# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.2 Quartic\1.2.2.2 (d x)^m (a+b x^2+c x^4)^p.txt"

lst:=[

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

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

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

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

# Integrands of the form (a^2+2 a b x^2+b^2 x^4)^(p/4)
[(a^2+2*a*b*x^2+b^2*x^4)^(3/4),x,4,1/4*x*(a^2+2*a*b*x^2+b^2*x^4)^(3/4)+3/8*a*x*(a^2+2*a*b*x^2+b^2*x^4)^(3/4)/(a+b*x^2)+3/8*(a^2+2*a*b*x^2+b^2*x^4)^(3/4)*arcsinh(x*sqrt(b)/sqrt(a))*sqrt(a)/((1+b*x^2/a)^(3/2)*sqrt(b))],
[(a^2+2*a*b*x^2+b^2*x^4)^(1/4),x,3,1/2*x*(a^2+2*a*b*x^2+b^2*x^4)^(1/4)+1/2*(a^2+2*a*b*x^2+b^2*x^4)^(1/4)*arcsinh(x*sqrt(b)/sqrt(a))*sqrt(a)/(sqrt(b)*sqrt(1+b*x^2/a))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(1/4),x,2,arcsinh(x*sqrt(b)/sqrt(a))*sqrt(a)*sqrt(1+b*x^2/a)/((a^2+2*a*b*x^2+b^2*x^4)^(1/4)*sqrt(b))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(3/4),x,2,x*(a+b*x^2)/(a*(a^2+2*a*b*x^2+b^2*x^4)^(3/4))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(5/4),x,3,1/3*x*(a+b*x^2)/(a*(a^2+2*a*b*x^2+b^2*x^4)^(5/4))+2/3*x/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(1/4)),2/3*x/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(1/4))+1/3*x/(a*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(1/4))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(7/4),x,4,1/5*x*(a+b*x^2)/(a*(a^2+2*a*b*x^2+b^2*x^4)^(7/4))+4/15*x/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/4))+8/15*x*(a+b*x^2)/(a^3*(a^2+2*a*b*x^2+b^2*x^4)^(3/4)),4/15*x/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/4))+1/5*x/(a*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/4))+8/15*x*(a+b*x^2)/(a^3*(a^2+2*a*b*x^2+b^2*x^4)^(3/4))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(9/4),x,5,1/7*x*(a+b*x^2)/(a*(a^2+2*a*b*x^2+b^2*x^4)^(9/4))+6/35*x/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(5/4))+8/35*x*(a+b*x^2)/(a^3*(a^2+2*a*b*x^2+b^2*x^4)^(5/4))+16/35*x/(a^4*(a^2+2*a*b*x^2+b^2*x^4)^(1/4)),16/35*x/(a^4*(a^2+2*a*b*x^2+b^2*x^4)^(1/4))+1/7*x/(a*(a+b*x^2)^3*(a^2+2*a*b*x^2+b^2*x^4)^(1/4))+6/35*x/(a^2*(a+b*x^2)^2*(a^2+2*a*b*x^2+b^2*x^4)^(1/4))+8/35*x/(a^3*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(1/4))],

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

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

#  Tests that NegativeQ[b^2-4*a*c] returns True to avoid I in antiderivative! 
[1/(1+a^2+2*a*x^2+x^4),x,9,-1/4*log(x^2+sqrt(1+a^2)-x*sqrt(2)*sqrt(-a+sqrt(1+a^2)))/(sqrt(2)*sqrt(1+a^2)*sqrt(-a+sqrt(1+a^2)))+1/4*log(x^2+sqrt(1+a^2)+x*sqrt(2)*sqrt(-a+sqrt(1+a^2)))/(sqrt(2)*sqrt(1+a^2)*sqrt(-a+sqrt(1+a^2)))-1/2*arctan((-x*sqrt(2)+sqrt(-a+sqrt(1+a^2)))/sqrt(a+sqrt(1+a^2)))/(sqrt(2)*sqrt(1+a^2)*sqrt(a+sqrt(1+a^2)))+1/2*arctan((x*sqrt(2)+sqrt(-a+sqrt(1+a^2)))/sqrt(a+sqrt(1+a^2)))/(sqrt(2)*sqrt(1+a^2)*sqrt(a+sqrt(1+a^2)))],
[1/(4-5*x^2+x^4),x,3,-1/6*arctanh(1/2*x)+1/3*arctanh(x)],
[1/(3+4*x^2+x^4),x,3,1/2*arctan(x)-1/2*arctan(x/sqrt(3))/sqrt(3)],
[1/(9+5*x^2+x^4),x,9,-1/12*log(3-x+x^2)+1/12*log(3+x+x^2)-1/6*arctan((1-2*x)/sqrt(11))/sqrt(11)+1/6*arctan((1+2*x)/sqrt(11))/sqrt(11)],
[1/(1-x^2+x^4),x,9,-1/2*arctan(-2*x+sqrt(3))+1/2*arctan(2*x+sqrt(3))-1/4*log(1+x^2-x*sqrt(3))/sqrt(3)+1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[1/(2+2*x^2+x^4),x,9,-1/8*log(x^2+sqrt(2)-x*sqrt(2*(-1+sqrt(2))))/sqrt(-1+sqrt(2))+1/8*log(x^2+sqrt(2)+x*sqrt(2*(-1+sqrt(2))))/sqrt(-1+sqrt(2))-1/4*arctan((-2*x+sqrt(2*(-1+sqrt(2))))/sqrt(2*(1+sqrt(2))))*sqrt(-1+sqrt(2))+1/4*arctan((2*x+sqrt(2*(-1+sqrt(2))))/sqrt(2*(1+sqrt(2))))*sqrt(-1+sqrt(2))],

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

# p>0

# p<0
[1/sqrt(2+5*x^2-3*x^4),x,2,EllipticF(x/sqrt(2),sqrt(-6))],
[1/sqrt(2+4*x^2-3*x^4),x,2,EllipticF(x*sqrt(1/2*(-2+sqrt(10))),sqrt(1/3*(-7-2*sqrt(10))))*sqrt(1/6*(2+sqrt(10)))],
[1/sqrt(2+3*x^2-3*x^4),x,2,EllipticF(x*sqrt(6/(3+sqrt(33))),sqrt(1/4*(-7-sqrt(33))))*sqrt(2/(-3+sqrt(33)))],
[1/sqrt(2+2*x^2-3*x^4),x,2,EllipticF(x*sqrt(3/(1+sqrt(7))),sqrt(1/3*(-4-sqrt(7))))/sqrt(-1+sqrt(7))],
[1/sqrt(2+x^2-3*x^4),x,2,EllipticF(x,sqrt(-3/2))/sqrt(2)],
[1/sqrt(2-3*x^4),x,1,EllipticF((3/2)^(1/4)*x,I)/6^(1/4)],
[1/sqrt(2-x^2-3*x^4),x,2,EllipticF(x*sqrt(3/2),sqrt(-2/3))/sqrt(3)],
[1/sqrt(2-2*x^2-3*x^4),x,2,EllipticF(x*sqrt(3/(-1+sqrt(7))),sqrt(1/3*(-4+sqrt(7))))/sqrt(1+sqrt(7))],
[1/sqrt(2-3*x^2-3*x^4),x,2,EllipticF(x*sqrt(6/(-3+sqrt(33))),sqrt(1/4*(-7+sqrt(33))))*sqrt(2/(3+sqrt(33)))],
[1/sqrt(2-4*x^2-3*x^4),x,2,EllipticF(x*sqrt(1/2*(2+sqrt(10))),sqrt(1/3*(-7+2*sqrt(10))))*sqrt(1/6*(-2+sqrt(10)))],
[1/sqrt(2-5*x^2-3*x^4),x,2,EllipticF(x*sqrt(3),sqrt(-1/6))/sqrt(6)],
[1/sqrt(3+7*x^2-2*x^4),x,2,EllipticF(2*x/sqrt(7+sqrt(73)),sqrt(1/12*(-61-7*sqrt(73))))*sqrt(2/(-7+sqrt(73)))],
[1/sqrt(3+6*x^2-2*x^4),x,2,EllipticF(x*sqrt(1/3*(-3+sqrt(15))),sqrt(-4-sqrt(15)))*sqrt(1/6*(3+sqrt(15)))],
[1/sqrt(3+5*x^2-2*x^4),x,2,EllipticF(x/sqrt(3),sqrt(-6))],
[1/sqrt(3+4*x^2-2*x^4),x,2,EllipticF(x*sqrt(2/(2+sqrt(10))),sqrt(1/3*(-7-2*sqrt(10))))/sqrt(-2+sqrt(10))],
[1/sqrt(3+3*x^2-2*x^4),x,2,EllipticF(2*x/sqrt(3+sqrt(33)),sqrt(1/4*(-7-sqrt(33))))*sqrt(2/(-3+sqrt(33)))],
[1/sqrt(3+2*x^2-2*x^4),x,2,EllipticF(x*sqrt(2/(1+sqrt(7))),sqrt(1/3*(-4-sqrt(7))))/sqrt(-1+sqrt(7))],
[1/sqrt(3+x^2-2*x^4),x,2,EllipticF(x*sqrt(2/3),sqrt(-3/2))/sqrt(2)],
[1/sqrt(3-2*x^4),x,1,EllipticF((2/3)^(1/4)*x,I)/6^(1/4)],
[1/sqrt(3-x^2-2*x^4),x,2,EllipticF(x,sqrt(-2/3))/sqrt(3)],
[1/sqrt(3-2*x^2-2*x^4),x,2,EllipticF(x*sqrt(2/(-1+sqrt(7))),sqrt(1/3*(-4+sqrt(7))))/sqrt(1+sqrt(7))],
[1/sqrt(3-3*x^2-2*x^4),x,2,EllipticF(2*x/sqrt(-3+sqrt(33)),sqrt(1/4*(-7+sqrt(33))))*sqrt(2/(3+sqrt(33)))],
[1/sqrt(3-4*x^2-2*x^4),x,2,EllipticF(x*sqrt(2/(-2+sqrt(10))),sqrt(1/3*(-7+2*sqrt(10))))/sqrt(2+sqrt(10))],
[1/sqrt(3-5*x^2-2*x^4),x,2,EllipticF(x*sqrt(2),sqrt(-1/6))/sqrt(6)],
[1/sqrt(3-6*x^2-2*x^4),x,2,EllipticF(x*sqrt(1/3*(3+sqrt(15))),sqrt(-4+sqrt(15)))*sqrt(1/6*(-3+sqrt(15)))],
[1/sqrt(3-7*x^2-2*x^4),x,2,EllipticF(2*x/sqrt(-7+sqrt(73)),sqrt(1/12*(-61+7*sqrt(73))))*sqrt(2/(7+sqrt(73)))],
[1/sqrt(-2+5*x^2+3*x^4),x,1,EllipticF(x*sqrt(7/2)/sqrt(-1+3*x^2),sqrt(6/7))*sqrt(2+x^2)*sqrt(-1+3*x^2)/(sqrt(7)*sqrt(-2+5*x^2+3*x^4))],
[1/sqrt(-2+4*x^2+3*x^4),x,1,1/2*EllipticF(2^(3/4)*5^(1/4)*x/sqrt(-2+x^2*(2+sqrt(10))),sqrt(1/10*(5+sqrt(10))))*sqrt((2-x^2*(2-sqrt(10)))/(2-x^2*(2+sqrt(10))))*sqrt(-2+x^2*(2+sqrt(10)))/(10^(1/4)*sqrt(-2+4*x^2+3*x^4)*sqrt(1/(2-x^2*(2+sqrt(10)))))],
[1/sqrt(-2+3*x^2+3*x^4),x,1,1/2*EllipticF(33^(1/4)*x*sqrt(2)/sqrt(-4+x^2*(3+sqrt(33))),sqrt(1/22*(11+sqrt(33))))*sqrt((4-x^2*(3-sqrt(33)))/(4-x^2*(3+sqrt(33))))*sqrt(-4+x^2*(3+sqrt(33)))/(33^(1/4)*sqrt(2)*sqrt(-2+3*x^2+3*x^4)*sqrt(1/(4-x^2*(3+sqrt(33)))))],
[1/sqrt(-2+2*x^2+3*x^4),x,1,1/2*EllipticF(7^(1/4)*x*sqrt(2)/sqrt(-2+x^2*(1+sqrt(7))),sqrt(1/14*(7+sqrt(7))))*sqrt((2-x^2*(1-sqrt(7)))/(2-x^2*(1+sqrt(7))))*sqrt(-2+x^2*(1+sqrt(7)))/(7^(1/4)*sqrt(-2+2*x^2+3*x^4)*sqrt(1/(2-x^2*(1+sqrt(7)))))],
[1/sqrt(-2+x^2+3*x^4),x,1,EllipticF(x*sqrt(5)/sqrt(-2+3*x^2),sqrt(3/5))*sqrt(1+x^2)*sqrt(-2+3*x^2)/(sqrt(5)*sqrt(-2+x^2+3*x^4))],
[1/sqrt(-2+3*x^4),x,1,1/2*EllipticF(2^(3/4)*3^(1/4)*x/sqrt(-2+x^2*sqrt(6)),sqrt(1/2))*sqrt(-2+x^2*sqrt(6))*sqrt((2+x^2*sqrt(6))/(2-x^2*sqrt(6)))/(6^(1/4)*sqrt(-2+3*x^4)*sqrt(1/(2-x^2*sqrt(6))))],
[1/sqrt(-2-x^2+3*x^4),x,1,EllipticF(x*sqrt(5/2)/sqrt(-1+x^2),sqrt(2/5))*sqrt(-1+x^2)*sqrt(2+3*x^2)/(sqrt(5)*sqrt(-2-x^2+3*x^4))],
[1/sqrt(-2-2*x^2+3*x^4),x,1,1/2*EllipticF(7^(1/4)*x*sqrt(2)/sqrt(-2-x^2*(1-sqrt(7))),sqrt(1/14*(7-sqrt(7))))*sqrt(-2-x^2*(1-sqrt(7)))*sqrt((2+x^2*(1+sqrt(7)))/(2+x^2*(1-sqrt(7))))/(7^(1/4)*sqrt(-2-2*x^2+3*x^4)*sqrt(1/(2+x^2*(1-sqrt(7)))))],
[1/sqrt(-2-3*x^2+3*x^4),x,1,1/2*EllipticF(33^(1/4)*x*sqrt(2)/sqrt(-4-x^2*(3-sqrt(33))),sqrt(1/22*(11-sqrt(33))))*sqrt(-4-x^2*(3-sqrt(33)))*sqrt((4+x^2*(3+sqrt(33)))/(4+x^2*(3-sqrt(33))))/(33^(1/4)*sqrt(2)*sqrt(-2-3*x^2+3*x^4)*sqrt(1/(4+x^2*(3-sqrt(33)))))],
[1/sqrt(-2-4*x^2+3*x^4),x,1,1/2*EllipticF(2^(3/4)*5^(1/4)*x/sqrt(-2-x^2*(2-sqrt(10))),sqrt(1/10*(5-sqrt(10))))*sqrt(-2-x^2*(2-sqrt(10)))*sqrt((2+x^2*(2+sqrt(10)))/(2+x^2*(2-sqrt(10))))/(10^(1/4)*sqrt(-2-4*x^2+3*x^4)*sqrt(1/(2+x^2*(2-sqrt(10)))))],
[1/sqrt(-2-5*x^2+3*x^4),x,1,EllipticF(x*sqrt(7)/sqrt(-2+x^2),sqrt(1/7))*sqrt(-2+x^2)*sqrt(1+3*x^2)/(sqrt(7)*sqrt(-2-5*x^2+3*x^4))],
[1/sqrt(-3+7*x^2+2*x^4),x,1,1/2*EllipticF(73^(1/4)*x*sqrt(2)/sqrt(-6+x^2*(7+sqrt(73))),sqrt(1/146*(73+7*sqrt(73))))*sqrt((6-x^2*(7-sqrt(73)))/(6-x^2*(7+sqrt(73))))*sqrt(-6+x^2*(7+sqrt(73)))/(73^(1/4)*sqrt(3)*sqrt(-3+7*x^2+2*x^4)*sqrt(1/(6-x^2*(7+sqrt(73)))))],
[1/sqrt(-3+6*x^2+2*x^4),x,1,EllipticF(15^(1/4)*x*sqrt(2)/sqrt(-3+x^2*(3+sqrt(15))),sqrt(1/10*(5+sqrt(15))))*sqrt((3-x^2*(3-sqrt(15)))/(3-x^2*(3+sqrt(15))))*sqrt(-3+x^2*(3+sqrt(15)))/(3^(3/4)*5^(1/4)*sqrt(2)*sqrt(-3+6*x^2+2*x^4)*sqrt(1/(3-x^2*(3+sqrt(15)))))],
[1/sqrt(-3+5*x^2+2*x^4),x,1,EllipticF(x*sqrt(7/3)/sqrt(-1+2*x^2),sqrt(6/7))*sqrt(3+x^2)*sqrt(-1+2*x^2)/(sqrt(7)*sqrt(-3+5*x^2+2*x^4))],
[1/sqrt(-3+4*x^2+2*x^4),x,1,EllipticF(2^(3/4)*5^(1/4)*x/sqrt(-3+x^2*(2+sqrt(10))),sqrt(1/10*(5+sqrt(10))))*sqrt((3-x^2*(2-sqrt(10)))/(3-x^2*(2+sqrt(10))))*sqrt(-3+x^2*(2+sqrt(10)))/(2^(3/4)*5^(1/4)*sqrt(3)*sqrt(-3+4*x^2+2*x^4)*sqrt(1/(3-x^2*(2+sqrt(10)))))],
[1/sqrt(-3+3*x^2+2*x^4),x,1,1/2*EllipticF(33^(1/4)*x*sqrt(2)/sqrt(-6+x^2*(3+sqrt(33))),sqrt(1/22*(11+sqrt(33))))*sqrt((6-x^2*(3-sqrt(33)))/(6-x^2*(3+sqrt(33))))*sqrt(-6+x^2*(3+sqrt(33)))/(3^(3/4)*11^(1/4)*sqrt(-3+3*x^2+2*x^4)*sqrt(1/(6-x^2*(3+sqrt(33)))))],
[1/sqrt(-3+2*x^2+2*x^4),x,1,EllipticF(7^(1/4)*x*sqrt(2)/sqrt(-3+x^2*(1+sqrt(7))),sqrt(1/14*(7+sqrt(7))))*sqrt((3-x^2*(1-sqrt(7)))/(3-x^2*(1+sqrt(7))))*sqrt(-3+x^2*(1+sqrt(7)))/(7^(1/4)*sqrt(6)*sqrt(-3+2*x^2+2*x^4)*sqrt(1/(3-x^2*(1+sqrt(7)))))],
[1/sqrt(-3+x^2+2*x^4),x,1,EllipticF(x*sqrt(5/3)/sqrt(-1+x^2),sqrt(3/5))*sqrt(-1+x^2)*sqrt(3+2*x^2)/(sqrt(5)*sqrt(-3+x^2+2*x^4))],
[1/sqrt(-3+2*x^4),x,1,EllipticF(2^(3/4)*3^(1/4)*x/sqrt(-3+x^2*sqrt(6)),sqrt(1/2))*sqrt(-3+x^2*sqrt(6))*sqrt((3+x^2*sqrt(6))/(3-x^2*sqrt(6)))/(6^(3/4)*sqrt(-3+2*x^4)*sqrt(1/(3-x^2*sqrt(6))))],
[1/sqrt(-3-x^2+2*x^4),x,1,EllipticF(x*sqrt(5)/sqrt(-3+2*x^2),sqrt(2/5))*sqrt(1+x^2)*sqrt(-3+2*x^2)/(sqrt(5)*sqrt(-3-x^2+2*x^4))],
[1/sqrt(-3-2*x^2+2*x^4),x,1,EllipticF(7^(1/4)*x*sqrt(2)/sqrt(-3-x^2*(1-sqrt(7))),sqrt(1/14*(7-sqrt(7))))*sqrt(-3-x^2*(1-sqrt(7)))*sqrt((3+x^2*(1+sqrt(7)))/(3+x^2*(1-sqrt(7))))/(7^(1/4)*sqrt(6)*sqrt(-3-2*x^2+2*x^4)*sqrt(1/(3+x^2*(1-sqrt(7)))))],
[1/sqrt(-3-3*x^2+2*x^4),x,1,1/2*EllipticF(33^(1/4)*x*sqrt(2)/sqrt(-6-x^2*(3-sqrt(33))),sqrt(1/22*(11-sqrt(33))))*sqrt(-6-x^2*(3-sqrt(33)))*sqrt((6+x^2*(3+sqrt(33)))/(6+x^2*(3-sqrt(33))))/(3^(3/4)*11^(1/4)*sqrt(-3-3*x^2+2*x^4)*sqrt(1/(6+x^2*(3-sqrt(33)))))],
[1/sqrt(-3-4*x^2+2*x^4),x,1,EllipticF(2^(3/4)*5^(1/4)*x/sqrt(-3-x^2*(2-sqrt(10))),sqrt(1/10*(5-sqrt(10))))*sqrt(-3-x^2*(2-sqrt(10)))*sqrt((3+x^2*(2+sqrt(10)))/(3+x^2*(2-sqrt(10))))/(2^(3/4)*5^(1/4)*sqrt(3)*sqrt(-3-4*x^2+2*x^4)*sqrt(1/(3+x^2*(2-sqrt(10)))))],
[1/sqrt(-3-5*x^2+2*x^4),x,1,EllipticF(x*sqrt(7)/sqrt(-3+x^2),sqrt(1/7))*sqrt(-3+x^2)*sqrt(1+2*x^2)/(sqrt(7)*sqrt(-3-5*x^2+2*x^4))],
[1/sqrt(2+5*x^2+3*x^4),x,1,(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-1/2))*sqrt((2+3*x^2)/(1+x^2))/(sqrt(2)*sqrt(2+5*x^2+3*x^4))],
[1/sqrt(2+4*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/2+(-1)/sqrt(6)))*(2+x^2*sqrt(6))*sqrt((2+4*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2+4*x^2+3*x^4))],
[1/sqrt(2+3*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/8*(4-sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2+3*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2+3*x^2+3*x^4))],
[1/sqrt(2+2*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/12*(6-sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2+2*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2+2*x^2+3*x^4))],
[1/sqrt(2+x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/24*(12-sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2+x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2+x^2+3*x^4))],
[1/sqrt(2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/2))*(2+x^2*sqrt(6))*sqrt((2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2+3*x^4))],
[1/sqrt(2-x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/24*(12+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2-x^2+3*x^4))],
[1/sqrt(2-2*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/12*(6+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-2*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2-2*x^2+3*x^4))],
[1/sqrt(2-3*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/8*(4+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-3*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2-3*x^2+3*x^4))],
[1/sqrt(2-4*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/2+1/sqrt(6)))*(2+x^2*sqrt(6))*sqrt((2-4*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2-4*x^2+3*x^4))],
[1/sqrt(2-5*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/24*(12+5*sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-5*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2-5*x^2+3*x^4))],
[1/sqrt(2-6*x^2+3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/4*(2+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-6*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(2-6*x^2+3*x^4))],
[1/sqrt(3+9*x^2+2*x^4),x,1,sqrt(cos(arctan(x*sqrt(1/6*(9+sqrt(57)))))^2)/cos(arctan(x*sqrt(1/6*(9+sqrt(57)))))*EllipticF(sin(arctan(x*sqrt(1/6*(9+sqrt(57))))),sqrt(1/4*(-19+3*sqrt(57))))*(6+x^2*(9+sqrt(57)))*sqrt((6+x^2*(9-sqrt(57)))/(6+x^2*(9+sqrt(57))))/(sqrt(3+9*x^2+2*x^4)*sqrt(6*(9+sqrt(57))))],
[1/sqrt(3+8*x^2+2*x^4),x,1,sqrt(cos(arctan(x*sqrt(1/3*(4+sqrt(10)))))^2)/cos(arctan(x*sqrt(1/3*(4+sqrt(10)))))*EllipticF(sin(arctan(x*sqrt(1/3*(4+sqrt(10))))),sqrt(-2/3*(5-2*sqrt(10))))*(3+x^2*(4+sqrt(10)))*sqrt((3+x^2*(4-sqrt(10)))/(3+x^2*(4+sqrt(10))))/(sqrt(3+8*x^2+2*x^4)*sqrt(3*(4+sqrt(10))))],
[1/sqrt(3+7*x^2+2*x^4),x,1,(1+2*x^2)*sqrt(cos(arctan(x*sqrt(2)))^2)/cos(arctan(x*sqrt(2)))*EllipticF(sin(arctan(x*sqrt(2))),sqrt(5/6))*sqrt((3+x^2)/(1+2*x^2))/(sqrt(6)*sqrt(3+7*x^2+2*x^4))],
[1/sqrt(3+6*x^2+2*x^4),x,1,sqrt(cos(arctan(x*sqrt(1/3*(3+sqrt(3)))))^2)/cos(arctan(x*sqrt(1/3*(3+sqrt(3)))))*EllipticF(sin(arctan(x*sqrt(1/3*(3+sqrt(3))))),sqrt(-1+sqrt(3)))*(3+x^2*(3+sqrt(3)))*sqrt((3+x^2*(3-sqrt(3)))/(3+x^2*(3+sqrt(3))))/(sqrt(3+6*x^2+2*x^4)*sqrt(3*(3+sqrt(3))))],
[1/sqrt(3+5*x^2+2*x^4),x,1,(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/3))*sqrt((3+2*x^2)/(1+x^2))/(sqrt(3)*sqrt(3+5*x^2+2*x^4))],
[1/sqrt(3+4*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/2+(-1)/sqrt(6)))*(3+x^2*sqrt(6))*sqrt((3+4*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3+4*x^2+2*x^4))],
[1/sqrt(3+3*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/8*(4-sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3+3*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3+3*x^2+2*x^4))],
[1/sqrt(3+2*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/12*(6-sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3+2*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3+2*x^2+2*x^4))],
[1/sqrt(3+x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/24*(12-sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3+x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3+x^2+2*x^4))],
[1/sqrt(3+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/2))*(3+x^2*sqrt(6))*sqrt((3+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3+2*x^4))],
[1/sqrt(3-x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/24*(12+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-x^2+2*x^4))],
[1/sqrt(3-2*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/12*(6+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-2*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-2*x^2+2*x^4))],
[1/sqrt(3-3*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/8*(4+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-3*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-3*x^2+2*x^4))],
[1/sqrt(3-4*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/2+1/sqrt(6)))*(3+x^2*sqrt(6))*sqrt((3-4*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-4*x^2+2*x^4))],
[1/sqrt(3-5*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/24*(12+5*sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-5*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-5*x^2+2*x^4))],
[1/sqrt(3-6*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/4*(2+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-6*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-6*x^2+2*x^4))],
[1/sqrt(3-7*x^2+2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/24*(12+7*sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-7*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(3-7*x^2+2*x^4))],
[1/sqrt(-3+7*x^2-2*x^4),x,2,-sqrt(cos(arccos(x/sqrt(3)))^2)/cos(arccos(x/sqrt(3)))*EllipticF(sin(arccos(x/sqrt(3))),sqrt(6/5))/sqrt(5)],
[1/sqrt(-3+6*x^2-2*x^4),x,2,-sqrt(cos(arccos(x*sqrt(1/3*(3-sqrt(3)))))^2)/cos(arccos(x*sqrt(1/3*(3-sqrt(3)))))*EllipticF(sin(arccos(x*sqrt(1/3*(3-sqrt(3))))),sqrt(1/2*(1+sqrt(3))))/(3^(1/4)*sqrt(2))],
[1/sqrt(-3+5*x^2-2*x^4),x,2,-sqrt(cos(arccos(x*sqrt(2/3)))^2)/cos(arccos(x*sqrt(2/3)))*EllipticF(sin(arccos(x*sqrt(2/3))),sqrt(3))],
[1/sqrt(-3+4*x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/2+1/sqrt(6)))*(3+x^2*sqrt(6))*sqrt((3-4*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3+4*x^2-2*x^4))],
[1/sqrt(-3+3*x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/8*(4+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-3*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3+3*x^2-2*x^4))],
[1/sqrt(-3+2*x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/12*(6+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-2*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3+2*x^2-2*x^4))],
[1/sqrt(-3+x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/24*(12+sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3-x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3+x^2-2*x^4))],
[1/sqrt(-3-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/2))*(3+x^2*sqrt(6))*sqrt((3+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3-2*x^4))],
[1/sqrt(-3-x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/24*(12-sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3+x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3-x^2-2*x^4))],
[1/sqrt(-3-2*x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/12*(6-sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3+2*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3-2*x^2-2*x^4))],
[1/sqrt(-3-3*x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/8*(4-sqrt(6))))*(3+x^2*sqrt(6))*sqrt((3+3*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3-3*x^2-2*x^4))],
[1/sqrt(-3-4*x^2-2*x^4),x,1,1/2*sqrt(cos(2*arctan((2/3)^(1/4)*x))^2)/cos(2*arctan((2/3)^(1/4)*x))*EllipticF(sin(2*arctan((2/3)^(1/4)*x)),sqrt(1/2+(-1)/sqrt(6)))*(3+x^2*sqrt(6))*sqrt((3+4*x^2+2*x^4)/(3+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-3-4*x^2-2*x^4))],
[1/sqrt(-3-5*x^2-2*x^4),x,2,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/3))*sqrt(3+2*x^2)/(sqrt(3)*sqrt(-1-x^2)*sqrt((3+2*x^2)/(1+x^2)))],
[1/sqrt(-2+6*x^2-3*x^4),x,2,-sqrt(cos(arccos(x*sqrt(3/(3+sqrt(3)))))^2)/cos(arccos(x*sqrt(3/(3+sqrt(3)))))*EllipticF(sin(arccos(x*sqrt(3/(3+sqrt(3))))),sqrt(1/2*(1+sqrt(3))))/(3^(1/4)*sqrt(2))],
[1/sqrt(-2+5*x^2-3*x^4),x,2,-sqrt(cos(arccos(x))^2)/cos(arccos(x))*EllipticF(sin(arccos(x)),sqrt(3))],
[1/sqrt(-2+4*x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/2+1/sqrt(6)))*(2+x^2*sqrt(6))*sqrt((2-4*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2+4*x^2-3*x^4))],
[1/sqrt(-2+3*x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/8*(4+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-3*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2+3*x^2-3*x^4))],
[1/sqrt(-2+2*x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/12*(6+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-2*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2+2*x^2-3*x^4))],
[1/sqrt(-2+x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/24*(12+sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2-x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2+x^2-3*x^4))],
[1/sqrt(-2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/2))*(2+x^2*sqrt(6))*sqrt((2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2-3*x^4))],
[1/sqrt(-2-x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/24*(12-sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2+x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2-x^2-3*x^4))],
[1/sqrt(-2-2*x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/12*(6-sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2+2*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2-2*x^2-3*x^4))],
[1/sqrt(-2-3*x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/8*(4-sqrt(6))))*(2+x^2*sqrt(6))*sqrt((2+3*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2-3*x^2-3*x^4))],
[1/sqrt(-2-4*x^2-3*x^4),x,1,1/2*sqrt(cos(2*arctan((3/2)^(1/4)*x))^2)/cos(2*arctan((3/2)^(1/4)*x))*EllipticF(sin(2*arctan((3/2)^(1/4)*x)),sqrt(1/2+(-1)/sqrt(6)))*(2+x^2*sqrt(6))*sqrt((2+4*x^2+3*x^4)/(2+x^2*sqrt(6))^2)/(6^(1/4)*sqrt(-2-4*x^2-3*x^4))],
[1/sqrt(-2-5*x^2-3*x^4),x,2,-sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-1/2))*sqrt(-2-3*x^2)/(sqrt(2)*sqrt(1+x^2)*sqrt((2+3*x^2)/(1+x^2)))],
[1/sqrt(2+5*x^2+5*x^4),x,1,1/2*sqrt(cos(2*arctan((5/2)^(1/4)*x))^2)/cos(2*arctan((5/2)^(1/4)*x))*EllipticF(sin(2*arctan((5/2)^(1/4)*x)),sqrt(1/8*(4-sqrt(10))))*(2+x^2*sqrt(10))*sqrt((2+5*x^2+5*x^4)/(2+x^2*sqrt(10))^2)/(10^(1/4)*sqrt(2+5*x^2+5*x^4))],
[1/sqrt(2+5*x^2+4*x^4),x,1,1/2*sqrt(cos(2*arctan(2^(1/4)*x))^2)/cos(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),sqrt(1/16*(8-5*sqrt(2))))*(1+x^2*sqrt(2))*sqrt((2+5*x^2+4*x^4)/(1+x^2*sqrt(2))^2)/(2^(3/4)*sqrt(2+5*x^2+4*x^4))],
[1/sqrt(2+5*x^2+3*x^4),x,1,(1+x^2)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-1/2))*sqrt((2+3*x^2)/(1+x^2))/(sqrt(2)*sqrt(2+5*x^2+3*x^4))],
[1/sqrt(2+5*x^2+2*x^4),x,1,1/2*(1+2*x^2)*sqrt(cos(arctan(x*sqrt(2)))^2)/cos(arctan(x*sqrt(2)))*EllipticF(sin(arctan(x*sqrt(2))),sqrt(3/4))*sqrt((2+x^2)/(1+2*x^2))/sqrt(2+5*x^2+2*x^4)],
[1/sqrt(2+5*x^2+x^4),x,1,1/2*sqrt(cos(arctan(1/2*x*sqrt(5+sqrt(17))))^2)/cos(arctan(1/2*x*sqrt(5+sqrt(17))))*EllipticF(sin(arctan(1/2*x*sqrt(5+sqrt(17)))),sqrt(1/4*(-17+5*sqrt(17))))*(4+x^2*(5+sqrt(17)))*sqrt((4+x^2*(5-sqrt(17)))/(4+x^2*(5+sqrt(17))))/(sqrt(2+5*x^2+x^4)*sqrt(5+sqrt(17)))],
[1/sqrt(2+5*x^2-x^4),x,2,EllipticF(x*sqrt(2/(5+sqrt(33))),sqrt(1/4*(-29-5*sqrt(33))))*sqrt(2/(-5+sqrt(33)))],
[1/sqrt(2+5*x^2-2*x^4),x,2,EllipticF(2*x/sqrt(5+sqrt(41)),sqrt(1/8*(-33-5*sqrt(41))))*sqrt(2/(-5+sqrt(41)))],
[1/sqrt(2+5*x^2-3*x^4),x,2,EllipticF(x/sqrt(2),sqrt(-6))],
[1/sqrt(2+5*x^2-4*x^4),x,2,EllipticF(2*x*sqrt(2/(5+sqrt(57))),sqrt(1/16*(-41-5*sqrt(57))))*sqrt(2/(-5+sqrt(57)))],
[1/sqrt(2+5*x^2-5*x^4),x,2,EllipticF(x*sqrt(10/(5+sqrt(65))),sqrt(1/4*(-9-sqrt(65))))*sqrt(2/(-5+sqrt(65)))],
[1/sqrt(2+5*x^2-6*x^4),x,2,EllipticF(2*x*sqrt(3/(5+sqrt(73))),sqrt(1/24*(-49-5*sqrt(73))))*sqrt(2/(-5+sqrt(73)))],
[1/sqrt(2+5*x^2-7*x^4),x,2,EllipticF(x,sqrt(-7/2))/sqrt(2)],
[1/sqrt(2+5*x^2-8*x^4),x,2,EllipticF(4*x/sqrt(5+sqrt(89)),sqrt(1/32*(-57-5*sqrt(89))))*sqrt(2/(-5+sqrt(89)))],
[1/sqrt(2+5*x^2-9*x^4),x,2,EllipticF(3*x*sqrt(2/(5+sqrt(97))),sqrt(1/36*(-61-5*sqrt(97))))*sqrt(2/(-5+sqrt(97)))],

