\[ \int \frac {x^4}{\left (a+\frac {b}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {8 b^{2} x}{5 a^{3} \sqrt {a +\frac {b}{x^{2}}}}-\frac {2 b \,x^{3}}{5 a^{2} \sqrt {a +\frac {b}{x^{2}}}}+\frac {x^{5}}{5 a \sqrt {a +\frac {b}{x^{2}}}}+\frac {16 b^{2} x \sqrt {a +\frac {b}{x^{2}}}}{5 a^{4}} \]
command
integrate(x^4/(a+b/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {16 \, b^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{5 \, a^{4}} + \frac {b^{3}}{\sqrt {a x^{2} + b} a^{4} \mathrm {sgn}\left (x\right )} + \frac {{\left (a x^{2} + b\right )}^{\frac {5}{2}} a^{16} - 5 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{16} b + 15 \, \sqrt {a x^{2} + b} a^{16} b^{2}}{5 \, a^{20} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________