Is there a nice closed-form solution to the following integral:
$$\int_0^1[\theta+\mu]^{a-1}[1-(\theta+\mu)]^{b-1}(e^{-\theta})^{S-1}(1-e^{-\theta})^{n-S}d\theta?$$
Here, $\mu,a,b>0$ and $S,n$ are positive integers with $n-S>0$.
Motivation: This comes from a beta-binomial prior model with parameter $e^{-\theta}$, where $\theta$ is assumed to be binomially distributed.
It looks like I can do a binomial expansion on some of the terms inside and then the answer will be a sum over beta functions (or really just one big hypergeometric sum). Is there a nicer expression for this?
Here's what I've tried (call the integral $I$ and for simplicity set $\mu=0$):
\begin{align*} I&=\sum_{i=0}^{n-S}\binom{n-S}{i}(-1)^i\int_0^1e^{-\theta (S+i)}\theta^{a-1}(1-\theta)^{b-1}d\theta\\ &=\sum_{i=0}^{n-S}\binom{n-S}{i}(-1)^i\ {}_1F_1( a;a+b;-\theta(S+i)),\end{align*}
where I expanded $(1-e^{-\theta})^{n-S}$ and used the moment generating function for the beta distribution. Unfortunately this doesn't seem to simplify. I've tried further expanding the confluent hypergeometric function and then swapping sums, but the result looks intractable.
I would actually be quite satisfied with a numerical method for efficiently calculating the integral, for $n$ on the order of 100, with typical values of $S/n\in(0,0.2)$.