# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.2 Quartic\1.2.2.6 P(x) (d x)^m (a+b x^2+c x^4)^p.txt"

lst:=[

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

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

# Integrands of the form P2[x] x^m (a+b x^2+c x^4)^p

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

# p<0
[x^4*(A+B*x+C*x^2)/(a+b*x^2+c*x^4),x,13,(A*c-b*C)*x/c^2+1/2*B*x^2/c+1/3*C*x^3/c-1/4*b*B*log(a+b*x^2+c*x^4)/c^2-1/2*B*(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))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(A*b*c-b^2*C+a*c*C+(-A*c*(b^2-2*a*c)+b*(b^2-3*a*c)*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)))*(A*b*c-b^2*C+a*c*C+(A*c*(b^2-2*a*c)-b*(b^2-3*a*c)*C)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^3*(A+B*x+C*x^2)/(a+b*x^2+c*x^4),x,12,B*x/c+1/2*C*x^2/c+1/4*(A*c-b*C)*log(a+b*x^2+c*x^4)/c^2+1/2*(A*b*c-b^2*C+2*a*c*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))-B*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)))-B*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+C*x^2)/(a+b*x^2+c*x^4),x,11,C*x/c+1/4*B*log(a+b*x^2+c*x^4)/c+1/2*b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(A*c-b*C+(-A*b*c+(b^2-2*a*c)*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)))*(A*c-b*C+(A*b*c-b^2*C+2*a*c*C)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x*(A+B*x+C*x^2)/(a+b*x^2+c*x^4),x,10,1/4*C*log(a+b*x^2+c*x^4)/c-1/2*(2*A*c-b*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c*sqrt(b^2-4*a*c))-B*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))+B*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))],
[(A+B*x+C*x^2)/(a+b*x^2+c*x^4),x,8,-B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(C+(2*A*c-b*C)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(C+(-2*A*c+b*C)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x+C*x^2)/(x*(a+b*x^2+c*x^4)),x,12,A*log(x)/a-1/4*A*log(a+b*x^2+c*x^4)/a+1/2*(A*b-2*a*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))+B*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)))-B*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)))],
[(A+B*x+C*x^2)/(x^2*(a+b*x^2+c*x^4)),x,13,-A/(a*x)+B*log(x)/a-1/4*B*log(a+b*x^2+c*x^4)/a+1/2*b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(A+(A*b-2*a*C)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(A+(-A*b+2*a*C)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x+C*x^2)/(x^3*(a+b*x^2+c*x^4)),x,13,-1/2*A/(a*x^2)-B/(a*x)-(A*b-a*C)*log(x)/a^2+1/4*(A*b-a*C)*log(a+b*x^2+c*x^4)/a^2-1/2*(A*(b^2-2*a*c)-a*b*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))-B*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)))-B*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)))],
[x^4*(A+B*x+C*x^2)/(a+b*x^2+c*x^4)^2,x,11,1/2*(2*A*c-b*C)*x/(c*(b^2-4*a*c))+1/2*B*x^2*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x^3*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+2*a*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(A*b*c+(b^2-6*a*c)*C+(-A*c*(b^2+4*a*c)-b*(b^2-8*a*c)*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)))*(A*b*c+(b^2-6*a*c)*C+(A*c*(b^2+4*a*c)+b*(b^2-8*a*c)*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^3*(A+B*x+C*x^2)/(a+b*x^2+c*x^4)^2,x,10,1/2*B*x*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(a*(2*A*c-b*C)+(A*b*c-b^2*C+2*a*c*C)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))-(A*b-2*a*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)+1/2*B*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*B*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+C*x^2)/(a+b*x^2+c*x^4)^2,x,10,1/2*B*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*A*c-b*C+(-4*A*b*c+(b^2+4*a*c)*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)))*(2*A*c-b*C+(4*A*b*c-(b^2+4*a*c)*C)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x*(A+B*x+C*x^2)/(a+b*x^2+c*x^4)^2,x,10,-1/2*B*x*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(-A*b+2*a*C-(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+(2*A*c-b*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)+B*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)))-B*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)))],
[(A+B*x+C*x^2)/(a+b*x^2+c*x^4)^2,x,10,-1/2*B*(b+2*c*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*x*(A*b^2-2*a*A*c-a*b*C+c*(A*b-2*a*C)*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+2*B*c*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(A*b-2*a*C+(A*(b^2-12*a*c)+4*a*b*C)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(A*b-2*a*C+(-A*b^2+12*a*A*c-4*a*b*C)/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x+C*x^2)/(x*(a+b*x^2+c*x^4)^2),x,14,1/2*B*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*(A*(b^2-2*a*c)-a*b*C+c*(A*b-2*a*C)*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(A*(b^3-6*a*b*c)+4*a^2*c*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2))+A*log(x)/a^2-1/4*A*log(a+b*x^2+c*x^4)/a^2+1/2*B*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*B*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)))],
[(A+B*x+C*x^2)/(x^2*(a+b*x^2+c*x^4)^2),x,15,1/2*(-3*A*b^2+10*a*A*c+a*b*C)/(a^2*(b^2-4*a*c)*x)+1/2*B*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(A*(b^2-2*a*c)-a*b*C+c*(A*b-2*a*C)*x^2)/(a*(b^2-4*a*c)*x*(a+b*x^2+c*x^4))+1/2*b*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))+B*log(x)/a^2-1/4*B*log(a+b*x^2+c*x^4)/a^2-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(-a*C*(b^2-12*a*c+b*sqrt(b^2-4*a*c))+A*(3*b^3-16*a*b*c+3*b^2*sqrt(b^2-4*a*c)-10*a*c*sqrt(b^2-4*a*c)))/(a^2*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(3*A*b^2-10*a*A*c-a*b*C+(-A*(3*b^3-16*a*b*c)+a*(b^2-12*a*c)*C)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A+B*x+C*x^2)/(x^3*(a+b*x^2+c*x^4)^2),x,15,1/2*(-2*A*b^2+6*a*A*c+a*b*C)/(a^2*(b^2-4*a*c)*x^2)-1/2*B*(3*b^2-10*a*c)/(a^2*(b^2-4*a*c)*x)+1/2*B*(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*(A*(b^2-2*a*c)-a*b*C+c*(A*b-2*a*C)*x^2)/(a*(b^2-4*a*c)*x^2*(a+b*x^2+c*x^4))-1/2*(2*A*(b^4-6*a*b^2*c+6*a^2*c^2)-a*b*(b^2-6*a*c)*C)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2))-(2*A*b-a*C)*log(x)/a^3+1/4*(2*A*b-a*C)*log(a+b*x^2+c*x^4)/a^3-1/2*B*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*B*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)))],

