How to integrate $\int\limits_{y = d}^\infty {\frac{{\exp \left( { - a \times y} \right)}}{{b \times y + c}}dy}$?

Question

How to integrate $\int\limits_{y = d}^\infty {\frac{{\exp \left( { - a \times y} \right)}}{{b \times y + c}}dy}$ (with respect to $y$) where $a,b,c,d>0$ or at least rewrite this integral as some form of special function such as bessel or Ei function ?

Answer 1

Firt quickly note that if $a = 0$, we have a simple logartihm, which tends to infinity.

Apart from the trivial case, this problem is very familiar to an exponential integral, so we can try to reshape it to the form where we can use it. Consider $$I(y) = \int \frac {e^{-ay}} {by + c}\,dy$$ Then we can make a variable change $$ay + \frac{ac}{b} = t\implies y = \frac t a - \frac {c}{b} \implies dy = \frac{dt}{a}$$ $$I(t(y)) = \int \frac{e^{-t + \frac{ac}{b}}}{\frac{b}{a}t} dt = \frac{a}{b}e^{\frac{ac}{b}}\int \frac{-e^t}{t} dt$$

So $$I(y=\infty) - I(y=d) = \frac{a}{b}e^{\frac{ac}{b}} Ei(t) = \frac{a}{b}e^{\frac{ac}{b}} Ei(ad + \frac{ac}{b})$$