\[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^5} \, dx \]
Optimal antiderivative \[ -\frac {b \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d^{3}}+\frac {c \left (-a d +b c \right )}{d^{3} \sqrt {c +\frac {d}{x^{2}}}}+\frac {\left (-a d +2 b c \right ) \sqrt {c +\frac {d}{x^{2}}}}{d^{3}} \]
command
integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (b c^{2} - a c d\right )} x}{\sqrt {c x^{2} + d} d^{3} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a \sqrt {c} d - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {3}{2}} d + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a \sqrt {c} d^{2} + 5 \, b c^{\frac {3}{2}} d^{2} - 3 \, a \sqrt {c} d^{3}\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{3} d^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {a + \frac {b}{x^{2}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{5}}\,{d x} \]________________________________________________________________________________________