c*d^2+a*e^2))*x^5+2/7*(b*c*d^2+b^2*d*e+2*a*c*d*e+a*b*e^2)*x^7+1/9*(c^2*d^2+b^2*e^2+2*c*e*(2*b*d+a*e))*x^9+2/11*c*e*(c*d+b*e)*x^11+1/13*c^2*e^2*x^13],
[(d+e*x^2)*(a+b*x^2+c*x^4)^2,x,2,a^2*d*x+1/3*a*(2*b*d+a*e)*x^3+1/5*(b^2*d+2*a*c*d+2*a*b*e)*x^5+1/7*(2*b*c*d+b^2*e+2*a*c*e)*x^7+1/9*c*(c*d+2*b*e)*x^9+1/11*c^2*e*x^11],
[(a+b*x^2+c*x^4)^2,x,2,a^2*x+2/3*a*b*x^3+1/5*(b^2+2*a*c)*x^5+2/7*b*c*x^7+1/9*c^2*x^9],
[(a+b*x^2+c*x^4)^2/(d+e*x^2),x,3,-(c*d-b*e)*(c*d^2-e*(b*d-2*a*e))*x/e^4+1/3*(c^2*d^2+b^2*e^2-2*c*e*(b*d-a*e))*x^3/e^3-1/5*c*(c*d-2*b*e)*x^5/e^2+1/7*c^2*x^7/e+(c*d^2-b*d*e+a*e^2)^2*arctan(x*sqrt(e)/sqrt(d))/(e^(9/2)*sqrt(d))],
[(a+b*x^2+c*x^4)^2/(d+e*x^2)^2,x,4,(3*c^2*d^2+b^2*e^2-2*c*e*(2*b*d-a*e))*x/e^4-2/3*c*(c*d-b*e)*x^3/e^3+1/5*c^2*x^5/e^2+1/2*(c*d^2-b*d*e+a*e^2)^2*x/(d*e^4*(d+e*x^2))-1/2*(c*d^2-b*d*e+a*e^2)*(7*c*d^2-e*(3*b*d+a*e))*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(9/2))],
[(a+b*x^2+c*x^4)^2/(d+e*x^2)^3,x,5,-c*(3*c*d-2*b*e)*x/e^4+1/3*c^2*x^3/e^3+1/4*(c*d^2-b*d*e+a*e^2)^2*x/(d*e^4*(d+e*x^2)^2)-1/8*(13*c*d^2-5*b*d*e-3*a*e^2)*(c*d^2-b*d*e+a*e^2)*x/(d^2*e^4*(d+e*x^2))+1/8*(35*c^2*d^4-6*c*d^2*e*(5*b*d-a*e)+e^2*(3*b^2*d^2+2*a*b*d*e+3*a^2*e^2))*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*e^(9/2))],
[(a+b*x^2+c*x^4)^2/(d+e*x^2)^4,x,5,c^2*x/e^4+1/6*(c*d^2-b*d*e+a*e^2)^2*x/(d*e^4*(d+e*x^2)^3)-1/24*(19*c*d^2-7*b*d*e-5*a*e^2)*(c*d^2-b*d*e+a*e^2)*x/(d^2*e^4*(d+e*x^2)^2)+1/16*(29*c^2*d^4-2*c*d^2*e*(11*b*d-a*e)+e^2*(b^2*d^2+2*a*b*d*e+5*a^2*e^2))*x/(d^3*e^4*(d+e*x^2))-1/16*(35*c^2*d^4-2*c*d^2*e*(5*b*d+a*e)-e^2*(b^2*d^2+2*a*b*d*e+5*a^2*e^2))*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*e^(9/2))],
[(a+b*x^2+c*x^4)^2/(d+e*x^2)^5,x,5,1/8*(c*d^2-b*d*e+a*e^2)^2*x/(d*e^4*(d+e*x^2)^4)-1/48*(25*c*d^2-9*b*d*e-7*a*e^2)*(c*d^2-b*d*e+a*e^2)*x/(d^2*e^4*(d+e*x^2)^3)+1/192*(163*c^2*d^4-2*c*d^2*e*(59*b*d-3*a*e)+e^2*(3*b^2*d^2+10*a*b*d*e+35*a^2*e^2))*x/(d^3*e^4*(d+e*x^2)^2)-1/128*(93*c^2*d^4-2*c*d^2*e*(5*b*d+3*a*e)-e^2*(3*b^2*d^2+10*a*b*d*e+35*a^2*e^2))*x/(d^4*e^4*(d+e*x^2))+1/128*(35*c^2*d^4+2*c*d^2*e*(5*b*d+3*a*e)+e^2*(3*b^2*d^2+10*a*b*d*e+35*a^2*e^2))*arctan(x*sqrt(e)/sqrt(d))/(d^(9/2)*e^(9/2))],

#  Following integrands are equal: 
[(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+x^2*(b+c*x^2))/(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))],

# p<0
[(d+e*x^2)^4/(a+b*x^2+c*x^4),x,5,e^2*(6*c^2*d^2+b^2*e^2-c*e*(4*b*d+a*e))*x/c^3+1/3*e^3*(4*c*d-b*e)*x^3/c^2+1/5*e^4*x^5/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(e*(2*c*d-b*e)*(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))+(2*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(b*d+a*e)-4*c^3*d^2*e*(b*d+3*a*e)+2*c^2*e^2*(3*b^2*d^2+6*a*b*d*e+a^2*e^2))/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(e*(2*c*d-b*e)*(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))+(-2*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(b*d+a*e)+4*c^3*d^2*e*(b*d+3*a*e)-2*c^2*e^2*(3*b^2*d^2+6*a*b*d*e+a^2*e^2))/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)^3/(a+b*x^2+c*x^4),x,5,e^2*(3*c*d-b*e)*x/c^2+1/3*e^3*x^3/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(e*(3*c^2*d^2+b^2*e^2-c*e*(3*b*d+a*e))+(2*c*d-b*e)*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))/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)))*(e*(3*c^2*d^2+b^2*e^2-c*e*(3*b*d+a*e))-(2*c*d-b*e)*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)^2/(a+b*x^2+c*x^4),x,5,e^2*x/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(e*(2*c*d-b*e)+(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))/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)))*(e*(2*c*d-b*e)+(-2*c^2*d^2-b^2*e^2+2*c*e*(b*d+a*e))/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*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)))*(e+(2*c*d-b*e)/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)))*(e+(-2*c*d+b*e)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(a+b*x^2+c*x^4),x,3,arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(2)*sqrt(c)/(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(2)*sqrt(c)/(sqrt(b^2-4*a*c)*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/((d+e*x^2)^2*(a+b*x^2+c*x^4)),x,8,1/2*e^2*x/(d*(c*d^2-b*d*e+a*e^2)*(d+e*x^2))+1/2*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c*d^2-b*d*e+a*e^2))+e^(3/2)*(2*c*d-b*e)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2-b*d*e+a*e^2)^2*sqrt(d))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(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*d^2-b*d*e+a*e^2)^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)*(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*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)^3/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(c*(b^2*d^3-2*a*d*(c*d^2-3*a*e^2)-a*b*e*(3*c*d^2+a*e^2)/c)-(a*b^2*e^3+2*a*c*e*(3*c*d^2-a*e^2)-b*c*d*(c*d^2+3*a*e^2))*x^2)/(a*c*(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)))*(a*b^3*e^3+6*a*c*(c*d^2+a*e^2)*(2*c*d+e*sqrt(b^2-4*a*c))-b^2*(c^2*d^3-3*a*c*d*e^2+a*e^3*sqrt(b^2-4*a*c))-b*c*(c*d^2*(12*a*e+d*sqrt(b^2-4*a*c))+a*e^2*(8*a*e+3*d*sqrt(b^2-4*a*c))))/(a*c^(3/2)*(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)))*(a*b^3*e^3+6*a*c*(c*d^2+a*e^2)*(2*c*d-e*sqrt(b^2-4*a*c))-b^2*(c^2*d^3-3*a*c*d*e^2-a*e^3*sqrt(b^2-4*a*c))+b*c*(c*d^2*(-12*a*e+d*sqrt(b^2-4*a*c))+a*e^2*(-8*a*e+3*d*sqrt(b^2-4*a*c))))/(a*c^(3/2)*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)^2/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(b^2*d^2-2*a*b*d*e-2*a*(c*d^2-a*e^2)+(b*c*d^2-4*a*c*d*e+a*b*e^2)*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)))*(b*c*d^2-4*a*c*d*e+a*b*e^2+(8*a*b*c*d*e+b^2*(c*d^2-a*e^2)-4*a*c*(3*c*d^2+a*e^2))/sqrt(b^2-4*a*c))/(a*(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*c*d^2-4*a*c*d*e+a*b*e^2+(-8*a*b*c*d*e-b^2*(c*d^2-a*e^2)+4*a*c*(3*c*d^2+a*e^2))/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2)/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(b^2*d-2*a*c*d-a*b*e+c*(b*d-2*a*e)*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)*(b*d-2*a*e+(b^2*d-12*a*c*d+4*a*b*e)/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)*(b*d-2*a*e+(-b^2*d+12*a*c*d-4*a*b*e)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(b^2-2*a*c+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)*(b^2-12*a*c+b*sqrt(b^2-4*a*c))/(a*(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)*(b^2-12*a*c-b*sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/((d+e*x^2)*(a+b*x^2+c*x^4)^2),x,10,1/2*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e+c*(b*c*d-b^2*e+2*a*c*e)*x^2)/(a*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x^2+c*x^4))+e^(7/2)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2-b*d*e+a*e^2)^2*sqrt(d))-e^2*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)^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)*(b*c*d-b^2*e+2*a*c*e+(b^2*c*d-12*a*c^2*d-b^3*e+8*a*b*c*e)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-e^2*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)^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)*(b*c*d-b^2*e+2*a*c*e+(-b^2*c*d+12*a*c^2*d+b^3*e-8*a*b*c*e)/sqrt(b^2-4*a*c))/(a*(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)^2*(a+b*x^2+c*x^4)^2),x,12,1/2*e^4*x/(d*(c*d^2-b*d*e+a*e^2)^2*(d+e*x^2))+1/2*x*(a*b*c*e*(2*c*d-b*e)+(b^2-2*a*c)*(c^2*d^2+b^2*e^2-c*e*(2*b*d+a*e))-c*(2*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-b*c*(c*d^2-3*a*e^2))*x^2)/(a*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*(a+b*x^2+c*x^4))+1/2*e^(7/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c*d^2-b*d*e+a*e^2)^2)+2*e^(7/2)*(2*c*d-b*e)*arctan(x*sqrt(e)/sqrt(d))/((c*d^2-b*d*e+a*e^2)^3*sqrt(d))+e^2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(2)*sqrt(c)*(3*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-c*e*(3*b*d+a*e+2*d*sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)^3*sqrt(b^2-4*a*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)))*sqrt(c)*(b^4*e^2-b^3*e*(2*c*d-e*sqrt(b^2-4*a*c))-4*a*c^2*(3*c*d^2-e*(3*a*e+d*sqrt(b^2-4*a*c)))-b*c*(3*a*e^2*sqrt(b^2-4*a*c)-c*d*(16*a*e+d*sqrt(b^2-4*a*c)))+b^2*c*(c*d^2-e*(9*a*e+2*d*sqrt(b^2-4*a*c))))/(a*(b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-e^2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(2)*sqrt(c)*(3*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-c*e*(3*b*d+a*e-2*d*sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)^3*sqrt(b^2-4*a*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)))*sqrt(c)*(b^4*e^2-b^3*e*(2*c*d+e*sqrt(b^2-4*a*c))+b*c*(3*a*e^2*sqrt(b^2-4*a*c)-c*d*(-16*a*e+d*sqrt(b^2-4*a*c)))-4*a*c^2*(3*c*d^2+e*(-3*a*e+d*sqrt(b^2-4*a*c)))+b^2*c*(c*d^2+e*(-9*a*e+2*d*sqrt(b^2-4*a*c))))/(a*(b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],

