exponential integration of mixed how to integrate 
$\frac{dx}{(1-e^{ax})(b+x)}$
where a and b are constants.
this is part of a wider chemistry problem that have never been solved. i solved the case where (b+x) does not appear in the equation but have (me and everyone else in the literature) hard time getting a way to solve the equation set in the question.
any help will be most appreciated 
thank you
 A: The antiderivative is not an elementary function. We could invent a new name for it or a close relative, but it can't be expressed in closed form using the usual algebraic, trigonometric, exponential, and logarithmic functions.
Where does that leave us? We can approximate the integral (over some specific interval, with specific values of the parameters $a$ and $b$) using numerical methods. We can look at how it changes depending on those parameters. We can look at limiting behavior as $x\to 0^+$ and as $x\to\infty$. Everything that matters, we can do - except that we can't write down a nice formula, because there isn't one.
A: Even if $b=0$, I do not think that there is an antiderivative even using special functions.
We could make the problem "simpler" using $x=b t$ to make
$$\int\frac{dx}{(1-e^{ax})(b+x)}=\int\frac{dt}{(1-e^{abx})(1+t)}=\int\frac{dt}{(1-e^{kt})(1+t)}$$
and, as said in answer and comments, we could perform a series expansion around $t=0$ to get
$$\frac{1}{(1-e^{kt})(1+t)}=-\frac{1}{k
   t}+\left(\frac{1}{k}+\frac{1}{2}\right)-\left(\frac{k}{12}+\frac{1}{k}+\frac{1}
   {2}\right) t+\left(\frac{k}{12}+\frac{1}{k}+\frac{1}{2}\right)
   t^2+O\left(t^3\right)$$ and integrate termwise.
