\[ \int \frac {x^3}{\sqrt {a x^3+b x^4}} \, dx \]
Optimal antiderivative \[ \frac {3 a^{2} \arctanh \left (\frac {x^{2} \sqrt {b}}{\sqrt {b \,x^{4}+x^{3} a}}\right )}{4 b^{\frac {5}{2}}}+\frac {\sqrt {b \,x^{4}+x^{3} a}}{2 b}-\frac {3 a \sqrt {b \,x^{4}+x^{3} a}}{4 b^{2} x} \]
command
integrate(x^3/(b*x^4+a*x^3)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (\frac {2 \, x}{b \mathrm {sgn}\left (x\right )} - \frac {3 \, a}{b^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {3 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, b^{\frac {5}{2}}} - \frac {3 \, a^{2} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{8 \, b^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{3}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]________________________________________________________________________________________