Solving integration involving exponentials and powers I am stuck at solving this integral.
$$\int_{x=0}^c x \exp(-x) (1-\exp(-x))^q (\exp(-x)-\exp(-c))^p dx$$
I tried shifting the two terms of $(1-\exp(-x))^q$ and $(\exp(-x)-\exp(-c))^p$ using combinatorials and converted these to $O(pq)$ summation where each integral could be directly solved. However, the sum of these $pq$ elements could not be simplified. A decent formula, maybe with 1 summation maybe helpful. 

First, performing the change of variable
$$\frac{1-e^{-x}}{1-e^{-c}}=t$$
your integral reads
$$\int_{0}^c x \,e^{-x} (1-e^{-x})^q (e^{-x}-e^{-c})^p dx=A^{p+q+1}\int_0^1-\ln(1-t\,A)t^q(1-t)^pdt\,\,\,\,\,\text{,}\,\,\,\,A=1-e^{-c}$$
Now, using the Taylor series of $\displaystyle -\ln(1-t\,A)=\sum_{n=0}^{\infty}\frac{A^{n+1}}{n+1}t^{n+1}$ that converges in $|t|<A^{-1}$ (greater than 1) we obtain 
$$A^{p+q+2}\sum_{n=0}^{\infty}\frac{A^{n}}{n+1}\int_0^1t^{q+n+1}(1-t)^pdt=A^{p+q+2}\sum_{n=0}^{\infty}\frac{A^{n}}{n+1}\frac{\Gamma(p+1)\Gamma(q+n+1)}{\Gamma(p+q+n+2)}$$
Finally, using the series definition of hypergeometric function ${}_3F_2$ we have

$\boxed{(1-e^{-c})^{p+q+2}\frac{\Gamma(p+1)\Gamma(q+2)}{\Gamma(p+q+3)}{}_3F_2\left(\left.\begin{matrix}1,1,q+2\\2,p+q+3\end{matrix}\right|1-e^{-c}\right)}$

