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