\[ \int \frac {x^5}{\left (a x^2+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 x^{3}}{b \sqrt {b \,x^{3}+a \,x^{2}}}+\frac {8 \sqrt {b \,x^{3}+a \,x^{2}}}{3 b^{2}}-\frac {16 a \sqrt {b \,x^{3}+a \,x^{2}}}{3 b^{3} x} \]
command
integrate(x^5/(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {16 \, a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, b^{3}} - \frac {2 \, a^{2}}{\sqrt {b x + a} b^{3} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x + a} a b^{6}\right )}}{3 \, b^{9} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{5}}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________