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

# Integrands of the form P2[x] (d x)^m (a+b x^2+c x^4)^p when m symbolic
[(d*x)^m*(A+B*x+C*x^2)*(a+b*x^2+c*x^4)^3,x,2,a^3*A*(d*x)^(1+m)/(d*(1+m))+a^3*B*(d*x)^(2+m)/(d^2*(2+m))+a^2*(3*A*b+a*C)*(d*x)^(3+m)/(d^3*(3+m))+3*a^2*b*B*(d*x)^(4+m)/(d^4*(4+m))+3*a*(A*(b^2+a*c)+a*b*C)*(d*x)^(5+m)/(d^5*(5+m))+3*a*B*(b^2+a*c)*(d*x)^(6+m)/(d^6*(6+m))+(A*(b^3+6*a*b*c)+3*a*(b^2+a*c)*C)*(d*x)^(7+m)/(d^7*(7+m))+b*B*(b^2+6*a*c)*(d*x)^(8+m)/(d^8*(8+m))+(3*A*c*(b^2+a*c)+b*(b^2+6*a*c)*C)*(d*x)^(9+m)/(d^9*(9+m))+3*B*c*(b^2+a*c)*(d*x)^(10+m)/(d^10*(10+m))+3*c*(A*b*c+(b^2+a*c)*C)*(d*x)^(11+m)/(d^11*(11+m))+3*b*B*c^2*(d*x)^(12+m)/(d^12*(12+m))+c^2*(A*c+3*b*C)*(d*x)^(13+m)/(d^13*(13+m))+B*c^3*(d*x)^(14+m)/(d^14*(14+m))+c^3*C*(d*x)^(15+m)/(d^15*(15+m))],
[(d*x)^m*(A+B*x+C*x^2)*(a+b*x^2+c*x^4)^2,x,2,a^2*A*(d*x)^(1+m)/(d*(1+m))+a^2*B*(d*x)^(2+m)/(d^2*(2+m))+a*(2*A*b+a*C)*(d*x)^(3+m)/(d^3*(3+m))+2*a*b*B*(d*x)^(4+m)/(d^4*(4+m))+(A*(b^2+2*a*c)+2*a*b*C)*(d*x)^(5+m)/(d^5*(5+m))+B*(b^2+2*a*c)*(d*x)^(6+m)/(d^6*(6+m))+(2*A*b*c+(b^2+2*a*c)*C)*(d*x)^(7+m)/(d^7*(7+m))+2*b*B*c*(d*x)^(8+m)/(d^8*(8+m))+c*(A*c+2*b*C)*(d*x)^(9+m)/(d^9*(9+m))+B*c^2*(d*x)^(10+m)/(d^10*(10+m))+c^2*C*(d*x)^(11+m)/(d^11*(11+m))],
[(d*x)^m*(A+B*x+C*x^2)*(a+b*x^2+c*x^4),x,2,a*A*(d*x)^(1+m)/(d*(1+m))+a*B*(d*x)^(2+m)/(d^2*(2+m))+(A*b+a*C)*(d*x)^(3+m)/(d^3*(3+m))+b*B*(d*x)^(4+m)/(d^4*(4+m))+(A*c+b*C)*(d*x)^(5+m)/(d^5*(5+m))+B*c*(d*x)^(6+m)/(d^6*(6+m))+c*C*(d*x)^(7+m)/(d^7*(7+m))],
[(d*x)^m*(A+B*x+C*x^2)/(a+b*x^2+c*x^4),x,8,(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)))*(C+(2*A*c-b*C)/sqrt(b^2-4*a*c))/(d*(1+m)*(b-sqrt(b^2-4*a*c)))+2*B*c*(d*x)^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-2*c*x^2/(b-sqrt(b^2-4*a*c)))/(d^2*(2+m)*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*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)))*(C+(-2*A*c+b*C)/sqrt(b^2-4*a*c))/(d*(1+m)*(b+sqrt(b^2-4*a*c)))-2*B*c*(d*x)^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-2*c*x^2/(b+sqrt(b^2-4*a*c)))/(d^2*(2+m)*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[(d*x)^m*(A+B*x+C*x^2)/(a+b*x^2+c*x^4)^2,x,10,1/2*B*(d*x)^(2+m)*(b^2-2*a*c+b*c*x^2)/(a*(b^2-4*a*c)*d^2*(a+b*x^2+c*x^4))+1/2*(d*x)^(1+m)*(A*(b^2-2*a*c)-a*b*C+c*(A*b-2*a*C)*x^2)/(a*(b^2-4*a*c)*d*(a+b*x^2+c*x^4))+1/2*B*c*(d*x)^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-2*c*x^2/(b+sqrt(b^2-4*a*c)))*(4*a*c*(2-m)+b*m*(b-sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*d^2*(2+m)*(b+sqrt(b^2-4*a*c)))-1/2*B*c*(d*x)^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-2*c*x^2/(b-sqrt(b^2-4*a*c)))*(4*a*c*(2-m)+b*m*(b+sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)^(3/2)*d^2*(2+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)))*(2*a*C*(2*b+(1-m)*sqrt(b^2-4*a*c))+A*(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)))*(2*a*C*(2*b-(1-m)*sqrt(b^2-4*a*c))+A*(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)))],

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

#  Note: The following integrands are equal. 
[x^2*(A+B*x+C*x^2)/(a+b*x^2+c*x^4)^2,x,10,1/2*B*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*A*c-b*C+(-4*A*b*c+(b^2+4*a*c)*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)))*(2*A*c-b*C+(4*A*b*c-(b^2+4*a*c)*C)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x*(A*x+B*x^2+C*x^3)/(a+b*x^2+c*x^4)^2,x,11,1/2*B*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*A*c-b*C+(-4*A*b*c+(b^2+4*a*c)*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)))*(2*A*c-b*C+(4*A*b*c-(b^2+4*a*c)*C)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A*x^2+B*x^3+C*x^4)/(a+b*x^2+c*x^4)^2,x,11,1/2*B*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*A*c-b*C+(-4*A*b*c+(b^2+4*a*c)*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)))*(2*A*c-b*C+(4*A*b*c-(b^2+4*a*c)*C)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A*x^3+B*x^4+C*x^5)/(x*(a+b*x^2+c*x^4)^2),x,11,1/2*B*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*A*c-b*C+(-4*A*b*c+(b^2+4*a*c)*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)))*(2*A*c-b*C+(4*A*b*c-(b^2+4*a*c)*C)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(A*x^4+B*x^5+C*x^6)/(x^2*(a+b*x^2+c*x^4)^2),x,11,1/2*B*(2*a+b*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*x*(A*b-2*a*C+(2*A*c-b*C)*x^2)/((b^2-4*a*c)*(a+b*x^2+c*x^4))-b*B*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*A*c-b*C+(-4*A*b*c+(b^2+4*a*c)*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)))*(2*A*c-b*C+(4*A*b*c-(b^2+4*a*c)*C)/sqrt(b^2-4*a*c))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],

