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