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In $\mathbb{R}^d$, I would like to estimate (to within some universal constant, if possible)

\begin{equation*} \int_{\mathcal{C}} \|x\|^{-\alpha}_2 dx, \end{equation*}

where $\alpha>0$, and $\mathcal{C} := [x_1+\ell_1]\times\cdots\times [x_d+\ell_d]$ (let's assume that $x_1,\ldots,x_d\geq 1$ and $\ell_1,\ldots,\ell_d>0$). If the integration is much simpler, you can replace the $\|\cdot\|_2$ norm with the $\|\cdot\|_1$ or $\|\cdot\|_\infty$ norms instead (of course, it's a finite-dimensional vector space...)

Since you are doing $d$ integrations, the result will involve exponents of $(d-a)$, but it would be nice to get a fairly compact estimate. For example, you can do the integrals explicitly for the $1$-norm, but you get $2^d$ terms in the final answer, which is not really easy to work with for my purposes.

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