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

# p>0
[(d+e*x^2)^(5/2)*(a+b*x^2+c*x^4),x,7,1/384*d*(3*c*d^2-10*b*d*e+80*a*e^2)*x*(d+e*x^2)^(3/2)/e^2+1/480*(3*c*d^2-10*b*d*e+80*a*e^2)*x*(d+e*x^2)^(5/2)/e^2-1/80*(3*c*d-10*b*e)*x*(d+e*x^2)^(7/2)/e^2+1/10*c*x^3*(d+e*x^2)^(7/2)/e+1/256*d^3*(3*c*d^2-10*b*d*e+80*a*e^2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)+1/256*d^2*(3*c*d^2-10*b*d*e+80*a*e^2)*x*sqrt(d+e*x^2)/e^2],
[(d+e*x^2)^(3/2)*(a+b*x^2+c*x^4),x,6,1/192*(3*c*d^2-8*b*d*e+48*a*e^2)*x*(d+e*x^2)^(3/2)/e^2-1/48*(3*c*d-8*b*e)*x*(d+e*x^2)^(5/2)/e^2+1/8*c*x^3*(d+e*x^2)^(5/2)/e+1/128*d^2*(3*c*d^2-8*b*d*e+48*a*e^2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)+1/128*d*(3*c*d^2-8*b*d*e+48*a*e^2)*x*sqrt(d+e*x^2)/e^2],
[(d+e*x^2)^(1/2)*(a+b*x^2+c*x^4),x,5,-1/8*(c*d-2*b*e)*x*(d+e*x^2)^(3/2)/e^2+1/6*c*x^3*(d+e*x^2)^(3/2)/e+1/16*d*(c*d^2-2*b*d*e+8*a*e^2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)+1/16*(c*d^2-2*b*d*e+8*a*e^2)*x*sqrt(d+e*x^2)/e^2],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(1/2),x,4,1/8*(3*c*d^2-4*b*d*e+8*a*e^2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)-1/8*(3*c*d-4*b*e)*x*sqrt(d+e*x^2)/e^2+1/4*c*x^3*sqrt(d+e*x^2)/e],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(3/2),x,4,-1/2*(3*c*d-2*b*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)+(a+d*(c*d-b*e)/e^2)*x/(d*sqrt(d+e*x^2))+1/2*c*x*sqrt(d+e*x^2)/e^2],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(5/2),x,4,1/3*(a+d*(c*d-b*e)/e^2)*x/(d*(d+e*x^2)^(3/2))+c*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)-1/3*(4*c*d^2-e*(b*d+2*a*e))*x/(d^2*e^2*sqrt(d+e*x^2))],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(7/2),x,4,a*x/(d*(d+e*x^2)^(5/2))+1/3*(b*d+4*a*e)*x^3/(d^2*(d+e*x^2)^(5/2))+1/15*(3*c*d^2+2*e*(b*d+4*a*e))*x^5/(d^3*(d+e*x^2)^(5/2))],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(9/2),x,5,a*x/(d*(d+e*x^2)^(7/2))+1/3*(b*d+6*a*e)*x^3/(d^2*(d+e*x^2)^(7/2))+1/15*(3*c*d^2+4*e*(b*d+6*a*e))*x^5/(d^3*(d+e*x^2)^(7/2))+2/105*e*(3*c*d^2+4*e*(b*d+6*a*e))*x^7/(d^4*(d+e*x^2)^(7/2))],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(11/2),x,6,a*x/(d*(d+e*x^2)^(9/2))+1/3*(b*d+8*a*e)*x^3/(d^2*(d+e*x^2)^(9/2))+1/5*(c*d^2+2*e*(b*d+8*a*e))*x^5/(d^3*(d+e*x^2)^(9/2))+4/35*e*(c*d^2+2*e*(b*d+8*a*e))*x^7/(d^4*(d+e*x^2)^(9/2))+8/315*e^2*(c*d^2+2*e*(b*d+8*a*e))*x^9/(d^5*(d+e*x^2)^(9/2))],
[(a+b*x^2+c*x^4)/(d+e*x^2)^(13/2),x,7,a*x/(d*(d+e*x^2)^(11/2))+1/3*(b*d+10*a*e)*x^3/(d^2*(d+e*x^2)^(11/2))+1/15*(3*c*d^2+8*e*(b*d+10*a*e))*x^5/(d^3*(d+e*x^2)^(11/2))+2/35*e*(3*c*d^2+8*e*(b*d+10*a*e))*x^7/(d^4*(d+e*x^2)^(11/2))+8/315*e^2*(3*c*d^2+8*e*(b*d+10*a*e))*x^9/(d^5*(d+e*x^2)^(11/2))+16/3465*e^3*(3*c*d^2+8*e*(b*d+10*a*e))*x^11/(d^6*(d+e*x^2)^(11/2))],

