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