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