I want to compute the following integral:
$$ \int_{\mathbb{R}^d} \Vert x+b\Vert^3 e^{-a\Vert x\Vert^2}dx$$
Where $\Vert x\Vert$ is the norm of $x$? If $b=0$ then from a paper, I have this $$ \int_{\mathbb{R}^d} \Vert x\Vert^3 e^{-a\Vert x\Vert^2}dx=a^{-\frac{M+3}{2}}\pi^{\frac{m}{2}}\frac{\Gamma(\frac{m+3}{2})}{\Gamma(\frac{m}{2})}$$
How to compute it? If integration is not possible, what is a tight upper bound for it?