\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx \]
Optimal antiderivative \[ \frac {2 a^{4}}{3 b^{5} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}+\frac {8 a \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{3 b^{5}}-\frac {2 \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}{5 b^{5}}-\frac {8 a^{3}}{b^{5} \sqrt {a +\frac {b}{x}}}-\frac {12 a^{2} \sqrt {a +\frac {b}{x}}}{b^{5}} \]
command
integrate(1/(a+b/x)^(5/2)/x^6,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
\[ -\frac {2 \, {\left (90 \, a^{2} \sqrt {\frac {a x + b}{x}} - \frac {20 \, {\left (a x + b\right )} a \sqrt {\frac {a x + b}{x}}}{x} - \frac {5 \, {\left (a^{4} - \frac {12 \, {\left (a x + b\right )} a^{3}}{x}\right )} x}{{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} + \frac {3 \, {\left (a x + b\right )}^{2} \sqrt {\frac {a x + b}{x}}}{x^{2}}\right )}}{15 \, b^{5}} \]