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?

$x^3-3=y^2 \Rightarrow x=(y^2+3)^{\frac 1 3} \Rightarrow dx= \dfrac {2y} 3 \cdot  {(y^2+3)^{\frac {-2} 3}}\cdot dy$
$$\int \sqrt{\dfrac {x^3-3} {x^{11}}}dx=\dfrac 2 3 \int\dfrac {y^2} {{(y^2+3)}^{\frac 5 2}}dy=\dfrac 23 \{\int \dfrac 1 {(y^2+3)^{\frac 3 2}}dy-3\int \dfrac 1 {(y^2+3)^{\frac 5 2}}dy \}= $$
$$=\dfrac 2 3 \{\dfrac {y} {3(y^2+3)^{\frac 1 2}}-\dfrac {y(2y^2+9)} {9(y^2+3)^{\frac 3 2}} \}+c=\dfrac {2y^3} {27(y^2+3)^{\frac 3 2}}+c=$$
$$=\dfrac {2(x^3-3)^{\frac 3 2}} {27x^{\frac 9 2}}+c$$
