\[ \int \frac {x^4}{\sqrt {a x^2+b x^3}} \, dx \]
Optimal antiderivative \[ \frac {16 a^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{35 b^{3}}-\frac {32 a^{3} \sqrt {b \,x^{3}+a \,x^{2}}}{35 b^{4} x}-\frac {12 a x \sqrt {b \,x^{3}+a \,x^{2}}}{35 b^{2}}+\frac {2 x^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{7 b} \]
command
integrate(x^4/(b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {32 \, a^{\frac {7}{2}} \mathrm {sgn}\left (x\right )}{35 \, b^{4}} + \frac {2 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )}}{35 \, b^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{4}}{\sqrt {b x^{3} + a x^{2}}}\,{d x} \]________________________________________________________________________________________