\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^6} \, dx \]
Optimal antiderivative \[ \frac {1}{3 b \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}} x^{3}}-\frac {\arctanh \left (\frac {\sqrt {b}}{x \sqrt {a +\frac {b}{x^{2}}}}\right )}{b^{\frac {5}{2}}}+\frac {1}{b^{2} x \sqrt {a +\frac {b}{x^{2}}}} \]
command
integrate(1/(a+b/x^2)^(5/2)/x^6,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b} b^{\frac {5}{2}}} + \frac {\arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {3 \, a x^{2} + 4 \, b}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________