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