\[ \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 b x}{3 a^{2} \sqrt {a +\frac {b}{x^{2}}}}+\frac {x^{3}}{3 a \sqrt {a +\frac {b}{x^{2}}}}-\frac {8 b x \sqrt {a +\frac {b}{x^{2}}}}{3 a^{3}} \]
command
integrate(x^2/(a+b/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {8 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a^{3}} - \frac {b^{2}}{\sqrt {a x^{2} + b} a^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{6} - 6 \, \sqrt {a x^{2} + b} a^{6} b}{3 \, a^{9} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________