\[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {15 b^{2} \arctanh \left (\frac {\sqrt {a +\frac {b}{x^{2}}}}{\sqrt {a}}\right )}{8 a^{\frac {7}{2}}}-\frac {15 b^{2}}{8 a^{3} \sqrt {a +\frac {b}{x^{2}}}}-\frac {5 b \,x^{2}}{8 a^{2} \sqrt {a +\frac {b}{x^{2}}}}+\frac {x^{4}}{4 a \sqrt {a +\frac {b}{x^{2}}}} \]
command
integrate(x^3/(a+b/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (x^{2} {\left (\frac {2 \, x^{2}}{a \mathrm {sgn}\left (x\right )} - \frac {5 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} - \frac {15 \, b^{2}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} x}{8 \, \sqrt {a x^{2} + b}} + \frac {15 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {7}{2}}} - \frac {15 \, b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{8 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________