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