\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{2} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{b^{5}}+\frac {8 a \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}{5 b^{5}}-\frac {2 \left (a +\frac {b}{x}\right )^{\frac {7}{2}}}{7 b^{5}}+\frac {2 a^{4}}{b^{5} \sqrt {a +\frac {b}{x}}}+\frac {8 a^{3} \sqrt {a +\frac {b}{x}}}{b^{5}} \]
command
integrate(1/(a+b/x)^(3/2)/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
\[ \frac {2 \, {\left (\frac {35 \, a^{4}}{\sqrt {\frac {a x + b}{x}}} + 140 \, a^{3} \sqrt {\frac {a x + b}{x}} - \frac {70 \, {\left (a x + b\right )} a^{2} \sqrt {\frac {a x + b}{x}}}{x} + \frac {28 \, {\left (a x + b\right )}^{2} a \sqrt {\frac {a x + b}{x}}}{x^{2}} - \frac {5 \, {\left (a x + b\right )}^{3} \sqrt {\frac {a x + b}{x}}}{x^{3}}\right )}}{35 \, b^{5}} \]