\[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 b x}{3 a^{2} \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}}}+\frac {x^{3}}{3 a \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}}}+\frac {8 b x}{3 a^{3} \sqrt {a +\frac {b}{x^{2}}}}-\frac {16 b x \sqrt {a +\frac {b}{x^{2}}}}{3 a^{4}} \]
command
integrate(x^2/(a+b/x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {16 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a^{4}} - \frac {9 \, {\left (a x^{2} + b\right )} b^{2} - b^{3}}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{4} \mathrm {sgn}\left (x\right )} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{8} - 9 \, \sqrt {a x^{2} + b} a^{8} b}{3 \, a^{12} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________