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

# Integrands of the form P2[x^2] x^m (a+b x^2+c x^4)^p

# p>0

# p<0
[x^7*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,7,1/2*(b^2*c*e-a*c^2*e-b^3*f-b*c*(c*d-2*a*f))*x^2/c^4+1/4*(c^2*d+b^2*f-c*(b*e+a*f))*x^4/c^3+1/6*(c*e-b*f)*x^6/c^2+1/8*f*x^8/c-1/4*(b^3*c*e-2*a*b*c^2*e-b^4*f-b^2*c*(c*d-3*a*f)+a*c^2*(c*d-a*f))*log(a+b*x^2+c*x^4)/c^5-1/2*(b^4*c*e-4*a*b^2*c^2*e+2*a^2*c^3*e-b^5*f-b^3*c*(c*d-5*a*f)+a*b*c^2*(3*c*d-5*a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^5*sqrt(b^2-4*a*c))],
[x^5*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,7,1/2*(c^2*d+b^2*f-c*(b*e+a*f))*x^2/c^3+1/4*(c*e-b*f)*x^4/c^2+1/6*f*x^6/c+1/4*(b^2*c*e-a*c^2*e-b^3*f-b*c*(c*d-2*a*f))*log(a+b*x^2+c*x^4)/c^4+1/2*(b^3*c*e-3*a*b*c^2*e-b^4*f-b^2*c*(c*d-4*a*f)+2*a*c^2*(c*d-a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^4*sqrt(b^2-4*a*c))],
[x^3*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,7,1/2*(c*e-b*f)*x^2/c^2+1/4*f*x^4/c+1/4*(c^2*d+b^2*f-c*(b*e+a*f))*log(a+b*x^2+c*x^4)/c^3-1/2*(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(c*d-3*a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[x*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,7,1/2*f*x^2/c+1/4*(c*e-b*f)*log(a+b*x^2+c*x^4)/c^2-1/2*(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[(d+e*x^2+f*x^4)/(x*(a+b*x^2+c*x^4)),x,7,d*log(x)/a-1/4*(c*d-a*f)*log(a+b*x^2+c*x^4)/(a*c)+1/2*(b*c*d-2*a*c*e+a*b*f)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a*c*sqrt(b^2-4*a*c))],
[(d+e*x^2+f*x^4)/(x^3*(a+b*x^2+c*x^4)),x,7,-1/2*d/(a*x^2)-(b*d-a*e)*log(x)/a^2+1/4*(b*d-a*e)*log(a+b*x^2+c*x^4)/a^2-1/2*(b^2*d-a*b*e-2*a*(c*d-a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c))],
[(d+e*x^2+f*x^4)/(x^5*(a+b*x^2+c*x^4)),x,7,-1/4*d/(a*x^4)+1/2*(b*d-a*e)/(a^2*x^2)+(b^2*d-a*b*e-a*(c*d-a*f))*log(x)/a^3-1/4*(b^2*d-a*b*e-a*(c*d-a*f))*log(a+b*x^2+c*x^4)/a^3+1/2*(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(3*c*d-a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*sqrt(b^2-4*a*c))],
[(d+e*x^2+f*x^4)/(x^7*(a+b*x^2+c*x^4)),x,7,-1/6*d/(a*x^6)+1/4*(b*d-a*e)/(a^2*x^4)+1/2*(-b^2*d+a*b*e+a*(c*d-a*f))/(a^3*x^2)-(b^3*d-a*b^2*e+a^2*c*e-a*b*(2*c*d-a*f))*log(x)/a^4+1/4*(b^3*d-a*b^2*e+a^2*c*e-a*b*(2*c*d-a*f))*log(a+b*x^2+c*x^4)/a^4-1/2*(b^4*d-a*b^3*e+3*a^2*b*c*e+2*a^2*c*(c*d-a*f)-a*b^2*(4*c*d-a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^4*sqrt(b^2-4*a*c))],
[x^4*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,5,(c^2*d+b^2*f-c*(b*e+a*f))*x/c^3+1/3*(c*e-b*f)*x^3/c^2+1/5*f*x^5/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b^2*c*e-a*c^2*e-b^3*f-b*c*(c*d-2*a*f)+(-b^3*c*e+3*a*b*c^2*e+b^4*f+b^2*c*(c*d-4*a*f)-2*a*c^2*(c*d-a*f))/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b^2*c*e-a*c^2*e-b^3*f-b*c*(c*d-2*a*f)+(b^3*c*e-3*a*b*c^2*e-b^4*f-b^2*c*(c*d-4*a*f)+2*a*c^2*(c*d-a*f))/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,5,(c*e-b*f)*x/c^2+1/3*f*x^3/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(c^2*d-b*c*e+b^2*f-a*c*f+(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(c*d-3*a*f))/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)))*(c^2*d-b*c*e+b^2*f-a*c*f+(-b^2*c*e+2*a*c^2*e+b^3*f+b*c*(c*d-3*a*f))/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4),x,5,f*x/c+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(c*e-b*f+(2*c^2*d+b^2*f-c*(b*e+2*a*f))/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)))*(c*e-b*f+(-2*c^2*d+b*c*e-b^2*f+2*a*c*f)/sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(x^2*(a+b*x^2+c*x^4)),x,5,-d/(a*x)-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(c*d-a*f+(b*c*d-2*a*c*e+a*b*f)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))-arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(c*d-a*f+(-b*c*d+2*a*c*e-a*b*f)/sqrt(b^2-4*a*c))/(a*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(x^4*(a+b*x^2+c*x^4)),x,5,-1/3*d/(a*x^3)+(b*d-a*e)/(a^2*x)+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b*d-a*e+(b^2*d-a*b*e-2*a*(c*d-a*f))/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^2*d-b*(a*e+d*sqrt(b^2-4*a*c))-a*(2*c*d-2*a*f-e*sqrt(b^2-4*a*c)))/(a^2*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(x^6*(a+b*x^2+c*x^4)),x,5,-1/5*d/(a*x^5)+1/3*(b*d-a*e)/(a^2*x^3)+(-b^2*d+a*b*e+a*(c*d-a*f))/(a^3*x)-arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*sqrt(c)*(b^2*d-a*b*e-a*(c*d-a*f)+(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(3*c*d-a*f))/sqrt(b^2-4*a*c))/(a^3*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^2*d-a*b*e-a*(c*d-a*f)+(-b^3*d+a*b^2*e-2*a^2*c*e+a*b*(3*c*d-a*f))/sqrt(b^2-4*a*c))/(a^3*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^7*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,8,1/2*(2*b^2*c*e-6*a*c^2*e-3*b^3*f-b*c*(c*d-11*a*f))*x^2/(c^3*(b^2-4*a*c))+1/4*(4*c^2*d+3*b^2*f-2*c*(b*e+4*a*f))*x^4/(c^2*(b^2-4*a*c))+1/2*x^6*(2*a*c*e-b*(c*d+a*f)-(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*(2*b^4*c*e-12*a*b^2*c^2*e+12*a^2*c^3*e-3*b^5*f-b^3*c*(c*d-20*a*f)+6*a*b*c^2*(c*d-5*a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^4*(b^2-4*a*c)^(3/2))+1/4*(c^2*d+3*b^2*f-2*c*(b*e+a*f))*log(a+b*x^2+c*x^4)/c^4],
[x^5*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,7,1/2*(2*c^2*d+2*b^2*f-c*(b*e+6*a*f))*x^2/(c^2*(b^2-4*a*c))+1/2*x^4*(2*a*c*e-b*(c*d+a*f)-(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*(12*a^2*c^2*f-b^3*(c*e-2*b*f)-2*a*c*(2*c^2*d-3*b*c*e+6*b^2*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*(b^2-4*a*c)^(3/2))+1/4*(c*e-2*b*f)*log(a+b*x^2+c*x^4)/c^3],
