Evaluating the integral $\int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx$

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.

flag	asked Jun 2 at 0:34 jack 1

Why? ${}{}{}{}$ – Will Jagy Jun 2 at 2:35

Feeding this integral into Mathematica returns it unevaluated. This means that it is unlikely that this integral can be exactly evaluated even in terms of such a general class of special functions as Meijer G functions. In other words, it is unlikely that the answer is in any kind of recognizable "closed form". In that case, any "simplification" that can be done to this integral depends very strongly on what you intend to do with it. Possibilities: asymptotic expansion in a parameters, satisfying a differential equation in parameters, suitability for numeric integration. Please clarify. – Igor Khavkine Jun 2 at 23:43

Maple 15 can't even do simple cases like $a=0,b=c=1$ and $a=b=1,x=0$. – Brendan McKay Jun 3 at 7:32

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