\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 b}{3 a^{2} \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}} x^{3}}-\frac {1}{a \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}} x} \]
command
integrate(1/(a+b/x^2)^(5/2)/x^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, \mathrm {sgn}\left (x\right )}{3 \, a^{2} \sqrt {b}} - \frac {3 \, a x^{2} + 2 \, b}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________