[x^3*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,6,1/2*x^2*(2*a*c*e-b*(c*d+a*f)-(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(4*a*c^2*e+b^3*f-2*b*c*(c*d+3*a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^2*(b^2-4*a*c)^(3/2))+1/4*f*log(a+b*x^2+c*x^4)/c^2],
[x*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,5,1/2*(2*a*c*e-b*(c*d+a*f)-(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))+(2*c*d-b*e+2*a*f)*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(b^2-4*a*c)^(3/2)],
[(d+e*x^2+f*x^4)/(x*(a+b*x^2+c*x^4)^2),x,8,1/2*(b^2*d-a*b*e-2*a*(c*d-a*f)+(b*c*d-2*a*c*e+a*b*f)*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(b^3*d+4*a^2*c*e-2*a*b*(3*c*d+a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)^(3/2))+d*log(x)/a^2-1/4*d*log(a+b*x^2+c*x^4)/a^2],
[(d+e*x^2+f*x^4)/(x^3*(a+b*x^2+c*x^4)^2),x,8,-1/2*d/(a^2*x^2)+1/2*(-b^3*d+a*b^2*e-2*a^2*c*e+a*b*(3*c*d-a*f)-c*(b^2*d-a*b*e-2*a*(c*d-a*f))*x^2)/(a^2*(b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*(2*b^4*d-12*a*b^2*c*d-a*b^3*e+6*a^2*b*c*e+4*a^2*c*(3*c*d-a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)^(3/2))-(2*b*d-a*e)*log(x)/a^3+1/4*(2*b*d-a*e)*log(a+b*x^2+c*x^4)/a^3],
[(d+e*x^2+f*x^4)/(x^5*(a+b*x^2+c*x^4)^2),x,8,-1/4*d/(a^2*x^4)+1/2*(2*b*d-a*e)/(a^3*x^2)+1/2*(b^4*d-a*b^3*e+3*a^2*b*c*e+2*a^2*c*(c*d-a*f)-a*b^2*(4*c*d-a*f)+c*(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(3*c*d-a*f))*x^2)/(a^3*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*(3*b^5*d-2*a*b^4*e+12*a^2*b^2*c*e-12*a^3*c^2*e+6*a^2*b*c*(5*c*d-a*f)-a*b^3*(20*c*d-a*f))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(a^4*(b^2-4*a*c)^(3/2))+(3*b^2*d-2*a*b*e-a*(2*c*d-a*f))*log(x)/a^4-1/4*(3*b^2*d-2*a*b*e-a*(2*c*d-a*f))*log(a+b*x^2+c*x^4)/a^4],
[x^6*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,6,(c*e-2*b*f)*x/c^3+1/3*f*x^3/c^2+1/2*x*(a*(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(c*d-3*a*f))+(b^3*c*e-3*a*b*c^2*e-b^4*f-b^2*c*(c*d-4*a*f)+2*a*c^2*(c*d-a*f))*x^2)/(c^3*(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*c*e-13*a*b*c^2*e-5*b^4*f-b^2*c*(c*d-24*a*f)+2*a*c^2*(3*c*d-7*a*f)+(-3*b^4*c*e+19*a*b^2*c^2*e-20*a^2*c^3*e+5*b^5*f+b^3*c*(c*d-34*a*f)-4*a*b*c^2*(2*c*d-13*a*f))/sqrt(b^2-4*a*c))/(c^(7/2)*(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*c*e-13*a*b*c^2*e-5*b^4*f-b^2*c*(c*d-24*a*f)+2*a*c^2*(3*c*d-7*a*f)+(3*b^4*c*e-19*a*b^2*c^2*e+20*a^2*c^3*e-5*b^5*f-b^3*c*(c*d-34*a*f)+4*a*b*c^2*(2*c*d-13*a*f))/sqrt(b^2-4*a*c))/(c^(7/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^4*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,6,f*x/c^2+1/2*x*(a*(2*c^2*d-b*c*e+b^2*f-2*a*c*f)-(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(c*d-3*a*f))*x^2)/(c^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*c*e-6*a*c^2*e-3*b^3*f+b*c*(c*d+13*a*f)+(-b^3*c*e+8*a*b*c^2*e+3*b^4*f-4*a*c^2*(c*d-5*a*f)-b^2*c*(c*d+19*a*f))/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)))*(b^2*c*e-6*a*c^2*e-3*b^3*f+b*c*(c*d+13*a*f)+(b^3*c*e-8*a*b*c^2*e-3*b^4*f+4*a*c^2*(c*d-5*a*f)+b^2*c*(c*d+19*a*f))/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^2*(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,4,-1/2*x*(b*c*d-2*a*c*e+a*b*f+(2*c^2*d-b*c*e+b^2*f-2*a*c*f)*x^2)/(c*(b^2-4*a*c)*(a+b*x^2+c*x^4))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(2*c*d-b*e+6*a*f-b^2*f/c+(b^2*c*e+4*a*c^2*e+b^3*f-4*b*c*(c*d+2*a*f))/(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)))*(2*c*d-b*e+6*a*f-b^2*f/c+(-b^2*c*e-4*a*c^2*e-b^3*f+4*b*c*(c*d+2*a*f))/(c*sqrt(b^2-4*a*c)))/((b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(b^2*d-a*b*e-2*a*(c*d-a*f)+(b*c*d-2*a*c*e+a*b*f)*x^2)/(a*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b*c*d-2*a*c*e+a*b*f+(4*a*b*c*e+b^2*(c*d-a*f)-4*a*c*(3*c*d+a*f))/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b*c*d-2*a*c*e+a*b*f+(-4*a*b*c*e-b^2*(c*d-a*f)+4*a*c*(3*c*d+a*f))/sqrt(b^2-4*a*c))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(x^2*(a+b*x^2+c*x^4)^2),x,6,-d/(a^2*x)-1/2*x*(a*(b^3*d/a-b*(3*c*d+b*e)+a*(2*c*e+b*f))+c*(b^2*d-a*b*e-2*a*(c*d-a*f))*x^2)/(a^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)))*sqrt(c)*(3*b^2*d-a*b*e-2*a*(5*c*d-a*f)+(3*b^3*d-a*b^2*e+12*a^2*c*e-4*a*b*(4*c*d+a*f))/sqrt(b^2-4*a*c))/(a^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)))*sqrt(c)*(3*b^2*d-a*b*e-2*a*(5*c*d-a*f)+(-3*b^3*d+a*b^2*e-12*a^2*c*e+4*a*b*(4*c*d+a*f))/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4)/(x^4*(a+b*x^2+c*x^4)^2),x,6,-1/3*d/(a^2*x^3)+(2*b*d-a*e)/(a^3*x)+1/2*x*(a^2*(b^4*d/a^2+2*c^2*d+3*b*c*e-b^2*(4*c*d+b*e)/a+b^2*f-2*a*c*f)+c*(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(3*c*d-a*f))*x^2)/(a^3*(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)*(5*b^4*d+b^3*(-3*a*e+5*d*sqrt(b^2-4*a*c))-a*b^2*(29*c*d-a*f+3*e*sqrt(b^2-4*a*c))+2*a^2*c*(14*c*d-6*a*f+5*e*sqrt(b^2-4*a*c))-a*b*(-16*a*c*e+19*c*d*sqrt(b^2-4*a*c)-a*f*sqrt(b^2-4*a*c)))/(a^3*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))-1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^4*d-b^3*(3*a*e+5*d*sqrt(b^2-4*a*c))+2*a^2*c*(14*c*d-6*a*f-5*e*sqrt(b^2-4*a*c))-a*b^2*(29*c*d-a*f-3*e*sqrt(b^2-4*a*c))+a*b*(16*a*c*e+19*c*d*sqrt(b^2-4*a*c)-a*f*sqrt(b^2-4*a*c)))/(a^3*(b^2-4*a*c)^(3/2)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],

