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