I want to evaluate or approximate a lower bound for the following integral, \begin{eqnarray} \int \exp\left[-q(\mathbf{x})-\sum_i \frac{(u_i- \mathbf{v}_i^\mathsf{T}\mathbf{x})^2}{1+c_i \Vert x\Vert^2}\right] d^n\mathbf{x}, \end{eqnarray} where $x\in \mathbb R^n$; $c_i>0$ and $\mathbf{v}_i\in \mathbb R^n$ are constants; $q(\mathbf{x}) = t\Vert x \Vert^2$, $t\in\mathbb R$; and $u_i\in \mathbb R$ are variables which I want to find the integral in terms of.
If there wasn't $\Vert x\Vert^2$ at the denominator of the term inside summation, we can easily solve this integral.
Is there any way to find a the answer to this integral or find a function which lower bound this function ? How do you face such a problem if it show up in your research? Is there any other choice of $q(\mathbf{x})$ which makes the problem simple?
Is it a good idea to replace the term $(u_i- \mathbf{v}_i^\mathsf{T}\mathbf{x})^2/(1+c_i \Vert x\Vert^2)$ with \begin{eqnarray} \frac{(u_i- \mathbf{v}_i^\mathsf{T}\mathbf{x})^2}{1+c_i \Vert x\Vert^2} = (u_i- \mathbf{v}_i^\mathsf{T}\mathbf{x})^2 \left[1-c_i\Vert x\Vert^2+(c_i\Vert x\Vert^2)^2-(c_i\Vert x\Vert^2)^3 +o(\epsilon)\right] \end{eqnarray} and if yes, how to continue?