\[ \int \frac {\left (a+\frac {b}{x^2}\right ) x^3}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 d \left (-5 a d +4 b c \right ) \arctanh \left (\frac {\sqrt {c +\frac {d}{x^{2}}}}{\sqrt {c}}\right )}{8 c^{\frac {7}{2}}}+\frac {3 d \left (-5 a d +4 b c \right )}{8 c^{3} \sqrt {c +\frac {d}{x^{2}}}}+\frac {\left (-5 a d +4 b c \right ) x^{2}}{8 c^{2} \sqrt {c +\frac {d}{x^{2}}}}+\frac {a \,x^{4}}{4 c \sqrt {c +\frac {d}{x^{2}}}} \]
command
integrate((a+b/x^2)*x^3/(c+d/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (x^{2} {\left (\frac {2 \, a x^{2}}{c \mathrm {sgn}\left (x\right )} + \frac {4 \, b c^{4} \mathrm {sgn}\left (x\right ) - 5 \, a c^{3} d \mathrm {sgn}\left (x\right )}{c^{5}}\right )} + \frac {3 \, {\left (4 \, b c^{3} d \mathrm {sgn}\left (x\right ) - 5 \, a c^{2} d^{2} \mathrm {sgn}\left (x\right )\right )}}{c^{5}}\right )} x}{8 \, \sqrt {c x^{2} + d}} - \frac {3 \, {\left (4 \, b c d \log \left ({\left | d \right |}\right ) - 5 \, a d^{2} \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{16 \, c^{\frac {7}{2}}} + \frac {3 \, {\left (4 \, b c d - 5 \, a d^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{8 \, c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________