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