\[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^9} \, dx \]
Optimal antiderivative \[ -\frac {c \left (-a d +2 b c \right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{d^{5}}+\frac {\left (-a d +4 b c \right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5 d^{5}}-\frac {b \left (c +\frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7 d^{5}}+\frac {c^{3} \left (-a d +b c \right )}{d^{5} \sqrt {c +\frac {d}{x^{2}}}}+\frac {c^{2} \left (-3 a d +4 b c \right ) \sqrt {c +\frac {d}{x^{2}}}}{d^{5}} \]
command
integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^9,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (b c^{4} - a c^{3} d\right )} x}{\sqrt {c x^{2} + d} d^{5} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {7}{2}} - 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {5}{2}} d - 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {7}{2}} d + 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {5}{2}} d^{2} + 1015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {7}{2}} d^{2} - 1015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {5}{2}} d^{3} - 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {7}{2}} d^{3} + 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {5}{2}} d^{4} + 1673 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {7}{2}} d^{4} - 1337 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {5}{2}} d^{5} - 616 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {7}{2}} d^{5} + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {5}{2}} d^{6} + 93 \, b c^{\frac {7}{2}} d^{6} - 77 \, a c^{\frac {5}{2}} d^{7}\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{7} d^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {a + \frac {b}{x^{2}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{9}}\,{d x} \]________________________________________________________________________________________