How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$? As the title says I am trying to compute the integral  $I=\displaystyle\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}$ where $a>0$ and $b$ is a complex number. For the special case of $b=-2a$, we have $I=-\displaystyle\frac{1}{2}\exp(a)\:\text{Ei}(-a)$ (where $\text{Ei}$ is the Exponential Integral function). However, for the general complex values of $b$ I am not sure how to compute this integral in terms of some known function.

Using
$${{\rm e}^{bx}}=\sum _{n=0}^{\infty }{\frac { \left( bx \right) ^{n}}{n
!}}
$$
and Maple I am obtaining the following result is terms of  Meijer G function
$$\int _{0}^{\infty }\!{\frac {{{\rm e}^{-a{x}^{2}+bx}}}{x+1}}{dx}=\sum 
_{n=0}^{\infty }{\frac {{b}^{n}
G^{3, 2}_{2, 3}\left(a\, \Big\vert\,^{1/2, 1}_{1, 1/2, n/2+1/2}\right)
}{2n!\,\pi \,{a}^{n/2+1/2}}}
$$
