\[ \int \frac {\left (a+\frac {b}{x^2}\right ) x^4}{\sqrt {c+\frac {d}{x^2}}} \, dx \]
Optimal antiderivative \[ -\frac {2 d \left (-4 a d +5 b c \right ) x \sqrt {c +\frac {d}{x^{2}}}}{15 c^{3}}+\frac {\left (-4 a d +5 b c \right ) x^{3} \sqrt {c +\frac {d}{x^{2}}}}{15 c^{2}}+\frac {a \,x^{5} \sqrt {c +\frac {d}{x^{2}}}}{5 c} \]
command
integrate((a+b/x^2)*x^4/(c+d/x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, b c d^{\frac {3}{2}} - 4 \, a d^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{15 \, c^{3}} - \frac {{\left (b c d - a d^{2}\right )} \sqrt {c x^{2} + d}}{c^{3} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a + 5 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c - 10 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a d}{15 \, c^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________