The Expected Mills Ratio where $c$ is Normally Distributed

Question

Suppose that $c$ is normally distributed with mean 0 and variance $\Sigma_c$. Is there a formula for the following: \begin{equation} \int_{-\infty}^{\infty} \frac{n(a+b\cdot c)}{1-N(a+b\cdot c)}\cdot\left(\frac{1}{\sqrt{2\cdot\pi\cdot\Sigma_c}}\cdot e^{-\frac{c^2}{2\cdot\Sigma_c}}\right)dc \end{equation} which, I believe, is effectively the expected value of a Mills Ratio evaluated at a linear transformation of c which is normally distributed: $E[E_c[x|x>c]]$ for x normally distributed with some nonzero mean, $m$, and variance $V$. Any insights on this would be greatly appreciated. Thank you!