\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^9} \, dx \]
Optimal antiderivative \[ \frac {a \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}}}{b^{4}}-\frac {\left (a +\frac {b}{x^{2}}\right )^{\frac {5}{2}}}{5 b^{4}}-\frac {a^{3}}{b^{4} \sqrt {a +\frac {b}{x^{2}}}}-\frac {3 a^{2} \sqrt {a +\frac {b}{x^{2}}}}{b^{4}} \]
command
integrate(1/(a+b/x^2)^(3/2)/x^9,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {a^{3} x}{\sqrt {a x^{2} + b} b^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{8} a^{\frac {5}{2}} - 30 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{6} a^{\frac {5}{2}} b + 80 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{4} a^{\frac {5}{2}} b^{2} - 50 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} a^{\frac {5}{2}} b^{3} + 11 \, a^{\frac {5}{2}} b^{4}\right )}}{5 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )}^{5} b^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{9}}\,{d x} \]________________________________________________________________________________________