\[ \int \frac {\left (a+\frac {b}{x^2}\right ) x}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 a d +2 b c \right ) \arctanh \left (\frac {\sqrt {c +\frac {d}{x^{2}}}}{\sqrt {c}}\right )}{2 c^{\frac {5}{2}}}+\frac {3 a d -2 b c}{2 c^{2} \sqrt {c +\frac {d}{x^{2}}}}+\frac {a \,x^{2}}{2 c \sqrt {c +\frac {d}{x^{2}}}} \]
command
integrate((a+b/x^2)*x/(c+d/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {x {\left (\frac {a x^{2}}{c \mathrm {sgn}\left (x\right )} - \frac {2 \, b c^{2} \mathrm {sgn}\left (x\right ) - 3 \, a c d \mathrm {sgn}\left (x\right )}{c^{3}}\right )}}{2 \, \sqrt {c x^{2} + d}} + \frac {{\left (2 \, b c \log \left ({\left | d \right |}\right ) - 3 \, a d \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{4 \, c^{\frac {5}{2}}} - \frac {{\left (2 \, b c - 3 \, a d\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{2 \, c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________