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