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