I need to compute the integral \begin{equation} I=\int_{a_1}^{b_1} \int_{a_2}^{b_2} \exp\left(c_1x_1+c_2x_2+d_{11}x_1^2+d_{22}x_2^2+d_{12}x_1x_2\right)dx_2 dx_1 \end{equation} for $x_{i} \in \mathbb{R}$, but $c_{i}, d_{ij} \in \mathbb{C}$. This can be written in more compact form as \begin{equation} I=\int_{a_1}^{b_1} \int_{a_2}^{b_2} \exp(c^T x + x^TDx)dx_2 dx_1 \end{equation} for real $x \in \mathbb{R}^{2}$, but complex $c \in \mathbb{C}^{2}, D \in \mathbb{C}^{2 \times 2}$.
The usual apprach is to assume that $D$ can be decomposed. Can this integral be evaluated explicitly without this assumption?
Thanks in advance for your help!
Chris T.