# Analytic Integration of product of exponential families

I'm happy to join your community and I hope you can help me solve this seemingly straightforward dilemma I am facing. For my thesis, I am trying to solve analytically a product of two distributions from the exponential families. After a few passages I end up with this (I got rid of all the remaining pieces either by assumption or simplification):

$\int{\exp\left\{\theta\left(h\right)g\left(\beta\right)\right\}\exp\left\{f\left(h\right)\gamma\left(\alpha\right)\right\}}dh$

I am wondering whether is it possible to obtain a measure that can be written using only the normalization factors:

$\tau\left(\alpha\right) = \int{\exp\left\{f\left(h\right)\gamma\left(\alpha\right)\right\}}dh$

and

$\tau\left(\beta\right) = \int{\exp\left\{\theta\left(h\right)g\left(\beta\right)\right\}}dh$