\[ \int \frac {x^9}{\sqrt {a x^2+b x^5}} \, dx \]
Optimal antiderivative \[ \frac {16 a^{2} \sqrt {b \,x^{5}+a \,x^{2}}}{45 b^{3} x}-\frac {8 a \,x^{2} \sqrt {b \,x^{5}+a \,x^{2}}}{45 b^{2}}+\frac {2 x^{5} \sqrt {b \,x^{5}+a \,x^{2}}}{15 b} \]
command
integrate(x^9/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {16 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{45 \, b^{3}} + \frac {2 \, \sqrt {b x^{3} + a} a^{2}}{3 \, b^{3} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a\right )}}{45 \, b^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{9}}{\sqrt {b x^{5} + a x^{2}}}\,{d x} \]________________________________________________________________________________________