I'm trying to solve the following definite integral:
$\mathcal{I} = \int_0^1dx\ x^{P+k/2-m}(1-x)^me^{-\sqrt{x}}, $
where $P\in\mathcal{N}$ (whole positive numbers and zero), $m\in\mathcal{N}$, $k\in\mathcal{N}$, $0\leq k\leq P$ and $0\leq m\leq P$.
It's similar to the Beta function except for the exponential factor. I've been trying different approaches and looking around with no luck.
Mathematica gives the answer as
$0.443113\ m! \left[2 \Gamma \left(\frac{k}{2}-m+\text{P}+1\right) \, _1\tilde{F}_2\left(\frac{k}{2}-m+\text{P}+1;\frac{1}{2},\frac{k}{2}+\text{P}+2;\frac{1}{4}\right)+\Gamma \left(\frac{1}{2} (k-2 m+3)+\text{P}\right) \, _1\tilde{F}_2\left(\frac{1}{2} (k-2 m+3)+\text{P};\frac{3}{2},\frac{k+5}{2}+\text{P};\frac{1}{4}\right)\right]$
where $\ _p\tilde{F}_q\Big(\{a_1,\ldots,a_p\};\{b_q,\ldots,b_q\};z\Big)\ $ is the regularized generalized hypergeometric function.
I'm hoping that there is a simpler solution that becomes apparent when solving the integral (if so that's probably faster than double checking Mathematica's answer and start going through identities to rewrite it).