\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^7} \, dx \]
Optimal antiderivative \[ \frac {a^{2}}{3 b^{3} \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}}}-\frac {2 a}{b^{3} \sqrt {a +\frac {b}{x^{2}}}}-\frac {\sqrt {a +\frac {b}{x^{2}}}}{b^{3}} \]
command
integrate(1/(a+b/x^2)^(5/2)/x^7,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x {\left (\frac {5 \, a^{2} x^{2}}{b^{3} \mathrm {sgn}\left (x\right )} + \frac {6 \, a}{b^{2} \mathrm {sgn}\left (x\right )}\right )}}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}}} + \frac {2 \, \sqrt {a}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )} b^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} x^{7}}\,{d x} \]________________________________________________________________________________________