The integral is
$\int_{0}^{1}x^a (1+x)^b e^{-cx^2}{_2F_1}(-\frac{n}{2},n-1,n-0.5,x)dx$,
where $a$, $b$ and $n$ are positive integers, $c$ is a positive real number.
Is there a closed-form result of this integral, or alternatively an upper bound?
I have tried several methods to obtain an upper bound, mainly by separating $e^{-cx^2}$ and the hypergeometric part (since there is no exisiting result of a integral with both, to best of my knowledge). But the monotonicity of the separated parts can not match.
You could consult the following paper where many explicit examples of the style of your integral are treated.
Letac, G. and Piccioni, M. (2012) 'Random continued fractions with beta hypergeometric distribution.' Ann. Probab. 40, number 3, 1105-1134.