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