\[ \int \frac {\left (a+\frac {b}{x^2}\right ) x^2}{\sqrt {c+\frac {d}{x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (-2 a d +3 b c \right ) x \sqrt {c +\frac {d}{x^{2}}}}{3 c^{2}}+\frac {a \,x^{3} \sqrt {c +\frac {d}{x^{2}}}}{3 c} \]
command
integrate((a+b/x^2)*x^2/(c+d/x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (3 \, b c \sqrt {d} - 2 \, a d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{3 \, c^{2}} + \frac {{\left (c x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, c^{2} \mathrm {sgn}\left (x\right )} + \frac {\sqrt {c x^{2} + d} {\left (b c - a d\right )}}{c^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________