# p<0

# Integrands of the form (7+5 x^2)^q (2+3 x^2+x^4)^(p/2)

# p>0
[(7+5*x^2)^3*sqrt(2+3*x^2+x^4),x,6,275/7*x*(2+3*x^2+x^4)^(3/2)+125/9*x^3*(2+3*x^2+x^4)^(3/2)+577/3*x*(2+x^2)/sqrt(2+3*x^2+x^4)-577/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+2945/21*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/21*x*(2608+757*x^2)*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)^2*sqrt(2+3*x^2+x^4),x,5,25/7*x*(2+3*x^2+x^4)^(3/2)+31*x*(2+x^2)/sqrt(2+3*x^2+x^4)-31*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+472/21*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/21*x*(407+114*x^2)*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)*sqrt(2+3*x^2+x^4),x,4,5*x*(2+x^2)/sqrt(2+3*x^2+x^4)-5*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+11/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/3*x*(10+3*x^2)*sqrt(2+3*x^2+x^4)],
[sqrt(2+3*x^2+x^4),x,4,x*(2+x^2)/sqrt(2+3*x^2+x^4)-(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+2/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/3*x*sqrt(2+3*x^2+x^4)],
[sqrt(2+3*x^2+x^4)/(7+5*x^2),x,8,1/5*x*(2+x^2)/sqrt(2+3*x^2+x^4)+3/35*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-1/5*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/5*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/sqrt(2+3*x^2+x^4),1/5*x*(2+x^2)/sqrt(2+3*x^2+x^4)+3/35*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-3/25*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-1/5*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+4/25*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)],
[sqrt(2+3*x^2+x^4)/(7+5*x^2)^2,x,8,-1/70*x*(2+x^2)/sqrt(2+3*x^2+x^4)-1/980*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+1/35*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+3/140*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+1/14*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],
[sqrt(2+3*x^2+x^4)/(7+5*x^2)^3,x,25,-11/11760*x*(2+x^2)/sqrt(2+3*x^2+x^4)-1201/164640*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+11/5880*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+81/7840*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+1/28*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)^2+11/2352*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],
[(7+5*x^2)^3*(2+3*x^2+x^4)^(3/2),x,7,1/3003*x*(208212+65345*x^2)*(2+3*x^2+x^4)^(3/2)+3825/143*x*(2+3*x^2+x^4)^(5/2)+125/13*x^3*(2+3*x^2+x^4)^(5/2)+20884/65*x*(2+x^2)/sqrt(2+3*x^2+x^4)-20884/65*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1171349/5005*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/5005*x*(1032541+297911*x^2)*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)^2*(2+3*x^2+x^4)^(3/2),x,6,1/693*x*(7281+2240*x^2)*(2+3*x^2+x^4)^(3/2)+25/11*x*(2+3*x^2+x^4)^(5/2)+742/15*x*(2+x^2)/sqrt(2+3*x^2+x^4)-742/15*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+13879/385*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/1155*x*(36783+10643*x^2)*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)*(2+3*x^2+x^4)^(3/2),x,5,1/63*x*(108+35*x^2)*(2+3*x^2+x^4)^(3/2)+116/15*x*(2+x^2)/sqrt(2+3*x^2+x^4)-116/15*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+197/35*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/105*x*(519+149*x^2)*sqrt(2+3*x^2+x^4)],
[(2+3*x^2+x^4)^(3/2),x,5,1/7*x*(2+3*x^2+x^4)^(3/2)+6/5*x*(2+x^2)/sqrt(2+3*x^2+x^4)-6/5*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+31/35*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/35*x*(29+9*x^2)*sqrt(2+3*x^2+x^4)],
[(2+3*x^2+x^4)^(3/2)/(7+5*x^2),x,13,24/125*x*(2+x^2)/sqrt(2+3*x^2+x^4)-9/875*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))*sqrt(2)/(sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-24/125*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+56/375*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/75*x*(11+3*x^2)*sqrt(2+3*x^2+x^4)],
[(2+3*x^2+x^4)^(3/2)/(7+5*x^2)^2,x,21,9/175*x*(2+x^2)/sqrt(2+3*x^2+x^4)-9/175*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+59/1050*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/sqrt(2+3*x^2+x^4)+9/2450*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/sqrt(2+3*x^2+x^4)+1/75*x*sqrt(2+3*x^2+x^4)-3/175*x*sqrt(2+3*x^2+x^4)/(7+5*x^2),9/175*x*(2+x^2)/sqrt(2+3*x^2+x^4)-39/12250*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+3/875*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))*sqrt(2)/(sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+81/8750*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-9/175*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+44/1875*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/75*x*sqrt(2+3*x^2+x^4)-3/175*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],
[(2+3*x^2+x^4)^(3/2)/(7+5*x^2)^3,x,27,3/392*x*(2+x^2)/sqrt(2+3*x^2+x^4)+141/27440*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-3/196*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/sqrt(2+3*x^2+x^4)+5/784*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/sqrt(2+3*x^2+x^4)-3/350*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)^2+17/9800*x*sqrt(2+3*x^2+x^4)/(7+5*x^2),3/392*x*(2+x^2)/sqrt(2+3*x^2+x^4)+141/27440*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-39/24500*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+5/784*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-6/875*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)-3/350*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)^2+17/9800*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],

# p<0
[(7+5*x^2)^3/sqrt(2+3*x^2+x^4),x,5,135*x*(2+x^2)/sqrt(2+3*x^2+x^4)+193*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-135*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+75*x*sqrt(2+3*x^2+x^4)+25*x^3*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)^2/sqrt(2+3*x^2+x^4),x,4,20*x*(2+x^2)/sqrt(2+3*x^2+x^4)+97/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-20*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+25/3*x*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)/sqrt(2+3*x^2+x^4),x,3,5*x*(2+x^2)/sqrt(2+3*x^2+x^4)+7*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-5*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)],
[1/sqrt(2+3*x^2+x^4),x,1,(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))],
[1/((7+5*x^2)*sqrt(2+3*x^2+x^4)),x,4,-5/14*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+1/2*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))],
[1/((7+5*x^2)^2*sqrt(2+3*x^2+x^4)),x,9,5/84*x*(2+x^2)/sqrt(2+3*x^2+x^4)-65/1176*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-5/42*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+9/56*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-25/84*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],
[1/((7+5*x^2)^3*sqrt(2+3*x^2+x^4)),x,10,65/4704*x*(2+x^2)/sqrt(2+3*x^2+x^4)-2525/65856*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-65/2352*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+631/9408*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-25/168*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)^2-325/4704*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],
[(7+5*x^2)^5/(2+3*x^2+x^4)^(3/2),x,6,7679/2*x*(2+x^2)/sqrt(2+3*x^2+x^4)-1/2*x*(115+179*x^2)/sqrt(2+3*x^2+x^4)-7679*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+15383/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+5000/3*x*sqrt(2+3*x^2+x^4)+625*x^3*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)^4/(2+3*x^2+x^4)^(3/2),x,5,637/2*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/2*x*(145+113*x^2)/sqrt(2+3*x^2+x^4)-637*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+1067/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+625/3*x*sqrt(2+3*x^2+x^4)],
[(7+5*x^2)^3/(2+3*x^2+x^4)^(3/2),x,4,1/2*x*(5-11*x^2)/sqrt(2+3*x^2+x^4)+261/2*x*(2+x^2)/sqrt(2+3*x^2+x^4)-261*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+169*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))],
[(7+5*x^2)^2/(2+3*x^2+x^4)^(3/2),x,4,-17/2*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/2*x*(25+17*x^2)/sqrt(2+3*x^2+x^4)+17*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+6*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)],
[(7+5*x^2)/(2+3*x^2+x^4)^(3/2),x,4,-1/2*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/2*x*(5+x^2)/sqrt(2+3*x^2+x^4)+(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))],
[1/(2+3*x^2+x^4)^(3/2),x,4,-3/2*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/2*x*(5+3*x^2)/sqrt(2+3*x^2+x^4)+3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)],
[1/((7+5*x^2)*(2+3*x^2+x^4)^(3/2)),x,9,1/6*x/sqrt(2+3*x^2+x^4)+125/84*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+1/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)-9/4*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/sqrt(2+3*x^2+x^4),-1/3*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/6*x*(5+2*x^2)/sqrt(2+3*x^2+x^4)+125/84*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))-9/4*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+1/3*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)],
[1/((7+5*x^2)^2*(2+3*x^2+x^4)^(3/2)),x,19,-31/56*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/36*x*(20+11*x^2)/sqrt(2+3*x^2+x^4)+375/784*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+31/28*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-463/336*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+625/504*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],
[1/((7+5*x^2)^3*(2+3*x^2+x^4)^(3/2)),x,29,-5797/28224*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/216*x*(50+23*x^2)/sqrt(2+3*x^2+x^4)+192625/395136*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),2/7,sqrt(1/2))/(sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+5797/14112*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-49907/56448*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))+625/1008*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)^2+41875/84672*x*sqrt(2+3*x^2+x^4)/(7+5*x^2)],

