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