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