# Integrands of the form (7+5 x^2)^q (2+x^2-x^4)^(p/2)

# p>0
[(7+5*x^2)^4*sqrt(2+x^2-x^4),x,8,-116100/77*x*(2+x^2-x^4)^(3/2)-14500/33*x^3*(2+x^2-x^4)^(3/2)-625/11*x^5*(2+x^2-x^4)^(3/2)+3764813/231*EllipticE(x/sqrt(2),sqrt(-2))-539419/77*EllipticF(x/sqrt(2),sqrt(-2))+1/231*x*(177953+717372*x^2)*sqrt(2+x^2-x^4)],
[(7+5*x^2)^3*sqrt(2+x^2-x^4),x,7,-1825/21*x*(2+x^2-x^4)^(3/2)-125/9*x^3*(2+x^2-x^4)^(3/2)+79411/63*EllipticE(x/sqrt(2),sqrt(-2))-8735/21*EllipticF(x/sqrt(2),sqrt(-2))+1/63*x*(5956+14691*x^2)*sqrt(2+x^2-x^4)],
[(7+5*x^2)^2*sqrt(2+x^2-x^4),x,6,-25/7*x*(2+x^2-x^4)^(3/2)+2045/21*EllipticE(x/sqrt(2),sqrt(-2))-79/7*EllipticF(x/sqrt(2),sqrt(-2))+1/21*x*(275+354*x^2)*sqrt(2+x^2-x^4)],
[(7+5*x^2)*sqrt(2+x^2-x^4),x,5,7*EllipticE(x/sqrt(2),sqrt(-2))+3*EllipticF(x/sqrt(2),sqrt(-2))+x*(2+x^2)*sqrt(2+x^2-x^4)],
[sqrt(2+x^2-x^4),x,5,1/3*EllipticE(x/sqrt(2),sqrt(-2))+EllipticF(x/sqrt(2),sqrt(-2))+1/3*x*sqrt(2+x^2-x^4)],
[sqrt(2+x^2-x^4)/(7+5*x^2),x,7,-1/5*EllipticE(x/sqrt(2),sqrt(-2))+17/25*EllipticF(x/sqrt(2),sqrt(-2))-34/175*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))],
[sqrt(2+x^2-x^4)/(7+5*x^2)^2,x,7,1/70*EllipticE(x/sqrt(2),sqrt(-2))-6/175*EllipticF(x/sqrt(2),sqrt(-2))+99/2450*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))+1/14*x*sqrt(2+x^2-x^4)/(7+5*x^2)],
[sqrt(2+x^2-x^4)/(7+5*x^2)^3,x,21,-31/66640*EllipticE(x/sqrt(2),sqrt(-2))-269/166600*EllipticF(x/sqrt(2),sqrt(-2))+16601/2332400*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))+1/28*x*sqrt(2+x^2-x^4)/(7+5*x^2)^2-31/13328*x*sqrt(2+x^2-x^4)/(7+5*x^2)],
[(7+5*x^2)^4*(2+x^2-x^4)^(3/2),x,9,-1/1001*x*(69817-1581440*x^2)*(2+x^2-x^4)^(3/2)-132300/143*x*(2+x^2-x^4)^(5/2)-11750/39*x^3*(2+x^2-x^4)^(5/2)-125/3*x^5*(2+x^2-x^4)^(5/2)+124141422/5005*EllipticE(x/sqrt(2),sqrt(-2))-50794416/5005*EllipticF(x/sqrt(2),sqrt(-2))+3/5005*x*(2193559+7837383*x^2)*sqrt(2+x^2-x^4)],
[(7+5*x^2)^3*(2+x^2-x^4)^(3/2),x,8,1/3003*x*(33792+374045*x^2)*(2+x^2-x^4)^(3/2)-7825/143*x*(2+x^2-x^4)^(5/2)-125/13*x^3*(2+x^2-x^4)^(5/2)+31072528/15015*EllipticE(x/sqrt(2),sqrt(-2))-3199778/5005*EllipticF(x/sqrt(2),sqrt(-2))+1/15015*x*(2512273+5712051*x^2)*sqrt(2+x^2-x^4)],
[(7+5*x^2)^2*(2+x^2-x^4)^(3/2),x,7,1/99*x*(363+920*x^2)*(2+x^2-x^4)^(3/2)-25/11*x*(2+x^2-x^4)^(5/2)+85942/495*EllipticE(x/sqrt(2),sqrt(-2))-3392/165*EllipticF(x/sqrt(2),sqrt(-2))+1/495*x*(11497+14889*x^2)*sqrt(2+x^2-x^4)],
[(7+5*x^2)*(2+x^2-x^4)^(3/2),x,6,1/63*x*(48+35*x^2)*(2+x^2-x^4)^(3/2)+4432/315*EllipticE(x/sqrt(2),sqrt(-2))+418/105*EllipticF(x/sqrt(2),sqrt(-2))+1/315*x*(1087+669*x^2)*sqrt(2+x^2-x^4)],
[(2+x^2-x^4)^(3/2),x,6,1/7*x*(2+x^2-x^4)^(3/2)+34/35*EllipticE(x/sqrt(2),sqrt(-2))+48/35*EllipticF(x/sqrt(2),sqrt(-2))+1/35*x*(19+3*x^2)*sqrt(2+x^2-x^4)],
[(2+x^2-x^4)^(3/2)/(7+5*x^2),x,13,92/375*EllipticE(x/sqrt(2),sqrt(-2))-178/625*EllipticF(x/sqrt(2),sqrt(-2))+1156/4375*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))+1/75*x*(13-3*x^2)*sqrt(2+x^2-x^4)],
[(2+x^2-x^4)^(3/2)/(7+5*x^2)^2,x,21,-97/525*EllipticE(x/sqrt(2),sqrt(-2))+458/875*EllipticF(x/sqrt(2),sqrt(-2))-1241/6125*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))-1/75*x*sqrt(2+x^2-x^4)-17/175*x*sqrt(2+x^2-x^4)/(7+5*x^2)],
[(2+x^2-x^4)^(3/2)/(7+5*x^2)^3,x,27,191/9800*EllipticE(x/sqrt(2),sqrt(-2))-1251/24500*EllipticF(x/sqrt(2),sqrt(-2))+9879/343000*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))-17/350*x*sqrt(2+x^2-x^4)/(7+5*x^2)^2+563/9800*x*sqrt(2+x^2-x^4)/(7+5*x^2)],

