\[ \int \frac {x^3}{\sqrt {a x^2+b x^3}} \, dx \]
Optimal antiderivative \[ -\frac {8 a \sqrt {b \,x^{3}+a \,x^{2}}}{15 b^{2}}+\frac {16 a^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{15 b^{3} x}+\frac {2 x \sqrt {b \,x^{3}+a \,x^{2}}}{5 b} \]
command
integrate(x^3/(b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {16 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{15 \, b^{3}} + \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )}}{15 \, b^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{3}}{\sqrt {b x^{3} + a x^{2}}}\,{d x} \]________________________________________________________________________________________