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