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