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

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

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

# p>0
[x^2*(b*x^2+c*x^4),x,2,1/5*b*x^5+1/7*c*x^7],
[x*(b*x^2+c*x^4),x,2,1/4*b*x^4+1/6*c*x^6],
[b*x^2+c*x^4,x,1,1/3*b*x^3+1/5*c*x^5],
[(b*x^2+c*x^4)/x,x,2,1/2*b*x^2+1/4*c*x^4],
[(b*x^2+c*x^4)/x^2,x,2,b*x+1/3*c*x^3],
[(b*x^2+c*x^4)/x^3,x,2,1/2*c*x^2+b*log(x)],
[(b*x^2+c*x^4)/x^4,x,2,-b/x+c*x],
[(b*x^2+c*x^4)/x^5,x,2,-1/2*b/x^2+c*log(x)],
[(b*x^2+c*x^4)/x^6,x,2,-1/3*b/x^3-c/x],
[(b*x^2+c*x^4)/x^7,x,2,-1/4*b/x^4-1/2*c/x^2],
[(b*x^2+c*x^4)/x^8,x,2,-1/5*b/x^5-1/3*c/x^3],
[(b*x^2+c*x^4)^2,x,3,1/5*b^2*x^5+2/7*b*c*x^7+1/9*c^2*x^9],
[(b*x^2+c*x^4)^2/x,x,4,1/4*b^2*x^4+1/3*b*c*x^6+1/8*c^2*x^8],
[(b*x^2+c*x^4)^2/x^2,x,3,1/3*b^2*x^3+2/5*b*c*x^5+1/7*c^2*x^7],
[(b*x^2+c*x^4)^2/x^3,x,2,1/6*(b+c*x^2)^3/c],
[(b*x^2+c*x^4)^2/x^4,x,3,b^2*x+2/3*b*c*x^3+1/5*c^2*x^5],
[(b*x^2+c*x^4)^2/x^5,x,4,b*c*x^2+1/4*c^2*x^4+b^2*log(x)],
[(b*x^2+c*x^4)^2/x^6,x,3,-b^2/x+2*b*c*x+1/3*c^2*x^3],
[(b*x^2+c*x^4)^2/x^7,x,4,-1/2*b^2/x^2+1/2*c^2*x^2+2*b*c*log(x)],
[(b*x^2+c*x^4)^2/x^8,x,3,-1/3*b^2/x^3-2*b*c/x+c^2*x],
[(b*x^2+c*x^4)^2/x^9,x,4,-1/4*b^2/x^4-b*c/x^2+c^2*log(x)],
[(b*x^2+c*x^4)^2/x^10,x,3,-1/5*b^2/x^5-2/3*b*c/x^3-c^2/x],
[(b*x^2+c*x^4)^2/x^11,x,2,-1/6*(b+c*x^2)^3/(b*x^6)],
[(b*x^2+c*x^4)^2/x^12,x,3,-1/7*b^2/x^7-2/5*b*c/x^5-1/3*c^2/x^3],
[(b*x^2+c*x^4)^3/x^2,x,3,1/5*b^3*x^5+3/7*b^2*c*x^7+1/3*b*c^2*x^9+1/11*c^3*x^11],
[(b*x^2+c*x^4)^3/x^3,x,4,-1/8*b*(b+c*x^2)^4/c^2+1/10*(b+c*x^2)^5/c^2],
[(b*x^2+c*x^4)^3/x^4,x,3,1/3*b^3*x^3+3/5*b^2*c*x^5+3/7*b*c^2*x^7+1/9*c^3*x^9],
[(b*x^2+c*x^4)^3/x^5,x,2,1/8*(b+c*x^2)^4/c],
[(b*x^2+c*x^4)^3/x^6,x,3,b^3*x+b^2*c*x^3+3/5*b*c^2*x^5+1/7*c^3*x^7],
[(b*x^2+c*x^4)^3/x^7,x,4,3/2*b^2*c*x^2+3/4*b*c^2*x^4+1/6*c^3*x^6+b^3*log(x)],
[(b*x^2+c*x^4)^3/x^8,x,3,-b^3/x+3*b^2*c*x+b*c^2*x^3+1/5*c^3*x^5],
[(b*x^2+c*x^4)^3/x^9,x,4,-1/2*b^3/x^2+3/2*b*c^2*x^2+1/4*c^3*x^4+3*b^2*c*log(x)],
[(b*x^2+c*x^4)^3/x^10,x,3,-1/3*b^3/x^3-3*b^2*c/x+3*b*c^2*x+1/3*c^3*x^3],
[(b*x^2+c*x^4)^3/x^11,x,4,-1/4*b^3/x^4-3/2*b^2*c/x^2+1/2*c^3*x^2+3*b*c^2*log(x)],
[(b*x^2+c*x^4)^3/x^12,x,3,-1/5*b^3/x^5-b^2*c/x^3-3*b*c^2/x+c^3*x],
[(b*x^2+c*x^4)^3/x^13,x,4,-1/6*b^3/x^6-3/4*b^2*c/x^4-3/2*b*c^2/x^2+c^3*log(x)],
[(b*x^2+c*x^4)^3/x^14,x,3,-1/7*b^3/x^7-3/5*b^2*c/x^5-b*c^2/x^3-c^3/x],
[(b*x^2+c*x^4)^3/x^15,x,2,-1/8*(b+c*x^2)^4/(b*x^8)],
[(b*x^2+c*x^4)^3/x^16,x,3,-1/9*b^3/x^9-3/7*b^2*c/x^7-3/5*b*c^2/x^5-1/3*c^3/x^3],
[(b*x^2+c*x^4)^3/x^17,x,4,-1/10*(b+c*x^2)^4/(b*x^10)+1/40*c*(b+c*x^2)^4/(b^2*x^8)],

# p<0
[x^10/(b*x^2+c*x^4),x,4,-b^3*x/c^4+1/3*b^2*x^3/c^3-1/5*b*x^5/c^2+1/7*x^7/c+b^(7/2)*arctan(x*sqrt(c)/sqrt(b))/c^(9/2)],
[x^9/(b*x^2+c*x^4),x,4,1/2*b^2*x^2/c^3-1/4*b*x^4/c^2+1/6*x^6/c-1/2*b^3*log(b+c*x^2)/c^4],
[x^8/(b*x^2+c*x^4),x,4,b^2*x/c^3-1/3*b*x^3/c^2+1/5*x^5/c-b^(5/2)*arctan(x*sqrt(c)/sqrt(b))/c^(7/2)],
[x^7/(b*x^2+c*x^4),x,4,-1/2*b*x^2/c^2+1/4*x^4/c+1/2*b^2*log(b+c*x^2)/c^3],
[x^6/(b*x^2+c*x^4),x,4,-b*x/c^2+1/3*x^3/c+b^(3/2)*arctan(x*sqrt(c)/sqrt(b))/c^(5/2)],
[x^5/(b*x^2+c*x^4),x,4,1/2*x^2/c-1/2*b*log(b+c*x^2)/c^2],
[x^4/(b*x^2+c*x^4),x,3,x/c-arctan(x*sqrt(c)/sqrt(b))*sqrt(b)/c^(3/2)],
[x^3/(b*x^2+c*x^4),x,2,1/2*log(b+c*x^2)/c],
[x^2/(b*x^2+c*x^4),x,2,arctan(x*sqrt(c)/sqrt(b))/(sqrt(b)*sqrt(c))],
[x/(b*x^2+c*x^4),x,5,log(x)/b-1/2*log(b+c*x^2)/b],
[1/(b*x^2+c*x^4),x,3,(-1)/(b*x)-arctan(x*sqrt(c)/sqrt(b))*sqrt(c)/b^(3/2)],
[1/(x*(b*x^2+c*x^4)),x,4,(-1/2)/(b*x^2)-c*log(x)/b^2+1/2*c*log(b+c*x^2)/b^2],
[1/(x^2*(b*x^2+c*x^4)),x,4,(-1/3)/(b*x^3)+c/(b^2*x)+c^(3/2)*arctan(x*sqrt(c)/sqrt(b))/b^(5/2)],
[1/(x^3*(b*x^2+c*x^4)),x,4,(-1/4)/(b*x^4)+1/2*c/(b^2*x^2)+c^2*log(x)/b^3-1/2*c^2*log(b+c*x^2)/b^3],
[1/(x^4*(b*x^2+c*x^4)),x,5,(-1/5)/(b*x^5)+1/3*c/(b^2*x^3)-c^2/(b^3*x)-c^(5/2)*arctan(x*sqrt(c)/sqrt(b))/b^(7/2)],
[1/(x^5*(b*x^2+c*x^4)),x,4,(-1/6)/(b*x^6)+1/4*c/(b^2*x^4)-1/2*c^2/(b^3*x^2)-c^3*log(x)/b^4+1/2*c^3*log(b+c*x^2)/b^4],
[x^12/(b*x^2+c*x^4)^2,x,5,7/2*b^2*x/c^4-7/6*b*x^3/c^3+7/10*x^5/c^2-1/2*x^7/(c*(b+c*x^2))-7/2*b^(5/2)*arctan(x*sqrt(c)/sqrt(b))/c^(9/2)],
[x^11/(b*x^2+c*x^4)^2,x,4,-b*x^2/c^3+1/4*x^4/c^2+1/2*b^3/(c^4*(b+c*x^2))+3/2*b^2*log(b+c*x^2)/c^4],
[x^10/(b*x^2+c*x^4)^2,x,5,-5/2*b*x/c^3+5/6*x^3/c^2-1/2*x^5/(c*(b+c*x^2))+5/2*b^(3/2)*arctan(x*sqrt(c)/sqrt(b))/c^(7/2)],
[x^9/(b*x^2+c*x^4)^2,x,4,1/2*x^2/c^2-1/2*b^2/(c^3*(b+c*x^2))-b*log(b+c*x^2)/c^3],
[x^8/(b*x^2+c*x^4)^2,x,4,3/2*x/c^2-1/2*x^3/(c*(b+c*x^2))-3/2*arctan(x*sqrt(c)/sqrt(b))*sqrt(b)/c^(5/2)],
[x^7/(b*x^2+c*x^4)^2,x,4,1/2*b/(c^2*(b+c*x^2))+1/2*log(b+c*x^2)/c^2],
[x^6/(b*x^2+c*x^4)^2,x,3,-1/2*x/(c*(b+c*x^2))+1/2*arctan(x*sqrt(c)/sqrt(b))/(c^(3/2)*sqrt(b))],
[x^5/(b*x^2+c*x^4)^2,x,2,(-1/2)/(c*(b+c*x^2))],
[x^4/(b*x^2+c*x^4)^2,x,3,1/2*x/(b*(b+c*x^2))+1/2*arctan(x*sqrt(c)/sqrt(b))/(b^(3/2)*sqrt(c))],
[x^3/(b*x^2+c*x^4)^2,x,4,1/2/(b*(b+c*x^2))+log(x)/b^2-1/2*log(b+c*x^2)/b^2],
[x^2/(b*x^2+c*x^4)^2,x,4,(-3/2)/(b^2*x)+1/2/(b*x*(b+c*x^2))-3/2*arctan(x*sqrt(c)/sqrt(b))*sqrt(c)/b^(5/2)],
[x/(b*x^2+c*x^4)^2,x,4,(-1/2)/(b^2*x^2)-1/2*c/(b^2*(b+c*x^2))-2*c*log(x)/b^3+c*log(b+c*x^2)/b^3],
[1/(b*x^2+c*x^4)^2,x,5,(-5/6)/(b^2*x^3)+5/2*c/(b^3*x)+1/2/(b*x^3*(b+c*x^2))+5/2*c^(3/2)*arctan(x*sqrt(c)/sqrt(b))/b^(7/2)],
[1/(x*(b*x^2+c*x^4)^2),x,4,(-1/4)/(b^2*x^4)+c/(b^3*x^2)+1/2*c^2/(b^3*(b+c*x^2))+3*c^2*log(x)/b^4-3/2*c^2*log(b+c*x^2)/b^4],
[1/(x^2*(b*x^2+c*x^4)^2),x,6,(-7/10)/(b^2*x^5)+7/6*c/(b^3*x^3)-7/2*c^2/(b^4*x)+1/2/(b*x^5*(b+c*x^2))-7/2*c^(5/2)*arctan(x*sqrt(c)/sqrt(b))/b^(9/2)],
[x^14/(b*x^2+c*x^4)^3,x,6,-35/8*b*x/c^4+35/24*x^3/c^3-1/4*x^7/(c*(b+c*x^2)^2)-7/8*x^5/(c^2*(b+c*x^2))+35/8*b^(3/2)*arctan(x*sqrt(c)/sqrt(b))/c^(9/2)],
[x^13/(b*x^2+c*x^4)^3,x,4,1/2*x^2/c^3+1/4*b^3/(c^4*(b+c*x^2)^2)-3/2*b^2/(c^4*(b+c*x^2))-3/2*b*log(b+c*x^2)/c^4],
[x^12/(b*x^2+c*x^4)^3,x,5,15/8*x/c^3-1/4*x^5/(c*(b+c*x^2)^2)-5/8*x^3/(c^2*(b+c*x^2))-15/8*arctan(x*sqrt(c)/sqrt(b))*sqrt(b)/c^(7/2)],
[x^11/(b*x^2+c*x^4)^3,x,4,-1/4*b^2/(c^3*(b+c*x^2)^2)+b/(c^3*(b+c*x^2))+1/2*log(b+c*x^2)/c^3],
[x^10/(b*x^2+c*x^4)^3,x,4,-1/4*x^3/(c*(b+c*x^2)^2)-3/8*x/(c^2*(b+c*x^2))+3/8*arctan(x*sqrt(c)/sqrt(b))/(c^(5/2)*sqrt(b))],
[x^9/(b*x^2+c*x^4)^3,x,2,1/4*x^4/(b*(b+c*x^2)^2)],
[x^8/(b*x^2+c*x^4)^3,x,4,-1/4*x/(c*(b+c*x^2)^2)+1/8*x/(b*c*(b+c*x^2))+1/8*arctan(x*sqrt(c)/sqrt(b))/(b^(3/2)*c^(3/2))],
[x^7/(b*x^2+c*x^4)^3,x,2,(-1/4)/(c*(b+c*x^2)^2)],
[x^6/(b*x^2+c*x^4)^3,x,4,1/4*x/(b*(b+c*x^2)^2)+3/8*x/(b^2*(b+c*x^2))+3/8*arctan(x*sqrt(c)/sqrt(b))/(b^(5/2)*sqrt(c))],
[x^5/(b*x^2+c*x^4)^3,x,4,1/4/(b*(b+c*x^2)^2)+1/2/(b^2*(b+c*x^2))+log(x)/b^3-1/2*log(b+c*x^2)/b^3],
[x^4/(b*x^2+c*x^4)^3,x,5,(-15/8)/(b^3*x)+1/4/(b*x*(b+c*x^2)^2)+5/8/(b^2*x*(b+c*x^2))-15/8*arctan(x*sqrt(c)/sqrt(b))*sqrt(c)/b^(7/2)],
[x^3/(b*x^2+c*x^4)^3,x,4,(-1/2)/(b^3*x^2)-1/4*c/(b^2*(b+c*x^2)^2)-c/(b^3*(b+c*x^2))-3*c*log(x)/b^4+3/2*c*log(b+c*x^2)/b^4],
[x^2/(b*x^2+c*x^4)^3,x,6,(-35/24)/(b^3*x^3)+35/8*c/(b^4*x)+1/4/(b*x^3*(b+c*x^2)^2)+7/8/(b^2*x^3*(b+c*x^2))+35/8*c^(3/2)*arctan(x*sqrt(c)/sqrt(b))/b^(9/2)],
[x/(b*x^2+c*x^4)^3,x,4,(-1/4)/(b^3*x^4)+3/2*c/(b^4*x^2)+1/4*c^2/(b^3*(b+c*x^2)^2)+3/2*c^2/(b^4*(b+c*x^2))+6*c^2*log(x)/b^5-3*c^2*log(b+c*x^2)/b^5],
[1/(b*x^2+c*x^4)^3,x,7,(-63/40)/(b^3*x^5)+21/8*c/(b^4*x^3)-63/8*c^2/(b^5*x)+1/4/(b*x^5*(b+c*x^2)^2)+9/8/(b^2*x^5*(b+c*x^2))-63/8*c^(5/2)*arctan(x*sqrt(c)/sqrt(b))/b^(11/2)],
[1/(x*(b*x^2+c*x^4)^3),x,4,(-1/6)/(b^3*x^6)+3/4*c/(b^4*x^4)-3*c^2/(b^5*x^2)-1/4*c^3/(b^4*(b+c*x^2)^2)-2*c^3/(b^5*(b+c*x^2))-10*c^3*log(x)/b^6+5*c^3*log(b+c*x^2)/b^6],

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

# p>0
[x^5*sqrt(b*x^2+c*x^4),x,6,-5/48*b*(b*x^2+c*x^4)^(3/2)/c^2+1/8*x^2*(b*x^2+c*x^4)^(3/2)/c-5/128*b^4*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(7/2)+5/128*b^2*(b+2*c*x^2)*sqrt(b*x^2+c*x^4)/c^3],
[x^3*sqrt(b*x^2+c*x^4),x,5,1/6*(b*x^2+c*x^4)^(3/2)/c+1/16*b^3*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(5/2)-1/16*b*(b+2*c*x^2)*sqrt(b*x^2+c*x^4)/c^2],
[x*sqrt(b*x^2+c*x^4),x,4,-1/8*b^2*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(3/2)+1/8*(b+2*c*x^2)*sqrt(b*x^2+c*x^4)/c],
[sqrt(b*x^2+c*x^4)/x,x,4,1/2*b*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/sqrt(c)+1/2*sqrt(b*x^2+c*x^4)],
[sqrt(b*x^2+c*x^4)/x^3,x,4,arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))*sqrt(c)-sqrt(b*x^2+c*x^4)/x^2],
[sqrt(b*x^2+c*x^4)/x^5,x,1,-1/3*(b*x^2+c*x^4)^(3/2)/(b*x^6)],
[sqrt(b*x^2+c*x^4)/x^7,x,2,-1/5*(b*x^2+c*x^4)^(3/2)/(b*x^8)+2/15*c*(b*x^2+c*x^4)^(3/2)/(b^2*x^6)],
[sqrt(b*x^2+c*x^4)/x^9,x,3,-1/7*(b*x^2+c*x^4)^(3/2)/(b*x^10)+4/35*c*(b*x^2+c*x^4)^(3/2)/(b^2*x^8)-8/105*c^2*(b*x^2+c*x^4)^(3/2)/(b^3*x^6)],
[sqrt(b*x^2+c*x^4)/x^11,x,4,-1/9*(b*x^2+c*x^4)^(3/2)/(b*x^12)+2/21*c*(b*x^2+c*x^4)^(3/2)/(b^2*x^10)-8/105*c^2*(b*x^2+c*x^4)^(3/2)/(b^3*x^8)+16/315*c^3*(b*x^2+c*x^4)^(3/2)/(b^4*x^6)],
[sqrt(b*x^2+c*x^4)/x^13,x,5,-1/11*(b*x^2+c*x^4)^(3/2)/(b*x^14)+8/99*c*(b*x^2+c*x^4)^(3/2)/(b^2*x^12)-16/231*c^2*(b*x^2+c*x^4)^(3/2)/(b^3*x^10)+64/1155*c^3*(b*x^2+c*x^4)^(3/2)/(b^4*x^8)-128/3465*c^4*(b*x^2+c*x^4)^(3/2)/(b^5*x^6)],
[x^4*sqrt(b*x^2+c*x^4),x,3,8/105*b^2*(b*x^2+c*x^4)^(3/2)/(c^3*x^3)-4/35*b*(b*x^2+c*x^4)^(3/2)/(c^2*x)+1/7*x*(b*x^2+c*x^4)^(3/2)/c],
[x^2*sqrt(b*x^2+c*x^4),x,2,-2/15*b*(b*x^2+c*x^4)^(3/2)/(c^2*x^3)+1/5*(b*x^2+c*x^4)^(3/2)/(c*x)],
[sqrt(b*x^2+c*x^4),x,1,1/3*(b*x^2+c*x^4)^(3/2)/(c*x^3)],
[sqrt(b*x^2+c*x^4)/x^2,x,3,-arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))*sqrt(b)+sqrt(b*x^2+c*x^4)/x],
[sqrt(b*x^2+c*x^4)/x^4,x,3,-1/2*c*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/sqrt(b)-1/2*sqrt(b*x^2+c*x^4)/x^3],
[sqrt(b*x^2+c*x^4)/x^6,x,4,1/8*c^2*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(3/2)-1/4*sqrt(b*x^2+c*x^4)/x^5-1/8*c*sqrt(b*x^2+c*x^4)/(b*x^3)],
[sqrt(b*x^2+c*x^4)/x^8,x,5,-1/16*c^3*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(5/2)-1/6*sqrt(b*x^2+c*x^4)/x^7-1/24*c*sqrt(b*x^2+c*x^4)/(b*x^5)+1/16*c^2*sqrt(b*x^2+c*x^4)/(b^2*x^3)],
[x^3*(b*x^2+c*x^4)^(3/2),x,6,-1/32*b*(b+2*c*x^2)*(b*x^2+c*x^4)^(3/2)/c^2+1/10*(b*x^2+c*x^4)^(5/2)/c-3/256*b^5*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(7/2)+3/256*b^3*(b+2*c*x^2)*sqrt(b*x^2+c*x^4)/c^3],
[x*(b*x^2+c*x^4)^(3/2),x,5,1/16*(b+2*c*x^2)*(b*x^2+c*x^4)^(3/2)/c+3/128*b^4*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(5/2)-3/128*b^2*(b+2*c*x^2)*sqrt(b*x^2+c*x^4)/c^2],
[(b*x^2+c*x^4)^(3/2)/x,x,5,1/6*(b*x^2+c*x^4)^(3/2)-1/16*b^3*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(3/2)+1/16*b*(b+2*c*x^2)*sqrt(b*x^2+c*x^4)/c],
[(b*x^2+c*x^4)^(3/2)/x^3,x,5,1/4*(b*x^2+c*x^4)^(3/2)/x^2+3/8*b^2*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/sqrt(c)+3/8*b*sqrt(b*x^2+c*x^4)],
[(b*x^2+c*x^4)^(3/2)/x^5,x,5,-(b*x^2+c*x^4)^(3/2)/x^4+3/2*b*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))*sqrt(c)+3/2*c*sqrt(b*x^2+c*x^4)],
[(b*x^2+c*x^4)^(3/2)/x^7,x,5,-1/3*(b*x^2+c*x^4)^(3/2)/x^6+c^(3/2)*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))-c*sqrt(b*x^2+c*x^4)/x^2],
[(b*x^2+c*x^4)^(3/2)/x^9,x,1,-1/5*(b*x^2+c*x^4)^(5/2)/(b*x^10)],
[(b*x^2+c*x^4)^(3/2)/x^11,x,2,-1/7*(b*x^2+c*x^4)^(5/2)/(b*x^12)+2/35*c*(b*x^2+c*x^4)^(5/2)/(b^2*x^10)],
[(b*x^2+c*x^4)^(3/2)/x^13,x,3,-1/9*(b*x^2+c*x^4)^(5/2)/(b*x^14)+4/63*c*(b*x^2+c*x^4)^(5/2)/(b^2*x^12)-8/315*c^2*(b*x^2+c*x^4)^(5/2)/(b^3*x^10)],
[(b*x^2+c*x^4)^(3/2)/x^15,x,4,-1/11*(b*x^2+c*x^4)^(5/2)/(b*x^16)+2/33*c*(b*x^2+c*x^4)^(5/2)/(b^2*x^14)-8/231*c^2*(b*x^2+c*x^4)^(5/2)/(b^3*x^12)+16/1155*c^3*(b*x^2+c*x^4)^(5/2)/(b^4*x^10)],
[(b*x^2+c*x^4)^(3/2)/x^17,x,5,-1/13*(b*x^2+c*x^4)^(5/2)/(b*x^18)+8/143*c*(b*x^2+c*x^4)^(5/2)/(b^2*x^16)-16/429*c^2*(b*x^2+c*x^4)^(5/2)/(b^3*x^14)+64/3003*c^3*(b*x^2+c*x^4)^(5/2)/(b^4*x^12)-128/15015*c^4*(b*x^2+c*x^4)^(5/2)/(b^5*x^10)],
[x^6*(b*x^2+c*x^4)^(3/2),x,5,128/15015*b^4*(b*x^2+c*x^4)^(5/2)/(c^5*x^5)-64/3003*b^3*(b*x^2+c*x^4)^(5/2)/(c^4*x^3)+16/429*b^2*(b*x^2+c*x^4)^(5/2)/(c^3*x)-8/143*b*x*(b*x^2+c*x^4)^(5/2)/c^2+1/13*x^3*(b*x^2+c*x^4)^(5/2)/c],
[x^4*(b*x^2+c*x^4)^(3/2),x,4,-16/1155*b^3*(b*x^2+c*x^4)^(5/2)/(c^4*x^5)+8/231*b^2*(b*x^2+c*x^4)^(5/2)/(c^3*x^3)-2/33*b*(b*x^2+c*x^4)^(5/2)/(c^2*x)+1/11*x*(b*x^2+c*x^4)^(5/2)/c],
[x^2*(b*x^2+c*x^4)^(3/2),x,3,8/315*b^2*(b*x^2+c*x^4)^(5/2)/(c^3*x^5)-4/63*b*(b*x^2+c*x^4)^(5/2)/(c^2*x^3)+1/9*(b*x^2+c*x^4)^(5/2)/(c*x)],
[(b*x^2+c*x^4)^(3/2),x,2,-2/35*b*(b*x^2+c*x^4)^(5/2)/(c^2*x^5)+1/7*(b*x^2+c*x^4)^(5/2)/(c*x^3)],
[(b*x^2+c*x^4)^(3/2)/x^2,x,1,1/5*(b*x^2+c*x^4)^(5/2)/(c*x^5)],
[(b*x^2+c*x^4)^(3/2)/x^4,x,4,1/3*(b*x^2+c*x^4)^(3/2)/x^3-b^(3/2)*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))+b*sqrt(b*x^2+c*x^4)/x],
[(b*x^2+c*x^4)^(3/2)/x^6,x,4,-1/2*(b*x^2+c*x^4)^(3/2)/x^5-3/2*c*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))*sqrt(b)+3/2*c*sqrt(b*x^2+c*x^4)/x],
[(b*x^2+c*x^4)^(3/2)/x^8,x,4,-1/4*(b*x^2+c*x^4)^(3/2)/x^7-3/8*c^2*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/sqrt(b)-3/8*c*sqrt(b*x^2+c*x^4)/x^3],
[(b*x^2+c*x^4)^(3/2)/x^10,x,5,-1/6*(b*x^2+c*x^4)^(3/2)/x^9+1/16*c^3*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(3/2)-1/8*c*sqrt(b*x^2+c*x^4)/x^5-1/16*c^2*sqrt(b*x^2+c*x^4)/(b*x^3)],
[(b*x^2+c*x^4)^(3/2)/x^12,x,6,-1/8*(b*x^2+c*x^4)^(3/2)/x^11-3/128*c^4*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(5/2)-1/16*c*sqrt(b*x^2+c*x^4)/x^7-1/64*c^2*sqrt(b*x^2+c*x^4)/(b*x^5)+3/128*c^3*sqrt(b*x^2+c*x^4)/(b^2*x^3)],
[(b*x^2+c*x^4)^(3/2)/x^14,x,7,-1/10*(b*x^2+c*x^4)^(3/2)/x^13+3/256*c^5*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(7/2)-3/80*c*sqrt(b*x^2+c*x^4)/x^9-1/160*c^2*sqrt(b*x^2+c*x^4)/(b*x^7)+1/128*c^3*sqrt(b*x^2+c*x^4)/(b^2*x^5)-3/256*c^4*sqrt(b*x^2+c*x^4)/(b^3*x^3)],

# p<0
[x^7/sqrt(b*x^2+c*x^4),x,6,-5/16*b^3*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(7/2)+5/16*b^2*sqrt(b*x^2+c*x^4)/c^3-5/24*b*x^2*sqrt(b*x^2+c*x^4)/c^2+1/6*x^4*sqrt(b*x^2+c*x^4)/c],
[x^5/sqrt(b*x^2+c*x^4),x,5,3/8*b^2*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(5/2)-3/8*b*sqrt(b*x^2+c*x^4)/c^2+1/4*x^2*sqrt(b*x^2+c*x^4)/c],
[x^3/sqrt(b*x^2+c*x^4),x,4,-1/2*b*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(3/2)+1/2*sqrt(b*x^2+c*x^4)/c],
[x/sqrt(b*x^2+c*x^4),x,3,arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/sqrt(c)],
[1/(x*sqrt(b*x^2+c*x^4)),x,1,-sqrt(b*x^2+c*x^4)/(b*x^2)],
[1/(x^3*sqrt(b*x^2+c*x^4)),x,2,-1/3*sqrt(b*x^2+c*x^4)/(b*x^4)+2/3*c*sqrt(b*x^2+c*x^4)/(b^2*x^2)],
[1/(x^5*sqrt(b*x^2+c*x^4)),x,3,-1/5*sqrt(b*x^2+c*x^4)/(b*x^6)+4/15*c*sqrt(b*x^2+c*x^4)/(b^2*x^4)-8/15*c^2*sqrt(b*x^2+c*x^4)/(b^3*x^2)],
[1/(x^7*sqrt(b*x^2+c*x^4)),x,4,-1/7*sqrt(b*x^2+c*x^4)/(b*x^8)+6/35*c*sqrt(b*x^2+c*x^4)/(b^2*x^6)-8/35*c^2*sqrt(b*x^2+c*x^4)/(b^3*x^4)+16/35*c^3*sqrt(b*x^2+c*x^4)/(b^4*x^2)],
[x^4/sqrt(b*x^2+c*x^4),x,2,-2/3*b*sqrt(b*x^2+c*x^4)/(c^2*x)+1/3*x*sqrt(b*x^2+c*x^4)/c],
[x^2/sqrt(b*x^2+c*x^4),x,1,sqrt(b*x^2+c*x^4)/(c*x)],
[1/sqrt(b*x^2+c*x^4),x,2,-arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/sqrt(b)],
[1/(x^2*sqrt(b*x^2+c*x^4)),x,3,1/2*c*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(3/2)-1/2*sqrt(b*x^2+c*x^4)/(b*x^3)],
[1/(x^4*sqrt(b*x^2+c*x^4)),x,4,-3/8*c^2*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(5/2)-1/4*sqrt(b*x^2+c*x^4)/(b*x^5)+3/8*c*sqrt(b*x^2+c*x^4)/(b^2*x^3)],
[x^9/(b*x^2+c*x^4)^(3/2),x,6,15/8*b^2*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(7/2)-x^6/(c*sqrt(b*x^2+c*x^4))-15/8*b*sqrt(b*x^2+c*x^4)/c^3+5/4*x^2*sqrt(b*x^2+c*x^4)/c^2],
[x^7/(b*x^2+c*x^4)^(3/2),x,5,-3/2*b*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(5/2)-x^4/(c*sqrt(b*x^2+c*x^4))+3/2*sqrt(b*x^2+c*x^4)/c^2],
[x^5/(b*x^2+c*x^4)^(3/2),x,4,arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(3/2)-x^2/(c*sqrt(b*x^2+c*x^4))],
[x^3/(b*x^2+c*x^4)^(3/2),x,1,x^2/(b*sqrt(b*x^2+c*x^4))],
[x/(b*x^2+c*x^4)^(3/2),x,2,(-b-2*c*x^2)/(b^2*sqrt(b*x^2+c*x^4))],
[1/(x*(b*x^2+c*x^4)^(3/2)),x,3,1/(b*x^2*sqrt(b*x^2+c*x^4))-4/3*sqrt(b*x^2+c*x^4)/(b^2*x^4)+8/3*c*sqrt(b*x^2+c*x^4)/(b^3*x^2)],
[1/(x^3*(b*x^2+c*x^4)^(3/2)),x,4,1/(b*x^4*sqrt(b*x^2+c*x^4))-6/5*sqrt(b*x^2+c*x^4)/(b^2*x^6)+8/5*c*sqrt(b*x^2+c*x^4)/(b^3*x^4)-16/5*c^2*sqrt(b*x^2+c*x^4)/(b^4*x^2)],
[1/(x^5*(b*x^2+c*x^4)^(3/2)),x,5,1/(b*x^6*sqrt(b*x^2+c*x^4))-8/7*sqrt(b*x^2+c*x^4)/(b^2*x^8)+48/35*c*sqrt(b*x^2+c*x^4)/(b^3*x^6)-64/35*c^2*sqrt(b*x^2+c*x^4)/(b^4*x^4)+128/35*c^3*sqrt(b*x^2+c*x^4)/(b^5*x^2)],
[x^6/(b*x^2+c*x^4)^(3/2),x,2,-x^3/(c*sqrt(b*x^2+c*x^4))+2*sqrt(b*x^2+c*x^4)/(c^2*x)],
[x^4/(b*x^2+c*x^4)^(3/2),x,1,-x/(c*sqrt(b*x^2+c*x^4))],
[x^2/(b*x^2+c*x^4)^(3/2),x,3,-arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(3/2)+x/(b*sqrt(b*x^2+c*x^4))],
[1/(b*x^2+c*x^4)^(3/2),x,4,3/2*c*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(5/2)+1/(b*x*sqrt(b*x^2+c*x^4))-3/2*sqrt(b*x^2+c*x^4)/(b^2*x^3)],
[1/(x^2*(b*x^2+c*x^4)^(3/2)),x,5,-15/8*c^2*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(7/2)+1/(b*x^3*sqrt(b*x^2+c*x^4))-5/4*sqrt(b*x^2+c*x^4)/(b^2*x^5)+15/8*c*sqrt(b*x^2+c*x^4)/(b^3*x^3)],
[x^3/sqrt(3*x^2-4*x^4),x,4,-3/32*arcsin(1-8/3*x^2)-1/8*sqrt(3*x^2-4*x^4)],
[x^3/sqrt(-3*x^2-4*x^4),x,4,-3/32*arcsin(1+8/3*x^2)-1/8*sqrt(-3*x^2-4*x^4)],
[x^3/sqrt(3*x^2+4*x^4),x,4,-3/16*arctanh(2*x^2/sqrt(3*x^2+4*x^4))+1/8*sqrt(3*x^2+4*x^4)],
[x^3/sqrt(-3*x^2+4*x^4),x,4,3/16*arctanh(2*x^2/sqrt(-3*x^2+4*x^4))+1/8*sqrt(-3*x^2+4*x^4)],
[x^3/sqrt(a*x^2+b*x^4),x,4,-1/2*a*arctanh(x^2*sqrt(b)/sqrt(a*x^2+b*x^4))/b^(3/2)+1/2*sqrt(a*x^2+b*x^4)/b],
[x^3/sqrt(a*x^2-b*x^4),x,4,1/2*a*arctan(x^2*sqrt(b)/sqrt(a*x^2-b*x^4))/b^(3/2)-1/2*sqrt(a*x^2-b*x^4)/b],

