\[ \int \frac {\left (a+\frac {b}{x^2}\right ) x^4}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 d \left (-6 a d +5 b c \right ) x}{15 c^{3} \sqrt {c +\frac {d}{x^{2}}}}+\frac {\left (-6 a d +5 b c \right ) x^{3}}{15 c^{2} \sqrt {c +\frac {d}{x^{2}}}}+\frac {a \,x^{5}}{5 c \sqrt {c +\frac {d}{x^{2}}}}-\frac {8 d \left (-6 a d +5 b c \right ) x \sqrt {c +\frac {d}{x^{2}}}}{15 c^{4}} \]
command
integrate((a+b/x^2)*x^4/(c+d/x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {8 \, {\left (5 \, b c d^{2} - 6 \, a d^{3}\right )} \mathrm {sgn}\left (x\right )}{15 \, c^{4} \sqrt {d}} - \frac {b c d^{2} - a d^{3}}{\sqrt {c x^{2} + d} c^{4} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a c^{16} + 5 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{17} - 15 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{16} d - 30 \, \sqrt {c x^{2} + d} b c^{17} d + 45 \, \sqrt {c x^{2} + d} a c^{16} d^{2}}{15 \, c^{20} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________