\[ \int \frac {x^3 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {B \arctanh \left (\frac {x^{2} \sqrt {c}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{c^{\frac {3}{2}}}-\frac {\left (-A c +b B \right ) x^{2}}{b c \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^3*(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
\[ \frac {B \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, c^{\frac {3}{2}}} - \frac {{\left (B b \mathrm {sgn}\left (x\right ) - A c \mathrm {sgn}\left (x\right )\right )} x}{\sqrt {c x^{2} + b} b c} - \frac {B \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________