## Distribution of sum of normal and gamma random variable

Let where and using this parametrization of the Gamma distribution. Also assume and are independent. Then what is the distribution (pdf) of ?

----------

This question doesn't seem as straightforward as it sounds. For example, I have tried using the convolution formulas here, but can't seem to find a closed form expression for the integral. I have also tried multiplying the moment generating functions (mgfs) of and , but it does not seem to match up to any known mgfs.

Does anyone have any ideas on how to find the distribution for ?