\[ \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx \]
Optimal antiderivative \[ -\frac {c \left (-4 a d +3 b c \right ) \arctanh \left (\frac {\sqrt {d}}{x \sqrt {c +\frac {d}{x^{2}}}}\right )}{8 d^{\frac {5}{2}}}-\frac {b \sqrt {c +\frac {d}{x^{2}}}}{4 d \,x^{3}}+\frac {\left (-4 a d +3 b c \right ) \sqrt {c +\frac {d}{x^{2}}}}{8 d^{2} x} \]
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
\[ \frac {\frac {{\left (3 \, b c^{3} - 4 \, a c^{2} d\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d} d^{2}} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{3} - 4 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{2} d - 5 \, \sqrt {c x^{2} + d} b c^{3} d + 4 \, \sqrt {c x^{2} + d} a c^{2} d^{2}}{c^{2} d^{2} x^{4}}}{8 \, c \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________