\[ \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^7} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +3 b c \right ) \left (c +\frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d^{4}}-\frac {b \left (c +\frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5 d^{4}}-\frac {c^{2} \left (-a d +b c \right )}{d^{4} \sqrt {c +\frac {d}{x^{2}}}}-\frac {c \left (-2 a d +3 b c \right ) \sqrt {c +\frac {d}{x^{2}}}}{d^{4}} \]
command
integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^7,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (b c^{3} - a c^{2} d\right )} x}{\sqrt {c x^{2} + d} d^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {5}{2}} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {3}{2}} d - 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {5}{2}} d + 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {3}{2}} d^{2} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {5}{2}} d^{2} - 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {3}{2}} d^{3} - 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {5}{2}} d^{3} + 110 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {3}{2}} d^{4} + 33 \, b c^{\frac {5}{2}} d^{4} - 25 \, a c^{\frac {3}{2}} d^{5}\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{5} d^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {a + \frac {b}{x^{2}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{7}}\,{d x} \]________________________________________________________________________________________