\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^8} \, dx \]
Optimal antiderivative \[ \frac {1}{3 b \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}} x^{5}}+\frac {5 a \arctanh \left (\frac {\sqrt {b}}{x \sqrt {a +\frac {b}{x^{2}}}}\right )}{2 b^{\frac {7}{2}}}+\frac {5}{3 b^{2} x^{3} \sqrt {a +\frac {b}{x^{2}}}}-\frac {5 \sqrt {a +\frac {b}{x^{2}}}}{2 b^{3} x} \]
command
integrate(1/(a+b/x^2)^(5/2)/x^8,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {5 \, a \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b} b^{3} \mathrm {sgn}\left (x\right )} - \frac {6 \, {\left (a x^{2} + b\right )} a + a b}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right )} - \frac {\sqrt {a x^{2} + b}}{2 \, b^{3} x^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________