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