\[ \int \frac {\left (a+\frac {b}{x^2}\right ) x^2}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-4 a d +3 b c \right ) x}{3 c^{2} \sqrt {c +\frac {d}{x^{2}}}}+\frac {a \,x^{3}}{3 c \sqrt {c +\frac {d}{x^{2}}}}+\frac {2 \left (-4 a d +3 b c \right ) x \sqrt {c +\frac {d}{x^{2}}}}{3 c^{3}} \]
command
integrate((a+b/x^2)*x^2/(c+d/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (3 \, b c d - 4 \, a d^{2}\right )} \mathrm {sgn}\left (x\right )}{3 \, c^{3} \sqrt {d}} + \frac {b c d - a d^{2}}{\sqrt {c x^{2} + d} c^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{6} + 3 \, \sqrt {c x^{2} + d} b c^{7} - 6 \, \sqrt {c x^{2} + d} a c^{6} d}{3 \, c^{9} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________