I was wondering if there is a generalization of the integral discussed here to a case like,

\begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm \vec{k}_2\vert ^{\nu_3}} \end{equation}

Does some similar Fourier convolution argument again go through?

Is there a generalization to arbitrary number of factors in the denominator?

(..I am using the symbol $\pm$ hoping that like there here too all the $4$ sign combinations probably give the same result..)