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

# p>0

# p<0
[x^9*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,7,-293/2*x^2+49/2*x^4-9/2*x^6+5/8*x^8+1/2*(414+415*x^2)/(2+3*x^2+x^4)+2*log(1+x^2)+392*log(2+x^2)],
[x^7*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,7,49*x^2-27/4*x^4+5/6*x^6+1/2*(-206-207*x^2)/(2+3*x^2+x^4)-5/2*log(1+x^2)-144*log(2+x^2)],
[x^5*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,7,-27/2*x^2+5/4*x^4+1/2*(102+103*x^2)/(2+3*x^2+x^4)+3*log(1+x^2)+46*log(2+x^2)],
[x^3*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,7,5/2*x^2+1/2*(-50-51*x^2)/(2+3*x^2+x^4)-7/2*log(1+x^2)-10*log(2+x^2)],
[x*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,5,1/2*(24+25*x^2)/(2+3*x^2+x^4)+4*log(1+x^2)-3/2*log(2+x^2)],
[(4+x^2+3*x^4+5*x^6)/(x*(2+3*x^2+x^4)^2),x,4,1/2*(-11-12*x^2)/(2+3*x^2+x^4)+log(x)-9/2*log(1+x^2)+4*log(2+x^2)],
[(4+x^2+3*x^4+5*x^6)/(x^3*(2+3*x^2+x^4)^2),x,4,(-1/2)/x^2+1/4*(9+11*x^2)/(2+3*x^2+x^4)-11/4*log(x)+5*log(1+x^2)-29/8*log(2+x^2)],
[(4+x^2+3*x^4+5*x^6)/(x^5*(2+3*x^2+x^4)^2),x,4,(-1/4)/x^4+11/8/x^2+1/8*(-5-9*x^2)/(2+3*x^2+x^4)+23/4*log(x)-11/2*log(1+x^2)+21/8*log(2+x^2)],
[x^8*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,6,-293*x+98/3*x^3-27/5*x^5+5/7*x^7-1/2*x*(206+207*x^2)/(2+3*x^2+x^4)+9/2*arctan(x)+340*arctan(x/sqrt(2))*sqrt(2)],
[x^6*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,6,98*x-9*x^3+x^5+1/2*x*(102+103*x^2)/(2+3*x^2+x^4)-11/2*arctan(x)-118*arctan(x/sqrt(2))*sqrt(2)],
[x^4*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,6,-27*x+5/3*x^3-1/2*x*(50+51*x^2)/(2+3*x^2+x^4)+13/2*arctan(x)+33*arctan(x/sqrt(2))*sqrt(2)],
[x^2*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,6,5*x+1/2*x*(24+25*x^2)/(2+3*x^2+x^4)-15/2*arctan(x)-7*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^2,x,4,-1/2*x*(11+12*x^2)/(2+3*x^2+x^4)+17/2*arctan(x)-19/2*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^2*(2+3*x^2+x^4)^2),x,5,(-1)/x+1/4*x*(9+11*x^2)/(2+3*x^2+x^4)-19/2*arctan(x)+45/4*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^4*(2+3*x^2+x^4)^2),x,5,(-1/3)/x^3+11/4/x-1/8*x*(5+9*x^2)/(2+3*x^2+x^4)+21/2*arctan(x)-71/8*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^6*(2+3*x^2+x^4)^2),x,5,(-1/5)/x^5+11/12/x^3+(-23/4)/x-1/16*x*(3-5*x^2)/(2+3*x^2+x^4)-23/2*arctan(x)+97/16*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^8*(2+3*x^2+x^4)^2),x,5,(-1/7)/x^7+11/20/x^5+(-23/12)/x^3+137/16/x+1/32*x*(19+3*x^2)/(2+3*x^2+x^4)+25/2*arctan(x)-123/32*arctan(x/sqrt(2))/sqrt(2)],
[x^10*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^3,x,7,214*x-14*x^3+x^5+1/4*x*(414+415*x^2)/(2+3*x^2+x^4)^2+1/8*x*(824+1669*x^2)/(2+3*x^2+x^4)+477/8*arctan(x)-351*arctan(x/sqrt(2))*sqrt(2)],
[x^8*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^3,x,7,-42*x+5/3*x^3-1/4*x*(206+207*x^2)/(2+3*x^2+x^4)^2+1/8*x*(24-409*x^2)/(2+3*x^2+x^4)-449/8*arctan(x)+219*arctan(x/sqrt(2))/sqrt(2)],
[x^6*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^3,x,7,5*x+1/4*x*(102+103*x^2)/(2+3*x^2+x^4)^2-1/8*x*(244+15*x^2)/(2+3*x^2+x^4)+413/8*arctan(x)-191/2*arctan(x/sqrt(2))/sqrt(2)],
[x^4*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^3,x,5,-1/4*x*(50+51*x^2)/(2+3*x^2+x^4)^2+1/8*x*(254+125*x^2)/(2+3*x^2+x^4)-369/8*arctan(x)+267/4*arctan(x/sqrt(2))/sqrt(2)],
[x^2*(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^3,x,5,1/4*x*(24+25*x^2)/(2+3*x^2+x^4)^2-1/8*x*(211+130*x^2)/(2+3*x^2+x^4)+317/8*arctan(x)-447/8*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(2+3*x^2+x^4)^3,x,5,-1/4*x*(11+12*x^2)/(2+3*x^2+x^4)^2+1/16*x*(335+217*x^2)/(2+3*x^2+x^4)-257/8*arctan(x)+731/16*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^2*(2+3*x^2+x^4)^3),x,6,(-1/2)/x+1/8*x*(9+11*x^2)/(2+3*x^2+x^4)^2-1/32*x*(547+347*x^2)/(2+3*x^2+x^4)+189/8*arctan(x)-1119/32*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^4*(2+3*x^2+x^4)^3),x,6,(-1/6)/x^3+17/8/x-1/16*x*(5+9*x^2)/(2+3*x^2+x^4)^2+1/64*x*(951+571*x^2)/(2+3*x^2+x^4)-113/8*arctan(x)+1611/64*arctan(x/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^6*(2+3*x^2+x^4)^3),x,6,(-1/10)/x^5+17/24/x^3+(-93/16)/x-1/32*x*(3-5*x^2)/(2+3*x^2+x^4)^2-1/128*x*(1771+999*x^2)/(2+3*x^2+x^4)+29/8*arctan(x)-2207/128*arctan(x/sqrt(2))/sqrt(2)],
[x^9*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,8,19*x^2+19/4*x^4-17/6*x^6+5/8*x^8-25/8*(15+7*x^2)/(3+2*x^2+x^4)-183/4*log(3+2*x^2+x^4)+201/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[x^7*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,8,19/2*x^2-17/4*x^4+5/6*x^6+25/8*(3+5*x^2)/(3+2*x^2+x^4)+19/2*log(3+2*x^2+x^4)-455/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[x^5*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,8,-17/2*x^2+5/4*x^4+25/8*(3-x^2)/(3+2*x^2+x^4)+19/4*log(3+2*x^2+x^4)+203/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[x^3*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,8,5/2*x^2-25/8*(3+x^2)/(3+2*x^2+x^4)-17/4*log(3+2*x^2+x^4)-17/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[x*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,6,25/8*(1+x^2)/(3+2*x^2+x^4)+5/4*log(3+2*x^2+x^4)-23/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x*(3+2*x^2+x^4)^2),x,8,25/24*(1-x^2)/(3+2*x^2+x^4)+4/9*log(x)-1/9*log(3+2*x^2+x^4)+89/72*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^3*(3+2*x^2+x^4)^2),x,8,(-2/9)/x^2-25/72*(5+x^2)/(3+2*x^2+x^4)-13/27*log(x)+13/108*log(3+2*x^2+x^4)-71/216*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^5*(3+2*x^2+x^4)^2),x,8,(-1/9)/x^4+13/54/x^2+25/216*(7+5*x^2)/(3+2*x^2+x^4)+13/27*log(x)-13/108*log(3+2*x^2+x^4)+125/216*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[(4+x^2+3*x^4+5*x^6)/(x^7*(3+2*x^2+x^4)^2),x,8,(-2/27)/x^6+13/108/x^4+(-13/54)/x^2+25/648*(1-7*x^2)/(3+2*x^2+x^4)+61/243*log(x)-61/972*log(3+2*x^2+x^4)-1237/1944*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[x^8*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,12,38*x+19/3*x^3-17/5*x^5+5/7*x^7+25/8*x*(3+5*x^2)/(3+2*x^2+x^4)-1/32*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/2*(-262771+618291*sqrt(3)))+1/32*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/2*(-262771+618291*sqrt(3)))+1/16*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/2*(262771+618291*sqrt(3)))-1/16*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/2*(262771+618291*sqrt(3)))],
[x^6*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,12,19*x-17/3*x^3+x^5+25/8*x*(3-x^2)/(3+2*x^2+x^4)+3/16*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(3/2*(-8669+5011*sqrt(3)))-3/16*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(3/2*(-8669+5011*sqrt(3)))+3/32*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(3/2*(8669+5011*sqrt(3)))-3/32*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(3/2*(8669+5011*sqrt(3)))],
[x^4*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,12,-17*x+5/3*x^3-25/8*x*(3+x^2)/(3+2*x^2+x^4)-1/32*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/2*(-14395+26499*sqrt(3)))+1/32*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/2*(-14395+26499*sqrt(3)))-1/16*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/2*(14395+26499*sqrt(3)))+1/16*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/2*(14395+26499*sqrt(3)))],
[x^2*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,12,5*x+25/8*x*(1+x^2)/(3+2*x^2+x^4)-1/32*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(-19291+12899*sqrt(3)))+1/32*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(-19291+12899*sqrt(3)))+1/16*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(19291+12899*sqrt(3)))-1/16*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(19291+12899*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^2,x,10,25/24*x*(1-x^2)/(3+2*x^2+x^4)-1/48*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(-11567+12897*sqrt(3)))+1/48*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(-11567+12897*sqrt(3)))+1/96*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(11567+12897*sqrt(3)))-1/96*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(11567+12897*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(x^2*(3+2*x^2+x^4)^2),x,12,(-4/9)/x-25/72*x*(5+x^2)/(3+2*x^2+x^4)+1/48*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(-965+699*sqrt(3)))-1/48*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(-965+699*sqrt(3)))-1/96*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(965+699*sqrt(3)))+1/96*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(965+699*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(x^4*(3+2*x^2+x^4)^2),x,12,(-4/27)/x^3+13/27/x+25/216*x*(7+5*x^2)/(3+2*x^2+x^4)+1/864*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(-6073+56673*sqrt(3)))-1/864*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(-6073+56673*sqrt(3)))-1/432*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(6073+56673*sqrt(3)))+1/432*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(6073+56673*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(x^6*(3+2*x^2+x^4)^2),x,12,(-4/45)/x^5+13/81/x^3+(-13/27)/x+25/648*x*(1-7*x^2)/(3+2*x^2+x^4)+1/1296*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(-1139381+688419*sqrt(3)))-1/1296*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/6*(-1139381+688419*sqrt(3)))-1/2592*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(1139381+688419*sqrt(3)))+1/2592*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/6*(1139381+688419*sqrt(3)))],
