\[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^6} \, dx \]
Optimal antiderivative \[ -\frac {3 c \left (-4 a d +5 b c \right ) \arctanh \left (\frac {\sqrt {d}}{x \sqrt {c +\frac {d}{x^{2}}}}\right )}{8 d^{\frac {7}{2}}}-\frac {b}{4 d \,x^{5} \sqrt {c +\frac {d}{x^{2}}}}+\frac {4 a d -5 b c}{4 d^{2} x^{3} \sqrt {c +\frac {d}{x^{2}}}}+\frac {3 \left (-4 a d +5 b c \right ) \sqrt {c +\frac {d}{x^{2}}}}{8 d^{3} x} \]
command
integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^6,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (5 \, b c^{2} - 4 \, a c d\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{8 \, \sqrt {-d} d^{3} \mathrm {sgn}\left (x\right )} + \frac {b c^{2} - a c d}{\sqrt {c x^{2} + d} d^{3} \mathrm {sgn}\left (x\right )} + \frac {7 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{2} - 4 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c d - 9 \, \sqrt {c x^{2} + d} b c^{2} d + 4 \, \sqrt {c x^{2} + d} a c d^{2}}{8 \, c^{2} d^{3} x^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________