# p<0
[(7+5*x^2)^3/sqrt(2+x^2-x^4),x,6,3905/3*EllipticE(x/sqrt(2),sqrt(-2))-542*EllipticF(x/sqrt(2),sqrt(-2))-625/3*x*sqrt(2+x^2-x^4)-25*x^3*sqrt(2+x^2-x^4)],
[(7+5*x^2)^2/sqrt(2+x^2-x^4),x,5,260/3*EllipticE(x/sqrt(2),sqrt(-2))-21*EllipticF(x/sqrt(2),sqrt(-2))-25/3*x*sqrt(2+x^2-x^4)],
[(7+5*x^2)/sqrt(2+x^2-x^4),x,4,5*EllipticE(x/sqrt(2),sqrt(-2))+2*EllipticF(x/sqrt(2),sqrt(-2))],
[1/sqrt(2+x^2-x^4),x,2,EllipticF(x/sqrt(2),sqrt(-2))],
[1/((7+5*x^2)*sqrt(2+x^2-x^4)),x,2,1/7*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))],
[1/((7+5*x^2)^2*sqrt(2+x^2-x^4)),x,8,-5/476*EllipticE(x/sqrt(2),sqrt(-2))-1/238*EllipticF(x/sqrt(2),sqrt(-2))+167/3332*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))-25/476*x*sqrt(2+x^2-x^4)/(7+5*x^2)],
[1/((7+5*x^2)^3*sqrt(2+x^2-x^4)),x,9,-2505/453152*EllipticE(x/sqrt(2),sqrt(-2))-263/226576*EllipticF(x/sqrt(2),sqrt(-2))+58915/3172064*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))-25/952*x*sqrt(2+x^2-x^4)/(7+5*x^2)^2-12525/453152*x*sqrt(2+x^2-x^4)/(7+5*x^2)],
[(7+5*x^2)^5/(2+x^2-x^4)^(3/2),x,7,-3482293/18*EllipticE(x/sqrt(2),sqrt(-2))+627857/6*EllipticF(x/sqrt(2),sqrt(-2))+1/18*x*(1419985+1419793*x^2)/sqrt(2+x^2-x^4)+27500/3*x*sqrt(2+x^2-x^4)+625*x^3*sqrt(2+x^2-x^4)],
[(7+5*x^2)^4/(2+x^2-x^4)^(3/2),x,6,-165239/18*EllipticE(x/sqrt(2),sqrt(-2))+31921/6*EllipticF(x/sqrt(2),sqrt(-2))+1/18*x*(83585+83489*x^2)/sqrt(2+x^2-x^4)+625/3*x*sqrt(2+x^2-x^4)],
[(7+5*x^2)^3/(2+x^2-x^4)^(3/2),x,5,-7147/18*EllipticE(x/sqrt(2),sqrt(-2))+1763/6*EllipticF(x/sqrt(2),sqrt(-2))+1/18*x*(4945+4897*x^2)/sqrt(2+x^2-x^4)],
[(7+5*x^2)^2/(2+x^2-x^4)^(3/2),x,5,-281/18*EllipticE(x/sqrt(2),sqrt(-2))+139/6*EllipticF(x/sqrt(2),sqrt(-2))+1/18*x*(305+281*x^2)/sqrt(2+x^2-x^4)],
[(7+5*x^2)/(2+x^2-x^4)^(3/2),x,5,-13/18*EllipticE(x/sqrt(2),sqrt(-2))+17/6*EllipticF(x/sqrt(2),sqrt(-2))+1/18*x*(25+13*x^2)/sqrt(2+x^2-x^4)],
[1/(2+x^2-x^4)^(3/2),x,5,1/18*EllipticE(x/sqrt(2),sqrt(-2))+1/6*EllipticF(x/sqrt(2),sqrt(-2))+1/18*x*(5-x^2)/sqrt(2+x^2-x^4)],
[1/((7+5*x^2)*(2+x^2-x^4)^(3/2)),x,8,8/153*EllipticE(x/sqrt(2),sqrt(-2))+1/102*EllipticF(x/sqrt(2),sqrt(-2))-25/238*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))+1/306*x*(35-16*x^2)/sqrt(2+x^2-x^4)],
[1/((7+5*x^2)^2*(2+x^2-x^4)^(3/2)),x,17,5143/145656*EllipticE(x/sqrt(2),sqrt(-2))+89/24276*EllipticF(x/sqrt(2),sqrt(-2))-10825/113288*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))+1/10404*x*(580-287*x^2)/sqrt(2+x^2-x^4)+625/16184*x*sqrt(2+x^2-x^4)/(7+5*x^2)],
[1/((7+5*x^2)^3*(2+x^2-x^4)^(3/2)),x,26,3086453/138664512*EllipticE(x/sqrt(2),sqrt(-2))+60409/23110752*EllipticF(x/sqrt(2),sqrt(-2))-6898575/107850176*EllipticPi(x/sqrt(2),-10/7,sqrt(-2))+1/353736*x*(9830-4909*x^2)/sqrt(2+x^2-x^4)+625/32368*x*sqrt(2+x^2-x^4)/(7+5*x^2)^2+645625/15407168*x*sqrt(2+x^2-x^4)/(7+5*x^2)],

# Integrands of the form (7+5 x^2)^q (4+3 x^2+x^4)^(p/2)

# p>0
[(7+5*x^2)^4*sqrt(4+3*x^2+x^4),x,7,3050/11*x*(4+3*x^2+x^4)^(3/2)+23500/99*x^3*(4+3*x^2+x^4)^(3/2)+625/11*x^5*(4+3*x^2+x^4)^(3/2)+51665/33*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/33*x*(18727+4516*x^2)*sqrt(4+3*x^2+x^4)+33159/11*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-51665/33*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)^3*sqrt(4+3*x^2+x^4),x,6,275/7*x*(4+3*x^2+x^4)^(3/2)+125/9*x^3*(4+3*x^2+x^4)^(3/2)+4717/21*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/21*x*(1708+407*x^2)*sqrt(4+3*x^2+x^4)+1301/3*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-4717/21*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)^2*sqrt(4+3*x^2+x^4),x,5,25/7*x*(4+3*x^2+x^4)^(3/2)+319/7*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/7*x*(119+38*x^2)*sqrt(4+3*x^2+x^4)+81*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-319/7*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)*sqrt(4+3*x^2+x^4),x,4,9*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/3*x*(10+3*x^2)*sqrt(4+3*x^2+x^4)+49/3*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-9*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[sqrt(4+3*x^2+x^4),x,4,1/3*x*sqrt(4+3*x^2+x^4)+x*sqrt(4+3*x^2+x^4)/(2+x^2)+7/3*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[sqrt(4+3*x^2+x^4)/(7+5*x^2),x,7,1/5*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(11/35)+1/5*x*sqrt(4+3*x^2+x^4)/(2+x^2)+9/25*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+187/525*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-1/5*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)-11/75*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[sqrt(4+3*x^2+x^4)/(7+5*x^2)^2,x,7,51/280*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))/sqrt(385)-1/70*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/14*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+1/35*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-1/35*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+289/9800*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[sqrt(4+3*x^2+x^4)/(7+5*x^2)^3,x,18,14999/344960*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))/sqrt(385)-139/86240*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/28*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)^2+139/17248*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+139/43120*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-23/2940*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+254983/36220800*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[(7+5*x^2)^4*(4+3*x^2+x^4)^(3/2),x,8,1/1287*x*(452001+131080*x^2)*(4+3*x^2+x^4)^(3/2)+92150/429*x*(4+3*x^2+x^4)^(5/2)+2250/13*x^3*(4+3*x^2+x^4)^(5/2)+125/3*x^5*(4+3*x^2+x^4)^(5/2)+12665086/2145*x*sqrt(4+3*x^2+x^4)/(2+x^2)+7/2145*x*(661429+174989*x^2)*sqrt(4+3*x^2+x^4)-12665086/2145*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+2383556/429*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)^3*(4+3*x^2+x^4)^(3/2),x,7,1/1001*x*(53504+15365*x^2)*(4+3*x^2+x^4)^(3/2)+3825/143*x*(4+3*x^2+x^4)^(5/2)+125/13*x^3*(4+3*x^2+x^4)^(5/2)+4525662/5005*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/5005*x*(1653701+435441*x^2)*sqrt(4+3*x^2+x^4)-4525662/5005*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+121826/143*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)^2*(4+3*x^2+x^4)^(3/2),x,6,1/693*x*(6831+2240*x^2)*(4+3*x^2+x^4)^(3/2)+25/11*x*(4+3*x^2+x^4)^(5/2)+175346/1155*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/1155*x*(64533+18253*x^2)*sqrt(4+3*x^2+x^4)-175346/1155*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+4628/33*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)*(4+3*x^2+x^4)^(3/2),x,5,1/63*x*(108+35*x^2)*(4+3*x^2+x^4)^(3/2)+2798/105*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/105*x*(1029+289*x^2)*sqrt(4+3*x^2+x^4)-2798/105*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+74/3*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(4+3*x^2+x^4)^(3/2),x,5,1/7*x*(4+3*x^2+x^4)^(3/2)+138/35*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/35*x*(49+9*x^2)*sqrt(4+3*x^2+x^4)-138/35*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+4*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(4+3*x^2+x^4)^(3/2)/(7+5*x^2),x,12,44/125*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(11/35)+94/125*x*sqrt(4+3*x^2+x^4)/(2+x^2)+1/75*x*(11+3*x^2)*sqrt(4+3*x^2+x^4)-94/125*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+54/125*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+4114/13125*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(4+3*x^2+x^4)^(3/2)/(7+5*x^2)^2,x,19,13/350*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(11/35)+1/75*x*sqrt(4+3*x^2+x^4)+4/175*x*sqrt(4+3*x^2+x^4)/(2+x^2)+22/175*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+2431/36750*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-4/175*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+4/175*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4),13/350*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(11/35)+1/75*x*sqrt(4+3*x^2+x^4)+4/175*x*sqrt(4+3*x^2+x^4)/(2+x^2)+22/175*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+6919/183750*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-4/175*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+4/175*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)+187/13125*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(4+3*x^2+x^4)^(3/2)/(7+5*x^2)^3,x,22,1347/7840*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))/sqrt(385)+9/1960*x*sqrt(4+3*x^2+x^4)/(2+x^2)+11/175*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)^2+167/9800*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+111/24500*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-817/91875*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+7633/274400*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-6/875*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)-22/13125*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],

