\[ \int \frac {x^4}{\sqrt {a+\frac {b}{x^2}}} \, dx \]
Optimal antiderivative \[ \frac {8 b^{2} x \sqrt {a +\frac {b}{x^{2}}}}{15 a^{3}}-\frac {4 b \,x^{3} \sqrt {a +\frac {b}{x^{2}}}}{15 a^{2}}+\frac {x^{5} \sqrt {a +\frac {b}{x^{2}}}}{5 a} \]
command
integrate(x^4/(a+b/x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {8 \, b^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{15 \, a^{3}} + \frac {\sqrt {a x^{2} + b} b^{2}}{a^{3} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (a x^{2} + b\right )}^{\frac {5}{2}} - 10 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b}{15 \, a^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________