[x^10*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^3,x,13,58*x-9*x^3+x^5-25/16*x*(15+7*x^2)/(3+2*x^2+x^4)^2+1/64*x*(3305+252*x^2)/(3+2*x^2+x^4)+3/256*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(-8595619+7678611*sqrt(3))-3/256*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(-8595619+7678611*sqrt(3))+3/512*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(8595619+7678611*sqrt(3))-3/512*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(8595619+7678611*sqrt(3))],
[x^8*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^3,x,13,-27*x+5/3*x^3+25/16*x*(3+5*x^2)/(3+2*x^2+x^4)^2-1/64*x*(1468+835*x^2)/(3+2*x^2+x^4)-21/512*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(-34271+22721*sqrt(3))+21/512*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(-34271+22721*sqrt(3))-21/256*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(34271+22721*sqrt(3))+21/256*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(34271+22721*sqrt(3))],
[x^6*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^3,x,13,5*x+25/16*x*(3-x^2)/(3+2*x^2+x^4)^2+7/64*x*(11+58*x^2)/(3+2*x^2+x^4)-1/512*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(-827621+1176531*sqrt(3))+1/512*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(-827621+1176531*sqrt(3))+1/256*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(827621+1176531*sqrt(3))-1/256*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(827621+1176531*sqrt(3))],
[x^4*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^3,x,11,-25/16*x*(3+x^2)/(3+2*x^2+x^4)^2+1/64*x*(238-59*x^2)/(3+2*x^2+x^4)-1/256*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(3*(-48835+32827*sqrt(3)))+1/256*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(3*(-48835+32827*sqrt(3)))+1/512*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(3*(48835+32827*sqrt(3)))-1/512*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(3*(48835+32827*sqrt(3)))],
[x^2*(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^3,x,11,25/16*x*(1+x^2)/(3+2*x^2+x^4)^2-1/192*x*(353+88*x^2)/(3+2*x^2+x^4)-11/768*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(-1825+1089*sqrt(3)))+11/768*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(-1825+1089*sqrt(3)))-11/1536*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(1825+1089*sqrt(3)))+11/1536*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(1825+1089*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(3+2*x^2+x^4)^3,x,11,25/48*x*(1-x^2)/(3+2*x^2+x^4)^2+1/192*x*(64+51*x^2)/(3+2*x^2+x^4)-1/256*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(-1291+1019*sqrt(3)))+1/256*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(-1291+1019*sqrt(3)))+1/512*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(1291+1019*sqrt(3)))-1/512*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(1291+1019*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(x^2*(3+2*x^2+x^4)^3),x,13,(-4/27)/x-25/144*x*(5+x^2)/(3+2*x^2+x^4)^2-1/1728*x*(325+242*x^2)/(3+2*x^2+x^4)-1/4608*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(-59711+55161*sqrt(3)))+1/4608*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(-59711+55161*sqrt(3)))+1/2304*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(59711+55161*sqrt(3)))-1/2304*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(59711+55161*sqrt(3)))],
[(4+x^2+3*x^4+5*x^6)/(x^4*(3+2*x^2+x^4)^3),x,13,(-4/81)/x^3+7/27/x+25/432*x*(7+5*x^2)/(3+2*x^2+x^4)^2+1/5184*x*(1474+1025*x^2)/(3+2*x^2+x^4)+1/41472*log(x^2+sqrt(3)-x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(-10004741+11240451*sqrt(3)))-1/41472*log(x^2+sqrt(3)+x*sqrt(2*(-1+sqrt(3))))*sqrt(1/3*(-10004741+11240451*sqrt(3)))-1/20736*arctan((-2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(10004741+11240451*sqrt(3)))+1/20736*arctan((2*x+sqrt(2*(-1+sqrt(3))))/sqrt(2*(1+sqrt(3))))*sqrt(1/3*(10004741+11240451*sqrt(3)))],
[x*(d+e*x^2+f*x^4+g*x^6)/(a+b*x^2+c*x^4),x,7,1/2*(c*f-b*g)*x^2/c^2+1/4*g*x^4/c+1/4*(c^2*e+b^2*g-c*(b*f+a*g))*log(a+b*x^2+c*x^4)/c^3-1/2*(2*c^3*d-c^2*(b*e+2*a*f)-b^3*g+b*c*(b*f+3*a*g))*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c))],
[x^4*(d+e*x^2+f*x^4+g*x^6)/(a+b*x^2+c*x^4)^2,x,6,(c*f-2*b*g)*x/c^3+1/3*g*x^3/c^2+1/2*x*(a*(2*c^3*d-c^2*(b*e+2*a*f)-b^3*g+b*c*(b*f+3*a*g))+(b^3*c*f+b*c^2*(c*d-3*a*f)-b^4*g-b^2*c*(c*e-4*a*g)+2*a*c^2*(c*e-a*g))*x^2)/(c^3*(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*c*f-b*c^2*(c*d+13*a*f)-5*b^4*g-b^2*c*(c*e-24*a*g)+2*a*c^2*(3*c*e-7*a*g)+(-3*b^4*c*f+4*a*c^3*(c*d-5*a*f)+b^2*c^2*(c*d+19*a*f)+5*b^5*g+b^3*c*(c*e-34*a*g)-4*a*b*c^2*(2*c*e-13*a*g))/sqrt(b^2-4*a*c))/(c^(7/2)*(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*c*f-b*c^2*(c*d+13*a*f)-5*b^4*g-b^2*c*(c*e-24*a*g)+2*a*c^2*(3*c*e-7*a*g)+(3*b^4*c*f-4*a*c^3*(c*d-5*a*f)-b^2*c^2*(c*d+19*a*f)-5*b^5*g-b^3*c*(c*e-34*a*g)+4*a*b*c^2*(2*c*e-13*a*g))/sqrt(b^2-4*a*c))/(c^(7/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2*(d+e*x^2+f*x^4+g*x^6)/(a+b*x^2+c*x^4)^2,x,6,g*x/c^2-1/2*x*(b*c*(c*d+a*f)-a*b^2*g-2*a*c*(c*e-a*g)+(2*c^3*d-c^2*(b*e+2*a*f)-b^3*g+b*c*(b*f+3*a*g))*x^2)/(c^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)))*(2*c^3*d-c^2*(b*e-6*a*f)+3*b^3*g-b*c*(b*f+13*a*g)+(b^3*c*f-4*b*c^2*(c*d+2*a*f)-3*b^4*g+4*a*c^2*(c*e-5*a*g)+b^2*c*(c*e+19*a*g))/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)))*(2*c^3*d-c^2*(b*e-6*a*f)+3*b^3*g-b*c*(b*f+13*a*g)+(-b^3*c*f+4*b*c^2*(c*d+2*a*f)+3*b^4*g-4*a*c^2*(c*e-5*a*g)-b^2*c*(c*e+19*a*g))/sqrt(b^2-4*a*c))/(c^(5/2)*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4+g*x^6)/(a+b*x^2+c*x^4)^2,x,4,1/2*x*(c*(b^2*d-2*a*(c*d-a*f)-a*b*(c*e+a*g)/c)+(b*c*(c*d+a*f)-a*b^2*g-2*a*c*(c*e-a*g))*x^2)/(a*c*(b^2-4*a*c)*(a+b*x^2+c*x^4))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(b*(c*d+a*f)+a*b^2*g/c-2*a*(c*e+3*a*g)+(b^2*c*(c*d-a*f)-4*a*c^2*(3*c*d+a*f)-a*b^3*g+4*a*b*c*(c*e+2*a*g))/(c*sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(b*(c*d+a*f)+a*b^2*g/c-2*a*(c*e+3*a*g)+(-b^2*c*(c*d-a*f)+4*a*c^2*(3*c*d+a*f)+a*b^3*g-4*a*b*c*(c*e+2*a*g))/(c*sqrt(b^2-4*a*c)))/(a*(b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4+g*x^6)/(x^2*(a+b*x^2+c*x^4)^2),x,6,-d/(a^2*x)-1/2*x*(a*(b^3*d/a-b*(3*c*d+b*e)+a*(2*c*e+b*f)-2*a^2*g)+(b^2*c*d-2*a*c*(c*d-a*f)-a*b*(c*e+a*g))*x^2)/(a^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^2*c*d-2*a*c*(5*c*d-a*f)-a*b*(c*e+a*g)+(3*b^3*c*d-4*a*b*c*(4*c*d+a*f)-a*b^2*(c*e-a*g)+4*a^2*c*(3*c*e+a*g))/sqrt(b^2-4*a*c))/(a^2*(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)))*(3*b^2*c*d-2*a*c*(5*c*d-a*f)-a*b*(c*e+a*g)+(-3*b^3*c*d+4*a*b*c*(4*c*d+a*f)+a*b^2*(c*e-a*g)-4*a^2*c*(3*c*e+a*g))/sqrt(b^2-4*a*c))/(a^2*(b^2-4*a*c)*sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[(d+e*x^2+f*x^4+g*x^6)/(x^4*(a+b*x^2+c*x^4)^2),x,6,-1/3*d/(a^2*x^3)+(2*b*d-a*e)/(a^3*x)+1/2*x*(a^2*(b^4*d/a^2+2*c^2*d+3*b*c*e-b^2*(4*c*d+b*e)/a+b^2*f-a*(2*c*f+b*g))+c*(b^3*d-a*b^2*e-a*b*(3*c*d-a*f)+2*a^2*(c*e-a*g))*x^2)/(a^3*(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)*(5*b^3*d-3*a*b^2*e-a*b*(19*c*d-a*f)+2*a^2*(5*c*e-a*g)+(5*b^4*d-3*a*b^3*e+4*a^2*c*(7*c*d-3*a*f)-a*b^2*(29*c*d-a*f)+4*a^2*b*(4*c*e+a*g))/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)*sqrt(2)*sqrt(b-sqrt(b^2-4*a*c)))+1/2*arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*sqrt(c)*(5*b^3*d-3*a*b^2*e-a*b*(19*c*d-a*f)+2*a^2*(5*c*e-a*g)+(-5*b^4*d+3*a*b^3*e-4*a^2*c*(7*c*d-3*a*f)+a*b^2*(29*c*d-a*f)-4*a^2*b*(4*c*e+a*g))/sqrt(b^2-4*a*c))/(a^3*(b^2-4*a*c)*sqrt(2)*sqrt(b+sqrt(b^2-4*a*c)))],