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

# p>0
[x^(7/2)*(b*x^2+c*x^4),x,2,2/13*b*x^(13/2)+2/17*c*x^(17/2)],
[x^(5/2)*(b*x^2+c*x^4),x,2,2/11*b*x^(11/2)+2/15*c*x^(15/2)],
[x^(3/2)*(b*x^2+c*x^4),x,2,2/9*b*x^(9/2)+2/13*c*x^(13/2)],
[x^(1/2)*(b*x^2+c*x^4),x,2,2/7*b*x^(7/2)+2/11*c*x^(11/2)],
[(b*x^2+c*x^4)/x^(1/2),x,2,2/5*b*x^(5/2)+2/9*c*x^(9/2)],
[(b*x^2+c*x^4)/x^(3/2),x,2,2/3*b*x^(3/2)+2/7*c*x^(7/2)],
[(b*x^2+c*x^4)/x^(5/2),x,2,2/5*c*x^(5/2)+2*b*sqrt(x)],
[(b*x^2+c*x^4)/x^(7/2),x,2,2/3*c*x^(3/2)-2*b/sqrt(x)],
[x^(7/2)*(b*x^2+c*x^4)^2,x,3,2/17*b^2*x^(17/2)+4/21*b*c*x^(21/2)+2/25*c^2*x^(25/2)],
[x^(5/2)*(b*x^2+c*x^4)^2,x,3,2/15*b^2*x^(15/2)+4/19*b*c*x^(19/2)+2/23*c^2*x^(23/2)],
[x^(3/2)*(b*x^2+c*x^4)^2,x,3,2/13*b^2*x^(13/2)+4/17*b*c*x^(17/2)+2/21*c^2*x^(21/2)],
[x^(1/2)*(b*x^2+c*x^4)^2,x,3,2/11*b^2*x^(11/2)+4/15*b*c*x^(15/2)+2/19*c^2*x^(19/2)],
[(b*x^2+c*x^4)^2/x^(1/2),x,3,2/9*b^2*x^(9/2)+4/13*b*c*x^(13/2)+2/17*c^2*x^(17/2)],
[(b*x^2+c*x^4)^2/x^(3/2),x,3,2/7*b^2*x^(7/2)+4/11*b*c*x^(11/2)+2/15*c^2*x^(15/2)],
[(b*x^2+c*x^4)^2/x^(5/2),x,3,2/5*b^2*x^(5/2)+4/9*b*c*x^(9/2)+2/13*c^2*x^(13/2)],
[(b*x^2+c*x^4)^2/x^(7/2),x,3,2/3*b^2*x^(3/2)+4/7*b*c*x^(7/2)+2/11*c^2*x^(11/2)],
[x^(7/2)*(b*x^2+c*x^4)^3,x,3,2/21*b^3*x^(21/2)+6/25*b^2*c*x^(25/2)+6/29*b*c^2*x^(29/2)+2/33*c^3*x^(33/2)],
[x^(5/2)*(b*x^2+c*x^4)^3,x,3,2/19*b^3*x^(19/2)+6/23*b^2*c*x^(23/2)+2/9*b*c^2*x^(27/2)+2/31*c^3*x^(31/2)],
[x^(3/2)*(b*x^2+c*x^4)^3,x,3,2/17*b^3*x^(17/2)+2/7*b^2*c*x^(21/2)+6/25*b*c^2*x^(25/2)+2/29*c^3*x^(29/2)],
[x^(1/2)*(b*x^2+c*x^4)^3,x,3,2/15*b^3*x^(15/2)+6/19*b^2*c*x^(19/2)+6/23*b*c^2*x^(23/2)+2/27*c^3*x^(27/2)],
[(b*x^2+c*x^4)^3/x^(1/2),x,3,2/13*b^3*x^(13/2)+6/17*b^2*c*x^(17/2)+2/7*b*c^2*x^(21/2)+2/25*c^3*x^(25/2)],
[(b*x^2+c*x^4)^3/x^(3/2),x,3,2/11*b^3*x^(11/2)+2/5*b^2*c*x^(15/2)+6/19*b*c^2*x^(19/2)+2/23*c^3*x^(23/2)],
[(b*x^2+c*x^4)^3/x^(5/2),x,3,2/9*b^3*x^(9/2)+6/13*b^2*c*x^(13/2)+6/17*b*c^2*x^(17/2)+2/21*c^3*x^(21/2)],
[(b*x^2+c*x^4)^3/x^(7/2),x,3,2/7*b^3*x^(7/2)+6/11*b^2*c*x^(11/2)+2/5*b*c^2*x^(15/2)+2/19*c^3*x^(19/2)],

# p<0
[x^(13/2)/(b*x^2+c*x^4),x,13,-2/3*b*x^(3/2)/c^2+2/7*x^(7/2)/c-b^(7/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(11/4)*sqrt(2))+b^(7/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(11/4)*sqrt(2))+1/2*b^(7/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*sqrt(2))-1/2*b^(7/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*sqrt(2))],
[x^(11/2)/(b*x^2+c*x^4),x,13,2/5*x^(5/2)/c-b^(5/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(9/4)*sqrt(2))+b^(5/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(9/4)*sqrt(2))-1/2*b^(5/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*sqrt(2))+1/2*b^(5/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*sqrt(2))-2*b*sqrt(x)/c^2],
[x^(9/2)/(b*x^2+c*x^4),x,12,2/3*x^(3/2)/c+b^(3/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(7/4)*sqrt(2))-b^(3/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(7/4)*sqrt(2))-1/2*b^(3/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*sqrt(2))+1/2*b^(3/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(7/4)*sqrt(2))],
[x^(7/2)/(b*x^2+c*x^4),x,12,b^(1/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(5/4)*sqrt(2))-b^(1/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(5/4)*sqrt(2))+1/2*b^(1/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*sqrt(2))-1/2*b^(1/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(5/4)*sqrt(2))+2*sqrt(x)/c],
[x^(5/2)/(b*x^2+c*x^4),x,11,-arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(1/4)*c^(3/4)*sqrt(2))+arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(1/4)*c^(3/4)*sqrt(2))+1/2*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*c^(3/4)*sqrt(2))-1/2*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*c^(3/4)*sqrt(2))],
[x^(3/2)/(b*x^2+c*x^4),x,11,-arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(3/4)*c^(1/4)*sqrt(2))+arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(3/4)*c^(1/4)*sqrt(2))-1/2*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*c^(1/4)*sqrt(2))+1/2*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*c^(1/4)*sqrt(2))],
[sqrt(x)/(b*x^2+c*x^4),x,12,c^(1/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(5/4)*sqrt(2))-c^(1/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(5/4)*sqrt(2))-1/2*c^(1/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*sqrt(2))+1/2*c^(1/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*sqrt(2))+(-2)/(b*sqrt(x))],
[1/((b*x^2+c*x^4)*sqrt(x)),x,12,(-2/3)/(b*x^(3/2))+c^(3/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(7/4)*sqrt(2))-c^(3/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(7/4)*sqrt(2))+1/2*c^(3/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*sqrt(2))-1/2*c^(3/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*sqrt(2))],
[1/(x^(3/2)*(b*x^2+c*x^4)),x,13,(-2/5)/(b*x^(5/2))-c^(5/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(9/4)*sqrt(2))+c^(5/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(9/4)*sqrt(2))+1/2*c^(5/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*sqrt(2))-1/2*c^(5/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*sqrt(2))+2*c/(b^2*sqrt(x))],
[1/(x^(5/2)*(b*x^2+c*x^4)),x,13,(-2/7)/(b*x^(7/2))+2/3*c/(b^2*x^(3/2))-c^(7/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(11/4)*sqrt(2))+c^(7/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(11/4)*sqrt(2))-1/2*c^(7/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*sqrt(2))+1/2*c^(7/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*sqrt(2))],
[1/(x^(7/2)*(b*x^2+c*x^4)),x,14,(-2/9)/(b*x^(9/2))+2/5*c/(b^2*x^(5/2))+c^(9/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(13/4)*sqrt(2))-c^(9/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(13/4)*sqrt(2))-1/2*c^(9/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))+1/2*c^(9/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))-2*c^2/(b^3*sqrt(x))],
[x^(19/2)/(b*x^2+c*x^4)^2,x,14,9/10*x^(5/2)/c^2-1/2*x^(9/2)/(c*(b+c*x^2))-9/4*b^(5/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(13/4)*sqrt(2))+9/4*b^(5/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(13/4)*sqrt(2))-9/8*b^(5/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*sqrt(2))+9/8*b^(5/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*sqrt(2))-9/2*b*sqrt(x)/c^3],
[x^(17/2)/(b*x^2+c*x^4)^2,x,13,7/6*x^(3/2)/c^2-1/2*x^(7/2)/(c*(b+c*x^2))+7/4*b^(3/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(11/4)*sqrt(2))-7/4*b^(3/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(11/4)*sqrt(2))-7/8*b^(3/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*sqrt(2))+7/8*b^(3/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(11/4)*sqrt(2))],
[x^(15/2)/(b*x^2+c*x^4)^2,x,13,-1/2*x^(5/2)/(c*(b+c*x^2))+5/4*b^(1/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(9/4)*sqrt(2))-5/4*b^(1/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(9/4)*sqrt(2))+5/8*b^(1/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*sqrt(2))-5/8*b^(1/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(9/4)*sqrt(2))+5/2*sqrt(x)/c^2],
[x^(13/2)/(b*x^2+c*x^4)^2,x,12,-1/2*x^(3/2)/(c*(b+c*x^2))-3/4*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(1/4)*c^(7/4)*sqrt(2))+3/4*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(1/4)*c^(7/4)*sqrt(2))+3/8*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*c^(7/4)*sqrt(2))-3/8*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*c^(7/4)*sqrt(2))],
[x^(11/2)/(b*x^2+c*x^4)^2,x,12,-1/4*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(3/4)*c^(5/4)*sqrt(2))+1/4*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(3/4)*c^(5/4)*sqrt(2))-1/8*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*c^(5/4)*sqrt(2))+1/8*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*c^(5/4)*sqrt(2))-1/2*sqrt(x)/(c*(b+c*x^2))],
[x^(9/2)/(b*x^2+c*x^4)^2,x,12,1/2*x^(3/2)/(b*(b+c*x^2))-1/4*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(5/4)*c^(3/4)*sqrt(2))+1/4*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(5/4)*c^(3/4)*sqrt(2))+1/8*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*c^(3/4)*sqrt(2))-1/8*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*c^(3/4)*sqrt(2))],
[x^(7/2)/(b*x^2+c*x^4)^2,x,12,-3/4*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(7/4)*c^(1/4)*sqrt(2))+3/4*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(7/4)*c^(1/4)*sqrt(2))-3/8*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*c^(1/4)*sqrt(2))+3/8*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*c^(1/4)*sqrt(2))+1/2*sqrt(x)/(b*(b+c*x^2))],
[x^(5/2)/(b*x^2+c*x^4)^2,x,13,5/4*c^(1/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(9/4)*sqrt(2))-5/4*c^(1/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(9/4)*sqrt(2))-5/8*c^(1/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*sqrt(2))+5/8*c^(1/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*sqrt(2))+(-5/2)/(b^2*sqrt(x))+1/2/(b*(b+c*x^2)*sqrt(x))],
[x^(3/2)/(b*x^2+c*x^4)^2,x,13,(-7/6)/(b^2*x^(3/2))+1/2/(b*x^(3/2)*(b+c*x^2))+7/4*c^(3/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(11/4)*sqrt(2))-7/4*c^(3/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(11/4)*sqrt(2))+7/8*c^(3/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*sqrt(2))-7/8*c^(3/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*sqrt(2))],
[sqrt(x)/(b*x^2+c*x^4)^2,x,14,(-9/10)/(b^2*x^(5/2))+1/2/(b*x^(5/2)*(b+c*x^2))-9/4*c^(5/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(13/4)*sqrt(2))+9/4*c^(5/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(13/4)*sqrt(2))+9/8*c^(5/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))-9/8*c^(5/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))+9/2*c/(b^3*sqrt(x))],
[1/((b*x^2+c*x^4)^2*sqrt(x)),x,14,(-11/14)/(b^2*x^(7/2))+11/6*c/(b^3*x^(3/2))+1/2/(b*x^(7/2)*(b+c*x^2))-11/4*c^(7/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(15/4)*sqrt(2))+11/4*c^(7/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(15/4)*sqrt(2))-11/8*c^(7/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(15/4)*sqrt(2))+11/8*c^(7/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(15/4)*sqrt(2))],
[1/(x^(3/2)*(b*x^2+c*x^4)^2),x,15,(-13/18)/(b^2*x^(9/2))+13/10*c/(b^3*x^(5/2))+1/2/(b*x^(9/2)*(b+c*x^2))+13/4*c^(9/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(17/4)*sqrt(2))-13/4*c^(9/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(17/4)*sqrt(2))-13/8*c^(9/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(17/4)*sqrt(2))+13/8*c^(9/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(17/4)*sqrt(2))-13/2*c^2/(b^4*sqrt(x))],
[x^(23/2)/(b*x^2+c*x^4)^3,x,14,-1/4*x^(9/2)/(c*(b+c*x^2)^2)-9/16*x^(5/2)/(c^2*(b+c*x^2))+45/32*b^(1/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(13/4)*sqrt(2))-45/32*b^(1/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(c^(13/4)*sqrt(2))+45/64*b^(1/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*sqrt(2))-45/64*b^(1/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(c^(13/4)*sqrt(2))+45/16*sqrt(x)/c^3],
[x^(21/2)/(b*x^2+c*x^4)^3,x,13,-1/4*x^(7/2)/(c*(b+c*x^2)^2)-7/16*x^(3/2)/(c^2*(b+c*x^2))-21/32*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(1/4)*c^(11/4)*sqrt(2))+21/32*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(1/4)*c^(11/4)*sqrt(2))+21/64*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*c^(11/4)*sqrt(2))-21/64*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(1/4)*c^(11/4)*sqrt(2))],
[x^(19/2)/(b*x^2+c*x^4)^3,x,13,-1/4*x^(5/2)/(c*(b+c*x^2)^2)-5/32*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(3/4)*c^(9/4)*sqrt(2))+5/32*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(3/4)*c^(9/4)*sqrt(2))-5/64*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*c^(9/4)*sqrt(2))+5/64*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(3/4)*c^(9/4)*sqrt(2))-5/16*sqrt(x)/(c^2*(b+c*x^2))],
[x^(17/2)/(b*x^2+c*x^4)^3,x,13,-1/4*x^(3/2)/(c*(b+c*x^2)^2)+3/16*x^(3/2)/(b*c*(b+c*x^2))-3/32*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(5/4)*c^(7/4)*sqrt(2))+3/32*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(5/4)*c^(7/4)*sqrt(2))+3/64*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*c^(7/4)*sqrt(2))-3/64*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(5/4)*c^(7/4)*sqrt(2))],
[x^(15/2)/(b*x^2+c*x^4)^3,x,13,-3/32*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(7/4)*c^(5/4)*sqrt(2))+3/32*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(7/4)*c^(5/4)*sqrt(2))-3/64*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*c^(5/4)*sqrt(2))+3/64*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(7/4)*c^(5/4)*sqrt(2))-1/4*sqrt(x)/(c*(b+c*x^2)^2)+1/16*sqrt(x)/(b*c*(b+c*x^2))],
[x^(13/2)/(b*x^2+c*x^4)^3,x,13,1/4*x^(3/2)/(b*(b+c*x^2)^2)+5/16*x^(3/2)/(b^2*(b+c*x^2))-5/32*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(9/4)*c^(3/4)*sqrt(2))+5/32*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(9/4)*c^(3/4)*sqrt(2))+5/64*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*c^(3/4)*sqrt(2))-5/64*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(9/4)*c^(3/4)*sqrt(2))],
[x^(11/2)/(b*x^2+c*x^4)^3,x,13,-21/32*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(11/4)*c^(1/4)*sqrt(2))+21/32*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(11/4)*c^(1/4)*sqrt(2))-21/64*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*c^(1/4)*sqrt(2))+21/64*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(11/4)*c^(1/4)*sqrt(2))+1/4*sqrt(x)/(b*(b+c*x^2)^2)+7/16*sqrt(x)/(b^2*(b+c*x^2))],
[x^(9/2)/(b*x^2+c*x^4)^3,x,14,45/32*c^(1/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(13/4)*sqrt(2))-45/32*c^(1/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(13/4)*sqrt(2))-45/64*c^(1/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))+45/64*c^(1/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(13/4)*sqrt(2))+(-45/16)/(b^3*sqrt(x))+1/4/(b*(b+c*x^2)^2*sqrt(x))+9/16/(b^2*(b+c*x^2)*sqrt(x))],
[x^(7/2)/(b*x^2+c*x^4)^3,x,14,(-77/48)/(b^3*x^(3/2))+1/4/(b*x^(3/2)*(b+c*x^2)^2)+11/16/(b^2*x^(3/2)*(b+c*x^2))+77/32*c^(3/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(15/4)*sqrt(2))-77/32*c^(3/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(15/4)*sqrt(2))+77/64*c^(3/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(15/4)*sqrt(2))-77/64*c^(3/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(15/4)*sqrt(2))],
[x^(5/2)/(b*x^2+c*x^4)^3,x,15,(-117/80)/(b^3*x^(5/2))+1/4/(b*x^(5/2)*(b+c*x^2)^2)+13/16/(b^2*x^(5/2)*(b+c*x^2))-117/32*c^(5/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(17/4)*sqrt(2))+117/32*c^(5/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(17/4)*sqrt(2))+117/64*c^(5/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(17/4)*sqrt(2))-117/64*c^(5/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(17/4)*sqrt(2))+117/16*c/(b^4*sqrt(x))],
[x^(3/2)/(b*x^2+c*x^4)^3,x,15,(-165/112)/(b^3*x^(7/2))+55/16*c/(b^4*x^(3/2))+1/4/(b*x^(7/2)*(b+c*x^2)^2)+15/16/(b^2*x^(7/2)*(b+c*x^2))-165/32*c^(7/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(19/4)*sqrt(2))+165/32*c^(7/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(19/4)*sqrt(2))-165/64*c^(7/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(19/4)*sqrt(2))+165/64*c^(7/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(19/4)*sqrt(2))],
[sqrt(x)/(b*x^2+c*x^4)^3,x,16,(-221/144)/(b^3*x^(9/2))+221/80*c/(b^4*x^(5/2))+1/4/(b*x^(9/2)*(b+c*x^2)^2)+17/16/(b^2*x^(9/2)*(b+c*x^2))+221/32*c^(9/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(21/4)*sqrt(2))-221/32*c^(9/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(21/4)*sqrt(2))-221/64*c^(9/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(21/4)*sqrt(2))+221/64*c^(9/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(21/4)*sqrt(2))-221/16*c^2/(b^5*sqrt(x))],
[1/((b*x^2+c*x^4)^3*sqrt(x)),x,16,(-285/176)/(b^3*x^(11/2))+285/112*c/(b^4*x^(7/2))-95/16*c^2/(b^5*x^(3/2))+1/4/(b*x^(11/2)*(b+c*x^2)^2)+19/16/(b^2*x^(11/2)*(b+c*x^2))+285/32*c^(11/4)*arctan(1-c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(23/4)*sqrt(2))-285/32*c^(11/4)*arctan(1+c^(1/4)*sqrt(2)*sqrt(x)/b^(1/4))/(b^(23/4)*sqrt(2))+285/64*c^(11/4)*log(sqrt(b)+x*sqrt(c)-b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(23/4)*sqrt(2))-285/64*c^(11/4)*log(sqrt(b)+x*sqrt(c)+b^(1/4)*c^(1/4)*sqrt(2)*sqrt(x))/(b^(23/4)*sqrt(2))],

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

# p>0
[x^(7/2)*sqrt(b*x^2+c*x^4),x,8,28/195*b^3*x^(3/2)*(b+c*x^2)/(c^(5/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+4/117*b*x^(5/2)*sqrt(b*x^2+c*x^4)/c+2/13*x^(9/2)*sqrt(b*x^2+c*x^4)-28/585*b^2*sqrt(x)*sqrt(b*x^2+c*x^4)/c^2-28/195*b^(13/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))+14/195*b^(13/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))],
[x^(5/2)*sqrt(b*x^2+c*x^4),x,6,4/77*b*x^(3/2)*sqrt(b*x^2+c*x^4)/c+2/11*x^(7/2)*sqrt(b*x^2+c*x^4)-20/231*b^2*sqrt(b*x^2+c*x^4)/(c^2*sqrt(x))+10/231*b^(11/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(9/4)*sqrt(b*x^2+c*x^4))],
[x^(3/2)*sqrt(b*x^2+c*x^4),x,7,-4/15*b^2*x^(3/2)*(b+c*x^2)/(c^(3/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+2/9*x^(5/2)*sqrt(b*x^2+c*x^4)+4/45*b*sqrt(x)*sqrt(b*x^2+c*x^4)/c+4/15*b^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))-2/15*b^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))],
[sqrt(x)*sqrt(b*x^2+c*x^4),x,5,2/7*x^(3/2)*sqrt(b*x^2+c*x^4)+4/21*b*sqrt(b*x^2+c*x^4)/(c*sqrt(x))-2/21*b^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(5/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/sqrt(x),x,6,4/5*b*x^(3/2)*(b+c*x^2)/(sqrt(c)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+2/5*sqrt(x)*sqrt(b*x^2+c*x^4)-4/5*b^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(3/4)*sqrt(b*x^2+c*x^4))+2/5*b^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(3/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/x^(3/2),x,4,2/3*sqrt(b*x^2+c*x^4)/sqrt(x)+2/3*b^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(1/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/x^(5/2),x,6,4*x^(3/2)*(b+c*x^2)*sqrt(c)/((sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2*sqrt(b*x^2+c*x^4)/x^(3/2)-4*b^(1/4)*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)+2*b^(1/4)*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)],
[sqrt(b*x^2+c*x^4)/x^(7/2),x,4,-2/3*sqrt(b*x^2+c*x^4)/x^(5/2)+2/3*c^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(1/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/x^(9/2),x,7,4/5*c^(3/2)*x^(3/2)*(b+c*x^2)/(b*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2/5*sqrt(b*x^2+c*x^4)/x^(7/2)-4/5*c*sqrt(b*x^2+c*x^4)/(b*x^(3/2))-4/5*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*sqrt(b*x^2+c*x^4))+2/5*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/x^(11/2),x,5,-2/7*sqrt(b*x^2+c*x^4)/x^(9/2)-4/21*c*sqrt(b*x^2+c*x^4)/(b*x^(5/2))-2/21*c^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(5/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/x^(13/2),x,8,-4/15*c^(5/2)*x^(3/2)*(b+c*x^2)/(b^2*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2/9*sqrt(b*x^2+c*x^4)/x^(11/2)-4/45*c*sqrt(b*x^2+c*x^4)/(b*x^(7/2))+4/15*c^2*sqrt(b*x^2+c*x^4)/(b^2*x^(3/2))+4/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))-2/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))],
[sqrt(b*x^2+c*x^4)/x^(15/2),x,6,-2/11*sqrt(b*x^2+c*x^4)/x^(13/2)-4/77*c*sqrt(b*x^2+c*x^4)/(b*x^(9/2))+20/231*c^2*sqrt(b*x^2+c*x^4)/(b^2*x^(5/2))+10/231*c^(11/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(9/4)*sqrt(b*x^2+c*x^4))],
[x^(3/2)*(b*x^2+c*x^4)^(3/2),x,9,2/17*x^(5/2)*(b*x^2+c*x^4)^(3/2)+56/1105*b^4*x^(3/2)*(b+c*x^2)/(c^(5/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+8/663*b^2*x^(5/2)*sqrt(b*x^2+c*x^4)/c+12/221*b*x^(9/2)*sqrt(b*x^2+c*x^4)-56/3315*b^3*sqrt(x)*sqrt(b*x^2+c*x^4)/c^2-56/1105*b^(17/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))+28/1105*b^(17/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)*sqrt(x),x,7,2/15*x^(3/2)*(b*x^2+c*x^4)^(3/2)+8/385*b^2*x^(3/2)*sqrt(b*x^2+c*x^4)/c+4/55*b*x^(7/2)*sqrt(b*x^2+c*x^4)-8/231*b^3*sqrt(b*x^2+c*x^4)/(c^2*sqrt(x))+4/231*b^(15/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(9/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/sqrt(x),x,8,2/13*(b*x^2+c*x^4)^(3/2)*sqrt(x)-8/65*b^3*x^(3/2)*(b+c*x^2)/(c^(3/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+4/39*b*x^(5/2)*sqrt(b*x^2+c*x^4)+8/195*b^2*sqrt(x)*sqrt(b*x^2+c*x^4)/c+8/65*b^(13/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))-4/65*b^(13/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(3/2),x,6,2/11*(b*x^2+c*x^4)^(3/2)/sqrt(x)+12/77*b*x^(3/2)*sqrt(b*x^2+c*x^4)+8/77*b^2*sqrt(b*x^2+c*x^4)/(c*sqrt(x))-4/77*b^(11/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(5/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(5/2),x,7,2/9*(b*x^2+c*x^4)^(3/2)/x^(3/2)+8/15*b^2*x^(3/2)*(b+c*x^2)/(sqrt(c)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+4/15*b*sqrt(x)*sqrt(b*x^2+c*x^4)-8/15*b^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(3/4)*sqrt(b*x^2+c*x^4))+4/15*b^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(3/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(7/2),x,5,2/7*(b*x^2+c*x^4)^(3/2)/x^(5/2)+4/7*b*sqrt(b*x^2+c*x^4)/sqrt(x)+4/7*b^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(1/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(9/2),x,7,-2*(b*x^2+c*x^4)^(3/2)/x^(7/2)+24/5*b*x^(3/2)*(b+c*x^2)*sqrt(c)/((sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+12/5*c*sqrt(x)*sqrt(b*x^2+c*x^4)-24/5*b^(5/4)*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)+12/5*b^(5/4)*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)],
[(b*x^2+c*x^4)^(3/2)/x^(11/2),x,5,-2/3*(b*x^2+c*x^4)^(3/2)/x^(9/2)+4/3*c*sqrt(b*x^2+c*x^4)/sqrt(x)+4/3*b^(3/4)*c^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)],
[(b*x^2+c*x^4)^(3/2)/x^(13/2),x,7,-2/5*(b*x^2+c*x^4)^(3/2)/x^(11/2)+24/5*c^(3/2)*x^(3/2)*(b+c*x^2)/((sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-12/5*c*sqrt(b*x^2+c*x^4)/x^(3/2)-24/5*b^(1/4)*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)+12/5*b^(1/4)*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/sqrt(b*x^2+c*x^4)],
[(b*x^2+c*x^4)^(3/2)/x^(15/2),x,5,-2/7*(b*x^2+c*x^4)^(3/2)/x^(13/2)-4/7*c*sqrt(b*x^2+c*x^4)/x^(5/2)+4/7*c^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(1/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(17/2),x,8,-2/9*(b*x^2+c*x^4)^(3/2)/x^(15/2)+8/15*c^(5/2)*x^(3/2)*(b+c*x^2)/(b*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-4/15*c*sqrt(b*x^2+c*x^4)/x^(7/2)-8/15*c^2*sqrt(b*x^2+c*x^4)/(b*x^(3/2))-8/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*sqrt(b*x^2+c*x^4))+4/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(19/2),x,6,-2/11*(b*x^2+c*x^4)^(3/2)/x^(17/2)-12/77*c*sqrt(b*x^2+c*x^4)/x^(9/2)-8/77*c^2*sqrt(b*x^2+c*x^4)/(b*x^(5/2))-4/77*c^(11/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(5/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(21/2),x,9,-2/13*(b*x^2+c*x^4)^(3/2)/x^(19/2)-8/65*c^(7/2)*x^(3/2)*(b+c*x^2)/(b^2*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-4/39*c*sqrt(b*x^2+c*x^4)/x^(11/2)-8/195*c^2*sqrt(b*x^2+c*x^4)/(b*x^(7/2))+8/65*c^3*sqrt(b*x^2+c*x^4)/(b^2*x^(3/2))+8/65*c^(13/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))-4/65*c^(13/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))],
[(b*x^2+c*x^4)^(3/2)/x^(23/2),x,7,-2/15*(b*x^2+c*x^4)^(3/2)/x^(21/2)-4/55*c*sqrt(b*x^2+c*x^4)/x^(13/2)-8/385*c^2*sqrt(b*x^2+c*x^4)/(b*x^(9/2))+8/231*c^3*sqrt(b*x^2+c*x^4)/(b^2*x^(5/2))+4/231*c^(15/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(9/4)*sqrt(b*x^2+c*x^4))],

# p<0
[x^(13/2)/sqrt(b*x^2+c*x^4),x,6,-18/77*b*x^(3/2)*sqrt(b*x^2+c*x^4)/c^2+2/11*x^(7/2)*sqrt(b*x^2+c*x^4)/c+30/77*b^2*sqrt(b*x^2+c*x^4)/(c^3*sqrt(x))-15/77*b^(11/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(13/4)*sqrt(b*x^2+c*x^4))],
[x^(11/2)/sqrt(b*x^2+c*x^4),x,7,14/15*b^2*x^(3/2)*(b+c*x^2)/(c^(5/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+2/9*x^(5/2)*sqrt(b*x^2+c*x^4)/c-14/45*b*sqrt(x)*sqrt(b*x^2+c*x^4)/c^2-14/15*b^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))+7/15*b^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))],
[x^(9/2)/sqrt(b*x^2+c*x^4),x,5,2/7*x^(3/2)*sqrt(b*x^2+c*x^4)/c-10/21*b*sqrt(b*x^2+c*x^4)/(c^2*sqrt(x))+5/21*b^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(9/4)*sqrt(b*x^2+c*x^4))],
[x^(7/2)/sqrt(b*x^2+c*x^4),x,6,-6/5*b*x^(3/2)*(b+c*x^2)/(c^(3/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+2/5*sqrt(x)*sqrt(b*x^2+c*x^4)/c+6/5*b^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))-3/5*b^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))],
[x^(5/2)/sqrt(b*x^2+c*x^4),x,4,2/3*sqrt(b*x^2+c*x^4)/(c*sqrt(x))-1/3*b^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(5/4)*sqrt(b*x^2+c*x^4))],
[x^(3/2)/sqrt(b*x^2+c*x^4),x,5,2*x^(3/2)*(b+c*x^2)/(sqrt(c)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2*b^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(3/4)*sqrt(b*x^2+c*x^4))+b^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(3/4)*sqrt(b*x^2+c*x^4))],
[sqrt(x)/sqrt(b*x^2+c*x^4),x,3,x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(1/4)*c^(1/4)*sqrt(b*x^2+c*x^4))],
[1/(sqrt(x)*sqrt(b*x^2+c*x^4)),x,6,2*x^(3/2)*(b+c*x^2)*sqrt(c)/(b*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2*sqrt(b*x^2+c*x^4)/(b*x^(3/2))-2*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*sqrt(b*x^2+c*x^4))+c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(3/2)*sqrt(b*x^2+c*x^4)),x,4,-2/3*sqrt(b*x^2+c*x^4)/(b*x^(5/2))-1/3*c^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(5/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(5/2)*sqrt(b*x^2+c*x^4)),x,7,-6/5*c^(3/2)*x^(3/2)*(b+c*x^2)/(b^2*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2/5*sqrt(b*x^2+c*x^4)/(b*x^(7/2))+6/5*c*sqrt(b*x^2+c*x^4)/(b^2*x^(3/2))+6/5*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))-3/5*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(7/2)*sqrt(b*x^2+c*x^4)),x,5,-2/7*sqrt(b*x^2+c*x^4)/(b*x^(9/2))+10/21*c*sqrt(b*x^2+c*x^4)/(b^2*x^(5/2))+5/21*c^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(9/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(9/2)*sqrt(b*x^2+c*x^4)),x,8,14/15*c^(5/2)*x^(3/2)*(b+c*x^2)/(b^3*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-2/9*sqrt(b*x^2+c*x^4)/(b*x^(11/2))+14/45*c*sqrt(b*x^2+c*x^4)/(b^2*x^(7/2))-14/15*c^2*sqrt(b*x^2+c*x^4)/(b^3*x^(3/2))-14/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(11/4)*sqrt(b*x^2+c*x^4))+7/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(11/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(11/2)*sqrt(b*x^2+c*x^4)),x,6,-2/11*sqrt(b*x^2+c*x^4)/(b*x^(13/2))+18/77*c*sqrt(b*x^2+c*x^4)/(b^2*x^(9/2))-30/77*c^2*sqrt(b*x^2+c*x^4)/(b^3*x^(5/2))-15/77*c^(11/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(13/4)*sqrt(b*x^2+c*x^4))],
[x^(17/2)/(b*x^2+c*x^4)^(3/2),x,6,-x^(11/2)/(c*sqrt(b*x^2+c*x^4))+9/7*x^(3/2)*sqrt(b*x^2+c*x^4)/c^2-15/7*b*sqrt(b*x^2+c*x^4)/(c^3*sqrt(x))+15/14*b^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(13/4)*sqrt(b*x^2+c*x^4))],
[x^(15/2)/(b*x^2+c*x^4)^(3/2),x,7,-x^(9/2)/(c*sqrt(b*x^2+c*x^4))-21/5*b*x^(3/2)*(b+c*x^2)/(c^(5/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+7/5*sqrt(x)*sqrt(b*x^2+c*x^4)/c^2+21/5*b^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))-21/10*b^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(11/4)*sqrt(b*x^2+c*x^4))],
[x^(13/2)/(b*x^2+c*x^4)^(3/2),x,5,-x^(7/2)/(c*sqrt(b*x^2+c*x^4))+5/3*sqrt(b*x^2+c*x^4)/(c^2*sqrt(x))-5/6*b^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(9/4)*sqrt(b*x^2+c*x^4))],
[x^(11/2)/(b*x^2+c*x^4)^(3/2),x,6,-x^(5/2)/(c*sqrt(b*x^2+c*x^4))+3*x^(3/2)*(b+c*x^2)/(c^(3/2)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-3*b^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))+3/2*b^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(c^(7/4)*sqrt(b*x^2+c*x^4))],
[x^(9/2)/(b*x^2+c*x^4)^(3/2),x,4,-x^(3/2)/(c*sqrt(b*x^2+c*x^4))+1/2*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(1/4)*c^(5/4)*sqrt(b*x^2+c*x^4))],
[x^(7/2)/(b*x^2+c*x^4)^(3/2),x,6,x^(5/2)/(b*sqrt(b*x^2+c*x^4))-x^(3/2)*(b+c*x^2)/(b*sqrt(c)*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*c^(3/4)*sqrt(b*x^2+c*x^4))-1/2*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(3/4)*c^(3/4)*sqrt(b*x^2+c*x^4))],
[x^(5/2)/(b*x^2+c*x^4)^(3/2),x,4,x^(3/2)/(b*sqrt(b*x^2+c*x^4))+1/2*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(5/4)*c^(1/4)*sqrt(b*x^2+c*x^4))],
[x^(3/2)/(b*x^2+c*x^4)^(3/2),x,7,3*x^(3/2)*(b+c*x^2)*sqrt(c)/(b^2*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))+sqrt(x)/(b*sqrt(b*x^2+c*x^4))-3*sqrt(b*x^2+c*x^4)/(b^2*x^(3/2))-3*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))+3/2*c^(1/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(7/4)*sqrt(b*x^2+c*x^4))],
[sqrt(x)/(b*x^2+c*x^4)^(3/2),x,5,1/(b*sqrt(x)*sqrt(b*x^2+c*x^4))-5/3*sqrt(b*x^2+c*x^4)/(b^2*x^(5/2))-5/6*c^(3/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(9/4)*sqrt(b*x^2+c*x^4))],
[1/((b*x^2+c*x^4)^(3/2)*sqrt(x)),x,8,1/(b*x^(3/2)*sqrt(b*x^2+c*x^4))-21/5*c^(3/2)*x^(3/2)*(b+c*x^2)/(b^3*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-7/5*sqrt(b*x^2+c*x^4)/(b^2*x^(7/2))+21/5*c*sqrt(b*x^2+c*x^4)/(b^3*x^(3/2))+21/5*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(11/4)*sqrt(b*x^2+c*x^4))-21/10*c^(5/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(11/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(3/2)*(b*x^2+c*x^4)^(3/2)),x,6,1/(b*x^(5/2)*sqrt(b*x^2+c*x^4))-9/7*sqrt(b*x^2+c*x^4)/(b^2*x^(9/2))+15/7*c*sqrt(b*x^2+c*x^4)/(b^3*x^(5/2))+15/14*c^(7/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(13/4)*sqrt(b*x^2+c*x^4))],
[1/(x^(5/2)*(b*x^2+c*x^4)^(3/2)),x,9,1/(b*x^(7/2)*sqrt(b*x^2+c*x^4))+77/15*c^(5/2)*x^(3/2)*(b+c*x^2)/(b^4*(sqrt(b)+x*sqrt(c))*sqrt(b*x^2+c*x^4))-11/9*sqrt(b*x^2+c*x^4)/(b^2*x^(11/2))+77/45*c*sqrt(b*x^2+c*x^4)/(b^3*x^(7/2))-77/15*c^2*sqrt(b*x^2+c*x^4)/(b^4*x^(3/2))-77/15*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(15/4)*sqrt(b*x^2+c*x^4))+77/30*c^(9/4)*x*sqrt(cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))^2)/cos(2*arctan(c^(1/4)*sqrt(x)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*sqrt(x)/b^(1/4))),sqrt(1/2))*(sqrt(b)+x*sqrt(c))*sqrt((b+c*x^2)/(sqrt(b)+x*sqrt(c))^2)/(b^(15/4)*sqrt(b*x^2+c*x^4))],

