Double Summation over inverse square roots

Question

I am working on an error estimate that requires me to compute the following expression: $$\sum_{i=-n}^{n}\sum_{j=-n}^{n}\frac{1}{\sqrt{\frac{i^4+j^4}{n^4a^4}+ \frac{1}{4}}}$$ where $a\in\mathbb{R}$ and $n\in\mathbb{N}$ are constants. I can surely evaluate this on the computer for given $a,n$, but I wonder if this can be brought into a closed-form expression. I have seen some posts here on the summation of inverse square roots, but the arguments are always easier than in this case.