# p<0
[(7+5*x^2)^3/sqrt(4+3*x^2+x^4),x,5,75*x*sqrt(4+3*x^2+x^4)+25*x^3*sqrt(4+3*x^2+x^4)-15*x*sqrt(4+3*x^2+x^4)/(2+x^2)+13/2*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+15*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)^2/sqrt(4+3*x^2+x^4),x,4,25/3*x*sqrt(4+3*x^2+x^4)+20*x*sqrt(4+3*x^2+x^4)/(2+x^2)+167/6*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-20*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[(7+5*x^2)/sqrt(4+3*x^2+x^4),x,3,5*x*sqrt(4+3*x^2+x^4)/(2+x^2)+17/2*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-5*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[1/sqrt(4+3*x^2+x^4),x,1,1/2*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[1/((7+5*x^2)*sqrt(4+3*x^2+x^4)),x,3,1/4*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(5/77)-1/6*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+17/84*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[1/((7+5*x^2)^2*sqrt(4+3*x^2+x^4)),x,6,37/2464*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(5/77)-5/616*x*sqrt(4+3*x^2+x^4)/(2+x^2)+25/616*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+5/308*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-1/42*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+629/51744*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[1/((7+5*x^2)^3*sqrt(4+3*x^2+x^4)),x,7,-3285/3035648*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(5/77)-555/758912*x*sqrt(4+3*x^2+x^4)/(2+x^2)+25/1232*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)^2+2775/758912*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+555/379456*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-1/8624*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-18615/21249536*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[(7+5*x^2)^5/(4+3*x^2+x^4)^(3/2),x,6,1/28*x*(99493+45779*x^2)/sqrt(4+3*x^2+x^4)+5000/3*x*sqrt(4+3*x^2+x^4)+625*x^3*sqrt(4+3*x^2+x^4)-220779/28*x*sqrt(4+3*x^2+x^4)/(2+x^2)+220779/14*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-130729/12*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[(7+5*x^2)^4/(4+3*x^2+x^4)^(3/2),x,5,1/28*x*(2719-4023*x^2)/sqrt(4+3*x^2+x^4)+625/3*x*sqrt(4+3*x^2+x^4)+14523/28*x*sqrt(4+3*x^2+x^4)/(2+x^2)-14523/14*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+4243/12*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[(7+5*x^2)^3/(4+3*x^2+x^4)^(3/2),x,4,-1/28*x*(2323+949*x^2)/sqrt(4+3*x^2+x^4)+4449/28*x*sqrt(4+3*x^2+x^4)/(2+x^2)-4449/14*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+973/4*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[(7+5*x^2)^2/(4+3*x^2+x^4)^(3/2),x,4,-1/28*x*(9-113*x^2)/sqrt(4+3*x^2+x^4)-113/28*x*sqrt(4+3*x^2+x^4)/(2+x^2)+113/14*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+9/4*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[(7+5*x^2)/(4+3*x^2+x^4)^(3/2),x,4,1/28*x*(53+19*x^2)/sqrt(4+3*x^2+x^4)-19/28*x*sqrt(4+3*x^2+x^4)/(2+x^2)+19/14*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-3/4*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[1/(4+3*x^2+x^4)^(3/2),x,4,-1/28*x*(1+3*x^2)/sqrt(4+3*x^2+x^4)+3/28*x*sqrt(4+3*x^2+x^4)/(2+x^2)-3/14*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+1/4*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],
[1/((7+5*x^2)*(4+3*x^2+x^4)^(3/2)),x,8,25/176*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(5/77)-1/308*x*(13+4*x^2)/sqrt(4+3*x^2+x^4)+1/77*x*sqrt(4+3*x^2+x^4)/(2+x^2)-1/12*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+425/3696*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-1/77*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[1/((7+5*x^2)^2*(4+3*x^2+x^4)^(3/2)),x,15,575/108416*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(5/77)+1/13552*x*(24+37*x^2)/sqrt(4+3*x^2+x^4)-199/27104*x*sqrt(4+3*x^2+x^4)/(2+x^2)+625/27104*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+199/13552*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+9775/2276736*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-2/231*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt(2)*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/sqrt(4+3*x^2+x^4)],
[1/((7+5*x^2)^3*(4+3*x^2+x^4)^(3/2)),x,22,-529425/133568512*arctan(2*x*sqrt(11/35)/sqrt(4+3*x^2+x^4))*sqrt(5/77)+1/596288*x*(548+139*x^2)/sqrt(4+3*x^2+x^4)-18159/33392128*x*sqrt(4+3*x^2+x^4)/(2+x^2)+625/54208*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)^2+51875/33392128*x*sqrt(4+3*x^2+x^4)/(7+5*x^2)+18159/16696064*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticE(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))+843/379456*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticF(sin(2*arctan(x/sqrt(2))),sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))-3000075/934979584*(2+x^2)*sqrt(cos(2*arctan(x/sqrt(2)))^2)/cos(2*arctan(x/sqrt(2)))*EllipticPi(sin(2*arctan(x/sqrt(2))),-9/280,sqrt(1/8))*sqrt((4+3*x^2+x^4)/(2+x^2)^2)/(sqrt(2)*sqrt(4+3*x^2+x^4))],

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

# p>0

# p<0
[(d+e*x^2)^3/sqrt(a+b*x^2+c*x^4),x,5,1/15*e^2*(15*c*d-4*b*e)*x*sqrt(a+b*x^2+c*x^4)/c^2+1/5*e^3*x^3*sqrt(a+b*x^2+c*x^4)/c+1/15*e*(45*c^2*d^2+8*b^2*e^2-3*c*e*(10*b*d+3*a*e))*x*sqrt(a+b*x^2+c*x^4)/(c^(5/2)*(sqrt(a)+x^2*sqrt(c)))-1/15*a^(1/4)*e*(45*c^2*d^2+8*b^2*e^2-3*c*e*(10*b*d+3*a*e))*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))*(e*(45*c^2*d^2+8*b^2*e^2-3*c*e*(10*b*d+3*a*e))+(15*c^2*d^3-15*a*c*d*e^2+4*a*b*e^3)*sqrt(c)/sqrt(a))*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))],
[(d+e*x^2)^2/sqrt(a+b*x^2+c*x^4),x,4,1/3*e^2*x*sqrt(a+b*x^2+c*x^4)/c+2/3*e*(3*c*d-b*e)*x*sqrt(a+b*x^2+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))-2/3*a^(1/4)*e*(3*c*d-b*e)*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*e*(3*c*d-b*e)+(3*c*d^2-a*e^2)*sqrt(c)/sqrt(a))*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))],
[(d+e*x^2)/sqrt(a+b*x^2+c*x^4),x,3,e*x*sqrt(a+b*x^2+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*e*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))*(e+d*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))],
[1/((d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,3,1/2*arctan(x*sqrt(c*d^2-b*d*e+a*e^2)/(sqrt(d)*sqrt(e)*sqrt(a+b*x^2+c*x^4)))*sqrt(e)/(sqrt(d)*sqrt(c*d^2-b*d*e+a*e^2))+1/2*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))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+b*x^2+c*x^4))-1/4*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(e+d*sqrt(c)/sqrt(a))^2*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*d*(c*d^2-a*e^2)*sqrt(a+b*x^2+c*x^4))],
[1/((d+e*x^2)^2*sqrt(a+b*x^2+c*x^4)),x,6,1/4*(3*c*d^2-e*(2*b*d-a*e))*arctan(x*sqrt(c*d^2-b*d*e+a*e^2)/(sqrt(d)*sqrt(e)*sqrt(a+b*x^2+c*x^4)))*sqrt(e)/(d^(3/2)*(c*d^2-b*d*e+a*e^2)^(3/2))+1/2*e^2*x*sqrt(a+b*x^2+c*x^4)/(d*(c*d^2-b*d*e+a*e^2)*(d+e*x^2))-1/2*e*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(d*(c*d^2-b*d*e+a*e^2)*(sqrt(a)+x^2*sqrt(c)))+1/2*a^(1/4)*c^(1/4)*e*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)/(d*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x^2+c*x^4))+1/2*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))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*d*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+b*x^2+c*x^4))-1/8*(3*c*d^2-e*(2*b*d-a*e))*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(e*sqrt(a)+d*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d^2*(c*d^2-b*d*e+a*e^2)*(-e*sqrt(a)+d*sqrt(c))*sqrt(a+b*x^2+c*x^4))],

