<s>[INST] Integrate $ \int \sqrt{\frac{x^3-3}{x^{11}}}\ dx$ $$ \int \sqrt{\frac{x^3-3}{x^{11}}}\ dx $$
It seems like substitution could not do it. Is there another way?
 [/INST] Let us write
$$
\sqrt{\frac{x^3-3}{x^{11}}}=\frac{1}{x^4}\sqrt{1-\frac{3}{x^3}}.
$$
Then, with $t=1-3/x^3$, we find that $dt=9/x^4\,dx$, and so
$$
\int \sqrt{\frac{x^3-3}{x^{11}}}\,dx = \frac{1}{9}\int\sqrt{t}\,dt=\frac{1}{9}\frac{2}{3}t^{3/2}+C=\frac{2}{27}\Bigl(1-\frac{3}{x^3}\Bigr)^{3/2}+C.
$$
</s>