Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of $$X^tAX+b^tX \quad ?$$
The distribution without the linear term is solved in the answer here(Transformation of multivariate normal sum of chi-squared) but I have not solved the problem to the more general question above.