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