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