#  {1/((d + e*x^2)^3*Sqrt[a + b*x^2 + c*x^4]), x, 0, 0} 
[(d+e*x^2)^3/sqrt(a+b*x^2-c*x^4),x,6,-1/15*e^2*(15*c*d+4*b*e)*x*sqrt(a+b*x^2-c*x^4)/c^2-1/5*e^3*x^3*sqrt(a+b*x^2-c*x^4)/c-1/30*e*(45*c^2*d^2+8*b^2*e^2+3*c*e*(10*b*d+3*a*e))*EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(b-sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(c^(7/2)*sqrt(2)*sqrt(a+b*x^2-c*x^4))+1/30*EllipticF(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(e*(45*c^2*d^2+8*b^2*e^2+3*c*e*(10*b*d+3*a*e))+2*c*(15*c^2*d^3+15*a*c*d*e^2+4*a*b*e^3)/(b-sqrt(b^2+4*a*c)))*(b-sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(c^(7/2)*sqrt(2)*sqrt(a+b*x^2-c*x^4))],
[(d+e*x^2)^2/sqrt(a+b*x^2-c*x^4),x,5,-1/3*e^2*x*sqrt(a+b*x^2-c*x^4)/c-1/3*e*(3*c*d+b*e)*EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(b-sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(c^(5/2)*sqrt(2)*sqrt(a+b*x^2-c*x^4))+1/3*EllipticF(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(3*c^2*d^2+b*e^2*(b-sqrt(b^2+4*a*c))+c*e*(3*b*d+a*e-3*d*sqrt(b^2+4*a*c)))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(c^(5/2)*sqrt(2)*sqrt(a+b*x^2-c*x^4))],
[(d+e*x^2)/sqrt(a+b*x^2-c*x^4),x,4,1/2*EllipticF(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(2*c*d+e*(b-sqrt(b^2+4*a*c)))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(c^(3/2)*sqrt(2)*sqrt(a+b*x^2-c*x^4))-1/2*e*EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(b-sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(c^(3/2)*sqrt(2)*sqrt(a+b*x^2-c*x^4))],
[1/((d+e*x^2)*sqrt(a+b*x^2-c*x^4)),x,2,EllipticPi(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),-1/2*e*(b+sqrt(b^2+4*a*c))/(c*d),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(d*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))],
[1/((d+e*x^2)^2*sqrt(a+b*x^2-c*x^4)),x,8,-1/2*e^2*x*sqrt(a+b*x^2-c*x^4)/(d*(c*d^2+b*d*e-a*e^2)*(d+e*x^2))+1/2*(3*c*d^2+e*(2*b*d-a*e))*EllipticPi(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),-1/2*e*(b+sqrt(b^2+4*a*c))/(c*d),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(d^2*(c*d^2+e*(b*d-a*e))*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))-1/4*EllipticF(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(2*c*d+e*(b-sqrt(b^2+4*a*c)))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(d*(c*d^2+e*(b*d-a*e))*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))+1/4*e*EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)),sqrt((b+sqrt(b^2+4*a*c))/(b-sqrt(b^2+4*a*c))))*(b-sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))*sqrt(1-2*c*x^2/(b+sqrt(b^2+4*a*c)))/(d*(c*d^2+e*(b*d-a*e))*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))],

#  {1/((d + e*x^2)^3*Sqrt[a + b*x^2 - c*x^4]), x, 0, 0} 
[(d+e*x^2)/sqrt(-a+b*x^2+c*x^4),x,5,1/2*e*x*(1+2*c*x^2/(b-sqrt(b^2+4*a*c)))*(b-sqrt(b^2+4*a*c))/(c*sqrt(-a+b*x^2+c*x^4))+d*sqrt(cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c))))^2)/cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c))))*EllipticF(sin(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)))),sqrt(-2*sqrt(b^2+4*a*c)/(b-sqrt(b^2+4*a*c))))*(1+2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))/(sqrt(2)*sqrt(c)*sqrt(-a+b*x^2+c*x^4)*sqrt((1+2*c*x^2/(b-sqrt(b^2+4*a*c)))/(1+2*c*x^2/(b+sqrt(b^2+4*a*c)))))-1/2*e*sqrt(cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c))))^2)/cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c))))*EllipticE(sin(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)))),sqrt(-2*sqrt(b^2+4*a*c)/(b-sqrt(b^2+4*a*c))))*(1+2*c*x^2/(b-sqrt(b^2+4*a*c)))*(b-sqrt(b^2+4*a*c))*sqrt(b+sqrt(b^2+4*a*c))/(c^(3/2)*sqrt(2)*sqrt(-a+b*x^2+c*x^4)*sqrt((1+2*c*x^2/(b-sqrt(b^2+4*a*c)))/(1+2*c*x^2/(b+sqrt(b^2+4*a*c)))))],
[1/((d+e*x^2)*sqrt(-a+b*x^2+c*x^4)),x,2,EllipticPi(x*sqrt(2)*sqrt(c)/sqrt(-b+sqrt(b^2+4*a*c)),1/2*e*(b-sqrt(b^2+4*a*c))/(c*d),sqrt((b-sqrt(b^2+4*a*c))/(b+sqrt(b^2+4*a*c))))*sqrt(1+2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(-b+sqrt(b^2+4*a*c))*sqrt(1+2*c*x^2/(b+sqrt(b^2+4*a*c)))/(d*sqrt(2)*sqrt(c)*sqrt(-a+b*x^2+c*x^4))],
[(d+e*x^2)/sqrt(-a+b*x^2-c*x^4),x,3,-e*x*sqrt(-a+b*x^2-c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*e*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))*(e+d*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))],
[1/((d+e*x^2)*sqrt(-a+b*x^2-c*x^4)),x,3,1/2*arctan(x*sqrt(-c*d^2-b*d*e-a*e^2)/(sqrt(d)*sqrt(e)*sqrt(-a+b*x^2-c*x^4)))*sqrt(e)/(sqrt(d)*sqrt(-c*d^2-b*d*e-a*e^2))+1/2*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))*sqrt((a-b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(-e*sqrt(a)+d*sqrt(c))*sqrt(-a+b*x^2-c*x^4))-1/4*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*sqrt(a)+d*sqrt(c))^2/(d*e*sqrt(a)*sqrt(c)),sqrt(1/4*(2+b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(e+d*sqrt(c)/sqrt(a))^2*sqrt((a-b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*d*(c*d^2-a*e^2)*sqrt(-a+b*x^2-c*x^4))],
[(d+e*x^2)^3/sqrt(2+3*x^2+x^4),x,5,3/5*e*(5*d^2-10*d*e+6*e^2)*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/5*(5*d^3-10*d*e^2+8*e^3)*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-3/5*e*(5*d^2-10*d*e+6*e^2)*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/5*(5*d-4*e)*e^2*x*sqrt(2+3*x^2+x^4)+1/5*e^3*x^3*sqrt(2+3*x^2+x^4)],
[(d+e*x^2)^2/sqrt(2+3*x^2+x^4),x,4,2*(d-e)*e*x*(2+x^2)/sqrt(2+3*x^2+x^4)+1/3*(3*d^2-2*e^2)*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-2*(d-e)*e*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)+1/3*e^2*x*sqrt(2+3*x^2+x^4)],
[(d+e*x^2)/sqrt(2+3*x^2+x^4),x,3,e*x*(2+x^2)/sqrt(2+3*x^2+x^4)+d*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(sqrt(2)*sqrt(2+3*x^2+x^4))-e*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt((2+x^2)/(1+x^2))/sqrt(2+3*x^2+x^4)],
[1/((d+e*x^2)*sqrt(2+3*x^2+x^4)),x,4,(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/((d-e)*sqrt(2)*sqrt(2+3*x^2+x^4))-e*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),1-e/d,sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(d*(d-e)*sqrt(2)*sqrt(2+3*x^2+x^4)),-e*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),1-e/d,sqrt(1/2))/(d*(d-e)*sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/((d-e)*sqrt(2)*sqrt(2+3*x^2+x^4))],
[1/((d+e*x^2)^2*sqrt(2+3*x^2+x^4)),x,9,-1/2*e*x*(2+x^2)/(d*(d^2-3*d*e+2*e^2)*sqrt(2+3*x^2+x^4))-1/2*e*(3*d^2-6*d*e+2*e^2)*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),1-e/d,sqrt(1/2))/(d^2*(d-2*e)*(d-e)^2*sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+e*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(d*(d-2*e)*(d-e)*sqrt(2)*sqrt(2+3*x^2+x^4))+1/2*(2*d-e)*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(2+2*x^2))/(d*(d-e)^2*sqrt(2+3*x^2+x^4))+1/2*e^2*x*sqrt(2+3*x^2+x^4)/(d*(d^2-3*d*e+2*e^2)*(d+e*x^2)),-1/2*e*x*(2+x^2)/(d*(d^2-3*d*e+2*e^2)*sqrt(2+3*x^2+x^4))-1/2*e*(3*d^2-6*d*e+2*e^2)*(2+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),1-e/d,sqrt(1/2))/(d^2*(d-2*e)*(d-e)^2*sqrt(2)*sqrt((2+x^2)/(1+x^2))*sqrt(2+3*x^2+x^4))+e*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(d*(d-2*e)*(d-e)*sqrt(2)*sqrt(2+3*x^2+x^4))-1/2*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/((d-2*e)*(d-e)*sqrt(2)*sqrt(2+3*x^2+x^4))+1/2*(3*d^2-6*d*e+2*e^2)*(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt((2+x^2)/(1+x^2))/(d*(d-2*e)*(d-e)^2*sqrt(2)*sqrt(2+3*x^2+x^4))+1/2*e^2*x*sqrt(2+3*x^2+x^4)/(d*(d^2-3*d*e+2*e^2)*(d+e*x^2))],