# Integrands of the form P2[x^2] x^m (a+b x^2+c x^4)^p with p symbolic
[x^2*(a+b*x^2+c*x^4)^p*(3*a+b*(5+2*p)*x^2+c*(7+4*p)*x^4),x,1,x^3*(a+b*x^2+c*x^4)^(1+p)],

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

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

# p>0

# p<0
[x^5*(a+b*x^2+c*x^4)/(sqrt(d-e*x)*sqrt(d+e*x)),x,5,1/3*d^2*(4*c*d^4+3*b*d^2*e^2+2*a*e^4)*(d-e*x)^(3/2)*(d+e*x)^(3/2)/e^10-1/5*(6*c*d^4+3*b*d^2*e^2+a*e^4)*(d-e*x)^(5/2)*(d+e*x)^(5/2)/e^10+1/7*(4*c*d^2+b*e^2)*(d-e*x)^(7/2)*(d+e*x)^(7/2)/e^10-1/9*c*(d-e*x)^(9/2)*(d+e*x)^(9/2)/e^10-d^4*(c*d^4+b*d^2*e^2+a*e^4)*sqrt(d-e*x)*sqrt(d+e*x)/e^10,-d^4*(c*d^4+b*d^2*e^2+a*e^4)*(d^2-e^2*x^2)/(e^10*sqrt(d-e*x)*sqrt(d+e*x))+1/3*d^2*(4*c*d^4+3*b*d^2*e^2+2*a*e^4)*(d^2-e^2*x^2)^2/(e^10*sqrt(d-e*x)*sqrt(d+e*x))-1/5*(6*c*d^4+3*b*d^2*e^2+a*e^4)*(d^2-e^2*x^2)^3/(e^10*sqrt(d-e*x)*sqrt(d+e*x))+1/7*(4*c*d^2+b*e^2)*(d^2-e^2*x^2)^4/(e^10*sqrt(d-e*x)*sqrt(d+e*x))-1/9*c*(d^2-e^2*x^2)^5/(e^10*sqrt(d-e*x)*sqrt(d+e*x))],
[x^3*(a+b*x^2+c*x^4)/(sqrt(d-e*x)*sqrt(d+e*x)),x,4,1/3*(3*c*d^4+2*b*d^2*e^2+a*e^4)*(d-e*x)^(3/2)*(d+e*x)^(3/2)/e^8-1/5*(3*c*d^2+b*e^2)*(d-e*x)^(5/2)*(d+e*x)^(5/2)/e^8+1/7*c*(d-e*x)^(7/2)*(d+e*x)^(7/2)/e^8-d^2*(c*d^4+b*d^2*e^2+a*e^4)*sqrt(d-e*x)*sqrt(d+e*x)/e^8,-d^2*(c*d^4+b*d^2*e^2+a*e^4)*(d^2-e^2*x^2)/(e^8*sqrt(d-e*x)*sqrt(d+e*x))+1/3*(3*c*d^4+2*b*d^2*e^2+a*e^4)*(d^2-e^2*x^2)^2/(e^8*sqrt(d-e*x)*sqrt(d+e*x))-1/5*(3*c*d^2+b*e^2)*(d^2-e^2*x^2)^3/(e^8*sqrt(d-e*x)*sqrt(d+e*x))+1/7*c*(d^2-e^2*x^2)^4/(e^8*sqrt(d-e*x)*sqrt(d+e*x))],
[x*(a+b*x^2+c*x^4)/(sqrt(d-e*x)*sqrt(d+e*x)),x,4,1/3*(2*c*d^2+b*e^2)*(d-e*x)^(3/2)*(d+e*x)^(3/2)/e^6-1/5*c*(d-e*x)^(5/2)*(d+e*x)^(5/2)/e^6-(c*d^4+b*d^2*e^2+a*e^4)*sqrt(d-e*x)*sqrt(d+e*x)/e^6,-(c*d^4+b*d^2*e^2+a*e^4)*(d^2-e^2*x^2)/(e^6*sqrt(d-e*x)*sqrt(d+e*x))+1/3*(2*c*d^2+b*e^2)*(d^2-e^2*x^2)^2/(e^6*sqrt(d-e*x)*sqrt(d+e*x))-1/5*c*(d^2-e^2*x^2)^3/(e^6*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x*sqrt(d-e*x)*sqrt(d+e*x)),x,6,1/3*c*(d-e*x)^(3/2)*(d+e*x)^(3/2)/e^4-a*arctanh(sqrt(d-e*x)*sqrt(d+e*x)/d)/d-(c*d^2+b*e^2)*sqrt(d-e*x)*sqrt(d+e*x)/e^4,-(c*d^2+b*e^2)*(d^2-e^2*x^2)/(e^4*sqrt(d-e*x)*sqrt(d+e*x))+1/3*c*(d^2-e^2*x^2)^2/(e^4*sqrt(d-e*x)*sqrt(d+e*x))-a*arctanh(sqrt(d^2-e^2*x^2)/d)*sqrt(d^2-e^2*x^2)/(d*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^3*sqrt(d-e*x)*sqrt(d+e*x)),x,6,-1/2*(2*b*d^2+a*e^2)*arctanh(sqrt(d-e*x)*sqrt(d+e*x)/d)/d^3-c*sqrt(d-e*x)*sqrt(d+e*x)/e^2-1/2*a*sqrt(d-e*x)*sqrt(d+e*x)/(d^2*x^2),-c*(d^2-e^2*x^2)/(e^2*sqrt(d-e*x)*sqrt(d+e*x))-1/2*a*(d^2-e^2*x^2)/(d^2*x^2*sqrt(d-e*x)*sqrt(d+e*x))-1/2*(2*b*d^2+a*e^2)*arctanh(sqrt(d^2-e^2*x^2)/d)*sqrt(d^2-e^2*x^2)/(d^3*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^5*sqrt(d-e*x)*sqrt(d+e*x)),x,6,-1/8*(8*c*d^4+4*b*d^2*e^2+3*a*e^4)*arctanh(sqrt(d-e*x)*sqrt(d+e*x)/d)/d^5-1/4*a*sqrt(d-e*x)*sqrt(d+e*x)/(d^2*x^4)-1/8*(4*b*d^2+3*a*e^2)*sqrt(d-e*x)*sqrt(d+e*x)/(d^4*x^2),-1/4*a*(d^2-e^2*x^2)/(d^2*x^4*sqrt(d-e*x)*sqrt(d+e*x))-1/8*(4*b*d^2+3*a*e^2)*(d^2-e^2*x^2)/(d^4*x^2*sqrt(d-e*x)*sqrt(d+e*x))-1/8*(8*c*d^4+4*b*d^2*e^2+3*a*e^4)*arctanh(sqrt(d^2-e^2*x^2)/d)*sqrt(d^2-e^2*x^2)/(d^5*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^7*sqrt(d-e*x)*sqrt(d+e*x)),x,7,-1/6*a*sqrt(d-e*x)*sqrt(d+e*x)/(d^2*x^6)-1/24*(6*b*d^2+5*a*e^2)*sqrt(d-e*x)*sqrt(d+e*x)/(d^4*x^4)-1/16*(8*c*d^4+6*b*d^2*e^2+5*a*e^4)*sqrt(d-e*x)*sqrt(d+e*x)/(d^6*x^2)-1/16*e^2*(8*c*d^4+6*b*d^2*e^2+5*a*e^4)*arctanh(sqrt(d^2-e^2*x^2)/d)*sqrt(d^2-e^2*x^2)/(d^7*sqrt(d-e*x)*sqrt(d+e*x)),-1/6*a*(d^2-e^2*x^2)/(d^2*x^6*sqrt(d-e*x)*sqrt(d+e*x))-1/24*(6*b*d^2+5*a*e^2)*(d^2-e^2*x^2)/(d^4*x^4*sqrt(d-e*x)*sqrt(d+e*x))-1/16*(8*c*d^4+6*b*d^2*e^2+5*a*e^4)*(d^2-e^2*x^2)/(d^6*x^2*sqrt(d-e*x)*sqrt(d+e*x))-1/16*e^2*(8*c*d^4+6*b*d^2*e^2+5*a*e^4)*arctanh(sqrt(d^2-e^2*x^2)/d)*sqrt(d^2-e^2*x^2)/(d^7*sqrt(d-e*x)*sqrt(d+e*x))],
