\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {a +\frac {b}{x^{2}}}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {1}{a \sqrt {a +\frac {b}{x^{2}}}} \]
command
integrate(1/(a+b/x^2)^(3/2)/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {3}{2}}} - \frac {x}{\sqrt {a x^{2} + b} a \mathrm {sgn}\left (x\right )} - \frac {\log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________