\[ \int \frac {x^4}{\sqrt {a x^3+b x^4}} \, dx \]
Optimal antiderivative \[ -\frac {5 a^{3} \arctanh \left (\frac {x^{2} \sqrt {b}}{\sqrt {b \,x^{4}+x^{3} a}}\right )}{8 b^{\frac {7}{2}}}-\frac {5 a \sqrt {b \,x^{4}+x^{3} a}}{12 b^{2}}+\frac {5 a^{2} \sqrt {b \,x^{4}+x^{3} a}}{8 b^{3} x}+\frac {x \sqrt {b \,x^{4}+x^{3} a}}{3 b} \]
command
integrate(x^4/(b*x^4+a*x^3)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{24} \, \sqrt {b x^{2} + a x} {\left (2 \, x {\left (\frac {4 \, x}{b \mathrm {sgn}\left (x\right )} - \frac {5 \, a}{b^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {15 \, a^{2}}{b^{3} \mathrm {sgn}\left (x\right )}\right )} - \frac {5 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, b^{\frac {7}{2}}} + \frac {5 \, a^{3} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right )}{16 \, b^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{4}}{\sqrt {b x^{4} + a x^{3}}}\,{d x} \]________________________________________________________________________________________