While the integral is doable in the given order of variables (factorize the denominator and expand into partial fractions), swapping the order make things easier.
[tex]\displaystyle \int_0^4 \int_{\sqrt x}^2 \frac{dy\,dx}{4+y^3} = \int_0^2 \int_0^{y^2} \frac{dx\,dy}{4+y^3} \\\\ ~~~~~~~~ = \int_0^2 \frac{y^2}{4+y^3} \, dy \\\\ ~~~~~~~~ = \frac13 \int_4^{12} \frac{du}u \\\\ ~~~~~~~~ = \frac13 (\ln(12) - \ln(4)) = \boxed{\frac{\ln(3)}3}[/tex]
