I'm trying to evaluate or simplify this integral:

$$I_{a,b,c} = \int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx $$

with $a,b,c \in \mathbb{R}_+^*$.
and $ Ei(x) =\int_{-\infty}^{x} \frac{\exp(t)}{t} \mathbb{d}t $ : *The Exponential integral function*

Any ideas, hints, directions would be highly appreciated.