I am interested in computing $$J(a,b):=\int_{\mathcal{I}(a,b)}\frac{dx \ dy\ dz}{|ax^3+ay^2+bz^3|^{2/3}}, $$ where $a,b$ are natural numbers and $$\mathcal{I}(a,b):=(0,1]^3\cap\{x,y,z \in \mathbb{R}:|ax^3+ay^2+bz^3|\leq b\}. $$
In case a closed formula cannot be found, I would be equally content with an upper bound of the correct order of magnitude as $a,b \to +\infty.$