\[ \int \frac {x^6}{\left (a x^2+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 x^{4}}{b \sqrt {b \,x^{3}+a \,x^{2}}}-\frac {16 a \sqrt {b \,x^{3}+a \,x^{2}}}{5 b^{3}}+\frac {32 a^{2} \sqrt {b \,x^{3}+a \,x^{2}}}{5 b^{4} x}+\frac {12 x \sqrt {b \,x^{3}+a \,x^{2}}}{5 b^{2}} \]
command
integrate(x^6/(b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {32 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{5 \, b^{4}} + \frac {2 \, a^{3}}{\sqrt {b x + a} b^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {5}{2}} b^{16} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{16} + 15 \, \sqrt {b x + a} a^{2} b^{16}\right )}}{5 \, b^{20} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{6}}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________