Expectation of an exponentiated quadratic form

Question

Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional vector? I know that $\mathbb{E}[X^TAX] = \mathbb{E}[X^T]A\mathbb{E}[X]+tr(A\Sigma)$, but how should I solve this problem?

Answer 1

$$ X\sim N(\mu,\Sigma)\quad\Rightarrow\quad \text{pdf}=f_X(X)=\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(X-\mu)^T\Sigma^{-1}(X-\mu)) $$ Therefore: $$ \begin{split} E[\exp(X^TAX+b^TX)]&=\int_{\mathbb R^n}\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(X-\mu)^T\Sigma^{-1}(X-\mu)+X^TAX+b^TX)dX\\ &=\int_{\mathbb R^n}\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(X-\Delta)^T\Sigma'^{-1}(X-\Delta)+\Lambda)dX \end{split} $$ where $$ \Sigma'=(\Sigma^{-1}-2A)^{-1}, \Delta=\Sigma'(b+\Sigma^{-1}\mu), \Lambda=\frac{1}{2}(\Delta^T\Sigma'^{-1}\Delta-\mu^T\Sigma^{-1}\mu). $$ Therefore: $$ \begin{split} E[\exp(X^TAX+b^TX)]&=\sqrt{\frac{|\Sigma'|}{|\Sigma|}}\exp(\Lambda)\int_{\mathbb R^n}\frac{1}{\sqrt{(2\pi)^n|\Sigma'|}}\exp(-\frac{1}{2}(X-\Delta)^T\Sigma'^{-1}(X-\Delta))dX\\ &=\sqrt{\frac{|\Sigma'|}{|\Sigma|}}\exp(\Lambda) \end{split} $$