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