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