\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx \]
Optimal antiderivative \[ \frac {20 a^{3} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{3 b^{6}}-\frac {4 a^{2} \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}{b^{6}}+\frac {10 a \left (a +\frac {b}{x}\right )^{\frac {7}{2}}}{7 b^{6}}-\frac {2 \left (a +\frac {b}{x}\right )^{\frac {9}{2}}}{9 b^{6}}-\frac {2 a^{5}}{b^{6} \sqrt {a +\frac {b}{x}}}-\frac {10 a^{4} \sqrt {a +\frac {b}{x}}}{b^{6}} \]
command
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {2 \, {\left (\frac {63 \, a^{5}}{\sqrt {\frac {a x + b}{x}}} + 315 \, a^{4} \sqrt {\frac {a x + b}{x}} - \frac {210 \, {\left (a x + b\right )} a^{3} \sqrt {\frac {a x + b}{x}}}{x} + \frac {126 \, {\left (a x + b\right )}^{2} a^{2} \sqrt {\frac {a x + b}{x}}}{x^{2}} - \frac {45 \, {\left (a x + b\right )}^{3} a \sqrt {\frac {a x + b}{x}}}{x^{3}} + \frac {7 \, {\left (a x + b\right )}^{4} \sqrt {\frac {a x + b}{x}}}{x^{4}}\right )}}{63 \, b^{6}} \]