[x^2*(a+b*x^2+c*x^4)/(sqrt(d-e*x)*sqrt(d+e*x)),x,6,1/6*c*x^5*(-d+e*x)*sqrt(d+e*x)/(e^2*sqrt(d-e*x))-1/16*(5*c*d^4+6*b*d^2*e^2+8*a*e^4)*x*sqrt(d-e*x)*sqrt(d+e*x)/e^6-1/24*(5*c*d^2+6*b*e^2)*x^3*sqrt(d-e*x)*sqrt(d+e*x)/e^4+1/16*d^2*(5*c*d^4+6*b*d^2*e^2+8*a*e^4)*arctan(e*x/sqrt(d^2-e^2*x^2))*sqrt(d^2-e^2*x^2)/(e^7*sqrt(d-e*x)*sqrt(d+e*x)),-1/16*(5*c*d^4+6*b*d^2*e^2+8*a*e^4)*x*(d^2-e^2*x^2)/(e^6*sqrt(d-e*x)*sqrt(d+e*x))-1/24*(5*c*d^2+6*b*e^2)*x^3*(d^2-e^2*x^2)/(e^4*sqrt(d-e*x)*sqrt(d+e*x))-1/6*c*x^5*(d^2-e^2*x^2)/(e^2*sqrt(d-e*x)*sqrt(d+e*x))+1/16*d^2*(5*c*d^4+6*b*d^2*e^2+8*a*e^4)*arctan(e*x/sqrt(d^2-e^2*x^2))*sqrt(d^2-e^2*x^2)/(e^7*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(sqrt(d-e*x)*sqrt(d+e*x)),x,5,-1/4*(3*c*d^4+4*b*d^2*e^2+8*a*e^4)*arctan(sqrt(d-e*x)/sqrt(d+e*x))/e^5+1/4*c*x^3*(-d+e*x)*sqrt(d+e*x)/(e^2*sqrt(d-e*x))-1/8*(3*c*d^2+4*b*e^2)*x*sqrt(d-e*x)*sqrt(d+e*x)/e^4,-1/8*(3*c*d^2+4*b*e^2)*x*(d^2-e^2*x^2)/(e^4*sqrt(d-e*x)*sqrt(d+e*x))-1/4*c*x^3*(d^2-e^2*x^2)/(e^2*sqrt(d-e*x)*sqrt(d+e*x))+1/8*(3*c*d^4+4*b*d^2*e^2+8*a*e^4)*arctan(e*x/sqrt(d^2-e^2*x^2))*sqrt(d^2-e^2*x^2)/(e^5*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^2*sqrt(d-e*x)*sqrt(d+e*x)),x,5,-(c*d^2+2*b*e^2)*arctan(sqrt(d-e*x)/sqrt(d+e*x))/e^3+1/2*c*x*(-d+e*x)*sqrt(d+e*x)/(e^2*sqrt(d-e*x))-a*sqrt(d-e*x)*sqrt(d+e*x)/(d^2*x),-a*(d^2-e^2*x^2)/(d^2*x*sqrt(d-e*x)*sqrt(d+e*x))-1/2*c*x*(d^2-e^2*x^2)/(e^2*sqrt(d-e*x)*sqrt(d+e*x))+1/2*(c*d^2+2*b*e^2)*arctan(e*x/sqrt(d^2-e^2*x^2))*sqrt(d^2-e^2*x^2)/(e^3*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^4*sqrt(d-e*x)*sqrt(d+e*x)),x,5,-1/3*a*(d^2-e^2*x^2)/(d^2*x^3*sqrt(d-e*x)*sqrt(d+e*x))-1/3*(3*b*d^2+2*a*e^2)*(d^2-e^2*x^2)/(d^4*x*sqrt(d-e*x)*sqrt(d+e*x))+c*arctan(e*x/sqrt(d^2-e^2*x^2))*sqrt(d^2-e^2*x^2)/(e*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^6*sqrt(d-e*x)*sqrt(d+e*x)),x,4,-1/5*a*(d^2-e^2*x^2)/(d^2*x^5*sqrt(d-e*x)*sqrt(d+e*x))-1/15*(5*b*d^2+4*a*e^2)*(d^2-e^2*x^2)/(d^4*x^3*sqrt(d-e*x)*sqrt(d+e*x))-1/15*(15*c*d^4+10*b*d^2*e^2+8*a*e^4)*(d^2-e^2*x^2)/(d^6*x*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^8*sqrt(d-e*x)*sqrt(d+e*x)),x,5,-1/7*a*(d^2-e^2*x^2)/(d^2*x^7*sqrt(d-e*x)*sqrt(d+e*x))-1/35*(7*b*d^2+6*a*e^2)*(d^2-e^2*x^2)/(d^4*x^5*sqrt(d-e*x)*sqrt(d+e*x))-1/105*(35*c*d^4+28*b*d^2*e^2+24*a*e^4)*(d^2-e^2*x^2)/(d^6*x^3*sqrt(d-e*x)*sqrt(d+e*x))-2/105*e^2*(35*c*d^4+28*b*d^2*e^2+24*a*e^4)*(d^2-e^2*x^2)/(d^8*x*sqrt(d-e*x)*sqrt(d+e*x))],
[(a+b*x^2+c*x^4)/(x^10*sqrt(d-e*x)*sqrt(d+e*x)),x,6,-1/9*a*(d^2-e^2*x^2)/(d^2*x^9*sqrt(d-e*x)*sqrt(d+e*x))-1/63*(9*b*d^2+8*a*e^2)*(d^2-e^2*x^2)/(d^4*x^7*sqrt(d-e*x)*sqrt(d+e*x))-1/105*(21*c*d^4+18*b*d^2*e^2+16*a*e^4)*(d^2-e^2*x^2)/(d^6*x^5*sqrt(d-e*x)*sqrt(d+e*x))-4/315*e^2*(21*c*d^4+18*b*d^2*e^2+16*a*e^4)*(d^2-e^2*x^2)/(d^8*x^3*sqrt(d-e*x)*sqrt(d+e*x))-8/315*e^4*(21*c*d^4+18*b*d^2*e^2+16*a*e^4)*(d^2-e^2*x^2)/(d^10*x*sqrt(d-e*x)*sqrt(d+e*x))]]:
