Let $n$ be a positive integer and $a$ a positive real number. Is the below number irrational?
$$ N=\displaystyle\int_{-a}^{a}{x^n}{n^x}d x. $$
The closed form of this integral is related to the $\Gamma$ function and $\log $ function as shown below in note, which assures me that is always irrational , the weaker luck is to get it rational. Then my question is:
When is $N=\displaystyle\int_{-a}^{a}{x^n}{n^x}d x\,$ a rational number ?
note: $N=\displaystyle\int_{-a}^{a}{x^n}{n^x}d x\,=\displaystyle[_{-a}^{a}\frac{x^n(-x\log n))^{-n}\Gamma(n+1,-x\log n))}{\log n}$