# Integrands of the form (d+e x^2)^q (a+b x^2+c x^4)^p with p symbolic
[(c+e*x^2)^q*(a+c*x^2+b*x^4)^p,x,0,Unintegrable((c+e*x^2)^q*(a+c*x^2+b*x^4)^p,x)],
[(c+e*x^2)^3*(a+c*x^2+b*x^4)^p,x,8,-c*e^2*(e*(5+2*p)-3*b*(7+4*p))*x*(a+c*x^2+b*x^4)^(1+p)/(b^2*(35+48*p+16*p^2))+e^3*x^3*(a+c*x^2+b*x^4)^(1+p)/(b*(7+4*p))+c*(a*e^3*(5+2*p)-3*a*b*e^2*(7+4*p)+b^2*c^2*(35+48*p+16*p^2))*x*(a+c*x^2+b*x^4)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/(b^2*(5+4*p)*(7+4*p)*(1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)+1/3*e*(c^2*e^2*(15+16*p+4*p^2)+3*b^2*c^2*(35+48*p+16*p^2)-3*b*e*(a*e*(5+4*p)+c^2*(21+26*p+8*p^2)))*x^3*(a+c*x^2+b*x^4)^p*AppellF1(3/2,-p,-p,5/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/(b^2*(5+4*p)*(7+4*p)*(1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)],
[(c+e*x^2)^2*(a+c*x^2+b*x^4)^p,x,7,e^2*x*(a+c*x^2+b*x^4)^(1+p)/(b*(5+4*p))-(a*e^2-b*c^2*(5+4*p))*x*(a+c*x^2+b*x^4)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/(b*(5+4*p)*(1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)+1/3*c*e*(10*b-3*e+8*b*p-2*e*p)*x^3*(a+c*x^2+b*x^4)^p*AppellF1(3/2,-p,-p,5/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/(b*(5+4*p)*(1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)],
[(c+e*x^2)*(a+c*x^2+b*x^4)^p,x,6,c*x*(a+c*x^2+b*x^4)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/((1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)+1/3*e*x^3*(a+c*x^2+b*x^4)^p*AppellF1(3/2,-p,-p,5/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/((1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)],
[(a+c*x^2+b*x^4)^p,x,2,x*(a+c*x^2+b*x^4)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-sqrt(-4*a*b+c^2)),-2*b*x^2/(c+sqrt(-4*a*b+c^2)))/((1+2*b*x^2/(c-sqrt(-4*a*b+c^2)))^p*(1+2*b*x^2/(c+sqrt(-4*a*b+c^2)))^p)],
[(a+c*x^2+b*x^4)^p/(c+e*x^2),x,0,Unintegrable((a+c*x^2+b*x^4)^p/(c+e*x^2),x)],
[(a+c*x^2+b*x^4)^p/(c+e*x^2)^2,x,0,Unintegrable((a+c*x^2+b*x^4)^p/(c+e*x^2)^2,x)],

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

# Integrands of the form (d+e x)^q (f+g x)^r (a+b x^2+c x^4)^p with b=0
[(f+g*x)/((d+e*x)*sqrt(a+c*x^4)),x,8,1/2*(e*f-d*g)*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))/sqrt(-c*d^4-a*e^4)-1/2*(e*f-d*g)*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))/sqrt(c*d^4+a*e^4)-1/4*(e*f-d*g)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d*e*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+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(1/2))*(e*g*sqrt(a)+d*f*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))],
[(f+g*x)/((d+e*x)*sqrt(-a+c*x^4)),x,10,1/2*(e*f-d*g)*arctanh((a*e^2-c*d^2*x^2)/(sqrt(c*d^4-a*e^4)*sqrt(-a+c*x^4)))/sqrt(c*d^4-a*e^4)+a^(1/4)*g*EllipticF(c^(1/4)*x/a^(1/4),I)*sqrt(1-c*x^4/a)/(c^(1/4)*e*sqrt(-a+c*x^4))+a^(1/4)*(e*f-d*g)*EllipticPi(c^(1/4)*x/a^(1/4),e^2*sqrt(a)/(d^2*sqrt(c)),I)*sqrt(1-c*x^4/a)/(c^(1/4)*d*e*sqrt(-a+c*x^4))],

# Integrands of the form (d+e x)^q (f+g x)^r (a+b x^2+c x^4)^p with e^2 f^2+4 d e f g+d^2 g^2=0, 4 c d e f+2 c d^2 g-b e^2 g=0 and 4 a e^5 f+c d^5 g+15 a d e^4 g=0
[(1+x-sqrt(3))/((1+x+sqrt(3))*sqrt(-4+x^4+4*x^2*sqrt(3))),x,2,1/3*arctanh((1+x-sqrt(3))^2/(sqrt(3*(-3+2*sqrt(3)))*sqrt(-4+x^4+4*x^2*sqrt(3))))*sqrt(-3+2*sqrt(3))],
[(1+x+sqrt(3))/((1+x-sqrt(3))*sqrt(-4+x^4-4*x^2*sqrt(3))),x,2,-1/3*arctan((1+x+sqrt(3))^2/(sqrt(3*(3+2*sqrt(3)))*sqrt(-4+x^4-4*x^2*sqrt(3))))*sqrt(3+2*sqrt(3))],
[(1+2*x-sqrt(3))/((1+2*x+sqrt(3))*sqrt(-1+4*x^4+4*x^2*sqrt(3))),x,2,1/3*arctanh(1/2*(1+2*x-sqrt(3))^2/(sqrt(3*(-3+2*sqrt(3)))*sqrt(-1+4*x^4+4*x^2*sqrt(3))))*sqrt(-3+2*sqrt(3))],
[(1+2*x+sqrt(3))/((1+2*x-sqrt(3))*sqrt(-1+4*x^4-4*x^2*sqrt(3))),x,2,-1/3*arctan(1/2*(1+2*x+sqrt(3))^2/(sqrt(3*(3+2*sqrt(3)))*sqrt(-1+4*x^4-4*x^2*sqrt(3))))*sqrt(3+2*sqrt(3))],

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

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

# Integrands of the form (d+e x)^q (f+g x)^r (a+b x^2+c x^4)^(p/2)
[(f+g*x)/((d+e*x)*sqrt(a+b*x^2+c*x^4)),x,8,1/2*(e*f-d*g)*arctan(x*sqrt(-c*d^4-b*d^2*e^2-a*e^4)/(d*e*sqrt(a+b*x^2+c*x^4)))/sqrt(-c*d^4-b*d^2*e^2-a*e^4)-1/2*(e*f-d*g)*arctanh(1/2*(b*d^2+2*a*e^2+(2*c*d^2+b*e^2)*x^2)/(sqrt(c*d^4+b*d^2*e^2+a*e^4)*sqrt(a+b*x^2+c*x^4)))/sqrt(c*d^4+b*d^2*e^2+a*e^4)-1/4*(e*f-d*g)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d*e*(e^2*sqrt(a)+d^2*sqrt(c))*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)))))*(e*g*sqrt(a)+d*f*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+b*x^2+c*x^4))],
[(f+g*x)/((d+e*x)*sqrt(-a+b*x^2+c*x^4)),x,10,-1/2*(e*f-d*g)*arctanh(1/2*(b*d^2-2*a*e^2+(2*c*d^2+b*e^2)*x^2)/(sqrt(c*d^4+b*d^2*e^2-a*e^4)*sqrt(-a+b*x^2+c*x^4)))/sqrt(c*d^4+b*d^2*e^2-a*e^4)+g*sqrt(cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c))))^2)/cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c))))*EllipticF(sin(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2+4*a*c)))),sqrt(-2*sqrt(b^2+4*a*c)/(b-sqrt(b^2+4*a*c))))*(1+2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(b+sqrt(b^2+4*a*c))/(e*sqrt(2)*sqrt(c)*sqrt(-a+b*x^2+c*x^4)*sqrt((1+2*c*x^2/(b-sqrt(b^2+4*a*c)))/(1+2*c*x^2/(b+sqrt(b^2+4*a*c)))))+(e*f-d*g)*EllipticPi(x*sqrt(2)*sqrt(c)/sqrt(-b+sqrt(b^2+4*a*c)),-1/2*e^2*(b-sqrt(b^2+4*a*c))/(c*d^2),sqrt((b-sqrt(b^2+4*a*c))/(b+sqrt(b^2+4*a*c))))*sqrt(1+2*c*x^2/(b-sqrt(b^2+4*a*c)))*sqrt(-b+sqrt(b^2+4*a*c))*sqrt(1+2*c*x^2/(b+sqrt(b^2+4*a*c)))/(d*e*sqrt(2)*sqrt(c)*sqrt(-a+b*x^2+c*x^4))]]:
