Are there any neat formulas to reduce something like $\sum_{i=0}^{x} i \binom{n}{i} p^i (1-p)^{n-i}$ where $x<n$?
This would be proportional to $\mathbb{E}(X\leq x)$ where $X$~$\text{Bin}(n,p)$. We'd just have to divide by $\Pr(X\leq x)$. I realize there are a lot of asymptotic results, but I'd like to be able to consider small $n$.
I don't know if you'd call it "neat", but it can be expressed using a hypergeometric function:
$$np - \left( x+1 \right) {n\choose x+1}{p}^{x+1} \left( 1-p \right) ^{n- x-1}{\mbox{$_2$F$_1$}\left([1,1-n+x],\;[x+1],\,{\frac {-p}{1-p}}\right)} $$