\[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-3 A c +2 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c^{2} x}+\frac {B x \sqrt {c \,x^{4}+b \,x^{2}}}{3 c} \]
command
integrate(x^2*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (2 \, B b^{\frac {3}{2}} - 3 \, A \sqrt {b} c\right )} \mathrm {sgn}\left (x\right )}{3 \, c^{2}} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} B}{3 \, c^{2} \mathrm {sgn}\left (x\right )} - \frac {\sqrt {c x^{2} + b} {\left (B b - A c\right )}}{c^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{2} + A\right )} x^{2}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]________________________________________________________________________________________