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

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

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

# p>0
[x^3*(a^2+2*a*b*x^2+b^2*x^4),x,2,1/4*a^2*x^4+1/3*a*b*x^6+1/8*b^2*x^8],
[x^2*(a^2+2*a*b*x^2+b^2*x^4),x,2,1/3*a^2*x^3+2/5*a*b*x^5+1/7*b^2*x^7],
[x*(a^2+2*a*b*x^2+b^2*x^4),x,2,1/2*a^2*x^2+1/2*a*b*x^4+1/6*b^2*x^6],
[a^2+2*a*b*x^2+b^2*x^4,x,1,a^2*x+2/3*a*b*x^3+1/5*b^2*x^5],
[(a^2+2*a*b*x^2+b^2*x^4)/x,x,2,a*b*x^2+1/4*b^2*x^4+a^2*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)/x^2,x,2,-a^2/x+2*a*b*x+1/3*b^2*x^3],
[(a^2+2*a*b*x^2+b^2*x^4)/x^3,x,2,-1/2*a^2/x^2+1/2*b^2*x^2+2*a*b*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)/x^4,x,2,-1/3*a^2/x^3-2*a*b/x+b^2*x],
[(a^2+2*a*b*x^2+b^2*x^4)/x^5,x,2,-1/4*a^2/x^4-a*b/x^2+b^2*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)/x^6,x,2,-1/5*a^2/x^5-2/3*a*b/x^3-b^2/x],
[(a^2+2*a*b*x^2+b^2*x^4)/x^7,x,2,-1/6*a^2/x^6-1/2*a*b/x^4-1/2*b^2/x^2],
[(a^2+2*a*b*x^2+b^2*x^4)/x^8,x,2,-1/7*a^2/x^7-2/5*a*b/x^5-1/3*b^2/x^3],
[x^6*(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,1/7*a^4*x^7+4/9*a^3*b*x^9+6/11*a^2*b^2*x^11+4/13*a*b^3*x^13+1/15*b^4*x^15],
[x^5*(a^2+2*a*b*x^2+b^2*x^4)^2,x,4,1/10*a^2*(a+b*x^2)^5/b^3-1/6*a*(a+b*x^2)^6/b^3+1/14*(a+b*x^2)^7/b^3],
[x^4*(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,1/5*a^4*x^5+4/7*a^3*b*x^7+2/3*a^2*b^2*x^9+4/11*a*b^3*x^11+1/13*b^4*x^13],
[x^3*(a^2+2*a*b*x^2+b^2*x^4)^2,x,4,-1/10*a*(a+b*x^2)^5/b^2+1/12*(a+b*x^2)^6/b^2],
[x^2*(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,1/3*a^4*x^3+4/5*a^3*b*x^5+6/7*a^2*b^2*x^7+4/9*a*b^3*x^9+1/11*b^4*x^11],
[x*(a^2+2*a*b*x^2+b^2*x^4)^2,x,2,1/10*(a+b*x^2)^5/b],
[(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,a^4*x+4/3*a^3*b*x^3+6/5*a^2*b^2*x^5+4/7*a*b^3*x^7+1/9*b^4*x^9],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x,x,4,2*a^3*b*x^2+3/2*a^2*b^2*x^4+2/3*a*b^3*x^6+1/8*b^4*x^8+a^4*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^2,x,3,-a^4/x+4*a^3*b*x+2*a^2*b^2*x^3+4/5*a*b^3*x^5+1/7*b^4*x^7],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^3,x,4,-1/2*a^4/x^2+3*a^2*b^2*x^2+a*b^3*x^4+1/6*b^4*x^6+4*a^3*b*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^4,x,3,-1/3*a^4/x^3-4*a^3*b/x+6*a^2*b^2*x+4/3*a*b^3*x^3+1/5*b^4*x^5],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^5,x,4,-1/4*a^4/x^4-2*a^3*b/x^2+2*a*b^3*x^2+1/4*b^4*x^4+6*a^2*b^2*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^6,x,3,-1/5*a^4/x^5-4/3*a^3*b/x^3-6*a^2*b^2/x+4*a*b^3*x+1/3*b^4*x^3],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^7,x,4,-1/6*a^4/x^6-a^3*b/x^4-3*a^2*b^2/x^2+1/2*b^4*x^2+4*a*b^3*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^8,x,3,-1/7*a^4/x^7-4/5*a^3*b/x^5-2*a^2*b^2/x^3-4*a*b^3/x+b^4*x],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^9,x,4,-1/8*a^4/x^8-2/3*a^3*b/x^6-3/2*a^2*b^2/x^4-2*a*b^3/x^2+b^4*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^10,x,3,-1/9*a^4/x^9-4/7*a^3*b/x^7-6/5*a^2*b^2/x^5-4/3*a*b^3/x^3-b^4/x],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^11,x,2,-1/10*(a+b*x^2)^5/(a*x^10)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^12,x,3,-1/11*a^4/x^11-4/9*a^3*b/x^9-6/7*a^2*b^2/x^7-4/5*a*b^3/x^5-1/3*b^4/x^3],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^13,x,4,-1/12*(a+b*x^2)^5/(a*x^12)+1/60*b*(a+b*x^2)^5/(a^2*x^10)],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^14,x,3,-1/13*a^4/x^13-4/11*a^3*b/x^11-2/3*a^2*b^2/x^9-4/7*a*b^3/x^7-1/5*b^4/x^5],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^15,x,4,-1/14*a^4/x^14-1/3*a^3*b/x^12-3/5*a^2*b^2/x^10-1/2*a*b^3/x^8-1/6*b^4/x^6],
[(a^2+2*a*b*x^2+b^2*x^4)^2/x^16,x,3,-1/15*a^4/x^15-4/13*a^3*b/x^13-6/11*a^2*b^2/x^11-4/9*a*b^3/x^9-1/7*b^4/x^7],
[x^8*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,1/9*a^6*x^9+6/11*a^5*b*x^11+15/13*a^4*b^2*x^13+4/3*a^3*b^3*x^15+15/17*a^2*b^4*x^17+6/19*a*b^5*x^19+1/21*b^6*x^21],
[x^7*(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,-1/14*a^3*(a+b*x^2)^7/b^4+3/16*a^2*(a+b*x^2)^8/b^4-1/6*a*(a+b*x^2)^9/b^4+1/20*(a+b*x^2)^10/b^4],
[x^6*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,1/7*a^6*x^7+2/3*a^5*b*x^9+15/11*a^4*b^2*x^11+20/13*a^3*b^3*x^13+a^2*b^4*x^15+6/17*a*b^5*x^17+1/19*b^6*x^19],
[x^5*(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,1/14*a^2*(a+b*x^2)^7/b^3-1/8*a*(a+b*x^2)^8/b^3+1/18*(a+b*x^2)^9/b^3],
[x^4*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,1/5*a^6*x^5+6/7*a^5*b*x^7+5/3*a^4*b^2*x^9+20/11*a^3*b^3*x^11+15/13*a^2*b^4*x^13+2/5*a*b^5*x^15+1/17*b^6*x^17],
[x^3*(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,-1/14*a*(a+b*x^2)^7/b^2+1/16*(a+b*x^2)^8/b^2],
[x^2*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,1/3*a^6*x^3+6/5*a^5*b*x^5+15/7*a^4*b^2*x^7+20/9*a^3*b^3*x^9+15/11*a^2*b^4*x^11+6/13*a*b^5*x^13+1/15*b^6*x^15],
[x*(a^2+2*a*b*x^2+b^2*x^4)^3,x,2,1/14*(a+b*x^2)^7/b],
[(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,a^6*x+2*a^5*b*x^3+3*a^4*b^2*x^5+20/7*a^3*b^3*x^7+5/3*a^2*b^4*x^9+6/11*a*b^5*x^11+1/13*b^6*x^13],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x,x,4,3*a^5*b*x^2+15/4*a^4*b^2*x^4+10/3*a^3*b^3*x^6+15/8*a^2*b^4*x^8+3/5*a*b^5*x^10+1/12*b^6*x^12+a^6*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^2,x,3,-a^6/x+6*a^5*b*x+5*a^4*b^2*x^3+4*a^3*b^3*x^5+15/7*a^2*b^4*x^7+2/3*a*b^5*x^9+1/11*b^6*x^11],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^3,x,4,-1/2*a^6/x^2+15/2*a^4*b^2*x^2+5*a^3*b^3*x^4+5/2*a^2*b^4*x^6+3/4*a*b^5*x^8+1/10*b^6*x^10+6*a^5*b*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^4,x,3,-1/3*a^6/x^3-6*a^5*b/x+15*a^4*b^2*x+20/3*a^3*b^3*x^3+3*a^2*b^4*x^5+6/7*a*b^5*x^7+1/9*b^6*x^9],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^5,x,4,-1/4*a^6/x^4-3*a^5*b/x^2+10*a^3*b^3*x^2+15/4*a^2*b^4*x^4+a*b^5*x^6+1/8*b^6*x^8+15*a^4*b^2*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^6,x,3,-1/5*a^6/x^5-2*a^5*b/x^3-15*a^4*b^2/x+20*a^3*b^3*x+5*a^2*b^4*x^3+6/5*a*b^5*x^5+1/7*b^6*x^7],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^7,x,4,-1/6*a^6/x^6-3/2*a^5*b/x^4-15/2*a^4*b^2/x^2+15/2*a^2*b^4*x^2+3/2*a*b^5*x^4+1/6*b^6*x^6+20*a^3*b^3*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^8,x,3,-1/7*a^6/x^7-6/5*a^5*b/x^5-5*a^4*b^2/x^3-20*a^3*b^3/x+15*a^2*b^4*x+2*a*b^5*x^3+1/5*b^6*x^5],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^9,x,4,-1/8*a^6/x^8-a^5*b/x^6-15/4*a^4*b^2/x^4-10*a^3*b^3/x^2+3*a*b^5*x^2+1/4*b^6*x^4+15*a^2*b^4*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^10,x,3,-1/9*a^6/x^9-6/7*a^5*b/x^7-3*a^4*b^2/x^5-20/3*a^3*b^3/x^3-15*a^2*b^4/x+6*a*b^5*x+1/3*b^6*x^3],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^11,x,4,-1/10*a^6/x^10-3/4*a^5*b/x^8-5/2*a^4*b^2/x^6-5*a^3*b^3/x^4-15/2*a^2*b^4/x^2+1/2*b^6*x^2+6*a*b^5*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^12,x,3,-1/11*a^6/x^11-2/3*a^5*b/x^9-15/7*a^4*b^2/x^7-4*a^3*b^3/x^5-5*a^2*b^4/x^3-6*a*b^5/x+b^6*x],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^13,x,4,-1/12*a^6/x^12-3/5*a^5*b/x^10-15/8*a^4*b^2/x^8-10/3*a^3*b^3/x^6-15/4*a^2*b^4/x^4-3*a*b^5/x^2+b^6*log(x)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^14,x,3,-1/13*a^6/x^13-6/11*a^5*b/x^11-5/3*a^4*b^2/x^9-20/7*a^3*b^3/x^7-3*a^2*b^4/x^5-2*a*b^5/x^3-b^6/x],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^15,x,2,-1/14*(a+b*x^2)^7/(a*x^14)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^16,x,3,-1/15*a^6/x^15-6/13*a^5*b/x^13-15/11*a^4*b^2/x^11-20/9*a^3*b^3/x^9-15/7*a^2*b^4/x^7-6/5*a*b^5/x^5-1/3*b^6/x^3],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^17,x,4,-1/16*(a+b*x^2)^7/(a*x^16)+1/112*b*(a+b*x^2)^7/(a^2*x^14)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^18,x,3,-1/17*a^6/x^17-2/5*a^5*b/x^15-15/13*a^4*b^2/x^13-20/11*a^3*b^3/x^11-5/3*a^2*b^4/x^9-6/7*a*b^5/x^7-1/5*b^6/x^5],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^19,x,5,-1/18*(a+b*x^2)^7/(a*x^18)+1/72*b*(a+b*x^2)^7/(a^2*x^16)-1/504*b^2*(a+b*x^2)^7/(a^3*x^14)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^20,x,3,-1/19*a^6/x^19-6/17*a^5*b/x^17-a^4*b^2/x^15-20/13*a^3*b^3/x^13-15/11*a^2*b^4/x^11-2/3*a*b^5/x^9-1/7*b^6/x^7],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^21,x,6,-1/20*(a+b*x^2)^7/(a*x^20)+1/60*b*(a+b*x^2)^7/(a^2*x^18)-1/240*b^2*(a+b*x^2)^7/(a^3*x^16)+1/1680*b^3*(a+b*x^2)^7/(a^4*x^14)],
[(a^2+2*a*b*x^2+b^2*x^4)^3/x^22,x,3,-1/21*a^6/x^21-6/19*a^5*b/x^19-15/17*a^4*b^2/x^17-4/3*a^3*b^3/x^15-15/13*a^2*b^4/x^13-6/11*a*b^5/x^11-1/9*b^6/x^9],

# p<0
[x^11/(a^2+2*a*b*x^2+b^2*x^4),x,4,-2*a^3*x^2/b^5+3/4*a^2*x^4/b^4-1/3*a*x^6/b^3+1/8*x^8/b^2+1/2*a^5/(b^6*(a+b*x^2))+5/2*a^4*log(a+b*x^2)/b^6],
[x^9/(a^2+2*a*b*x^2+b^2*x^4),x,4,3/2*a^2*x^2/b^4-1/2*a*x^4/b^3+1/6*x^6/b^2-1/2*a^4/(b^5*(a+b*x^2))-2*a^3*log(a+b*x^2)/b^5],
[x^7/(a^2+2*a*b*x^2+b^2*x^4),x,4,-a*x^2/b^3+1/4*x^4/b^2+1/2*a^3/(b^4*(a+b*x^2))+3/2*a^2*log(a+b*x^2)/b^4],
[x^5/(a^2+2*a*b*x^2+b^2*x^4),x,4,1/2*x^2/b^2-1/2*a^2/(b^3*(a+b*x^2))-a*log(a+b*x^2)/b^3],
[x^3/(a^2+2*a*b*x^2+b^2*x^4),x,4,1/2*a/(b^2*(a+b*x^2))+1/2*log(a+b*x^2)/b^2],
[x/(a^2+2*a*b*x^2+b^2*x^4),x,2,(-1/2)/(b*(a+b*x^2))],
[1/(x*(a^2+2*a*b*x^2+b^2*x^4)),x,4,1/2/(a*(a+b*x^2))+log(x)/a^2-1/2*log(a+b*x^2)/a^2],
[1/(x^3*(a^2+2*a*b*x^2+b^2*x^4)),x,4,(-1/2)/(a^2*x^2)-1/2*b/(a^2*(a+b*x^2))-2*b*log(x)/a^3+b*log(a+b*x^2)/a^3],
[1/(x^5*(a^2+2*a*b*x^2+b^2*x^4)),x,4,(-1/4)/(a^2*x^4)+b/(a^3*x^2)+1/2*b^2/(a^3*(a+b*x^2))+3*b^2*log(x)/a^4-3/2*b^2*log(a+b*x^2)/a^4],
[x^10/(a^2+2*a*b*x^2+b^2*x^4),x,5,-9/2*a^3*x/b^5+3/2*a^2*x^3/b^4-9/10*a*x^5/b^3+9/14*x^7/b^2-1/2*x^9/(b*(a+b*x^2))+9/2*a^(7/2)*arctan(x*sqrt(b)/sqrt(a))/b^(11/2)],
[x^8/(a^2+2*a*b*x^2+b^2*x^4),x,5,7/2*a^2*x/b^4-7/6*a*x^3/b^3+7/10*x^5/b^2-1/2*x^7/(b*(a+b*x^2))-7/2*a^(5/2)*arctan(x*sqrt(b)/sqrt(a))/b^(9/2)],
[x^6/(a^2+2*a*b*x^2+b^2*x^4),x,5,-5/2*a*x/b^3+5/6*x^3/b^2-1/2*x^5/(b*(a+b*x^2))+5/2*a^(3/2)*arctan(x*sqrt(b)/sqrt(a))/b^(7/2)],
[x^4/(a^2+2*a*b*x^2+b^2*x^4),x,4,3/2*x/b^2-1/2*x^3/(b*(a+b*x^2))-3/2*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(5/2)],
[x^2/(a^2+2*a*b*x^2+b^2*x^4),x,3,-1/2*x/(b*(a+b*x^2))+1/2*arctan(x*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))],
[1/(a^2+2*a*b*x^2+b^2*x^4),x,3,1/2*x/(a*(a+b*x^2))+1/2*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*sqrt(b))],
[1/(x^2*(a^2+2*a*b*x^2+b^2*x^4)),x,4,(-3/2)/(a^2*x)+1/2/(a*x*(a+b*x^2))-3/2*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(5/2)],
[1/(x^4*(a^2+2*a*b*x^2+b^2*x^4)),x,5,(-5/6)/(a^2*x^3)+5/2*b/(a^3*x)+1/2/(a*x^3*(a+b*x^2))+5/2*b^(3/2)*arctan(x*sqrt(b)/sqrt(a))/a^(7/2)],
[1/(x^6*(a^2+2*a*b*x^2+b^2*x^4)),x,6,(-7/10)/(a^2*x^5)+7/6*b/(a^3*x^3)-7/2*b^2/(a^4*x)+1/2/(a*x^5*(a+b*x^2))-7/2*b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/a^(9/2)],
[x^11/(a^2+2*a*b*x^2+b^2*x^4)^2,x,4,-2*a*x^2/b^5+1/4*x^4/b^4+1/6*a^5/(b^6*(a+b*x^2)^3)-5/4*a^4/(b^6*(a+b*x^2)^2)+5*a^3/(b^6*(a+b*x^2))+5*a^2*log(a+b*x^2)/b^6],
[x^9/(a^2+2*a*b*x^2+b^2*x^4)^2,x,4,1/2*x^2/b^4-1/6*a^4/(b^5*(a+b*x^2)^3)+a^3/(b^5*(a+b*x^2)^2)-3*a^2/(b^5*(a+b*x^2))-2*a*log(a+b*x^2)/b^5],
[x^7/(a^2+2*a*b*x^2+b^2*x^4)^2,x,4,1/6*a^3/(b^4*(a+b*x^2)^3)-3/4*a^2/(b^4*(a+b*x^2)^2)+3/2*a/(b^4*(a+b*x^2))+1/2*log(a+b*x^2)/b^4],
[x^5/(a^2+2*a*b*x^2+b^2*x^4)^2,x,2,1/6*x^6/(a*(a+b*x^2)^3)],
[x^3/(a^2+2*a*b*x^2+b^2*x^4)^2,x,4,1/6*a/(b^2*(a+b*x^2)^3)+(-1/4)/(b^2*(a+b*x^2)^2)],
[x/(a^2+2*a*b*x^2+b^2*x^4)^2,x,2,(-1/6)/(b*(a+b*x^2)^3)],
[1/(x*(a^2+2*a*b*x^2+b^2*x^4)^2),x,4,1/6/(a*(a+b*x^2)^3)+1/4/(a^2*(a+b*x^2)^2)+1/2/(a^3*(a+b*x^2))+log(x)/a^4-1/2*log(a+b*x^2)/a^4],
[1/(x^3*(a^2+2*a*b*x^2+b^2*x^4)^2),x,4,(-1/2)/(a^4*x^2)-1/6*b/(a^2*(a+b*x^2)^3)-1/2*b/(a^3*(a+b*x^2)^2)-3/2*b/(a^4*(a+b*x^2))-4*b*log(x)/a^5+2*b*log(a+b*x^2)/a^5],
[1/(x^5*(a^2+2*a*b*x^2+b^2*x^4)^2),x,4,(-1/4)/(a^4*x^4)+2*b/(a^5*x^2)+1/6*b^2/(a^3*(a+b*x^2)^3)+3/4*b^2/(a^4*(a+b*x^2)^2)+3*b^2/(a^5*(a+b*x^2))+10*b^2*log(x)/a^6-5*b^2*log(a+b*x^2)/a^6],
[x^12/(a^2+2*a*b*x^2+b^2*x^4)^2,x,7,231/16*a^2*x/b^6-77/16*a*x^3/b^5+231/80*x^5/b^4-1/6*x^11/(b*(a+b*x^2)^3)-11/24*x^9/(b^2*(a+b*x^2)^2)-33/16*x^7/(b^3*(a+b*x^2))-231/16*a^(5/2)*arctan(x*sqrt(b)/sqrt(a))/b^(13/2)],
[x^10/(a^2+2*a*b*x^2+b^2*x^4)^2,x,7,-105/16*a*x/b^5+35/16*x^3/b^4-1/6*x^9/(b*(a+b*x^2)^3)-3/8*x^7/(b^2*(a+b*x^2)^2)-21/16*x^5/(b^3*(a+b*x^2))+105/16*a^(3/2)*arctan(x*sqrt(b)/sqrt(a))/b^(11/2)],
[x^8/(a^2+2*a*b*x^2+b^2*x^4)^2,x,6,35/16*x/b^4-1/6*x^7/(b*(a+b*x^2)^3)-7/24*x^5/(b^2*(a+b*x^2)^2)-35/48*x^3/(b^3*(a+b*x^2))-35/16*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(9/2)],
[x^6/(a^2+2*a*b*x^2+b^2*x^4)^2,x,5,-1/6*x^5/(b*(a+b*x^2)^3)-5/24*x^3/(b^2*(a+b*x^2)^2)-5/16*x/(b^3*(a+b*x^2))+5/16*arctan(x*sqrt(b)/sqrt(a))/(b^(7/2)*sqrt(a))],
[x^4/(a^2+2*a*b*x^2+b^2*x^4)^2,x,5,-1/6*x^3/(b*(a+b*x^2)^3)-1/8*x/(b^2*(a+b*x^2)^2)+1/16*x/(a*b^2*(a+b*x^2))+1/16*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(5/2))],
[x^2/(a^2+2*a*b*x^2+b^2*x^4)^2,x,5,-1/6*x/(b*(a+b*x^2)^3)+1/24*x/(a*b*(a+b*x^2)^2)+1/16*x/(a^2*b*(a+b*x^2))+1/16*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^2,x,5,1/6*x/(a*(a+b*x^2)^3)+5/24*x/(a^2*(a+b*x^2)^2)+5/16*x/(a^3*(a+b*x^2))+5/16*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*sqrt(b))],
[1/(x^2*(a^2+2*a*b*x^2+b^2*x^4)^2),x,6,(-35/16)/(a^4*x)+1/6/(a*x*(a+b*x^2)^3)+7/24/(a^2*x*(a+b*x^2)^2)+35/48/(a^3*x*(a+b*x^2))-35/16*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(9/2)],
[1/(x^4*(a^2+2*a*b*x^2+b^2*x^4)^2),x,7,(-35/16)/(a^4*x^3)+105/16*b/(a^5*x)+1/6/(a*x^3*(a+b*x^2)^3)+3/8/(a^2*x^3*(a+b*x^2)^2)+21/16/(a^3*x^3*(a+b*x^2))+105/16*b^(3/2)*arctan(x*sqrt(b)/sqrt(a))/a^(11/2)],
[1/(x^6*(a^2+2*a*b*x^2+b^2*x^4)^2),x,8,(-231/80)/(a^4*x^5)+77/16*b/(a^5*x^3)-231/16*b^2/(a^6*x)+1/6/(a*x^5*(a+b*x^2)^3)+11/24/(a^2*x^5*(a+b*x^2)^2)+33/16/(a^3*x^5*(a+b*x^2))-231/16*b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/a^(13/2)],
[x^15/(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,-3*a*x^2/b^7+1/4*x^4/b^6+1/10*a^7/(b^8*(a+b*x^2)^5)-7/8*a^6/(b^8*(a+b*x^2)^4)+7/2*a^5/(b^8*(a+b*x^2)^3)-35/4*a^4/(b^8*(a+b*x^2)^2)+35/2*a^3/(b^8*(a+b*x^2))+21/2*a^2*log(a+b*x^2)/b^8],
[x^13/(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,1/2*x^2/b^6-1/10*a^6/(b^7*(a+b*x^2)^5)+3/4*a^5/(b^7*(a+b*x^2)^4)-5/2*a^4/(b^7*(a+b*x^2)^3)+5*a^3/(b^7*(a+b*x^2)^2)-15/2*a^2/(b^7*(a+b*x^2))-3*a*log(a+b*x^2)/b^7],
[x^11/(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,1/10*a^5/(b^6*(a+b*x^2)^5)-5/8*a^4/(b^6*(a+b*x^2)^4)+5/3*a^3/(b^6*(a+b*x^2)^3)-5/2*a^2/(b^6*(a+b*x^2)^2)+5/2*a/(b^6*(a+b*x^2))+1/2*log(a+b*x^2)/b^6],
[x^9/(a^2+2*a*b*x^2+b^2*x^4)^3,x,2,1/10*x^10/(a*(a+b*x^2)^5)],
[x^7/(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,1/10*x^8/(a*(a+b*x^2)^5)+1/40*x^8/(a^2*(a+b*x^2)^4)],
[x^5/(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,-1/10*a^2/(b^3*(a+b*x^2)^5)+1/4*a/(b^3*(a+b*x^2)^4)+(-1/6)/(b^3*(a+b*x^2)^3)],
[x^3/(a^2+2*a*b*x^2+b^2*x^4)^3,x,4,1/10*a/(b^2*(a+b*x^2)^5)+(-1/8)/(b^2*(a+b*x^2)^4)],
[x/(a^2+2*a*b*x^2+b^2*x^4)^3,x,2,(-1/10)/(b*(a+b*x^2)^5)],
[1/(x*(a^2+2*a*b*x^2+b^2*x^4)^3),x,4,1/10/(a*(a+b*x^2)^5)+1/8/(a^2*(a+b*x^2)^4)+1/6/(a^3*(a+b*x^2)^3)+1/4/(a^4*(a+b*x^2)^2)+1/2/(a^5*(a+b*x^2))+log(x)/a^6-1/2*log(a+b*x^2)/a^6],
[1/(x^3*(a^2+2*a*b*x^2+b^2*x^4)^3),x,4,(-1/2)/(a^6*x^2)-1/10*b/(a^2*(a+b*x^2)^5)-1/4*b/(a^3*(a+b*x^2)^4)-1/2*b/(a^4*(a+b*x^2)^3)-b/(a^5*(a+b*x^2)^2)-5/2*b/(a^6*(a+b*x^2))-6*b*log(x)/a^7+3*b*log(a+b*x^2)/a^7],
[1/(x^5*(a^2+2*a*b*x^2+b^2*x^4)^3),x,4,(-1/4)/(a^6*x^4)+3*b/(a^7*x^2)+1/10*b^2/(a^3*(a+b*x^2)^5)+3/8*b^2/(a^4*(a+b*x^2)^4)+b^2/(a^5*(a+b*x^2)^3)+5/2*b^2/(a^6*(a+b*x^2)^2)+15/2*b^2/(a^7*(a+b*x^2))+21*b^2*log(x)/a^8-21/2*b^2*log(a+b*x^2)/a^8],
[x^16/(a^2+2*a*b*x^2+b^2*x^4)^3,x,9,9009/256*a^2*x/b^8-3003/256*a*x^3/b^7+9009/1280*x^5/b^6-1/10*x^15/(b*(a+b*x^2)^5)-3/16*x^13/(b^2*(a+b*x^2)^4)-13/32*x^11/(b^3*(a+b*x^2)^3)-143/128*x^9/(b^4*(a+b*x^2)^2)-1287/256*x^7/(b^5*(a+b*x^2))-9009/256*a^(5/2)*arctan(x*sqrt(b)/sqrt(a))/b^(17/2)],
[x^14/(a^2+2*a*b*x^2+b^2*x^4)^3,x,9,-3003/256*a*x/b^7+1001/256*x^3/b^6-1/10*x^13/(b*(a+b*x^2)^5)-13/80*x^11/(b^2*(a+b*x^2)^4)-143/480*x^9/(b^3*(a+b*x^2)^3)-429/640*x^7/(b^4*(a+b*x^2)^2)-3003/1280*x^5/(b^5*(a+b*x^2))+3003/256*a^(3/2)*arctan(x*sqrt(b)/sqrt(a))/b^(15/2)],
[x^12/(a^2+2*a*b*x^2+b^2*x^4)^3,x,8,693/256*x/b^6-1/10*x^11/(b*(a+b*x^2)^5)-11/80*x^9/(b^2*(a+b*x^2)^4)-33/160*x^7/(b^3*(a+b*x^2)^3)-231/640*x^5/(b^4*(a+b*x^2)^2)-231/256*x^3/(b^5*(a+b*x^2))-693/256*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/b^(13/2)],
[x^10/(a^2+2*a*b*x^2+b^2*x^4)^3,x,7,-1/10*x^9/(b*(a+b*x^2)^5)-9/80*x^7/(b^2*(a+b*x^2)^4)-21/160*x^5/(b^3*(a+b*x^2)^3)-21/128*x^3/(b^4*(a+b*x^2)^2)-63/256*x/(b^5*(a+b*x^2))+63/256*arctan(x*sqrt(b)/sqrt(a))/(b^(11/2)*sqrt(a))],
[x^8/(a^2+2*a*b*x^2+b^2*x^4)^3,x,7,-1/10*x^7/(b*(a+b*x^2)^5)-7/80*x^5/(b^2*(a+b*x^2)^4)-7/96*x^3/(b^3*(a+b*x^2)^3)-7/128*x/(b^4*(a+b*x^2)^2)+7/256*x/(a*b^4*(a+b*x^2))+7/256*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(9/2))],
[x^6/(a^2+2*a*b*x^2+b^2*x^4)^3,x,7,-1/10*x^5/(b*(a+b*x^2)^5)-1/16*x^3/(b^2*(a+b*x^2)^4)-1/32*x/(b^3*(a+b*x^2)^3)+1/128*x/(a*b^3*(a+b*x^2)^2)+3/256*x/(a^2*b^3*(a+b*x^2))+3/256*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(7/2))],
[x^4/(a^2+2*a*b*x^2+b^2*x^4)^3,x,7,-1/10*x^3/(b*(a+b*x^2)^5)-3/80*x/(b^2*(a+b*x^2)^4)+1/160*x/(a*b^2*(a+b*x^2)^3)+1/128*x/(a^2*b^2*(a+b*x^2)^2)+3/256*x/(a^3*b^2*(a+b*x^2))+3/256*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*b^(5/2))],
[x^2/(a^2+2*a*b*x^2+b^2*x^4)^3,x,7,-1/10*x/(b*(a+b*x^2)^5)+1/80*x/(a*b*(a+b*x^2)^4)+7/480*x/(a^2*b*(a+b*x^2)^3)+7/384*x/(a^3*b*(a+b*x^2)^2)+7/256*x/(a^4*b*(a+b*x^2))+7/256*arctan(x*sqrt(b)/sqrt(a))/(a^(9/2)*b^(3/2))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^3,x,7,1/10*x/(a*(a+b*x^2)^5)+9/80*x/(a^2*(a+b*x^2)^4)+21/160*x/(a^3*(a+b*x^2)^3)+21/128*x/(a^4*(a+b*x^2)^2)+63/256*x/(a^5*(a+b*x^2))+63/256*arctan(x*sqrt(b)/sqrt(a))/(a^(11/2)*sqrt(b))],
[1/(x^2*(a^2+2*a*b*x^2+b^2*x^4)^3),x,8,(-693/256)/(a^6*x)+1/10/(a*x*(a+b*x^2)^5)+11/80/(a^2*x*(a+b*x^2)^4)+33/160/(a^3*x*(a+b*x^2)^3)+231/640/(a^4*x*(a+b*x^2)^2)+231/256/(a^5*x*(a+b*x^2))-693/256*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/a^(13/2)],
[1/(x^4*(a^2+2*a*b*x^2+b^2*x^4)^3),x,9,(-1001/256)/(a^6*x^3)+3003/256*b/(a^7*x)+1/10/(a*x^3*(a+b*x^2)^5)+13/80/(a^2*x^3*(a+b*x^2)^4)+143/480/(a^3*x^3*(a+b*x^2)^3)+429/640/(a^4*x^3*(a+b*x^2)^2)+3003/1280/(a^5*x^3*(a+b*x^2))+3003/256*b^(3/2)*arctan(x*sqrt(b)/sqrt(a))/a^(15/2)],
[1/(x^6*(a^2+2*a*b*x^2+b^2*x^4)^3),x,10,(-9009/1280)/(a^6*x^5)+3003/256*b/(a^7*x^3)-9009/256*b^2/(a^8*x)+1/10/(a*x^5*(a+b*x^2)^5)+3/16/(a^2*x^5*(a+b*x^2)^4)+13/32/(a^3*x^5*(a+b*x^2)^3)+143/128/(a^4*x^5*(a+b*x^2)^2)+1287/256/(a^5*x^5*(a+b*x^2))-9009/256*b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/a^(17/2)],
[1/(1+2*x^2+x^4),x,3,1/2*x/(1+x^2)+1/2*arctan(x)],
[x/(1+2*x^2+x^4),x,2,(-1/2)/(1+x^2)],
[x^2/(1+2*x^2+x^4),x,3,-1/2*x/(1+x^2)+1/2*arctan(x)],
[x^3/(1+2*x^2+x^4),x,4,1/2/(1+x^2)+1/2*log(1+x^2)],
[x/(81-18*x^2+x^4),x,2,1/2/(9-x^2)],
[x^3/(16-8*x^2+x^4),x,4,2/(4-x^2)+1/2*log(4-x^2)],

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

# p>0
[x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,4,1/6*a*x^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/8*b*x^8*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,1/6*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/b^2-1/4*a*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^2],
[x*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,2,1/4*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x,x,3,1/2*b*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^3,x,3,-1/2*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+b*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^5,x,3,-1/4*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a*x^4)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^7,x,1,1/12*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(a^2*x^6)-1/4*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a*x^6)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^9,x,4,-1/8*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))-1/6*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^6*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^11,x,4,-1/10*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))-1/8*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))],
[x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,1/5*a*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/7*b*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,1/3*a*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/5*b*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4),x,2,a*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*b*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^2,x,3,-a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+b*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^4,x,3,-1/3*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^6,x,3,-1/5*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-1/3*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^8,x,3,-1/7*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-1/5*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/x^10,x,3,-1/9*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-1/7*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))],
[x^9*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,1/10*a^3*x^10*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/4*a^2*b*x^12*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/14*a*b^2*x^14*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/16*b^3*x^16*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^7*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,1/8*a^3*x^8*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/10*a^2*b*x^10*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/4*a*b^2*x^12*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/14*b^3*x^14*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^5*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/8*a^2*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/b^3-1/5*a*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/b^3+1/12*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/b^3,1/8*a^2*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^3-1/5*a*(a+b*x^2)^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^3+1/12*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^3],
[x^3*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,-1/8*a*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/b^2+1/10*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/b^2],
[x*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,2,1/8*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/b],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x,x,4,3/2*a^2*b*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/4*a*b^2*x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/6*b^3*x^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a^3*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^3,x,4,-1/2*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+3/2*a*b^2*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/4*b^3*x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3*a^2*b*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^5,x,4,-1/4*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^4*(a+b*x^2))-3/2*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+1/2*b^3*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3*a*b^2*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^7,x,4,-1/6*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^6*(a+b*x^2))-3/4*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^4*(a+b*x^2))-3/2*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+b^3*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^9,x,3,-1/8*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a*x^8)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^11,x,1,-1/8*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(a*x^10)+1/40*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a^2*x^10)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^13,x,4,-1/12*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^12*(a+b*x^2))-3/10*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))-3/8*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))-1/6*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^6*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^15,x,4,-1/14*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^14*(a+b*x^2))-1/4*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^12*(a+b*x^2))-3/10*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))-1/8*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^17,x,4,-1/16*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^16*(a+b*x^2))-3/14*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^14*(a+b*x^2))-1/4*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^12*(a+b*x^2))-1/10*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))],
[x^8*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/9*a^3*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/11*a^2*b*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/13*a*b^2*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/15*b^3*x^15*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^6*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/7*a^3*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*a^2*b*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/11*a*b^2*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/13*b^3*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^4*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/5*a^3*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/7*a^2*b*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*a*b^2*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/11*b^3*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/3*a^3*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/5*a^2*b*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+3/7*a*b^2*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/9*b^3*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,a^3*x*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(a+b*x^2)^3+a^2*b*x^3*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(a+b*x^2)^3+3/5*a*b^2*x^5*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(a+b*x^2)^3+1/7*b^3*x^7*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(a+b*x^2)^3],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^2,x,3,-a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+3*a^2*b*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a*b^2*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/5*b^3*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^4,x,3,-1/3*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-3*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+3*a*b^2*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*b^3*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^6,x,3,-1/5*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-3*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+b^3*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^8,x,3,-1/7*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-3/5*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^10,x,3,-1/9*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-3/7*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-3/5*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-1/3*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^12,x,3,-1/11*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-1/3*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-3/7*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-1/5*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^14,x,3,-1/13*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-3/11*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-1/3*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-1/7*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/x^16,x,3,-1/15*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^15*(a+b*x^2))-3/13*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-3/11*a*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-1/9*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))],
[x^13*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,4,1/14*a^5*x^14*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/16*a^4*b*x^16*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/9*a^3*b^2*x^18*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/2*a^2*b^3*x^20*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/22*a*b^4*x^22*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/24*b^5*x^24*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^11*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,4,1/12*a^5*x^12*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/14*a^4*b*x^14*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/8*a^3*b^2*x^16*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/9*a^2*b^3*x^18*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/4*a*b^4*x^20*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/22*b^5*x^22*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^9*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/12*a^4*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^5-2/7*a^3*(a+b*x^2)^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^5+3/8*a^2*(a+b*x^2)^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^5-2/9*a*(a+b*x^2)^8*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^5+1/20*(a+b*x^2)^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^5],
[x^7*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,4,-1/12*a^3*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^4+3/14*a^2*(a+b*x^2)^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^4-3/16*a*(a+b*x^2)^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^4+1/18*(a+b*x^2)^8*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^4],
[x^5*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,4,1/12*a^2*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^3-1/7*a*(a+b*x^2)^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^3+1/16*(a+b*x^2)^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^3],
[x^3*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,-1/12*a*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/b^2+1/14*(a^2+2*a*b*x^2+b^2*x^4)^(7/2)/b^2],
[x*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,2,1/12*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/b],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x,x,4,5/2*a^4*b*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/2*a^3*b^2*x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/3*a^2*b^3*x^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/8*a*b^4*x^8*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/10*b^5*x^10*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a^5*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^3,x,4,-1/2*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+5*a^3*b^2*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/2*a^2*b^3*x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/6*a*b^4*x^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/8*b^5*x^8*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5*a^4*b*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^5,x,4,-1/4*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^4*(a+b*x^2))-5/2*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+5*a^2*b^3*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/4*a*b^4*x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/6*b^5*x^6*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10*a^3*b^2*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^7,x,4,-1/6*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^6*(a+b*x^2))-5/4*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^4*(a+b*x^2))-5*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+5/2*a*b^4*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/4*b^5*x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10*a^2*b^3*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^9,x,4,-1/8*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))-5/6*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^6*(a+b*x^2))-5/2*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^4*(a+b*x^2))-5*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+1/2*b^5*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5*a*b^4*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^11,x,4,-1/10*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))-5/8*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))-5/3*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^6*(a+b*x^2))-5/2*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^4*(a+b*x^2))-5/2*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^2*(a+b*x^2))+b^5*log(x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^13,x,3,-1/12*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a*x^12)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^15,x,1,-1/12*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a*x^14)+1/84*(a^2+2*a*b*x^2+b^2*x^4)^(7/2)/(a^2*x^14)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^17,x,5,-1/16*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a*x^16)+1/56*b*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a^2*x^14)-1/336*b^2*(a+b*x^2)^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a^3*x^12)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^19,x,4,-1/18*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^18*(a+b*x^2))-5/16*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^16*(a+b*x^2))-5/7*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^14*(a+b*x^2))-5/6*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^12*(a+b*x^2))-1/2*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))-1/8*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^8*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^21,x,4,-1/20*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^20*(a+b*x^2))-5/18*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^18*(a+b*x^2))-5/8*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^16*(a+b*x^2))-5/7*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^14*(a+b*x^2))-5/12*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^12*(a+b*x^2))-1/10*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^10*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^23,x,4,-1/22*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^22*(a+b*x^2))-1/4*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^20*(a+b*x^2))-5/9*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^18*(a+b*x^2))-5/8*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^16*(a+b*x^2))-5/14*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^14*(a+b*x^2))-1/12*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^12*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^25,x,4,-1/24*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^24*(a+b*x^2))-5/22*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^22*(a+b*x^2))-1/2*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^20*(a+b*x^2))-5/9*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^18*(a+b*x^2))-5/16*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^16*(a+b*x^2))-1/14*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^14*(a+b*x^2))],
[x^12*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/13*a^5*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*a^4*b*x^15*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/17*a^3*b^2*x^17*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/19*a^2*b^3*x^19*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/21*a*b^4*x^21*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/23*b^5*x^23*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^10*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/11*a^5*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/13*a^4*b*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+2/3*a^3*b^2*x^15*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/17*a^2*b^3*x^17*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/19*a*b^4*x^19*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/21*b^5*x^21*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^8*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/9*a^5*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/11*a^4*b*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/13*a^3*b^2*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+2/3*a^2*b^3*x^15*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/17*a*b^4*x^17*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/19*b^5*x^19*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^6*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/7*a^5*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/9*a^4*b*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/11*a^3*b^2*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/13*a^2*b^3*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*a*b^4*x^15*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/17*b^5*x^17*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^4*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/5*a^5*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/7*a^4*b*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/9*a^3*b^2*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/11*a^2*b^3*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/13*a*b^4*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/15*b^5*x^15*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[x^2*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/3*a^5*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a^4*b*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/7*a^3*b^2*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/9*a^2*b^3*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/11*a*b^4*x^11*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/13*b^5*x^13*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,a^5*x*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a+b*x^2)^5+5/3*a^4*b*x^3*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a+b*x^2)^5+2*a^3*b^2*x^5*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a+b*x^2)^5+10/7*a^2*b^3*x^7*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a+b*x^2)^5+5/9*a*b^4*x^9*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a+b*x^2)^5+1/11*b^5*x^11*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(a+b*x^2)^5],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^2,x,3,-a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+5*a^4*b*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/3*a^3*b^2*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+2*a^2*b^3*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/7*a*b^4*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/9*b^5*x^9*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^4,x,3,-1/3*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-5*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+10*a^3*b^2*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+10/3*a^2*b^3*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+a*b^4*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/7*b^5*x^7*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^6,x,3,-1/5*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-5/3*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-10*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+10*a^2*b^3*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+5/3*a*b^4*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/5*b^5*x^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^8,x,3,-1/7*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-10/3*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-10*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+5*a*b^4*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)+1/3*b^5*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^10,x,3,-1/9*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-5/7*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-2*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-10/3*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-5*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))+b^5*x*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(a+b*x^2)],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^12,x,3,-1/11*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-5/9*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-10/7*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-2*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-5/3*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))-b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^14,x,3,-1/13*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-5/11*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-10/9*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-10/7*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))-1/3*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^3*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^16,x,3,-1/15*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^15*(a+b*x^2))-5/13*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-10/11*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-10/9*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-5/7*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))-1/5*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^5*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^18,x,3,-1/17*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^17*(a+b*x^2))-1/3*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^15*(a+b*x^2))-10/13*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-10/11*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-5/9*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))-1/7*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^7*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^20,x,3,-1/19*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^19*(a+b*x^2))-5/17*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^17*(a+b*x^2))-2/3*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^15*(a+b*x^2))-10/13*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-5/11*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))-1/9*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^9*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^22,x,3,-1/21*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^21*(a+b*x^2))-5/19*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^19*(a+b*x^2))-10/17*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^17*(a+b*x^2))-2/3*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^15*(a+b*x^2))-5/13*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))-1/11*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^11*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/x^24,x,3,-1/23*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^23*(a+b*x^2))-5/21*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^21*(a+b*x^2))-10/19*a^3*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^19*(a+b*x^2))-10/17*a^2*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^17*(a+b*x^2))-1/3*a*b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^15*(a+b*x^2))-1/13*b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(x^13*(a+b*x^2))],

# p<0
[x^5/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,4,-1/2*a*x^2*(a+b*x^2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4*x^4*(a+b*x^2)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*a^2*(a+b*x^2)*log(a+b*x^2)/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^3/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,4,-1/2*a*(a+b*x^2)*log(a+b*x^2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*sqrt(a^2+2*a*b*x^2+b^2*x^4)/b^2],
[x/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,1/2*(a+b*x^2)*log(a+b*x^2)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,5,(a+b*x^2)*log(x)/(a*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*(a+b*x^2)*log(a+b*x^2)/(a*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,4,1/2*(-a-b*x^2)/(a*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-b*(a+b*x^2)*log(x)/(a^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*b*(a+b*x^2)*log(a+b*x^2)/(a^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^4/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,4,-a*x*(a+b*x^2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/3*x^3*(a+b*x^2)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))+a^(3/2)*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(b^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^2/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,x*(a+b*x^2)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/(b^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,2,(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,3,(-a-b*x^2)/(a*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))-(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^4*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,4,1/3*(-a-b*x^2)/(a*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+b*(a+b*x^2)/(a^2*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))+b^(3/2)*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^7/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,-3/2*a^2/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4*a^3/(b^4*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*x^2*(a+b*x^2)/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/2*a*(a+b*x^2)*log(a+b*x^2)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^5/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,a/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*a^2/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*(a+b*x^2)*log(a+b*x^2)/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^3/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,1/4*x^4/(a*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)),(-1/2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4*a/(b^2*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,2,(-1/4)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,4,1/2/(a^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+(a+b*x^2)*log(x)/(a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*(a+b*x^2)*log(a+b*x^2)/(a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^3*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,4,-b/(a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*b/(a^2*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*(-a-b*x^2)/(a^3*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3*b*(a+b*x^2)*log(x)/(a^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/2*b*(a+b*x^2)*log(a+b*x^2)/(a^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^4/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,-3/8*x/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*x^3/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/8*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(b^(5/2)*sqrt(a)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^2/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,1/8*x/(a*b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*x/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,4,1/4*x*(a+b*x^2)/(a*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))+3/8*x*(a+b*x^2)^2/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))+3/8*(a+b*x^2)^3*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)*sqrt(b))],
[1/(x^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,5,5/8/(a^2*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a*x*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-15/8*(a+b*x^2)/(a^3*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))-15/8*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^4*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,6,7/8/(a^2*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a*x^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-35/24*(a+b*x^2)/(a^3*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/8*b*(a+b*x^2)/(a^4*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/8*b^(3/2)*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^11/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,4,-5*a^2/(b^6*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8*a^5/(b^6*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/6*a^4/(b^6*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/2*a^3/(b^6*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*x^2*(a+b*x^2)/(b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/2*a*(a+b*x^2)*log(a+b*x^2)/(b^6*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^9/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,4,2*a/(b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*a^4/(b^5*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+2/3*a^3/(b^5*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/2*a^2/(b^5*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*(a+b*x^2)*log(a+b*x^2)/(b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^7/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,1/8*x^8/(a*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^5/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,1,1/24*x^6/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))+1/8*x^6/(a*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))],
[x^3/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,(-1/6)/(b^2*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))+1/8*a/(b^2*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))],
[x/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,2,(-1/8)/(b*(a+b*x^2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2))],
[1/(x*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,4,1/2/(a^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8/(a*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/6/(a^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+(a+b*x^2)*log(x)/(a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*(a+b*x^2)*log(a+b*x^2)/(a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^3*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,4,-2*b/(a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*b/(a^2*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/3*b/(a^3*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/4*b/(a^4*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*(-a-b*x^2)/(a^5*x^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5*b*(a+b*x^2)*log(x)/(a^6*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/2*b*(a+b*x^2)*log(a+b*x^2)/(a^6*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^6/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,6,5/128*x/(a*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*x^5/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/48*x^3/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/64*x/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/128*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^4/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,6,3/128*x/(a^2*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*x^3/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/16*x/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/64*x/(a*b^2*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/128*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[x^2/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,6,5/128*x/(a^3*b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*x/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/48*x/(a*b*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/192*x/(a^2*b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/128*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(7/2)*b^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,6,1/8*x*(a+b*x^2)/(a*(a^2+2*a*b*x^2+b^2*x^4)^(5/2))+7/48*x*(a+b*x^2)^2/(a^2*(a^2+2*a*b*x^2+b^2*x^4)^(5/2))+35/192*x*(a+b*x^2)^3/(a^3*(a^2+2*a*b*x^2+b^2*x^4)^(5/2))+35/128*x*(a+b*x^2)^4/(a^4*(a^2+2*a*b*x^2+b^2*x^4)^(5/2))+35/128*(a+b*x^2)^5*arctan(x*sqrt(b)/sqrt(a))/(a^(9/2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)*sqrt(b))],
[1/(x^2*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,7,105/128/(a^4*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8/(a*x*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/16/(a^2*x*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+21/64/(a^3*x*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-315/128*(a+b*x^2)/(a^5*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))-315/128*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(x^4*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,8,231/128/(a^4*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8/(a*x^3*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+11/48/(a^2*x^3*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+33/64/(a^3*x^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-385/128*(a+b*x^2)/(a^5*x^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1155/128*b*(a+b*x^2)/(a^6*x*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1155/128*b^(3/2)*(a+b*x^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(13/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],

# Integrands of the form x^m (a^2+2 a b x^2+b^2 x^4)^(p/3)

# p>0

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

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

# p>0
[(d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4),x,2,2/7*a^2*(d*x)^(7/2)/d+4/11*a*b*(d*x)^(11/2)/d^3+2/15*b^2*(d*x)^(15/2)/d^5],
[(d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4),x,2,2/5*a^2*(d*x)^(5/2)/d+4/9*a*b*(d*x)^(9/2)/d^3+2/13*b^2*(d*x)^(13/2)/d^5],
[(a^2+2*a*b*x^2+b^2*x^4)*sqrt(d*x),x,2,2/3*a^2*(d*x)^(3/2)/d+4/7*a*b*(d*x)^(7/2)/d^3+2/11*b^2*(d*x)^(11/2)/d^5],
[(a^2+2*a*b*x^2+b^2*x^4)/sqrt(d*x),x,2,4/5*a*b*(d*x)^(5/2)/d^3+2/9*b^2*(d*x)^(9/2)/d^5+2*a^2*sqrt(d*x)/d],
[(a^2+2*a*b*x^2+b^2*x^4)/(d*x)^(3/2),x,2,4/3*a*b*(d*x)^(3/2)/d^3+2/7*b^2*(d*x)^(7/2)/d^5-2*a^2/(d*sqrt(d*x))],
[(a^2+2*a*b*x^2+b^2*x^4)/(d*x)^(5/2),x,2,-2/3*a^2/(d*(d*x)^(3/2))+2/5*b^2*(d*x)^(5/2)/d^5+4*a*b*sqrt(d*x)/d^3],
[(a^2+2*a*b*x^2+b^2*x^4)/(d*x)^(7/2),x,2,-2/5*a^2/(d*(d*x)^(5/2))+2/3*b^2*(d*x)^(3/2)/d^5-4*a*b/(d^3*sqrt(d*x))],
[(d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,2/7*a^4*(d*x)^(7/2)/d+8/11*a^3*b*(d*x)^(11/2)/d^3+4/5*a^2*b^2*(d*x)^(15/2)/d^5+8/19*a*b^3*(d*x)^(19/2)/d^7+2/23*b^4*(d*x)^(23/2)/d^9],
[(d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,2/5*a^4*(d*x)^(5/2)/d+8/9*a^3*b*(d*x)^(9/2)/d^3+12/13*a^2*b^2*(d*x)^(13/2)/d^5+8/17*a*b^3*(d*x)^(17/2)/d^7+2/21*b^4*(d*x)^(21/2)/d^9],
[(a^2+2*a*b*x^2+b^2*x^4)^2*sqrt(d*x),x,3,2/3*a^4*(d*x)^(3/2)/d+8/7*a^3*b*(d*x)^(7/2)/d^3+12/11*a^2*b^2*(d*x)^(11/2)/d^5+8/15*a*b^3*(d*x)^(15/2)/d^7+2/19*b^4*(d*x)^(19/2)/d^9],
[(a^2+2*a*b*x^2+b^2*x^4)^2/sqrt(d*x),x,3,8/5*a^3*b*(d*x)^(5/2)/d^3+4/3*a^2*b^2*(d*x)^(9/2)/d^5+8/13*a*b^3*(d*x)^(13/2)/d^7+2/17*b^4*(d*x)^(17/2)/d^9+2*a^4*sqrt(d*x)/d],
[(a^2+2*a*b*x^2+b^2*x^4)^2/(d*x)^(3/2),x,3,8/3*a^3*b*(d*x)^(3/2)/d^3+12/7*a^2*b^2*(d*x)^(7/2)/d^5+8/11*a*b^3*(d*x)^(11/2)/d^7+2/15*b^4*(d*x)^(15/2)/d^9-2*a^4/(d*sqrt(d*x))],
[(a^2+2*a*b*x^2+b^2*x^4)^2/(d*x)^(5/2),x,3,-2/3*a^4/(d*(d*x)^(3/2))+12/5*a^2*b^2*(d*x)^(5/2)/d^5+8/9*a*b^3*(d*x)^(9/2)/d^7+2/13*b^4*(d*x)^(13/2)/d^9+8*a^3*b*sqrt(d*x)/d^3],
[(a^2+2*a*b*x^2+b^2*x^4)^2/(d*x)^(7/2),x,3,-2/5*a^4/(d*(d*x)^(5/2))+4*a^2*b^2*(d*x)^(3/2)/d^5+8/7*a*b^3*(d*x)^(7/2)/d^7+2/11*b^4*(d*x)^(11/2)/d^9-8*a^3*b/(d^3*sqrt(d*x))],
[(d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,2/7*a^6*(d*x)^(7/2)/d+12/11*a^5*b*(d*x)^(11/2)/d^3+2*a^4*b^2*(d*x)^(15/2)/d^5+40/19*a^3*b^3*(d*x)^(19/2)/d^7+30/23*a^2*b^4*(d*x)^(23/2)/d^9+4/9*a*b^5*(d*x)^(27/2)/d^11+2/31*b^6*(d*x)^(31/2)/d^13],
[(d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,2/5*a^6*(d*x)^(5/2)/d+4/3*a^5*b*(d*x)^(9/2)/d^3+30/13*a^4*b^2*(d*x)^(13/2)/d^5+40/17*a^3*b^3*(d*x)^(17/2)/d^7+10/7*a^2*b^4*(d*x)^(21/2)/d^9+12/25*a*b^5*(d*x)^(25/2)/d^11+2/29*b^6*(d*x)^(29/2)/d^13],
[(a^2+2*a*b*x^2+b^2*x^4)^3*sqrt(d*x),x,3,2/3*a^6*(d*x)^(3/2)/d+12/7*a^5*b*(d*x)^(7/2)/d^3+30/11*a^4*b^2*(d*x)^(11/2)/d^5+8/3*a^3*b^3*(d*x)^(15/2)/d^7+30/19*a^2*b^4*(d*x)^(19/2)/d^9+12/23*a*b^5*(d*x)^(23/2)/d^11+2/27*b^6*(d*x)^(27/2)/d^13],
[(a^2+2*a*b*x^2+b^2*x^4)^3/sqrt(d*x),x,3,12/5*a^5*b*(d*x)^(5/2)/d^3+10/3*a^4*b^2*(d*x)^(9/2)/d^5+40/13*a^3*b^3*(d*x)^(13/2)/d^7+30/17*a^2*b^4*(d*x)^(17/2)/d^9+4/7*a*b^5*(d*x)^(21/2)/d^11+2/25*b^6*(d*x)^(25/2)/d^13+2*a^6*sqrt(d*x)/d],
[(a^2+2*a*b*x^2+b^2*x^4)^3/(d*x)^(3/2),x,3,4*a^5*b*(d*x)^(3/2)/d^3+30/7*a^4*b^2*(d*x)^(7/2)/d^5+40/11*a^3*b^3*(d*x)^(11/2)/d^7+2*a^2*b^4*(d*x)^(15/2)/d^9+12/19*a*b^5*(d*x)^(19/2)/d^11+2/23*b^6*(d*x)^(23/2)/d^13-2*a^6/(d*sqrt(d*x))],
[(a^2+2*a*b*x^2+b^2*x^4)^3/(d*x)^(5/2),x,3,-2/3*a^6/(d*(d*x)^(3/2))+6*a^4*b^2*(d*x)^(5/2)/d^5+40/9*a^3*b^3*(d*x)^(9/2)/d^7+30/13*a^2*b^4*(d*x)^(13/2)/d^9+12/17*a*b^5*(d*x)^(17/2)/d^11+2/21*b^6*(d*x)^(21/2)/d^13+12*a^5*b*sqrt(d*x)/d^3],
[(a^2+2*a*b*x^2+b^2*x^4)^3/(d*x)^(7/2),x,3,-2/5*a^6/(d*(d*x)^(5/2))+10*a^4*b^2*(d*x)^(3/2)/d^5+40/7*a^3*b^3*(d*x)^(7/2)/d^7+30/11*a^2*b^4*(d*x)^(11/2)/d^9+4/5*a*b^5*(d*x)^(15/2)/d^11+2/19*b^6*(d*x)^(19/2)/d^13-12*a^5*b/(d^3*sqrt(d*x))],

# p<0
[(d*x)^(11/2)/(a^2+2*a*b*x^2+b^2*x^4),x,14,9/10*d^3*(d*x)^(5/2)/b^2-1/2*d*(d*x)^(9/2)/(b*(a+b*x^2))-9/4*a^(5/4)*d^(11/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(13/4)*sqrt(2))+9/4*a^(5/4)*d^(11/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(13/4)*sqrt(2))-9/8*a^(5/4)*d^(11/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(13/4)*sqrt(2))+9/8*a^(5/4)*d^(11/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(13/4)*sqrt(2))-9/2*a*d^5*sqrt(d*x)/b^3],
[(d*x)^(9/2)/(a^2+2*a*b*x^2+b^2*x^4),x,13,7/6*d^3*(d*x)^(3/2)/b^2-1/2*d*(d*x)^(7/2)/(b*(a+b*x^2))+7/4*a^(3/4)*d^(9/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(11/4)*sqrt(2))-7/4*a^(3/4)*d^(9/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(11/4)*sqrt(2))-7/8*a^(3/4)*d^(9/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(11/4)*sqrt(2))+7/8*a^(3/4)*d^(9/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(11/4)*sqrt(2))],
[(d*x)^(7/2)/(a^2+2*a*b*x^2+b^2*x^4),x,13,-1/2*d*(d*x)^(5/2)/(b*(a+b*x^2))+5/4*a^(1/4)*d^(7/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(9/4)*sqrt(2))-5/4*a^(1/4)*d^(7/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(9/4)*sqrt(2))+5/8*a^(1/4)*d^(7/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(9/4)*sqrt(2))-5/8*a^(1/4)*d^(7/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(9/4)*sqrt(2))+5/2*d^3*sqrt(d*x)/b^2],
[(d*x)^(5/2)/(a^2+2*a*b*x^2+b^2*x^4),x,12,-1/2*d*(d*x)^(3/2)/(b*(a+b*x^2))-3/4*d^(5/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(7/4)*sqrt(2))+3/4*d^(5/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(7/4)*sqrt(2))+3/8*d^(5/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(7/4)*sqrt(2))-3/8*d^(5/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(7/4)*sqrt(2))],
[(d*x)^(3/2)/(a^2+2*a*b*x^2+b^2*x^4),x,12,-1/4*d^(3/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(5/4)*sqrt(2))+1/4*d^(3/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(5/4)*sqrt(2))-1/8*d^(3/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(5/4)*sqrt(2))+1/8*d^(3/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(5/4)*sqrt(2))-1/2*d*sqrt(d*x)/(b*(a+b*x^2))],
[sqrt(d*x)/(a^2+2*a*b*x^2+b^2*x^4),x,12,1/2*(d*x)^(3/2)/(a*d*(a+b*x^2))-1/4*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(5/4)*b^(3/4)*sqrt(2))+1/4*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(5/4)*b^(3/4)*sqrt(2))+1/8*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(5/4)*b^(3/4)*sqrt(2))-1/8*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(5/4)*b^(3/4)*sqrt(2))],
[1/((a^2+2*a*b*x^2+b^2*x^4)*sqrt(d*x)),x,12,-3/4*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(1/4)*sqrt(2)*sqrt(d))+3/4*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(1/4)*sqrt(2)*sqrt(d))-3/8*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(1/4)*sqrt(2)*sqrt(d))+3/8*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(1/4)*sqrt(2)*sqrt(d))+1/2*sqrt(d*x)/(a*d*(a+b*x^2))],
[1/((d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)),x,13,5/4*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*d^(3/2)*sqrt(2))-5/4*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*d^(3/2)*sqrt(2))-5/8*b^(1/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*d^(3/2)*sqrt(2))+5/8*b^(1/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*d^(3/2)*sqrt(2))+(-5/2)/(a^2*d*sqrt(d*x))+1/2/(a*d*(a+b*x^2)*sqrt(d*x))],
[1/((d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)),x,13,(-7/6)/(a^2*d*(d*x)^(3/2))+1/2/(a*d*(d*x)^(3/2)*(a+b*x^2))+7/4*b^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*d^(5/2)*sqrt(2))-7/4*b^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*d^(5/2)*sqrt(2))+7/8*b^(3/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*d^(5/2)*sqrt(2))-7/8*b^(3/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*d^(5/2)*sqrt(2))],
[1/((d*x)^(7/2)*(a^2+2*a*b*x^2+b^2*x^4)),x,14,(-9/10)/(a^2*d*(d*x)^(5/2))+1/2/(a*d*(d*x)^(5/2)*(a+b*x^2))-9/4*b^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*d^(7/2)*sqrt(2))+9/4*b^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*d^(7/2)*sqrt(2))+9/8*b^(5/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*d^(7/2)*sqrt(2))-9/8*b^(5/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*d^(7/2)*sqrt(2))+9/2*b/(a^3*d^3*sqrt(d*x))],
[(d*x)^(19/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,16,663/320*d^7*(d*x)^(5/2)/b^4-1/6*d*(d*x)^(17/2)/(b*(a+b*x^2)^3)-17/48*d^3*(d*x)^(13/2)/(b^2*(a+b*x^2)^2)-221/192*d^5*(d*x)^(9/2)/(b^3*(a+b*x^2))-663/128*a^(5/4)*d^(19/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(21/4)*sqrt(2))+663/128*a^(5/4)*d^(19/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(21/4)*sqrt(2))-663/256*a^(5/4)*d^(19/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(21/4)*sqrt(2))+663/256*a^(5/4)*d^(19/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(21/4)*sqrt(2))-663/64*a*d^9*sqrt(d*x)/b^5],
[(d*x)^(17/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,15,385/192*d^7*(d*x)^(3/2)/b^4-1/6*d*(d*x)^(15/2)/(b*(a+b*x^2)^3)-5/16*d^3*(d*x)^(11/2)/(b^2*(a+b*x^2)^2)-55/64*d^5*(d*x)^(7/2)/(b^3*(a+b*x^2))+385/128*a^(3/4)*d^(17/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(19/4)*sqrt(2))-385/128*a^(3/4)*d^(17/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(19/4)*sqrt(2))-385/256*a^(3/4)*d^(17/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(19/4)*sqrt(2))+385/256*a^(3/4)*d^(17/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(19/4)*sqrt(2))],
[(d*x)^(15/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,15,-1/6*d*(d*x)^(13/2)/(b*(a+b*x^2)^3)-13/48*d^3*(d*x)^(9/2)/(b^2*(a+b*x^2)^2)-39/64*d^5*(d*x)^(5/2)/(b^3*(a+b*x^2))+195/128*a^(1/4)*d^(15/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(17/4)*sqrt(2))-195/128*a^(1/4)*d^(15/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(17/4)*sqrt(2))+195/256*a^(1/4)*d^(15/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(17/4)*sqrt(2))-195/256*a^(1/4)*d^(15/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(17/4)*sqrt(2))+195/64*d^7*sqrt(d*x)/b^4],
[(d*x)^(13/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,-1/6*d*(d*x)^(11/2)/(b*(a+b*x^2)^3)-11/48*d^3*(d*x)^(7/2)/(b^2*(a+b*x^2)^2)-77/192*d^5*(d*x)^(3/2)/(b^3*(a+b*x^2))-77/128*d^(13/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(15/4)*sqrt(2))+77/128*d^(13/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(15/4)*sqrt(2))+77/256*d^(13/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(15/4)*sqrt(2))-77/256*d^(13/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(15/4)*sqrt(2))],
[(d*x)^(11/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,-1/6*d*(d*x)^(9/2)/(b*(a+b*x^2)^3)-3/16*d^3*(d*x)^(5/2)/(b^2*(a+b*x^2)^2)-15/128*d^(11/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(13/4)*sqrt(2))+15/128*d^(11/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(13/4)*sqrt(2))-15/256*d^(11/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(13/4)*sqrt(2))+15/256*d^(11/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(13/4)*sqrt(2))-15/64*d^5*sqrt(d*x)/(b^3*(a+b*x^2))],
[(d*x)^(9/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,-1/6*d*(d*x)^(7/2)/(b*(a+b*x^2)^3)-7/48*d^3*(d*x)^(3/2)/(b^2*(a+b*x^2)^2)+7/64*d^3*(d*x)^(3/2)/(a*b^2*(a+b*x^2))-7/128*d^(9/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(11/4)*sqrt(2))+7/128*d^(9/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(11/4)*sqrt(2))+7/256*d^(9/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(11/4)*sqrt(2))-7/256*d^(9/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(11/4)*sqrt(2))],
[(d*x)^(7/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,-1/6*d*(d*x)^(5/2)/(b*(a+b*x^2)^3)-5/128*d^(7/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(9/4)*sqrt(2))+5/128*d^(7/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(9/4)*sqrt(2))-5/256*d^(7/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(9/4)*sqrt(2))+5/256*d^(7/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(9/4)*sqrt(2))-5/48*d^3*sqrt(d*x)/(b^2*(a+b*x^2)^2)+5/192*d^3*sqrt(d*x)/(a*b^2*(a+b*x^2))],
[(d*x)^(5/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,-1/6*d*(d*x)^(3/2)/(b*(a+b*x^2)^3)+1/16*d*(d*x)^(3/2)/(a*b*(a+b*x^2)^2)+5/64*d*(d*x)^(3/2)/(a^2*b*(a+b*x^2))-5/128*d^(5/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*b^(7/4)*sqrt(2))+5/128*d^(5/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*b^(7/4)*sqrt(2))+5/256*d^(5/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*b^(7/4)*sqrt(2))-5/256*d^(5/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*b^(7/4)*sqrt(2))],
[(d*x)^(3/2)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,-7/128*d^(3/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(5/4)*sqrt(2))+7/128*d^(3/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(5/4)*sqrt(2))-7/256*d^(3/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(5/4)*sqrt(2))+7/256*d^(3/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(5/4)*sqrt(2))-1/6*d*sqrt(d*x)/(b*(a+b*x^2)^3)+1/48*d*sqrt(d*x)/(a*b*(a+b*x^2)^2)+7/192*d*sqrt(d*x)/(a^2*b*(a+b*x^2))],
[sqrt(d*x)/(a^2+2*a*b*x^2+b^2*x^4)^2,x,14,1/6*(d*x)^(3/2)/(a*d*(a+b*x^2)^3)+3/16*(d*x)^(3/2)/(a^2*d*(a+b*x^2)^2)+15/64*(d*x)^(3/2)/(a^3*d*(a+b*x^2))-15/128*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(13/4)*b^(3/4)*sqrt(2))+15/128*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(13/4)*b^(3/4)*sqrt(2))+15/256*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(13/4)*b^(3/4)*sqrt(2))-15/256*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(13/4)*b^(3/4)*sqrt(2))],
[1/((a^2+2*a*b*x^2+b^2*x^4)^2*sqrt(d*x)),x,14,-77/128*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*b^(1/4)*sqrt(2)*sqrt(d))+77/128*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*b^(1/4)*sqrt(2)*sqrt(d))-77/256*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*b^(1/4)*sqrt(2)*sqrt(d))+77/256*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*b^(1/4)*sqrt(2)*sqrt(d))+1/6*sqrt(d*x)/(a*d*(a+b*x^2)^3)+11/48*sqrt(d*x)/(a^2*d*(a+b*x^2)^2)+77/192*sqrt(d*x)/(a^3*d*(a+b*x^2))],
[1/((d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^2),x,15,195/128*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(17/4)*d^(3/2)*sqrt(2))-195/128*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(17/4)*d^(3/2)*sqrt(2))-195/256*b^(1/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(17/4)*d^(3/2)*sqrt(2))+195/256*b^(1/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(17/4)*d^(3/2)*sqrt(2))+(-195/64)/(a^4*d*sqrt(d*x))+1/6/(a*d*(a+b*x^2)^3*sqrt(d*x))+13/48/(a^2*d*(a+b*x^2)^2*sqrt(d*x))+39/64/(a^3*d*(a+b*x^2)*sqrt(d*x))],
[1/((d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^2),x,15,(-385/192)/(a^4*d*(d*x)^(3/2))+1/6/(a*d*(d*x)^(3/2)*(a+b*x^2)^3)+5/16/(a^2*d*(d*x)^(3/2)*(a+b*x^2)^2)+55/64/(a^3*d*(d*x)^(3/2)*(a+b*x^2))+385/128*b^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(19/4)*d^(5/2)*sqrt(2))-385/128*b^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(19/4)*d^(5/2)*sqrt(2))+385/256*b^(3/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(19/4)*d^(5/2)*sqrt(2))-385/256*b^(3/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(19/4)*d^(5/2)*sqrt(2))],
[1/((d*x)^(7/2)*(a^2+2*a*b*x^2+b^2*x^4)^2),x,16,(-663/320)/(a^4*d*(d*x)^(5/2))+1/6/(a*d*(d*x)^(5/2)*(a+b*x^2)^3)+17/48/(a^2*d*(d*x)^(5/2)*(a+b*x^2)^2)+221/192/(a^3*d*(d*x)^(5/2)*(a+b*x^2))-663/128*b^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(21/4)*d^(7/2)*sqrt(2))+663/128*b^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(21/4)*d^(7/2)*sqrt(2))+663/256*b^(5/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(21/4)*d^(7/2)*sqrt(2))-663/256*b^(5/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(21/4)*d^(7/2)*sqrt(2))+663/64*b/(a^5*d^3*sqrt(d*x))],
[(d*x)^(27/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,18,13923/4096*d^11*(d*x)^(5/2)/b^6-1/10*d*(d*x)^(25/2)/(b*(a+b*x^2)^5)-5/32*d^3*(d*x)^(21/2)/(b^2*(a+b*x^2)^4)-35/128*d^5*(d*x)^(17/2)/(b^3*(a+b*x^2)^3)-595/1024*d^7*(d*x)^(13/2)/(b^4*(a+b*x^2)^2)-7735/4096*d^9*(d*x)^(9/2)/(b^5*(a+b*x^2))-69615/8192*a^(5/4)*d^(27/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(29/4)*sqrt(2))+69615/8192*a^(5/4)*d^(27/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(29/4)*sqrt(2))-69615/16384*a^(5/4)*d^(27/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(29/4)*sqrt(2))+69615/16384*a^(5/4)*d^(27/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(29/4)*sqrt(2))-69615/4096*a*d^13*sqrt(d*x)/b^7],
[(d*x)^(25/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,17,33649/12288*d^11*(d*x)^(3/2)/b^6-1/10*d*(d*x)^(23/2)/(b*(a+b*x^2)^5)-23/160*d^3*(d*x)^(19/2)/(b^2*(a+b*x^2)^4)-437/1920*d^5*(d*x)^(15/2)/(b^3*(a+b*x^2)^3)-437/1024*d^7*(d*x)^(11/2)/(b^4*(a+b*x^2)^2)-4807/4096*d^9*(d*x)^(7/2)/(b^5*(a+b*x^2))+33649/8192*a^(3/4)*d^(25/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(27/4)*sqrt(2))-33649/8192*a^(3/4)*d^(25/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(27/4)*sqrt(2))-33649/16384*a^(3/4)*d^(25/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(27/4)*sqrt(2))+33649/16384*a^(3/4)*d^(25/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(27/4)*sqrt(2))],
[(d*x)^(23/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,17,-1/10*d*(d*x)^(21/2)/(b*(a+b*x^2)^5)-21/160*d^3*(d*x)^(17/2)/(b^2*(a+b*x^2)^4)-119/640*d^5*(d*x)^(13/2)/(b^3*(a+b*x^2)^3)-1547/5120*d^7*(d*x)^(9/2)/(b^4*(a+b*x^2)^2)-13923/20480*d^9*(d*x)^(5/2)/(b^5*(a+b*x^2))+13923/8192*a^(1/4)*d^(23/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(25/4)*sqrt(2))-13923/8192*a^(1/4)*d^(23/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(25/4)*sqrt(2))+13923/16384*a^(1/4)*d^(23/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(25/4)*sqrt(2))-13923/16384*a^(1/4)*d^(23/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(25/4)*sqrt(2))+13923/4096*d^11*sqrt(d*x)/b^6],
[(d*x)^(21/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(19/2)/(b*(a+b*x^2)^5)-19/160*d^3*(d*x)^(15/2)/(b^2*(a+b*x^2)^4)-19/128*d^5*(d*x)^(11/2)/(b^3*(a+b*x^2)^3)-209/1024*d^7*(d*x)^(7/2)/(b^4*(a+b*x^2)^2)-1463/4096*d^9*(d*x)^(3/2)/(b^5*(a+b*x^2))-4389/8192*d^(21/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(23/4)*sqrt(2))+4389/8192*d^(21/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(23/4)*sqrt(2))+4389/16384*d^(21/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(23/4)*sqrt(2))-4389/16384*d^(21/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(23/4)*sqrt(2))],
[(d*x)^(19/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(17/2)/(b*(a+b*x^2)^5)-17/160*d^3*(d*x)^(13/2)/(b^2*(a+b*x^2)^4)-221/1920*d^5*(d*x)^(9/2)/(b^3*(a+b*x^2)^3)-663/5120*d^7*(d*x)^(5/2)/(b^4*(a+b*x^2)^2)-663/8192*d^(19/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(21/4)*sqrt(2))+663/8192*d^(19/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(21/4)*sqrt(2))-663/16384*d^(19/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(21/4)*sqrt(2))+663/16384*d^(19/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(21/4)*sqrt(2))-663/4096*d^9*sqrt(d*x)/(b^5*(a+b*x^2))],
[(d*x)^(17/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(15/2)/(b*(a+b*x^2)^5)-3/32*d^3*(d*x)^(11/2)/(b^2*(a+b*x^2)^4)-11/128*d^5*(d*x)^(7/2)/(b^3*(a+b*x^2)^3)-77/1024*d^7*(d*x)^(3/2)/(b^4*(a+b*x^2)^2)+231/4096*d^7*(d*x)^(3/2)/(a*b^4*(a+b*x^2))-231/8192*d^(17/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(19/4)*sqrt(2))+231/8192*d^(17/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(19/4)*sqrt(2))+231/16384*d^(17/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(19/4)*sqrt(2))-231/16384*d^(17/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(19/4)*sqrt(2))],
[(d*x)^(15/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(13/2)/(b*(a+b*x^2)^5)-13/160*d^3*(d*x)^(9/2)/(b^2*(a+b*x^2)^4)-39/640*d^5*(d*x)^(5/2)/(b^3*(a+b*x^2)^3)-117/8192*d^(15/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(17/4)*sqrt(2))+117/8192*d^(15/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(17/4)*sqrt(2))-117/16384*d^(15/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(17/4)*sqrt(2))+117/16384*d^(15/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(17/4)*sqrt(2))-39/1024*d^7*sqrt(d*x)/(b^4*(a+b*x^2)^2)+39/4096*d^7*sqrt(d*x)/(a*b^4*(a+b*x^2))],
[(d*x)^(13/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(11/2)/(b*(a+b*x^2)^5)-11/160*d^3*(d*x)^(7/2)/(b^2*(a+b*x^2)^4)-77/1920*d^5*(d*x)^(3/2)/(b^3*(a+b*x^2)^3)+77/5120*d^5*(d*x)^(3/2)/(a*b^3*(a+b*x^2)^2)+77/4096*d^5*(d*x)^(3/2)/(a^2*b^3*(a+b*x^2))-77/8192*d^(13/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*b^(15/4)*sqrt(2))+77/8192*d^(13/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*b^(15/4)*sqrt(2))+77/16384*d^(13/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*b^(15/4)*sqrt(2))-77/16384*d^(13/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*b^(15/4)*sqrt(2))],
[(d*x)^(11/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(9/2)/(b*(a+b*x^2)^5)-9/160*d^3*(d*x)^(5/2)/(b^2*(a+b*x^2)^4)-63/8192*d^(11/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(13/4)*sqrt(2))+63/8192*d^(11/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(13/4)*sqrt(2))-63/16384*d^(11/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(13/4)*sqrt(2))+63/16384*d^(11/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(13/4)*sqrt(2))-3/128*d^5*sqrt(d*x)/(b^3*(a+b*x^2)^3)+3/1024*d^5*sqrt(d*x)/(a*b^3*(a+b*x^2)^2)+21/4096*d^5*sqrt(d*x)/(a^2*b^3*(a+b*x^2))],
[(d*x)^(9/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(7/2)/(b*(a+b*x^2)^5)-7/160*d^3*(d*x)^(3/2)/(b^2*(a+b*x^2)^4)+7/640*d^3*(d*x)^(3/2)/(a*b^2*(a+b*x^2)^3)+63/5120*d^3*(d*x)^(3/2)/(a^2*b^2*(a+b*x^2)^2)+63/4096*d^3*(d*x)^(3/2)/(a^3*b^2*(a+b*x^2))-63/8192*d^(9/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*b^(11/4)*sqrt(2))+63/8192*d^(9/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*b^(11/4)*sqrt(2))+63/16384*d^(9/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*b^(11/4)*sqrt(2))-63/16384*d^(9/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*b^(11/4)*sqrt(2))],
[(d*x)^(7/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(5/2)/(b*(a+b*x^2)^5)-77/8192*d^(7/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*b^(9/4)*sqrt(2))+77/8192*d^(7/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*b^(9/4)*sqrt(2))-77/16384*d^(7/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*b^(9/4)*sqrt(2))+77/16384*d^(7/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*b^(9/4)*sqrt(2))-1/32*d^3*sqrt(d*x)/(b^2*(a+b*x^2)^4)+1/384*d^3*sqrt(d*x)/(a*b^2*(a+b*x^2)^3)+11/3072*d^3*sqrt(d*x)/(a^2*b^2*(a+b*x^2)^2)+77/12288*d^3*sqrt(d*x)/(a^3*b^2*(a+b*x^2))],
[(d*x)^(5/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-1/10*d*(d*x)^(3/2)/(b*(a+b*x^2)^5)+3/160*d*(d*x)^(3/2)/(a*b*(a+b*x^2)^4)+13/640*d*(d*x)^(3/2)/(a^2*b*(a+b*x^2)^3)+117/5120*d*(d*x)^(3/2)/(a^3*b*(a+b*x^2)^2)+117/4096*d*(d*x)^(3/2)/(a^4*b*(a+b*x^2))-117/8192*d^(5/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(17/4)*b^(7/4)*sqrt(2))+117/8192*d^(5/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(17/4)*b^(7/4)*sqrt(2))+117/16384*d^(5/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(17/4)*b^(7/4)*sqrt(2))-117/16384*d^(5/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(17/4)*b^(7/4)*sqrt(2))],
[(d*x)^(3/2)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,-231/8192*d^(3/2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(19/4)*b^(5/4)*sqrt(2))+231/8192*d^(3/2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(19/4)*b^(5/4)*sqrt(2))-231/16384*d^(3/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(19/4)*b^(5/4)*sqrt(2))+231/16384*d^(3/2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(19/4)*b^(5/4)*sqrt(2))-1/10*d*sqrt(d*x)/(b*(a+b*x^2)^5)+1/160*d*sqrt(d*x)/(a*b*(a+b*x^2)^4)+1/128*d*sqrt(d*x)/(a^2*b*(a+b*x^2)^3)+11/1024*d*sqrt(d*x)/(a^3*b*(a+b*x^2)^2)+77/4096*d*sqrt(d*x)/(a^4*b*(a+b*x^2))],
[sqrt(d*x)/(a^2+2*a*b*x^2+b^2*x^4)^3,x,16,1/10*(d*x)^(3/2)/(a*d*(a+b*x^2)^5)+17/160*(d*x)^(3/2)/(a^2*d*(a+b*x^2)^4)+221/1920*(d*x)^(3/2)/(a^3*d*(a+b*x^2)^3)+663/5120*(d*x)^(3/2)/(a^4*d*(a+b*x^2)^2)+663/4096*(d*x)^(3/2)/(a^5*d*(a+b*x^2))-663/8192*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(21/4)*b^(3/4)*sqrt(2))+663/8192*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(21/4)*b^(3/4)*sqrt(2))+663/16384*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(21/4)*b^(3/4)*sqrt(2))-663/16384*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(21/4)*b^(3/4)*sqrt(2))],
[1/((a^2+2*a*b*x^2+b^2*x^4)^3*sqrt(d*x)),x,16,-4389/8192*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(23/4)*b^(1/4)*sqrt(2)*sqrt(d))+4389/8192*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(23/4)*b^(1/4)*sqrt(2)*sqrt(d))-4389/16384*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(23/4)*b^(1/4)*sqrt(2)*sqrt(d))+4389/16384*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(23/4)*b^(1/4)*sqrt(2)*sqrt(d))+1/10*sqrt(d*x)/(a*d*(a+b*x^2)^5)+19/160*sqrt(d*x)/(a^2*d*(a+b*x^2)^4)+19/128*sqrt(d*x)/(a^3*d*(a+b*x^2)^3)+209/1024*sqrt(d*x)/(a^4*d*(a+b*x^2)^2)+1463/4096*sqrt(d*x)/(a^5*d*(a+b*x^2))],
[1/((d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^3),x,17,13923/8192*b^(1/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(25/4)*d^(3/2)*sqrt(2))-13923/8192*b^(1/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(25/4)*d^(3/2)*sqrt(2))-13923/16384*b^(1/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(25/4)*d^(3/2)*sqrt(2))+13923/16384*b^(1/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(25/4)*d^(3/2)*sqrt(2))+(-13923/4096)/(a^6*d*sqrt(d*x))+1/10/(a*d*(a+b*x^2)^5*sqrt(d*x))+21/160/(a^2*d*(a+b*x^2)^4*sqrt(d*x))+119/640/(a^3*d*(a+b*x^2)^3*sqrt(d*x))+1547/5120/(a^4*d*(a+b*x^2)^2*sqrt(d*x))+13923/20480/(a^5*d*(a+b*x^2)*sqrt(d*x))],
[1/((d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^3),x,17,(-33649/12288)/(a^6*d*(d*x)^(3/2))+1/10/(a*d*(d*x)^(3/2)*(a+b*x^2)^5)+23/160/(a^2*d*(d*x)^(3/2)*(a+b*x^2)^4)+437/1920/(a^3*d*(d*x)^(3/2)*(a+b*x^2)^3)+437/1024/(a^4*d*(d*x)^(3/2)*(a+b*x^2)^2)+4807/4096/(a^5*d*(d*x)^(3/2)*(a+b*x^2))+33649/8192*b^(3/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(27/4)*d^(5/2)*sqrt(2))-33649/8192*b^(3/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(27/4)*d^(5/2)*sqrt(2))+33649/16384*b^(3/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(27/4)*d^(5/2)*sqrt(2))-33649/16384*b^(3/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(27/4)*d^(5/2)*sqrt(2))],
[1/((d*x)^(7/2)*(a^2+2*a*b*x^2+b^2*x^4)^3),x,18,(-13923/4096)/(a^6*d*(d*x)^(5/2))+1/10/(a*d*(d*x)^(5/2)*(a+b*x^2)^5)+5/32/(a^2*d*(d*x)^(5/2)*(a+b*x^2)^4)+35/128/(a^3*d*(d*x)^(5/2)*(a+b*x^2)^3)+595/1024/(a^4*d*(d*x)^(5/2)*(a+b*x^2)^2)+7735/4096/(a^5*d*(d*x)^(5/2)*(a+b*x^2))-69615/8192*b^(5/4)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(29/4)*d^(7/2)*sqrt(2))+69615/8192*b^(5/4)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(29/4)*d^(7/2)*sqrt(2))+69615/16384*b^(5/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(29/4)*d^(7/2)*sqrt(2))-69615/16384*b^(5/4)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(29/4)*d^(7/2)*sqrt(2))+69615/4096*b/(a^7*d^3*sqrt(d*x))],

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

# p>0
[(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,2/7*a*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+2/11*b*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))],
[(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,2/5*a*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+2/9*b*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))],
[sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4),x,3,2/3*a*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+2/7*b*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/sqrt(d*x),x,3,2/5*b*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+2*a*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*x)^(3/2),x,3,2/3*b*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))-2*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2)*sqrt(d*x))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*x)^(5/2),x,3,-2/3*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(d*x)^(3/2)*(a+b*x^2))+2*b*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))],
[sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*x)^(7/2),x,3,-2/5*a*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(d*x)^(5/2)*(a+b*x^2))-2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2)*sqrt(d*x))],
[(d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,2/7*a^3*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+6/11*a^2*b*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+2/5*a*b^2*(d*x)^(15/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/19*b^3*(d*x)^(19/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))],
[(d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,2/5*a^3*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+2/3*a^2*b*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+6/13*a*b^2*(d*x)^(13/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/17*b^3*(d*x)^(17/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)*sqrt(d*x),x,3,2/3*a^3*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+6/7*a^2*b*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+6/11*a*b^2*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/15*b^3*(d*x)^(15/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/sqrt(d*x),x,3,6/5*a^2*b*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+2/3*a*b^2*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/13*b^3*(d*x)^(13/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+2*a^3*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(d*x)^(3/2),x,3,2*a^2*b*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+6/7*a*b^2*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/11*b^3*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))-2*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2)*sqrt(d*x))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(d*x)^(5/2),x,3,-2/3*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(d*x)^(3/2)*(a+b*x^2))+6/5*a*b^2*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/9*b^3*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+6*a^2*b*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(3/2)/(d*x)^(7/2),x,3,-2/5*a^3*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(d*x)^(5/2)*(a+b*x^2))+2*a*b^2*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+2/7*b^3*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))-6*a^2*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2)*sqrt(d*x))],
[(d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,2/7*a^5*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+10/11*a^4*b*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+4/3*a^3*b^2*(d*x)^(15/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+20/19*a^2*b^3*(d*x)^(19/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+10/23*a*b^4*(d*x)^(23/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/27*b^5*(d*x)^(27/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))],
[(d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,2/5*a^5*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+10/9*a^4*b*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+20/13*a^3*b^2*(d*x)^(13/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+20/17*a^2*b^3*(d*x)^(17/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+10/21*a*b^4*(d*x)^(21/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/25*b^5*(d*x)^(25/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)*sqrt(d*x),x,3,2/3*a^5*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))+10/7*a^4*b*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+20/11*a^3*b^2*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+4/3*a^2*b^3*(d*x)^(15/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+10/19*a*b^4*(d*x)^(19/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/23*b^5*(d*x)^(23/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/sqrt(d*x),x,3,2*a^4*b*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+20/9*a^3*b^2*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+20/13*a^2*b^3*(d*x)^(13/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+10/17*a*b^4*(d*x)^(17/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/21*b^5*(d*x)^(21/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))+2*a^5*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(d*x)^(3/2),x,3,10/3*a^4*b*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))+20/7*a^3*b^2*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+20/11*a^2*b^3*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+2/3*a*b^4*(d*x)^(15/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/19*b^5*(d*x)^(19/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))-2*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(a+b*x^2)*sqrt(d*x))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(d*x)^(5/2),x,3,-2/3*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(d*x)^(3/2)*(a+b*x^2))+4*a^3*b^2*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+20/9*a^2*b^3*(d*x)^(9/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+10/13*a*b^4*(d*x)^(13/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/17*b^5*(d*x)^(17/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))+10*a^4*b*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2))],
[(a^2+2*a*b*x^2+b^2*x^4)^(5/2)/(d*x)^(7/2),x,3,-2/5*a^5*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(d*x)^(5/2)*(a+b*x^2))+20/3*a^3*b^2*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(a+b*x^2))+20/7*a^2*b^3*(d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(a+b*x^2))+10/11*a*b^4*(d*x)^(11/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(a+b*x^2))+2/15*b^5*(d*x)^(15/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(a+b*x^2))-10*a^4*b*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(a+b*x^2)*sqrt(d*x))],

# p<0
[(d*x)^(7/2)/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,13,2/5*d*(d*x)^(5/2)*(a+b*x^2)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-a^(5/4)*d^(7/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+a^(5/4)*d^(7/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*a^(5/4)*d^(7/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*a^(5/4)*d^(7/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-2*a*d^3*(a+b*x^2)*sqrt(d*x)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(5/2)/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,12,2/3*d*(d*x)^(3/2)*(a+b*x^2)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))+a^(3/4)*d^(5/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-a^(3/4)*d^(5/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*a^(3/4)*d^(5/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*a^(3/4)*d^(5/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(3/2)/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,12,a^(1/4)*d^(3/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-a^(1/4)*d^(3/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*a^(1/4)*d^(3/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*a^(1/4)*d^(3/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+2*d*(a+b*x^2)*sqrt(d*x)/(b*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[sqrt(d*x)/sqrt(a^2+2*a*b*x^2+b^2*x^4),x,11,-(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(1/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(1/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(1/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(1/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/(sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,11,-(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,12,b^(1/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-b^(1/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*b^(1/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*b^(1/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-2*(a+b*x^2)/(a*d*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,12,-2/3*(a+b*x^2)/(a*d*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+b^(3/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-b^(3/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*b^(3/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*b^(3/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(7/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4)),x,13,-2/5*(a+b*x^2)/(a*d*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-b^(5/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+b^(5/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/2*b^(5/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/2*b^(5/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+2*b*(a+b*x^2)/(a^2*d^3*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(15/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,15,-13/16*d^3*(d*x)^(9/2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*d*(d*x)^(13/2)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+117/80*d^5*(d*x)^(5/2)*(a+b*x^2)/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-117/32*a^(5/4)*d^(15/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+117/32*a^(5/4)*d^(15/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-117/64*a^(5/4)*d^(15/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+117/64*a^(5/4)*d^(15/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-117/16*a*d^7*(a+b*x^2)*sqrt(d*x)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(13/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,14,-11/16*d^3*(d*x)^(7/2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*d*(d*x)^(11/2)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/48*d^5*(d*x)^(3/2)*(a+b*x^2)/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/32*a^(3/4)*d^(13/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/32*a^(3/4)*d^(13/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/64*a^(3/4)*d^(13/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/64*a^(3/4)*d^(13/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(11/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,14,-9/16*d^3*(d*x)^(5/2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*d*(d*x)^(9/2)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/32*a^(1/4)*d^(11/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/32*a^(1/4)*d^(11/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/64*a^(1/4)*d^(11/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/64*a^(1/4)*d^(11/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/16*d^5*(a+b*x^2)*sqrt(d*x)/(b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(9/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,13,-7/16*d^3*(d*x)^(3/2)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*d*(d*x)^(7/2)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-21/32*d^(9/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+21/32*d^(9/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+21/64*d^(9/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-21/64*d^(9/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(7/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,13,-1/4*d*(d*x)^(5/2)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/32*d^(7/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/32*d^(7/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/64*d^(7/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/64*d^(7/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/16*d^3*sqrt(d*x)/(b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(5/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,13,3/16*d*(d*x)^(3/2)/(a*b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*d*(d*x)^(3/2)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/32*d^(5/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/32*d^(5/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/64*d^(5/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/64*d^(5/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(3/2)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,13,-3/32*d^(3/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/32*d^(3/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/64*d^(3/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3/64*d^(3/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/16*d*sqrt(d*x)/(a*b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/4*d*sqrt(d*x)/(b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[sqrt(d*x)/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,13,5/16*(d*x)^(3/2)/(a^2*d*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4*(d*x)^(3/2)/(a*d*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/32*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(9/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/32*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(9/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/64*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(9/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/64*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(9/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((a^2+2*a*b*x^2+b^2*x^4)^(3/2)*sqrt(d*x)),x,13,-21/32*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+21/32*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-21/64*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+21/64*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7/16*sqrt(d*x)/(a^2*d*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4*sqrt(d*x)/(a*d*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,14,45/32*b^(1/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/32*b^(1/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/64*b^(1/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/64*b^(1/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+9/16/(a^2*d*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a*d*(a+b*x^2)*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/16*(a+b*x^2)/(a^3*d*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,14,11/16/(a^2*d*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a*d*(d*x)^(3/2)*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/48*(a+b*x^2)/(a^3*d*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/32*b^(3/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/32*b^(3/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/64*b^(3/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/64*b^(3/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(7/2)*(a^2+2*a*b*x^2+b^2*x^4)^(3/2)),x,15,13/16/(a^2*d*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/4/(a*d*(d*x)^(5/2)*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-117/80*(a+b*x^2)/(a^3*d*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-117/32*b^(5/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(17/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+117/32*b^(5/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(17/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+117/64*b^(5/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(17/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-117/64*b^(5/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(17/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+117/16*b*(a+b*x^2)/(a^4*d^3*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(23/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,17,-1547/1024*d^7*(d*x)^(9/2)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(21/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7/32*d^3*(d*x)^(17/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-119/256*d^5*(d*x)^(13/2)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13923/5120*d^9*(d*x)^(5/2)*(a+b*x^2)/(b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13923/2048*a^(5/4)*d^(23/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(25/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13923/2048*a^(5/4)*d^(23/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(25/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13923/4096*a^(5/4)*d^(23/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(25/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13923/4096*a^(5/4)*d^(23/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(25/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13923/1024*a*d^11*(a+b*x^2)*sqrt(d*x)/(b^6*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(21/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,16,-1045/1024*d^7*(d*x)^(7/2)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(19/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-19/96*d^3*(d*x)^(15/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-95/256*d^5*(d*x)^(11/2)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7315/3072*d^9*(d*x)^(3/2)*(a+b*x^2)/(b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7315/2048*a^(3/4)*d^(21/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(23/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7315/2048*a^(3/4)*d^(21/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(23/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7315/4096*a^(3/4)*d^(21/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(23/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7315/4096*a^(3/4)*d^(21/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(23/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(19/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,16,-663/1024*d^7*(d*x)^(5/2)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(17/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-17/96*d^3*(d*x)^(13/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-221/768*d^5*(d*x)^(9/2)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3315/2048*a^(1/4)*d^(19/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(21/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3315/2048*a^(1/4)*d^(19/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(b^(21/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3315/4096*a^(1/4)*d^(19/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(21/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3315/4096*a^(1/4)*d^(19/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(b^(21/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3315/1024*d^9*(a+b*x^2)*sqrt(d*x)/(b^5*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(17/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,-385/1024*d^7*(d*x)^(3/2)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(15/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/32*d^3*(d*x)^(11/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-55/256*d^5*(d*x)^(7/2)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1155/2048*d^(17/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(19/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1155/2048*d^(17/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(1/4)*b^(19/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1155/4096*d^(17/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(19/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1155/4096*d^(17/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(1/4)*b^(19/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(15/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,-1/8*d*(d*x)^(13/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13/96*d^3*(d*x)^(9/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-39/256*d^5*(d*x)^(5/2)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-195/2048*d^(15/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+195/2048*d^(15/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(3/4)*b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-195/4096*d^(15/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+195/4096*d^(15/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(3/4)*b^(17/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-195/1024*d^7*sqrt(d*x)/(b^4*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(13/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,77/1024*d^5*(d*x)^(3/2)/(a*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(11/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-11/96*d^3*(d*x)^(7/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/768*d^5*(d*x)^(3/2)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/2048*d^(13/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/2048*d^(13/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(5/4)*b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/4096*d^(13/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/4096*d^(13/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(5/4)*b^(15/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(11/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,-1/8*d*(d*x)^(9/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3/32*d^3*(d*x)^(5/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/2048*d^(11/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/2048*d^(11/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(7/4)*b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/4096*d^(11/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/4096*d^(11/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(7/4)*b^(13/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+15/1024*d^5*sqrt(d*x)/(a*b^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-15/256*d^5*sqrt(d*x)/(b^3*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(9/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,35/1024*d^3*(d*x)^(3/2)/(a^2*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(7/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7/96*d^3*(d*x)^(3/2)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7/256*d^3*(d*x)^(3/2)/(a*b^2*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-35/2048*d^(9/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/2048*d^(9/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(9/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/4096*d^(9/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-35/4096*d^(9/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(9/4)*b^(11/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(7/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,-1/8*d*(d*x)^(5/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))-35/2048*d^(7/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/2048*d^(7/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(11/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-35/4096*d^(7/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/4096*d^(7/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(11/4)*b^(9/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+35/3072*d^3*sqrt(d*x)/(a^2*b^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))-5/96*d^3*sqrt(d*x)/(b^2*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/768*d^3*sqrt(d*x)/(a*b^2*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(5/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,45/1024*d*(d*x)^(3/2)/(a^3*b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*(d*x)^(3/2)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/32*d*(d*x)^(3/2)/(a*b*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+9/256*d*(d*x)^(3/2)/(a^2*b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/2048*d^(5/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/2048*d^(5/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(13/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+45/4096*d^(5/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-45/4096*d^(5/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(13/4)*b^(7/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^(3/2)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,-77/2048*d^(3/2)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/2048*d^(3/2)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(15/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-77/4096*d^(3/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/4096*d^(3/2)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(15/4)*b^(5/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+77/3072*d*sqrt(d*x)/(a^3*b*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1/8*d*sqrt(d*x)/(b*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/96*d*sqrt(d*x)/(a*b*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+11/768*d*sqrt(d*x)/(a^2*b*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[sqrt(d*x)/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,15,195/1024*(d*x)^(3/2)/(a^4*d*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8*(d*x)^(3/2)/(a*d*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13/96*(d*x)^(3/2)/(a^2*d*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+39/256*(d*x)^(3/2)/(a^3*d*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-195/2048*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(17/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+195/2048*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))*sqrt(d)/(a^(17/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+195/4096*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(17/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-195/4096*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))*sqrt(d)/(a^(17/4)*b^(3/4)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((a^2+2*a*b*x^2+b^2*x^4)^(5/2)*sqrt(d*x)),x,15,-1155/2048*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(19/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1155/2048*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(19/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-1155/4096*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(19/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1155/4096*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(19/4)*b^(1/4)*sqrt(2)*sqrt(d)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+385/1024*sqrt(d*x)/(a^4*d*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8*sqrt(d*x)/(a*d*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+5/32*sqrt(d*x)/(a^2*d*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+55/256*sqrt(d*x)/(a^3*d*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(3/2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,16,3315/2048*b^(1/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(21/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3315/2048*b^(1/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(21/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3315/4096*b^(1/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(21/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+3315/4096*b^(1/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(21/4)*d^(3/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+663/1024/(a^4*d*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8/(a*d*(a+b*x^2)^3*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+17/96/(a^2*d*(a+b*x^2)^2*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+221/768/(a^3*d*(a+b*x^2)*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-3315/1024*(a+b*x^2)/(a^5*d*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(5/2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,16,1045/1024/(a^4*d*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8/(a*d*(d*x)^(3/2)*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+19/96/(a^2*d*(d*x)^(3/2)*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+95/256/(a^3*d*(d*x)^(3/2)*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7315/3072*(a+b*x^2)/(a^5*d*(d*x)^(3/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7315/2048*b^(3/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(23/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7315/2048*b^(3/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(23/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7315/4096*b^(3/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(23/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-7315/4096*b^(3/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(23/4)*d^(5/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[1/((d*x)^(7/2)*(a^2+2*a*b*x^2+b^2*x^4)^(5/2)),x,17,1547/1024/(a^4*d*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+1/8/(a*d*(d*x)^(5/2)*(a+b*x^2)^3*sqrt(a^2+2*a*b*x^2+b^2*x^4))+7/32/(a^2*d*(d*x)^(5/2)*(a+b*x^2)^2*sqrt(a^2+2*a*b*x^2+b^2*x^4))+119/256/(a^3*d*(d*x)^(5/2)*(a+b*x^2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13923/5120*(a+b*x^2)/(a^5*d*(d*x)^(5/2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13923/2048*b^(5/4)*(a+b*x^2)*arctan(1-b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(25/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13923/2048*b^(5/4)*(a+b*x^2)*arctan(1+b^(1/4)*sqrt(2)*sqrt(d*x)/(a^(1/4)*sqrt(d)))/(a^(25/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13923/4096*b^(5/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)-a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(25/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))-13923/4096*b^(5/4)*(a+b*x^2)*log(sqrt(a)*sqrt(d)+x*sqrt(b)*sqrt(d)+a^(1/4)*b^(1/4)*sqrt(2)*sqrt(d*x))/(a^(25/4)*d^(7/2)*sqrt(2)*sqrt(a^2+2*a*b*x^2+b^2*x^4))+13923/1024*b*(a+b*x^2)/(a^6*d^3*sqrt(d*x)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],

# Integrands of the form (d x)^m (a^2+2 a b x^2+b^2 x^4)^p with m symbolic
[(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^3,x,3,a^6*(d*x)^(1+m)/(d*(1+m))+6*a^5*b*(d*x)^(3+m)/(d^3*(3+m))+15*a^4*b^2*(d*x)^(5+m)/(d^5*(5+m))+20*a^3*b^3*(d*x)^(7+m)/(d^7*(7+m))+15*a^2*b^4*(d*x)^(9+m)/(d^9*(9+m))+6*a*b^5*(d*x)^(11+m)/(d^11*(11+m))+b^6*(d*x)^(13+m)/(d^13*(13+m))],
[(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^2,x,3,a^4*(d*x)^(1+m)/(d*(1+m))+4*a^3*b*(d*x)^(3+m)/(d^3*(3+m))+6*a^2*b^2*(d*x)^(5+m)/(d^5*(5+m))+4*a*b^3*(d*x)^(7+m)/(d^7*(7+m))+b^4*(d*x)^(9+m)/(d^9*(9+m))],
[(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4),x,2,a^2*(d*x)^(1+m)/(d*(1+m))+2*a*b*(d*x)^(3+m)/(d^3*(3+m))+b^2*(d*x)^(5+m)/(d^5*(5+m))],
[(d*x)^m/(a^2+2*a*b*x^2+b^2*x^4),x,2,(d*x)^(1+m)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^2*d*(1+m))],
[(d*x)^m/(a^2+2*a*b*x^2+b^2*x^4)^2,x,2,(d*x)^(1+m)*hypergeom([4,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^4*d*(1+m))],
[(d*x)^m/(a^2+2*a*b*x^2+b^2*x^4)^3,x,2,(d*x)^(1+m)*hypergeom([6,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^6*d*(1+m))],
[(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,3,a^5*(d*x)^(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(1+m)*(a+b*x^2))+5*a^4*b*(d*x)^(3+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(3+m)*(a+b*x^2))+10*a^3*b^2*(d*x)^(5+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(5+m)*(a+b*x^2))+10*a^2*b^3*(d*x)^(7+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(7+m)*(a+b*x^2))+5*a*b^4*(d*x)^(9+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^9*(9+m)*(a+b*x^2))+b^5*(d*x)^(11+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^11*(11+m)*(a+b*x^2))],
[(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,3,a^3*(d*x)^(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(1+m)*(a+b*x^2))+3*a^2*b*(d*x)^(3+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(3+m)*(a+b*x^2))+3*a*b^2*(d*x)^(5+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^5*(5+m)*(a+b*x^2))+b^3*(d*x)^(7+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^7*(7+m)*(a+b*x^2))],
[(d*x)^m*(a^2+2*a*b*x^2+b^2*x^4)^(1/2),x,3,a*(d*x)^(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d*(1+m)*(a+b*x^2))+b*(d*x)^(3+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4)/(d^3*(3+m)*(a+b*x^2))],
[(d*x)^m/(a^2+2*a*b*x^2+b^2*x^4)^(1/2),x,2,(d*x)^(1+m)*(a+b*x^2)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a*d*(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^m/(a^2+2*a*b*x^2+b^2*x^4)^(3/2),x,2,(d*x)^(1+m)*(a+b*x^2)*hypergeom([3,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^3*d*(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],
[(d*x)^m/(a^2+2*a*b*x^2+b^2*x^4)^(5/2),x,2,(d*x)^(1+m)*(a+b*x^2)*hypergeom([5,1/2*(1+m)],[1/2*(3+m)],-b*x^2/a)/(a^5*d*(1+m)*sqrt(a^2+2*a*b*x^2+b^2*x^4))],

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

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

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

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

# p<0
[x^7/(a+b*x^2+c*x^4),x,7,-1/2*b*x^2/c^2+1/4*x^4/c+1/4*(b^2-a*c)*log(a+b*x^2+c*x^4)/c^3+1/2*b*(b^2-3*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[x^5/(a+b*x^2+c*x^4),x,6,1/2*x^2/c-1/4*b*log(a+b*x^2+c*x^4)/c^2-1/2*(b^2-2*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[x^3/(a+b*x^2+c*x^4),x,5,1/4*log(a+b*x^2+c*x^4)/c+1/2*b*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[x/(a+b*x^2+c*x^4),x,3,-arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[1/(x*(a+b*x^2+c*x^4)),x,7,log(x)/a-1/4*log(a+b*x^2+c*x^4)/a+1/2*b*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[1/(x^3*(a+b*x^2+c*x^4)),x,8,(-1/2)/(a*x^2)-b*log(x)/a^2+1/4*b*log(a+b*x^2+c*x^4)/a^2-1/2*(b^2-2*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[1/(x^5*(a+b*x^2+c*x^4)),x,8,(-1/4)/(a*x^4)+1/2*b/(a^2*x^2)+(b^2-a*c)*log(x)/a^3-1/4*(b^2-a*c)*log(a+b*x^2+c*x^4)/a^3+1/2*b*(b^2-3*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*sqrt(b^2-4*a*c))],
[x^6/(a+b*x^2+c*x^4),x,5,-b*x/c^2+1/3*x^3/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2-a*c-b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2-a*c+b*(b^2-3*a*c)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^4/(a+b*x^2+c*x^4),x,4,x/c-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[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)))*sqrt(b-sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[1/(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/(x^2*(a+b*x^2+c*x^4)),x,4,(-1)/(a*x)-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(1+b/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(1-b/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(x^4*(a+b*x^2+c*x^4)),x,5,(-1/3)/(a*x^3)+b/(a^2*x)+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^7/(a+b*x^2+c*x^4)^2,x,7,-1/2*b*x^2/(c*(b^2-4*a*c))+1/2*x^4*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*b*(b^2-6*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/2))+1/4*log(a+b*x^2+c*x^4)/c^2],
[x^5/(a+b*x^2+c*x^4)^2,x,4,1/2*x^2*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+2*a*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[x^3/(a+b*x^2+c*x^4)^2,x,4,1/2*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[x/(a+b*x^2+c*x^4)^2,x,4,1/2*(-b-2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+2*c*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[1/(x*(a+b*x^2+c*x^4)^2),x,8,1/2*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*b*(b^2-6*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2))+log(x)/a^2-1/4*log(a+b*x^2+c*x^4)/a^2],
[1/(x^3*(a+b*x^2+c*x^4)^2),x,8,(-b^2+3*a*c)/(a^2*(b^2-4*a*c)*x^2)+1/2*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x^2*(a+b*x^2+c*x^4))-(b^4-6*a*b^2*c+6*a^2*c^2)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2))-2*b*log(x)/a^3+1/2*b*log(a+b*x^2+c*x^4)/a^3],
[x^8/(a+b*x^2+c*x^4)^2,x,6,1/2*(3*b^2-10*a*c)*x/(c^2*(b^2-4*a*c))-1/2*b*x^3/(c*(b^2-4*a*c))+1/2*x^5*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(3*b^3-13*a*b*c+(-3*b^4+19*a*b^2*c-20*a^2*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(3*b^3-13*a*b*c+(3*b^4-19*a*b^2*c+20*a^2*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^6/(a+b*x^2+c*x^4)^2,x,5,-1/2*b*x/(c*(b^2-4*a*c))+1/2*x^3*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2-6*a*c-b*(b^2-8*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2-6*a*c+b*(b^2-8*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^4/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2-4*a*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2+4*a*c+b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2/(a+b*x^2+c*x^4)^2,x,4,-1/2*x*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(2*b-sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*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)*(2*b+sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*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/(x^2*(a+b*x^2+c*x^4)^2),x,5,1/2*(-3*b^2+10*a*c)/(a^2*(b^2-4*a*c)*x)+1/2*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x*(a+b*x^2+c*x^4))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^3-16*a*b*c+(3*b^2-10*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^3-16*a*b*c-(3*b^2-10*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^11/(a+b*x^2+c*x^4)^3,x,8,-1/2*b*(b^2-7*a*c)*x^2/(c^2*(b^2-4*a*c)^2)+1/4*x^8*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/4*x^4*(a*(b^2-16*a*c)+b*(b^2-10*a*c)*x^2)/(c*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/2*b*(b^4-10*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(5/2))+1/4*log(a+b*x^2+c*x^4)/c^3],
[x^9/(a+b*x^2+c*x^4)^3,x,5,1/4*x^6*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-3/2*a*x^2*(2*a+b*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))-6*a^2*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x^7/(a+b*x^2+c*x^4)^3,x,5,-1/4*x^6*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/4*b*x^2*(2*a+b*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3*a*b*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x^5/(a+b*x^2+c*x^4)^3,x,5,1/4*x^2*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/2*(3*a*b+(b^2+2*a*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))-(b^2+2*a*c)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x^3/(a+b*x^2+c*x^4)^3,x,5,1/4*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-3/4*b*(b+2*c*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3*b*c*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[x/(a+b*x^2+c*x^4)^3,x,5,1/4*(-b-2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/2*c*(b+2*c*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))-6*c^2*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(5/2)],
[1/(x*(a+b*x^2+c*x^4)^3),x,9,1/4*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/4*(2*b^4-15*a*b^2*c+16*a^2*c^2+2*b*c*(b^2-7*a*c)*x^2)/(a^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/2*b*(b^4-10*a*b^2*c+30*a^2*c^2)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(5/2))+log(x)/a^3-1/4*log(a+b*x^2+c*x^4)/a^3],
[1/(x^3*(a+b*x^2+c*x^4)^3),x,9,-3/2*(b^2-5*a*c)*(b^2-2*a*c)/(a^3*(b^2-4*a*c)^2*x^2)+1/4*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x^2*(a+b*x^2+c*x^4)^2)+1/4*(3*b^4-20*a*b^2*c+20*a^2*c^2+3*b*c*(b^2-6*a*c)*x^2)/(a^2*(b^2-4*a*c)^2*x^2*(a+b*x^2+c*x^4))-3/2*(b^6-10*a*b^4*c+30*a^2*b^2*c^2-20*a^3*c^3)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(5/2))-3*b*log(x)/a^4+3/4*b*log(a+b*x^2+c*x^4)/a^4],
[x^10/(a+b*x^2+c*x^4)^3,x,7,-3/8*b*(b^2-8*a*c)*x/(c^2*(b^2-4*a*c)^2)+1/8*(b^2-28*a*c)*x^3/(c*(b^2-4*a*c)^2)+1/4*x^7*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/8*x^5*(12*a*b-(b^2-28*a*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^4-9*a*b^2*c+28*a^2*c^2+(-b^5+11*a*b^3*c-44*a^2*b*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^4-9*a*b^2*c+28*a^2*c^2+(b^5-11*a*b^3*c+44*a^2*b*c^2)/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^8/(a+b*x^2+c*x^4)^3,x,6,-1/8*(b^2+20*a*c)*x/(c*(b^2-4*a*c)^2)+1/4*x^5*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/8*x^3*(12*a*b+(b^2+20*a*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^3-16*a*b*c+(-b^4+18*a*b^2*c+40*a^2*c^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^3-16*a*b*c+(b^4-18*a*b^2*c-40*a^2*c^2)/sqrt(b^2-4*a*c))/(c^(3/2)*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^6/(a+b*x^2+c*x^4)^3,x,5,1/4*x^3*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/8*x*(4*a*b+(b^2+4*a*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2+4*a*c-b*(b^2+12*a*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^2*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2+4*a*c+b*(b^2+12*a*c)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^2*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^4/(a+b*x^2+c*x^4)^3,x,5,1/4*x*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/8*x*(7*b^2-4*a*c+12*b*c*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/4*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^2+4*a*c-2*b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/4*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*b^2+4*a*c+2*b*sqrt(b^2-4*a*c))/((b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2/(a+b*x^2+c*x^4)^3,x,5,-1/4*x*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/8*x*(b*(b^2+8*a*c)+c*(b^2+20*a*c)*x^2)/(a*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^2+20*a*c+b*(b^2-52*a*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b^2+20*a*c-b*(b^2-52*a*c)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(a+b*x^2+c*x^4)^3,x,5,1/4*x*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/8*x*((b^2-7*a*c)*(3*b^2-4*a*c)+3*b*c*(b^2-8*a*c)*x^2)/(a^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^4-10*a*b^2*c+56*a^2*c^2+b*(b^2-8*a*c)*sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(5/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(b^3-8*a*b*c+(-b^4+10*a*b^2*c-56*a^2*c^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[1/(x^2*(a+b*x^2+c*x^4)^3),x,6,-3/8*(5*b^2-12*a*c)*(b^2-5*a*c)/(a^3*(b^2-4*a*c)^2*x)+1/4*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x*(a+b*x^2+c*x^4)^2)+1/8*(5*b^4-35*a*b^2*c+36*a^2*c^2+b*c*(5*b^2-32*a*c)*x^2)/(a^2*(b^2-4*a*c)^2*x*(a+b*x^2+c*x^4))-3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*((5*b^2-12*a*c)*(b^2-5*a*c)+b*(5*b^4-47*a*b^2*c+124*a^2*c^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^2*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-3/8*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*((5*b^2-12*a*c)*(b^2-5*a*c)+(-5*b^5+47*a*b^3*c-124*a^2*b*c^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^2*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^5/(a-b*x^2+c*x^4),x,6,1/2*x^2/c+1/4*b*log(a-b*x^2+c*x^4)/c^2+1/2*(b^2-2*a*c)*arctanh((b-2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[x^3/(a-b*x^2+c*x^4),x,5,1/4*log(a-b*x^2+c*x^4)/c+1/2*b*arctanh((b-2*c*x^2)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))],
[x/(a-b*x^2+c*x^4),x,3,arctanh((b-2*c*x^2)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[1/(x*(a-b*x^2+c*x^4)),x,7,log(x)/a-1/4*log(a-b*x^2+c*x^4)/a+1/2*b*arctanh((b-2*c*x^2)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))],
[1/(x^3*(a-b*x^2+c*x^4)),x,8,(-1/2)/(a*x^2)+b*log(x)/a^2-1/4*b*log(a-b*x^2+c*x^4)/a^2+1/2*(b^2-2*a*c)*arctanh((b-2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[x^4/(a-b*x^2+c*x^4),x,4,x/c-arctanh(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctanh(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2/(a-b*x^2+c*x^4),x,3,arctanh(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(b-sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))-arctanh(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b^2-4*a*c))],
[1/(a-b*x^2+c*x^4),x,3,arctanh(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)))-arctanh(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/(x^2*(a-b*x^2+c*x^4)),x,4,(-1)/(a*x)+arctanh(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(1+b/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctanh(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(1-b/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^5/(a-b+2*a*x^2+a*x^4),x,6,1/2*x^2/a-1/2*log(a-b+2*a*x^2+a*x^4)/a-1/2*(a+b)*arctanh((1+x^2)*sqrt(a)/sqrt(b))/(a^(3/2)*sqrt(b))],
[x^3/(a-b+2*a*x^2+a*x^4),x,5,1/4*log(a-b+2*a*x^2+a*x^4)/a+1/2*arctanh((1+x^2)*sqrt(a)/sqrt(b))/(sqrt(a)*sqrt(b))],
[x/(a-b+2*a*x^2+a*x^4),x,3,-1/2*arctanh((1+x^2)*sqrt(a)/sqrt(b))/(sqrt(a)*sqrt(b))],
[1/(x*(a-b+2*a*x^2+a*x^4)),x,7,log(x)/(a-b)-1/4*log(a-b+2*a*x^2+a*x^4)/(a-b)+1/2*arctanh((1+x^2)*sqrt(a)/sqrt(b))*sqrt(a)/((a-b)*sqrt(b))],
[1/(x^3*(a-b+2*a*x^2+a*x^4)),x,8,(-1/2)/((a-b)*x^2)-2*a*log(x)/(a-b)^2+1/2*a*log(a-b+2*a*x^2+a*x^4)/(a-b)^2-1/2*(a+b)*arctanh((1+x^2)*sqrt(a)/sqrt(b))*sqrt(a)/((a-b)^2*sqrt(b))],
[x^4/(a-b+2*a*x^2+a*x^4),x,4,x/a+1/2*arctan(a^(1/4)*x/sqrt(sqrt(a)-sqrt(b)))*(sqrt(a)-sqrt(b))^(3/2)/(a^(5/4)*sqrt(b))-1/2*arctan(a^(1/4)*x/sqrt(sqrt(a)+sqrt(b)))*(sqrt(a)+sqrt(b))^(3/2)/(a^(5/4)*sqrt(b))],
[x^2/(a-b+2*a*x^2+a*x^4),x,3,-1/2*arctan(a^(1/4)*x/sqrt(sqrt(a)-sqrt(b)))*sqrt(sqrt(a)-sqrt(b))/(a^(3/4)*sqrt(b))+1/2*arctan(a^(1/4)*x/sqrt(sqrt(a)+sqrt(b)))*sqrt(sqrt(a)+sqrt(b))/(a^(3/4)*sqrt(b))],
[1/(a-b+2*a*x^2+a*x^4),x,3,1/2*arctan(a^(1/4)*x/sqrt(sqrt(a)-sqrt(b)))/(a^(1/4)*sqrt(b)*sqrt(sqrt(a)-sqrt(b)))-1/2*arctan(a^(1/4)*x/sqrt(sqrt(a)+sqrt(b)))/(a^(1/4)*sqrt(b)*sqrt(sqrt(a)+sqrt(b)))],
[1/(x^2*(a-b+2*a*x^2+a*x^4)),x,4,(-1)/((a-b)*x)-1/2*a^(1/4)*arctan(a^(1/4)*x/sqrt(sqrt(a)-sqrt(b)))/((sqrt(a)-sqrt(b))^(3/2)*sqrt(b))+1/2*a^(1/4)*arctan(a^(1/4)*x/sqrt(sqrt(a)+sqrt(b)))/(sqrt(b)*(sqrt(a)+sqrt(b))^(3/2))],
[x^5/(a+b+2*a*x^2+a*x^4),x,6,1/2*x^2/a-1/2*log(a+b+2*a*x^2+a*x^4)/a+1/2*(a-b)*arctan((1+x^2)*sqrt(a)/sqrt(b))/(a^(3/2)*sqrt(b))],
[x^3/(a+b+2*a*x^2+a*x^4),x,5,1/4*log(a+b+2*a*x^2+a*x^4)/a-1/2*arctan((1+x^2)*sqrt(a)/sqrt(b))/(sqrt(a)*sqrt(b))],
[x/(a+b+2*a*x^2+a*x^4),x,3,1/2*arctan((1+x^2)*sqrt(a)/sqrt(b))/(sqrt(a)*sqrt(b))],
[1/(x*(a+b+2*a*x^2+a*x^4)),x,7,log(x)/(a+b)-1/4*log(a+b+2*a*x^2+a*x^4)/(a+b)-1/2*arctan((1+x^2)*sqrt(a)/sqrt(b))*sqrt(a)/((a+b)*sqrt(b))],
[1/(x^3*(a+b+2*a*x^2+a*x^4)),x,8,(-1/2)/((a+b)*x^2)-2*a*log(x)/(a+b)^2+1/2*a*log(a+b+2*a*x^2+a*x^4)/(a+b)^2+1/2*(a-b)*arctan((1+x^2)*sqrt(a)/sqrt(b))*sqrt(a)/((a+b)^2*sqrt(b))],
[x^4/(a+b+2*a*x^2+a*x^4),x,10,x/a+1/4*log(x^2*sqrt(a)+sqrt(a+b)-a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))*(a+b-2*sqrt(a)*sqrt(a+b))/(a^(5/4)*sqrt(2)*sqrt(a+b)*sqrt(-sqrt(a)+sqrt(a+b)))-1/4*log(x^2*sqrt(a)+sqrt(a+b)+a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))*(a+b-2*sqrt(a)*sqrt(a+b))/(a^(5/4)*sqrt(2)*sqrt(a+b)*sqrt(-sqrt(a)+sqrt(a+b)))+1/2*arctan((-a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))*(a+b+2*sqrt(a)*sqrt(a+b))/(a^(5/4)*sqrt(2)*sqrt(a+b)*sqrt(sqrt(a)+sqrt(a+b)))-1/2*arctan((a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))*(a+b+2*sqrt(a)*sqrt(a+b))/(a^(5/4)*sqrt(2)*sqrt(a+b)*sqrt(sqrt(a)+sqrt(a+b)))],
[x^2/(a+b+2*a*x^2+a*x^4),x,9,1/4*log(x^2*sqrt(a)+sqrt(a+b)-a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))/(a^(3/4)*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))-1/4*log(x^2*sqrt(a)+sqrt(a+b)+a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))/(a^(3/4)*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))-1/2*arctan((-a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))/(a^(3/4)*sqrt(2)*sqrt(sqrt(a)+sqrt(a+b)))+1/2*arctan((a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))/(a^(3/4)*sqrt(2)*sqrt(sqrt(a)+sqrt(a+b)))],
[1/(a+b+2*a*x^2+a*x^4),x,9,-1/4*log(x^2*sqrt(a)+sqrt(a+b)-a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))/(a^(1/4)*sqrt(2)*sqrt(a+b)*sqrt(-sqrt(a)+sqrt(a+b)))+1/4*log(x^2*sqrt(a)+sqrt(a+b)+a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))/(a^(1/4)*sqrt(2)*sqrt(a+b)*sqrt(-sqrt(a)+sqrt(a+b)))-1/2*arctan((-a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))/(a^(1/4)*sqrt(2)*sqrt(a+b)*sqrt(sqrt(a)+sqrt(a+b)))+1/2*arctan((a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))/(a^(1/4)*sqrt(2)*sqrt(a+b)*sqrt(sqrt(a)+sqrt(a+b)))],
[1/(x^2*(a+b+2*a*x^2+a*x^4)),x,10,(-1)/((a+b)*x)+1/4*a^(1/4)*log(x^2*sqrt(a)+sqrt(a+b)-a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))*(2*sqrt(a)-sqrt(a+b))/((a+b)^(3/2)*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))-1/4*a^(1/4)*log(x^2*sqrt(a)+sqrt(a+b)+a^(1/4)*x*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))*(2*sqrt(a)-sqrt(a+b))/((a+b)^(3/2)*sqrt(2)*sqrt(-sqrt(a)+sqrt(a+b)))+1/2*a^(1/4)*arctan((-a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))*(2*sqrt(a)+sqrt(a+b))/((a+b)^(3/2)*sqrt(2)*sqrt(sqrt(a)+sqrt(a+b)))-1/2*a^(1/4)*arctan((a^(1/4)*x*sqrt(2)+sqrt(-sqrt(a)+sqrt(a+b)))/sqrt(sqrt(a)+sqrt(a+b)))*(2*sqrt(a)+sqrt(a+b))/((a+b)^(3/2)*sqrt(2)*sqrt(sqrt(a)+sqrt(a+b)))],
[x/(1+x^2+x^4),x,3,arctan((1+2*x^2)/sqrt(3))/sqrt(3)],
[x/(10+2*x^2+x^4),x,3,1/6*arctan(1/3*(1+x^2))],
[x^2/(20+9*x^2+x^4),x,3,-2*arctan(1/2*x)+arctan(x/sqrt(5))*sqrt(5)],
[x^2/(1-x^2+x^4),x,9,-1/2*arctan(-2*x+sqrt(3))+1/2*arctan(2*x+sqrt(3))+1/4*log(1+x^2-x*sqrt(3))/sqrt(3)-1/4*log(1+x^2+x*sqrt(3))/sqrt(3)],
[x^2/(2-2*x^2+x^4),x,9,-1/2*arctan((-2*x+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+1/2*arctan((2*x+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+1/4*log(x^2+sqrt(2)-x*sqrt(2*(1+sqrt(2))))/sqrt(2*(1+sqrt(2)))-1/4*log(x^2+sqrt(2)+x*sqrt(2*(1+sqrt(2))))/sqrt(2*(1+sqrt(2)))],

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

# p>0
[x^7*sqrt(a+b*x^2+c*x^4),x,6,1/10*x^4*(a+b*x^2+c*x^4)^(3/2)/c+1/480*(35*b^2-32*a*c-42*b*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c^3+1/512*b*(7*b^2-12*a*c)*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(9/2)-1/256*b*(7*b^2-12*a*c)*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^4],
[x^5*sqrt(a+b*x^2+c*x^4),x,6,-5/48*b*(a+b*x^2+c*x^4)^(3/2)/c^2+1/8*x^2*(a+b*x^2+c*x^4)^(3/2)/c-1/256*(b^2-4*a*c)*(5*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(7/2)+1/128*(5*b^2-4*a*c)*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^3],
[x^3*sqrt(a+b*x^2+c*x^4),x,5,1/6*(a+b*x^2+c*x^4)^(3/2)/c+1/32*b*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(5/2)-1/16*b*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^2],
[x*sqrt(a+b*x^2+c*x^4),x,4,-1/16*(b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(3/2)+1/8*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c],
[sqrt(a+b*x^2+c*x^4)/x,x,7,-1/2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))*sqrt(a)+1/4*b*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/sqrt(c)+1/2*sqrt(a+b*x^2+c*x^4)],
[sqrt(a+b*x^2+c*x^4)/x^3,x,7,-1/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/sqrt(a)+1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))*sqrt(c)-1/2*sqrt(a+b*x^2+c*x^4)/x^2],
[sqrt(a+b*x^2+c*x^4)/x^5,x,4,1/16*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(3/2)-1/8*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a*x^4)],
[sqrt(a+b*x^2+c*x^4)/x^7,x,5,-1/6*(a+b*x^2+c*x^4)^(3/2)/(a*x^6)-1/32*b*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(5/2)+1/16*b*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a^2*x^4)],
[sqrt(a+b*x^2+c*x^4)/x^9,x,6,-1/8*(a+b*x^2+c*x^4)^(3/2)/(a*x^8)+5/48*b*(a+b*x^2+c*x^4)^(3/2)/(a^2*x^6)+1/256*(b^2-4*a*c)*(5*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(7/2)-1/128*(5*b^2-4*a*c)*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a^3*x^4)],
[sqrt(a+b*x^2+c*x^4)/x^11,x,7,-1/10*(a+b*x^2+c*x^4)^(3/2)/(a*x^10)+7/80*b*(a+b*x^2+c*x^4)^(3/2)/(a^2*x^8)-1/480*(35*b^2-32*a*c)*(a+b*x^2+c*x^4)^(3/2)/(a^3*x^6)-1/512*b*(7*b^2-12*a*c)*(b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(9/2)+1/256*b*(7*b^2-12*a*c)*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a^4*x^4)],
[x^4*sqrt(a+b*x^2+c*x^4),x,5,-2/105*(2*b^2-5*a*c)*x*sqrt(a+b*x^2+c*x^4)/c^2+1/35*x^3*(b+5*c*x^2)*sqrt(a+b*x^2+c*x^4)/c+1/105*b*(8*b^2-29*a*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(5/2)*(sqrt(a)+x^2*sqrt(c)))-1/105*a^(1/4)*b*(8*b^2-29*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(11/4)*sqrt(a+b*x^2+c*x^4))+1/210*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(8*b^3-29*a*b*c+2*(2*b^2-5*a*c)*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(11/4)*sqrt(a+b*x^2+c*x^4))],
[x^2*sqrt(a+b*x^2+c*x^4),x,4,1/15*x*(b+3*c*x^2)*sqrt(a+b*x^2+c*x^4)/c-2/15*(b^2-3*a*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))+2/15*a^(1/4)*(b^2-3*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))-1/30*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))*(2*b^2-6*a*c+b*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))],
[sqrt(a+b*x^2+c*x^4),x,4,1/3*x*sqrt(a+b*x^2+c*x^4)+1/3*b*x*sqrt(a+b*x^2+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-1/3*a^(1/4)*b*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+b*x^2+c*x^4))+1/6*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))*(b+2*sqrt(a)*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))],
[sqrt(a+b*x^2+c*x^4)/x^2,x,4,-sqrt(a+b*x^2+c*x^4)/x+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((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))],
[sqrt(a+b*x^2+c*x^4)/x^4,x,5,-1/3*sqrt(a+b*x^2+c*x^4)/x^3-1/3*b*sqrt(a+b*x^2+c*x^4)/(a*x)+1/3*b*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a*(sqrt(a)+x^2*sqrt(c)))-1/3*b*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+b*x^2+c*x^4))+1/6*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(b+2*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+b*x^2+c*x^4))],
[sqrt(a+b*x^2+c*x^4)/x^6,x,6,-1/5*sqrt(a+b*x^2+c*x^4)/x^5-1/15*b*sqrt(a+b*x^2+c*x^4)/(a*x^3)+2/15*(b^2-3*a*c)*sqrt(a+b*x^2+c*x^4)/(a^2*x)-2/15*(b^2-3*a*c)*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a^2*(sqrt(a)+x^2*sqrt(c)))+2/15*c^(1/4)*(b^2-3*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))-1/30*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*b^2-6*a*c+b*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))],
[x^7*(a+b*x^2+c*x^4)^(3/2),x,7,-1/256*b*(3*b^2-4*a*c)*(b+2*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c^4+1/14*x^4*(a+b*x^2+c*x^4)^(5/2)/c+1/560*(21*b^2-16*a*c-30*b*c*x^2)*(a+b*x^2+c*x^4)^(5/2)/c^3-3/4096*b*(b^2-4*a*c)^2*(3*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(11/2)+3/2048*b*(b^2-4*a*c)*(3*b^2-4*a*c)*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^5],
[x^5*(a+b*x^2+c*x^4)^(3/2),x,7,1/384*(7*b^2-4*a*c)*(b+2*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c^3-7/120*b*(a+b*x^2+c*x^4)^(5/2)/c^2+1/12*x^2*(a+b*x^2+c*x^4)^(5/2)/c+1/2048*(b^2-4*a*c)^2*(7*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(9/2)-1/1024*(b^2-4*a*c)*(7*b^2-4*a*c)*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^4],
[x^3*(a+b*x^2+c*x^4)^(3/2),x,6,-1/32*b*(b+2*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c^2+1/10*(a+b*x^2+c*x^4)^(5/2)/c-3/512*b*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(7/2)+3/256*b*(b^2-4*a*c)*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^3],
[x*(a+b*x^2+c*x^4)^(3/2),x,5,1/16*(b+2*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c+3/256*(b^2-4*a*c)^2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(5/2)-3/128*(b^2-4*a*c)*(b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^2],
[(a+b*x^2+c*x^4)^(3/2)/x,x,8,1/6*(a+b*x^2+c*x^4)^(3/2)-1/2*a^(3/2)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))-1/32*b*(b^2-12*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(3/2)+1/16*(b^2+8*a*c+2*b*c*x^2)*sqrt(a+b*x^2+c*x^4)/c],
[(a+b*x^2+c*x^4)^(3/2)/x^3,x,8,-1/2*(a+b*x^2+c*x^4)^(3/2)/x^2-3/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))*sqrt(a)+3/16*(b^2+4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/sqrt(c)+3/8*(3*b+2*c*x^2)*sqrt(a+b*x^2+c*x^4)],
[(a+b*x^2+c*x^4)^(3/2)/x^5,x,8,-1/4*(a+b*x^2+c*x^4)^(3/2)/x^4-3/16*(b^2+4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/sqrt(a)+3/4*b*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))*sqrt(c)-3/8*(b-2*c*x^2)*sqrt(a+b*x^2+c*x^4)/x^2],
[(a+b*x^2+c*x^4)^(3/2)/x^7,x,8,-1/6*(a+b*x^2+c*x^4)^(3/2)/x^6+1/32*b*(b^2-12*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(3/2)+1/2*c^(3/2)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))-1/16*(2*a*b+(b^2+8*a*c)*x^2)*sqrt(a+b*x^2+c*x^4)/(a*x^4)],
[(a+b*x^2+c*x^4)^(3/2)/x^9,x,5,-1/16*(2*a+b*x^2)*(a+b*x^2+c*x^4)^(3/2)/(a*x^8)-3/256*(b^2-4*a*c)^2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(5/2)+3/128*(b^2-4*a*c)*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a^2*x^4)],
[(a+b*x^2+c*x^4)^(3/2)/x^11,x,6,1/32*b*(2*a+b*x^2)*(a+b*x^2+c*x^4)^(3/2)/(a^2*x^8)-1/10*(a+b*x^2+c*x^4)^(5/2)/(a*x^10)+3/512*b*(b^2-4*a*c)^2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(7/2)-3/256*b*(b^2-4*a*c)*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a^3*x^4)],
[(a+b*x^2+c*x^4)^(3/2)/x^13,x,7,-1/384*(7*b^2-4*a*c)*(2*a+b*x^2)*(a+b*x^2+c*x^4)^(3/2)/(a^3*x^8)-1/12*(a+b*x^2+c*x^4)^(5/2)/(a*x^12)+7/120*b*(a+b*x^2+c*x^4)^(5/2)/(a^2*x^10)-1/2048*(b^2-4*a*c)^2*(7*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(9/2)+1/1024*(b^2-4*a*c)*(7*b^2-4*a*c)*(2*a+b*x^2)*sqrt(a+b*x^2+c*x^4)/(a^4*x^4)],
[x^4*(a+b*x^2+c*x^4)^(3/2),x,6,1/33*x^3*(b+3*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c+1/1155*(8*b^4-51*a*b^2*c+60*a^2*c^2)*x*sqrt(a+b*x^2+c*x^4)/c^3-1/385*x^3*(b*(2*b^2+a*c)+10*c*(b^2-3*a*c)*x^2)*sqrt(a+b*x^2+c*x^4)/c^2-8/1155*b*(2*b^2-9*a*c)*(b^2-3*a*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(7/2)*(sqrt(a)+x^2*sqrt(c)))+8/1155*a^(1/4)*b*(2*b^2-9*a*c)*(b^2-3*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(15/4)*sqrt(a+b*x^2+c*x^4))-1/2310*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(8*b*(2*b^2-9*a*c)*(b^2-3*a*c)+(8*b^4-51*a*b^2*c+60*a^2*c^2)*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(15/4)*sqrt(a+b*x^2+c*x^4))],
[x^2*(a+b*x^2+c*x^4)^(3/2),x,5,1/63*x*(3*b+7*c*x^2)*(a+b*x^2+c*x^4)^(3/2)/c-1/315*x*(b*(4*b^2-9*a*c)+6*c*(2*b^2-7*a*c)*x^2)*sqrt(a+b*x^2+c*x^4)/c^2+1/315*(8*b^4-57*a*b^2*c+84*a^2*c^2)*x*sqrt(a+b*x^2+c*x^4)/(c^(5/2)*(sqrt(a)+x^2*sqrt(c)))-1/315*a^(1/4)*(8*b^4-57*a*b^2*c+84*a^2*c^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(1/4*(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))+1/630*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(8*b^4-57*a*b^2*c+84*a^2*c^2+4*b*(b^2-6*a*c)*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(11/4)*sqrt(a+b*x^2+c*x^4))],
[(a+b*x^2+c*x^4)^(3/2),x,5,1/7*x*(a+b*x^2+c*x^4)^(3/2)+1/35*x*(b^2+10*a*c+3*b*c*x^2)*sqrt(a+b*x^2+c*x^4)/c-2/35*b*(b^2-8*a*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))+2/35*a^(1/4)*b*(b^2-8*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))-1/70*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*b*(b^2-8*a*c)+(b^2-20*a*c)*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))],
[(a+b*x^2+c*x^4)^(3/2)/x^2,x,5,-(a+b*x^2+c*x^4)^(3/2)/x+1/5*x*(7*b+6*c*x^2)*sqrt(a+b*x^2+c*x^4)+1/5*(b^2+12*a*c)*x*sqrt(a+b*x^2+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-1/5*a^(1/4)*(b^2+12*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+b*x^2+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(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(b^2+12*a*c+8*b*sqrt(a)*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))],
[(a+b*x^2+c*x^4)^(3/2)/x^4,x,5,-1/3*(a+b*x^2+c*x^4)^(3/2)/x^3-1/3*(3*b-2*c*x^2)*sqrt(a+b*x^2+c*x^4)/x+8/3*b*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))-8/3*a^(1/4)*b*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/sqrt(a+b*x^2+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(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(3*b^2+4*a*c+8*b*sqrt(a)*sqrt(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))],
[(a+b*x^2+c*x^4)^(3/2)/x^6,x,6,-1/5*(a+b*x^2+c*x^4)^(3/2)/x^5-1/5*(b^2+12*a*c)*sqrt(a+b*x^2+c*x^4)/(a*x)-1/5*(b-6*c*x^2)*sqrt(a+b*x^2+c*x^4)/x^3+1/5*(b^2+12*a*c)*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a*(sqrt(a)+x^2*sqrt(c)))-1/5*c^(1/4)*(b^2+12*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+b*x^2+c*x^4))+1/10*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))*(b^2+12*a*c+8*b*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+b*x^2+c*x^4))],
[(a+b*x^2+c*x^4)^(3/2)/x^8,x,7,-1/7*(a+b*x^2+c*x^4)^(3/2)/x^7-1/35*(b^2-20*a*c)*sqrt(a+b*x^2+c*x^4)/(a*x^3)+2/35*b*(b^2-8*a*c)*sqrt(a+b*x^2+c*x^4)/(a^2*x)-3/35*(b+10*c*x^2)*sqrt(a+b*x^2+c*x^4)/x^5-2/35*b*(b^2-8*a*c)*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a^2*(sqrt(a)+x^2*sqrt(c)))+2/35*b*c^(1/4)*(b^2-8*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))-1/70*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*b*(b^2-8*a*c)+(b^2-20*a*c)*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))],
[sqrt(3-2*x^2-x^4),x,5,-2*EllipticE(x,sqrt(-1/3))/sqrt(3)+4*EllipticF(x,sqrt(-1/3))/sqrt(3)+1/3*x*sqrt(3-2*x^2-x^4)],

# p<0
[x^7/sqrt(a+b*x^2+c*x^4),x,5,-1/32*b*(5*b^2-12*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(7/2)+1/6*x^4*sqrt(a+b*x^2+c*x^4)/c+1/48*(15*b^2-16*a*c-10*b*c*x^2)*sqrt(a+b*x^2+c*x^4)/c^3],
[x^5/sqrt(a+b*x^2+c*x^4),x,5,1/16*(3*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(5/2)-3/8*b*sqrt(a+b*x^2+c*x^4)/c^2+1/4*x^2*sqrt(a+b*x^2+c*x^4)/c],
[x^3/sqrt(a+b*x^2+c*x^4),x,4,-1/4*b*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(3/2)+1/2*sqrt(a+b*x^2+c*x^4)/c],
[x/sqrt(a+b*x^2+c*x^4),x,3,1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/sqrt(c)],
[1/(x*sqrt(a+b*x^2+c*x^4)),x,3,-1/2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/sqrt(a)],
[1/(x^3*sqrt(a+b*x^2+c*x^4)),x,4,1/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(3/2)-1/2*sqrt(a+b*x^2+c*x^4)/(a*x^2)],
[1/(x^5*sqrt(a+b*x^2+c*x^4)),x,5,-1/16*(3*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(5/2)-1/4*sqrt(a+b*x^2+c*x^4)/(a*x^4)+3/8*b*sqrt(a+b*x^2+c*x^4)/(a^2*x^2)],
[1/(x^7*sqrt(a+b*x^2+c*x^4)),x,6,1/32*b*(5*b^2-12*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(7/2)-1/6*sqrt(a+b*x^2+c*x^4)/(a*x^6)+5/24*b*sqrt(a+b*x^2+c*x^4)/(a^2*x^4)-1/48*(15*b^2-16*a*c)*sqrt(a+b*x^2+c*x^4)/(a^3*x^2)],
[x^4/sqrt(a+b*x^2+c*x^4),x,4,1/3*x*sqrt(a+b*x^2+c*x^4)/c-2/3*b*x*sqrt(a+b*x^2+c*x^4)/(c^(3/2)*(sqrt(a)+x^2*sqrt(c)))+2/3*a^(1/4)*b*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))-1/6*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*b+sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*sqrt(a+b*x^2+c*x^4))],
[x^2/sqrt(a+b*x^2+c*x^4),x,3,x*sqrt(a+b*x^2+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-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)))*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))*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/sqrt(a+b*x^2+c*x^4),x,1,1/2*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))*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))],
[1/(x^2*sqrt(a+b*x^2+c*x^4)),x,5,-sqrt(a+b*x^2+c*x^4)/(a*x)+x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a*(sqrt(a)+x^2*sqrt(c)))-c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*sqrt(a+b*x^2+c*x^4))+1/2*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^(3/4)*sqrt(a+b*x^2+c*x^4))],
[1/(x^4*sqrt(a+b*x^2+c*x^4)),x,5,-1/3*sqrt(a+b*x^2+c*x^4)/(a*x^3)+2/3*b*sqrt(a+b*x^2+c*x^4)/(a^2*x)-2/3*b*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a^2*(sqrt(a)+x^2*sqrt(c)))+2/3*b*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))-1/6*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*(2*b+sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*sqrt(a+b*x^2+c*x^4))],
[x^7/sqrt(a+b*x^2-c*x^4),x,5,-1/32*b*(5*b^2+12*a*c)*arctan(1/2*(b-2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2-c*x^4)))/c^(7/2)-1/6*x^4*sqrt(a+b*x^2-c*x^4)/c-1/48*(15*b^2+16*a*c+10*b*c*x^2)*sqrt(a+b*x^2-c*x^4)/c^3],
[x^5/sqrt(a+b*x^2-c*x^4),x,5,-1/16*(3*b^2+4*a*c)*arctan(1/2*(b-2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2-c*x^4)))/c^(5/2)-3/8*b*sqrt(a+b*x^2-c*x^4)/c^2-1/4*x^2*sqrt(a+b*x^2-c*x^4)/c],
[x^3/sqrt(a+b*x^2-c*x^4),x,4,-1/4*b*arctan(1/2*(b-2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2-c*x^4)))/c^(3/2)-1/2*sqrt(a+b*x^2-c*x^4)/c],
[x/sqrt(a+b*x^2-c*x^4),x,3,-1/2*arctan(1/2*(b-2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2-c*x^4)))/sqrt(c)],
[1/(x*sqrt(-a+b*x^2+c*x^4)),x,3,-1/2*arctan(1/2*(2*a-b*x^2)/(sqrt(a)*sqrt(-a+b*x^2+c*x^4)))/sqrt(a)],
[1/(x^3*sqrt(-a+b*x^2+c*x^4)),x,4,-1/4*b*arctan(1/2*(2*a-b*x^2)/(sqrt(a)*sqrt(-a+b*x^2+c*x^4)))/a^(3/2)+1/2*sqrt(-a+b*x^2+c*x^4)/(a*x^2)],
[1/(x^5*sqrt(-a+b*x^2+c*x^4)),x,5,-1/16*(3*b^2+4*a*c)*arctan(1/2*(2*a-b*x^2)/(sqrt(a)*sqrt(-a+b*x^2+c*x^4)))/a^(5/2)+1/4*sqrt(-a+b*x^2+c*x^4)/(a*x^4)+3/8*b*sqrt(-a+b*x^2+c*x^4)/(a^2*x^2)],
[1/(x^7*sqrt(-a+b*x^2+c*x^4)),x,6,-1/32*b*(5*b^2+12*a*c)*arctan(1/2*(2*a-b*x^2)/(sqrt(a)*sqrt(-a+b*x^2+c*x^4)))/a^(7/2)+1/6*sqrt(-a+b*x^2+c*x^4)/(a*x^6)+5/24*b*sqrt(-a+b*x^2+c*x^4)/(a^2*x^4)+1/48*(15*b^2+16*a*c)*sqrt(-a+b*x^2+c*x^4)/(a^3*x^2)],
[x^4/sqrt(a+b*x^2-c*x^4),x,5,-1/3*x*sqrt(a+b*x^2-c*x^4)/c-1/3*b*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))))*(b^2+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))],
[x^2/sqrt(a+b*x^2-c*x^4),x,4,-1/2*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))+1/2*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))))*(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/sqrt(a+b*x^2-c*x^4),x,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))))*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)))/(sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))],
[1/(x^2*sqrt(a+b*x^2-c*x^4)),x,6,-sqrt(a+b*x^2-c*x^4)/(a*x)+1/2*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)))/(a*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*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))))*(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)))/(a*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))],
[1/(x^4*sqrt(a+b*x^2-c*x^4)),x,6,-1/3*sqrt(a+b*x^2-c*x^4)/(a*x^3)+2/3*b*sqrt(a+b*x^2-c*x^4)/(a^2*x)-1/3*b*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)))/(a^2*sqrt(2)*sqrt(c)*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))))*(b^2+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)))/(a^2*sqrt(2)*sqrt(c)*sqrt(a+b*x^2-c*x^4))],
[x^9/(a+b*x^2+c*x^4)^(3/2),x,6,3/16*(5*b^2-4*a*c)*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(7/2)+x^6*(2*a+b*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))-b*x^4*sqrt(a+b*x^2+c*x^4)/(c*(b^2-4*a*c))-1/8*(b*(15*b^2-52*a*c)-2*c*(5*b^2-12*a*c)*x^2)*sqrt(a+b*x^2+c*x^4)/(c^3*(b^2-4*a*c))],
[x^7/(a+b*x^2+c*x^4)^(3/2),x,5,-3/4*b*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(5/2)+x^4*(2*a+b*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))+1/2*(3*b^2-8*a*c-2*b*c*x^2)*sqrt(a+b*x^2+c*x^4)/(c^2*(b^2-4*a*c))],
[x^5/(a+b*x^2+c*x^4)^(3/2),x,5,1/2*arctanh(1/2*(b+2*c*x^2)/(sqrt(c)*sqrt(a+b*x^2+c*x^4)))/c^(3/2)+x^2*(2*a+b*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))-b*sqrt(a+b*x^2+c*x^4)/(c*(b^2-4*a*c))],
[x^3/(a+b*x^2+c*x^4)^(3/2),x,2,(2*a+b*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))],
[x/(a+b*x^2+c*x^4)^(3/2),x,2,(-b-2*c*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))],
[1/(x*(a+b*x^2+c*x^4)^(3/2)),x,5,-1/2*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(3/2)+(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))],
[1/(x^3*(a+b*x^2+c*x^4)^(3/2)),x,5,3/4*b*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(5/2)+(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x^2*sqrt(a+b*x^2+c*x^4))-1/2*(3*b^2-8*a*c)*sqrt(a+b*x^2+c*x^4)/(a^2*(b^2-4*a*c)*x^2)],
[1/(x^5*(a+b*x^2+c*x^4)^(3/2)),x,6,-3/16*(5*b^2-4*a*c)*arctanh(1/2*(2*a+b*x^2)/(sqrt(a)*sqrt(a+b*x^2+c*x^4)))/a^(7/2)+(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x^4*sqrt(a+b*x^2+c*x^4))-1/4*(5*b^2-12*a*c)*sqrt(a+b*x^2+c*x^4)/(a^2*(b^2-4*a*c)*x^4)+1/8*b*(15*b^2-52*a*c)*sqrt(a+b*x^2+c*x^4)/(a^3*(b^2-4*a*c)*x^2)],
[x^6/(a+b*x^2+c*x^4)^(3/2),x,5,x^3*(2*a+b*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))-b*x*sqrt(a+b*x^2+c*x^4)/(c*(b^2-4*a*c))+2*(b^2-3*a*c)*x*sqrt(a+b*x^2+c*x^4)/(c^(3/2)*(b^2-4*a*c)*(sqrt(a)+x^2*sqrt(c)))-2*a^(1/4)*(b^2-3*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*(b^2-4*a*c)*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))*(2*b^2-6*a*c+b*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(7/4)*(b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))],
[x^4/(a+b*x^2+c*x^4)^(3/2),x,4,x*(2*a+b*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))-b*x*sqrt(a+b*x^2+c*x^4)/((b^2-4*a*c)*sqrt(c)*(sqrt(a)+x^2*sqrt(c)))+a^(1/4)*b*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*(b^2-4*a*c)*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))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*(b-2*sqrt(a)*sqrt(c))*sqrt(a+b*x^2+c*x^4))],
[x^2/(a+b*x^2+c*x^4)^(3/2),x,4,-x*(b+2*c*x^2)/((b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))+2*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/((b^2-4*a*c)*(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)/((b^2-4*a*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)))))*(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)*(b-2*sqrt(a)*sqrt(c))*sqrt(a+b*x^2+c*x^4))],
[1/(a+b*x^2+c*x^4)^(3/2),x,4,x*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))-b*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a*(b^2-4*a*c)*(sqrt(a)+x^2*sqrt(c)))+b*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*(b^2-4*a*c)*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^(3/4)*(b-2*sqrt(a)*sqrt(c))*sqrt(a+b*x^2+c*x^4))],
[1/(x^2*(a+b*x^2+c*x^4)^(3/2)),x,5,(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*x*sqrt(a+b*x^2+c*x^4))-2*(b^2-3*a*c)*sqrt(a+b*x^2+c*x^4)/(a^2*(b^2-4*a*c)*x)+2*(b^2-3*a*c)*x*sqrt(c)*sqrt(a+b*x^2+c*x^4)/(a^2*(b^2-4*a*c)*(sqrt(a)+x^2*sqrt(c)))-2*c^(1/4)*(b^2-3*a*c)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/4*(2-b/(sqrt(a)*sqrt(c)))))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*(b^2-4*a*c)*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))*(2*b^2-6*a*c+b*sqrt(a)*sqrt(c))*sqrt((a+b*x^2+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(7/4)*(b^2-4*a*c)*sqrt(a+b*x^2+c*x^4))],

#  Must check to ensure constant term is not zero! 
[x^4/sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4),x,3,-2/3*b*sqrt(b*x^2+c*x^4)/(c^2*x)+1/3*x*sqrt(b*x^2+c*x^4)/c],
[x^3/sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4),x,5,-1/2*b*arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/c^(3/2)+1/2*sqrt(b*x^2+c*x^4)/c],
[x^2/sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4),x,2,sqrt(b*x^2+c*x^4)/(c*x)],
[x/sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4),x,4,arctanh(x^2*sqrt(c)/sqrt(b*x^2+c*x^4))/sqrt(c)],
[1/sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4),x,3,-arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/sqrt(b)],
[1/(x*sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4)),x,2,-sqrt(b*x^2+c*x^4)/(b*x^2)],
[1/(x^2*sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4)),x,4,1/2*c*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(3/2)-1/2*sqrt(b*x^2+c*x^4)/(b*x^3)],
[1/(x^3*sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4)),x,3,-1/3*sqrt(b*x^2+c*x^4)/(b*x^4)+2/3*c*sqrt(b*x^2+c*x^4)/(b^2*x^2)],
[1/(x^4*sqrt(2+2*a-2*(1+a)+b*x^2+c*x^4)),x,5,-3/8*c^2*arctanh(x*sqrt(b)/sqrt(b*x^2+c*x^4))/b^(5/2)-1/4*sqrt(b*x^2+c*x^4)/(b*x^5)+3/8*c*sqrt(b*x^2+c*x^4)/(b^2*x^3)],

#  Must check to ensure coefficient is not zero! 
[x^4/sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4),x,3,1/3*x*sqrt(a+c*x^4)/c-1/6*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)))*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)/(c^(5/4)*sqrt(a+c*x^4))],
[x^3/sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4),x,2,1/2*sqrt(a+c*x^4)/c],
[x^2/sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4),x,4,x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-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)))*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))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[x/sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4),x,4,1/2*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)],
[1/sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4),x,2,1/2*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))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*sqrt(a+c*x^4))],
[1/(x*sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4)),x,4,-1/2*arctanh(sqrt(a+c*x^4)/sqrt(a))/sqrt(a)],
[1/(x^2*sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4)),x,5,-sqrt(a+c*x^4)/(a*x)+x*sqrt(c)*sqrt(a+c*x^4)/(a*(sqrt(a)+x^2*sqrt(c)))-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(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*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^(3/4)*sqrt(a+c*x^4))],
[1/(x^3*sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4)),x,2,-1/2*sqrt(a+c*x^4)/(a*x^2)],
[1/(x^4*sqrt(a+(2+2*b-2*(1+b))*x^2+c*x^4)),x,3,-1/3*sqrt(a+c*x^4)/(a*x^3)-1/6*c^(3/4)*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^(5/4)*sqrt(a+c*x^4))],

#  Must check to ensure coefficient is not zero! 
[x^4/sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4),x,5,3/8*a^2*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)-3/8*a*x*sqrt(a+b*x^2)/b^2+1/4*x^3*sqrt(a+b*x^2)/b],
[x^3/sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4),x,4,1/3*(a+b*x^2)^(3/2)/b^2-a*sqrt(a+b*x^2)/b^2],
[x^2/sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4),x,4,-1/2*a*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/2*x*sqrt(a+b*x^2)/b],
[x/sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4),x,2,sqrt(a+b*x^2)/b],
[1/sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4),x,3,arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)],
[1/(x*sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4)),x,4,-arctanh(sqrt(a+b*x^2)/sqrt(a))/sqrt(a)],
[1/(x^2*sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4)),x,2,-sqrt(a+b*x^2)/(a*x)],
[1/(x^3*sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4)),x,5,1/2*b*arctanh(sqrt(a+b*x^2)/sqrt(a))/a^(3/2)-1/2*sqrt(a+b*x^2)/(a*x^2)],
[1/(x^4*sqrt(a+b*x^2+(2+2*c-2*(1+c))*x^4)),x,3,-1/3*sqrt(a+b*x^2)/(a*x^3)+2/3*b*sqrt(a+b*x^2)/(a^2*x)],

#  Must check to ensure constant term is not zero! 
[x^4/sqrt(2+2*a-2*(1+a)+c*x^4),x,3,1/3*x^5/sqrt(c*x^4)],
[x^3/sqrt(2+2*a-2*(1+a)+c*x^4),x,3,1/2*x^4/sqrt(c*x^4)],
[x^2/sqrt(2+2*a-2*(1+a)+c*x^4),x,3,x^3/sqrt(c*x^4)],
[x/sqrt(2+2*a-2*(1+a)+c*x^4),x,3,x^2*log(x)/sqrt(c*x^4)],
[1/sqrt(2+2*a-2*(1+a)+c*x^4),x,3,-x/sqrt(c*x^4)],
[1/(x*sqrt(2+2*a-2*(1+a)+c*x^4)),x,3,(-1/2)/sqrt(c*x^4)],
[1/(x^2*sqrt(2+2*a-2*(1+a)+c*x^4)),x,3,(-1/3)/(x*sqrt(c*x^4))],
[1/(x^3*sqrt(2+2*a-2*(1+a)+c*x^4)),x,3,(-1/4)/(x^2*sqrt(c*x^4))],
[1/(x^4*sqrt(2+2*a-2*(1+a)+c*x^4)),x,3,(-1/5)/(x^3*sqrt(c*x^4))],

#  Must check to ensure coefficient is not zero! 
[x^4/sqrt(a+(2+2*c-2*(1+c))*x^4),x,3,1/5*x^5/sqrt(a)],
[x^3/sqrt(a+(2+2*c-2*(1+c))*x^4),x,3,1/4*x^4/sqrt(a)],
[x^2/sqrt(a+(2+2*c-2*(1+c))*x^4),x,3,1/3*x^3/sqrt(a)],
[x/sqrt(a+(2+2*c-2*(1+c))*x^4),x,3,1/2*x^2/sqrt(a)],
[1/sqrt(a+(2+2*c-2*(1+c))*x^4),x,2,x/sqrt(a)],
[1/(x*sqrt(a+(2+2*c-2*(1+c))*x^4)),x,3,log(x)/sqrt(a)],
[1/(x^2*sqrt(a+(2+2*c-2*(1+c))*x^4)),x,3,(-1)/(x*sqrt(a))],
[1/(x^3*sqrt(a+(2+2*c-2*(1+c))*x^4)),x,3,(-1/2)/(x^2*sqrt(a))],
[1/(x^4*sqrt(a+(2+2*c-2*(1+c))*x^4)),x,3,(-1/3)/(x^3*sqrt(a))],
[1/sqrt(3-2*x^2-x^4),x,2,EllipticF(x,sqrt(-1/3))/sqrt(3)],
[1/sqrt(-1+5*x^2-x^4),x,2,-sqrt(cos(arccos(x*sqrt(2/(5+sqrt(21)))))^2)/cos(arccos(x*sqrt(2/(5+sqrt(21)))))*EllipticF(sin(arccos(x*sqrt(2/(5+sqrt(21))))),sqrt(1/42*(21+5*sqrt(21))))/21^(1/4)],

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

# p>0
[x^(5/2)*(a+b*x^2+c*x^4),x,2,2/7*a*x^(7/2)+2/11*b*x^(11/2)+2/15*c*x^(15/2)],
[x^(3/2)*(a+b*x^2+c*x^4),x,2,2/5*a*x^(5/2)+2/9*b*x^(9/2)+2/13*c*x^(13/2)],
[x^(1/2)*(a+b*x^2+c*x^4),x,2,2/3*a*x^(3/2)+2/7*b*x^(7/2)+2/11*c*x^(11/2)],
[(a+b*x^2+c*x^4)/x^(1/2),x,2,2/5*b*x^(5/2)+2/9*c*x^(9/2)+2*a*sqrt(x)],
[(a+b*x^2+c*x^4)/x^(3/2),x,2,2/3*b*x^(3/2)+2/7*c*x^(7/2)-2*a/sqrt(x)],
[(a+b*x^2+c*x^4)/x^(5/2),x,2,-2/3*a/x^(3/2)+2/5*c*x^(5/2)+2*b*sqrt(x)],
[(a+b*x^2+c*x^4)/x^(7/2),x,2,-2/5*a/x^(5/2)+2/3*c*x^(3/2)-2*b/sqrt(x)],
[x^(5/2)*(a+b*x^2+c*x^4)^2,x,2,2/7*a^2*x^(7/2)+4/11*a*b*x^(11/2)+2/15*(b^2+2*a*c)*x^(15/2)+4/19*b*c*x^(19/2)+2/23*c^2*x^(23/2)],
[x^(3/2)*(a+b*x^2+c*x^4)^2,x,2,2/5*a^2*x^(5/2)+4/9*a*b*x^(9/2)+2/13*(b^2+2*a*c)*x^(13/2)+4/17*b*c*x^(17/2)+2/21*c^2*x^(21/2)],
[x^(1/2)*(a+b*x^2+c*x^4)^2,x,2,2/3*a^2*x^(3/2)+4/7*a*b*x^(7/2)+2/11*(b^2+2*a*c)*x^(11/2)+4/15*b*c*x^(15/2)+2/19*c^2*x^(19/2)],
[(a+b*x^2+c*x^4)^2/x^(1/2),x,2,4/5*a*b*x^(5/2)+2/9*(b^2+2*a*c)*x^(9/2)+4/13*b*c*x^(13/2)+2/17*c^2*x^(17/2)+2*a^2*sqrt(x)],
[(a+b*x^2+c*x^4)^2/x^(3/2),x,2,4/3*a*b*x^(3/2)+2/7*(b^2+2*a*c)*x^(7/2)+4/11*b*c*x^(11/2)+2/15*c^2*x^(15/2)-2*a^2/sqrt(x)],
[(a+b*x^2+c*x^4)^2/x^(5/2),x,2,-2/3*a^2/x^(3/2)+2/5*(b^2+2*a*c)*x^(5/2)+4/9*b*c*x^(9/2)+2/13*c^2*x^(13/2)+4*a*b*sqrt(x)],
[(a+b*x^2+c*x^4)^2/x^(7/2),x,2,-2/5*a^2/x^(5/2)+2/3*(b^2+2*a*c)*x^(3/2)+4/7*b*c*x^(7/2)+2/11*c^2*x^(11/2)-4*a*b/sqrt(x)],
[x^(5/2)*(a+b*x^2+c*x^4)^3,x,2,2/7*a^3*x^(7/2)+6/11*a^2*b*x^(11/2)+2/5*a*(b^2+a*c)*x^(15/2)+2/19*b*(b^2+6*a*c)*x^(19/2)+6/23*c*(b^2+a*c)*x^(23/2)+2/9*b*c^2*x^(27/2)+2/31*c^3*x^(31/2)],
[x^(3/2)*(a+b*x^2+c*x^4)^3,x,2,2/5*a^3*x^(5/2)+2/3*a^2*b*x^(9/2)+6/13*a*(b^2+a*c)*x^(13/2)+2/17*b*(b^2+6*a*c)*x^(17/2)+2/7*c*(b^2+a*c)*x^(21/2)+6/25*b*c^2*x^(25/2)+2/29*c^3*x^(29/2)],
[x^(1/2)*(a+b*x^2+c*x^4)^3,x,2,2/3*a^3*x^(3/2)+6/7*a^2*b*x^(7/2)+6/11*a*(b^2+a*c)*x^(11/2)+2/15*b*(b^2+6*a*c)*x^(15/2)+6/19*c*(b^2+a*c)*x^(19/2)+6/23*b*c^2*x^(23/2)+2/27*c^3*x^(27/2)],
[(a+b*x^2+c*x^4)^3/x^(1/2),x,2,6/5*a^2*b*x^(5/2)+2/3*a*(b^2+a*c)*x^(9/2)+2/13*b*(b^2+6*a*c)*x^(13/2)+6/17*c*(b^2+a*c)*x^(17/2)+2/7*b*c^2*x^(21/2)+2/25*c^3*x^(25/2)+2*a^3*sqrt(x)],
[(a+b*x^2+c*x^4)^3/x^(3/2),x,2,2*a^2*b*x^(3/2)+6/7*a*(b^2+a*c)*x^(7/2)+2/11*b*(b^2+6*a*c)*x^(11/2)+2/5*c*(b^2+a*c)*x^(15/2)+6/19*b*c^2*x^(19/2)+2/23*c^3*x^(23/2)-2*a^3/sqrt(x)],
[(a+b*x^2+c*x^4)^3/x^(5/2),x,2,-2/3*a^3/x^(3/2)+6/5*a*(b^2+a*c)*x^(5/2)+2/9*b*(b^2+6*a*c)*x^(9/2)+6/13*c*(b^2+a*c)*x^(13/2)+6/17*b*c^2*x^(17/2)+2/21*c^3*x^(21/2)+6*a^2*b*sqrt(x)],
[(a+b*x^2+c*x^4)^3/x^(7/2),x,2,-2/5*a^3/x^(5/2)+2*a*(b^2+a*c)*x^(3/2)+2/7*b*(b^2+6*a*c)*x^(7/2)+6/11*c*(b^2+a*c)*x^(11/2)+2/5*b*c^2*x^(15/2)+2/19*c^3*x^(19/2)-6*a^2*b/sqrt(x)],

# p<0
[x^(9/2)/(a+b*x^2+c*x^4),x,9,2/3*x^(3/2)/c-arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b-sqrt(b^2-4*a*c))^(1/4))+arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b-sqrt(b^2-4*a*c))^(1/4))-arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b+sqrt(b^2-4*a*c))^(1/4))+arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[x^(7/2)/(a+b*x^2+c*x^4),x,9,arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))+arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b-sqrt(b^2-4*a*c))^(3/4))+arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))+arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(-b+sqrt(b^2-4*a*c))^(3/4))+2*sqrt(x)/c],
[x^(5/2)/(a+b*x^2+c*x^4),x,8,-arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))+arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))+arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))-arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(3/4)/(2^(3/4)*c^(3/4)*sqrt(b^2-4*a*c))],
[x^(3/2)/(a+b*x^2+c*x^4),x,8,arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))+arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(-b-sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))-arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))-arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(-b+sqrt(b^2-4*a*c))^(1/4)/(2^(1/4)*c^(1/4)*sqrt(b^2-4*a*c))],
[x^(1/2)/(a+b*x^2+c*x^4),x,8,-2^(1/4)*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))/((-b-sqrt(b^2-4*a*c))^(1/4)*sqrt(b^2-4*a*c))+2^(1/4)*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))/((-b-sqrt(b^2-4*a*c))^(1/4)*sqrt(b^2-4*a*c))+2^(1/4)*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))/(sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))-2^(1/4)*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))/(sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[1/(x^(1/2)*(a+b*x^2+c*x^4)),x,8,2^(3/4)*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))/((-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c))+2^(3/4)*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))/((-b-sqrt(b^2-4*a*c))^(3/4)*sqrt(b^2-4*a*c))-2^(3/4)*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))/(sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))-2^(3/4)*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))/(sqrt(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))],
[1/(x^(3/2)*(a+b*x^2+c*x^4)),x,9,-c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b-sqrt(b^2-4*a*c))^(1/4))+c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b-sqrt(b^2-4*a*c))^(1/4))-c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b+sqrt(b^2-4*a*c))^(1/4))+c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(3/4)*a*(-b+sqrt(b^2-4*a*c))^(1/4))+(-2)/(a*sqrt(x))],
[1/(x^(5/2)*(a+b*x^2+c*x^4)),x,9,(-2/3)/(a*x^(3/2))+c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b-sqrt(b^2-4*a*c))^(3/4))+c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(1-b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b-sqrt(b^2-4*a*c))^(3/4))+c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b+sqrt(b^2-4*a*c))^(3/4))+c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(1+b/sqrt(b^2-4*a*c))/(2^(1/4)*a*(-b+sqrt(b^2-4*a*c))^(3/4))],
[1/(x^(7/2)*(a+b*x^2+c*x^4)),x,10,(-2/5)/(a*x^(5/2))+c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(-b-sqrt(b^2-4*a*c))^(1/4))-c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(-b-sqrt(b^2-4*a*c))^(1/4))+c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(-b+sqrt(b^2-4*a*c))^(1/4))-c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(-b+sqrt(b^2-4*a*c))^(1/4))+2*b/(a^2*sqrt(x))],
[x^(13/2)/(a+b*x^2+c*x^4)^2,x,10,-1/2*b*x^(3/2)/(c*(b^2-4*a*c))+1/2*x^(7/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3*b^3-20*a*b*c-(3*b^2-14*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(1/4))+1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3*b^3-20*a*b*c-(3*b^2-14*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(1/4))+1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3*b^3-20*a*b*c+(3*b^2-14*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3*b^3-20*a*b*c+(3*b^2-14*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*c^(7/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))],
[x^(11/2)/(a+b*x^2+c*x^4)^2,x,10,1/2*x^(5/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2-10*a*c+b*(b^2-12*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2-10*a*c+b*(b^2-12*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^2-10*a*c-b*(b^2-12*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^2-10*a*c-b*(b^2-12*a*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/2*b*sqrt(x)/(c*(b^2-4*a*c))],
[x^(9/2)/(a+b*x^2+c*x^4)^2,x,9,1/2*x^(3/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2-12*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))-1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2-12*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))+1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2+12*a*c+b*sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2+12*a*c+b*sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))],
[x^(7/2)/(a+b*x^2+c*x^4)^2,x,9,1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3*b^2+4*a*c-3*b*sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3*b^2+4*a*c-3*b*sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/4*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3*b^2+4*a*c+3*b*sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/4*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3*b^2+4*a*c+3*b*sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*(2*a+b*x^2)*sqrt(x)/((b^2-4*a*c)*(a+b*x^2+c*x^4))],
[x^(5/2)/(a+b*x^2+c*x^4)^2,x,9,-1/2*x^(3/2)*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(4*b-sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(1/4))-1/2*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(4*b-sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(1/4))-1/2*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(4*b+sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))+1/2*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(4*b+sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))],
[x^(3/2)/(a+b*x^2+c*x^4)^2,x,9,1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3+4*b/sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3+4*b/sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3-4*b/sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3-4*b/sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/2*(b+2*c*x^2)*sqrt(x)/((b^2-4*a*c)*(a+b*x^2+c*x^4))],
[x^(1/2)/(a+b*x^2+c*x^4)^2,x,9,1/2*x^(3/2)*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/4*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+20*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/4*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b+(-b^2+20*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)*(-b-sqrt(b^2-4*a*c))^(1/4))+1/4*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-20*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))-1/4*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b+(b^2-20*a*c)/sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[1/(x^(1/2)*(a+b*x^2+c*x^4)^2),x,9,1/4*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3*b^2-28*a*c-3*b*sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/4*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(3*b^2-28*a*c-3*b*sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(3/4))-1/4*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3*b^2-28*a*c+3*b*sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/4*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(3*b^2-28*a*c+3*b*sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/2*(b^2-2*a*c+b*c*x^2)*sqrt(x)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))],
[1/(x^(3/2)*(a+b*x^2+c*x^4)^2),x,10,1/4*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(5*b^3-28*a*b*c-(5*b^2-18*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/4*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(5*b^3-28*a*b*c-(5*b^2-18*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(3/2)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/4*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(5*b^3-28*a*b*c+(5*b^2-18*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(1/4))+1/4*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(5*b^3-28*a*b*c+(5*b^2-18*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(3/2)*(-b+sqrt(b^2-4*a*c))^(1/4))+1/2*(-5*b^2+18*a*c)/(a^2*(b^2-4*a*c)*sqrt(x))+1/2*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)*sqrt(x))],
[x^(15/2)/(a+b*x^2+c*x^4)^3,x,11,1/4*x^(9/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/16*x^(5/2)*(8*a*b+(b^2+12*a*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))-3/32*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^3-28*a*b*c+(b^4-30*a*b^2*c-24*a^2*c^2)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)^2*(-b-sqrt(b^2-4*a*c))^(3/4))-3/32*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^3-28*a*b*c+(b^4-30*a*b^2*c-24*a^2*c^2)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)^2*(-b-sqrt(b^2-4*a*c))^(3/4))-3/32*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^3-28*a*b*c+(-b^4+30*a*b^2*c+24*a^2*c^2)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)^2*(-b+sqrt(b^2-4*a*c))^(3/4))-3/32*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^3-28*a*b*c+(-b^4+30*a*b^2*c+24*a^2*c^2)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(5/4)*(b^2-4*a*c)^2*(-b+sqrt(b^2-4*a*c))^(3/4))-3/16*(b^2+12*a*c)*sqrt(x)/(c*(b^2-4*a*c)^2)],
[x^(13/2)/(a+b*x^2+c*x^4)^3,x,10,1/4*x^(7/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/16*x^(3/2)*(24*a*b+(5*b^2+28*a*c)*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+1/32*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(5*b^2+28*a*c+(-5*b^3-172*a*b*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)^2*(-b+sqrt(b^2-4*a*c))^(1/4))-1/32*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(5*b^2+28*a*c+(-5*b^3-172*a*b*c)/sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)^2*(-b+sqrt(b^2-4*a*c))^(1/4))+1/32*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(5*b^3+172*a*b*c+(5*b^2+28*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(1/4))-1/32*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(5*b^3+172*a*b*c+(5*b^2+28*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*c^(3/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(1/4))],
[x^(11/2)/(a+b*x^2+c*x^4)^3,x,10,1/4*x^(5/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-3/32*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(7*b^2+20*a*c+(-7*b^3-36*a*b*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^2*(-b+sqrt(b^2-4*a*c))^(3/4))-3/32*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(7*b^2+20*a*c+(-7*b^3-36*a*b*c)/sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^2*(-b+sqrt(b^2-4*a*c))^(3/4))-3/32*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(7*b^3+36*a*b*c+(7*b^2+20*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(3/4))-3/32*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(7*b^3+36*a*b*c+(7*b^2+20*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*c^(1/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/16*(24*a*b+(7*b^2+20*a*c)*x^2)*sqrt(x)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))],
[x^(9/2)/(a+b*x^2+c*x^4)^3,x,10,1/4*x^(3/2)*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-3/16*x^(3/2)*(5*b^2-4*a*c+8*b*c*x^2)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/16*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(11*b^2+20*a*c-4*b*sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(1/4))-3/16*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(11*b^2+20*a*c-4*b*sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(1/4))-3/16*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(11*b^2+20*a*c+4*b*sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(1/4))+3/16*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(11*b^2+20*a*c+4*b*sqrt(b^2-4*a*c))/(2^(3/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(1/4))],
[x^(7/2)/(a+b*x^2+c*x^4)^3,x,10,-1/16*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(41*b^2+28*a*c-36*b*sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/16*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(41*b^2+28*a*c-36*b*sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/16*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(41*b^2+28*a*c+36*b*sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/16*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(41*b^2+28*a*c+36*b*sqrt(b^2-4*a*c))/(2^(1/4)*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(3/4))+1/4*(2*a+b*x^2)*sqrt(x)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)-1/16*(13*b^2-4*a*c+24*b*c*x^2)*sqrt(x)/((b^2-4*a*c)^2*(a+b*x^2+c*x^4))],
[x^(5/2)/(a+b*x^2+c*x^4)^3,x,10,-1/4*x^(3/2)*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+3/16*x^(3/2)*(b*(b^2+4*a*c)+c*(b^2+12*a*c)*x^2)/(a*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))+3/32*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2+12*a*c-b^3/sqrt(b^2-4*a*c)+68*a*b*c/sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)^2*(-b-sqrt(b^2-4*a*c))^(1/4))-3/32*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2+12*a*c-b^3/sqrt(b^2-4*a*c)+68*a*b*c/sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)^2*(-b-sqrt(b^2-4*a*c))^(1/4))+3/32*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^3-68*a*b*c+(b^2+12*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(1/4))-3/32*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^3-68*a*b*c+(b^2+12*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[x^(3/2)/(a+b*x^2+c*x^4)^3,x,10,-3/32*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2+44*a*c-b^3/sqrt(b^2-4*a*c)+68*a*b*c/sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^2*(-b-sqrt(b^2-4*a*c))^(3/4))-3/32*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(b^2+44*a*c-b^3/sqrt(b^2-4*a*c)+68*a*b*c/sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^2*(-b-sqrt(b^2-4*a*c))^(3/4))-3/32*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^3-68*a*b*c+(b^2+44*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(3/4))-3/32*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(b^3-68*a*b*c+(b^2+44*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*a*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(3/4))-1/4*(b+2*c*x^2)*sqrt(x)/((b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/16*(b*(b^2+20*a*c)+c*(b^2+44*a*c)*x^2)*sqrt(x)/(a*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))],
[x^(1/2)/(a+b*x^2+c*x^4)^3,x,10,1/4*x^(3/2)*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/16*x^(3/2)*(5*b^4-45*a*b^2*c+52*a^2*c^2+b*c*(5*b^2-44*a*c)*x^2)/(a^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))-1/32*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2-b*(5*b^2-44*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(1/4))+1/32*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2-b*(5*b^2-44*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(1/4))+1/32*c^(1/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2+b*(5*b^2-44*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(1/4))-1/32*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2+b*(5*b^2-44*a*c)*sqrt(b^2-4*a*c))/(2^(3/4)*a^2*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(1/4))],
[1/(x^(1/2)*(a+b*x^2+c*x^4)^3),x,10,3/32*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(7*b^4-66*a*b^2*c+280*a^2*c^2-b*(7*b^2-52*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*a^2*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(3/4))+3/32*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b-sqrt(b^2-4*a*c))^(1/4))*(7*b^4-66*a*b^2*c+280*a^2*c^2-b*(7*b^2-52*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*a^2*(b^2-4*a*c)^(5/2)*(-b-sqrt(b^2-4*a*c))^(3/4))-3/32*c^(3/4)*arctan(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(7*b^4-66*a*b^2*c+280*a^2*c^2+b*(7*b^2-52*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*a^2*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(3/4))-3/32*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*sqrt(x)/(-b+sqrt(b^2-4*a*c))^(1/4))*(7*b^4-66*a*b^2*c+280*a^2*c^2+b*(7*b^2-52*a*c)*sqrt(b^2-4*a*c))/(2^(1/4)*a^2*(b^2-4*a*c)^(5/2)*(-b+sqrt(b^2-4*a*c))^(3/4))+1/4*(b^2-2*a*c+b*c*x^2)*sqrt(x)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4)^2)+1/16*(7*b^4-55*a*b^2*c+60*a^2*c^2+b*c*(7*b^2-52*a*c)*x^2)*sqrt(x)/(a^2*(b^2-4*a*c)^2*(a+b*x^2+c*x^4))],

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

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

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

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

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