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