Closed form of $\sum_{n=1}^\infty (n+k)!(a/n)^n$

Question

I got this equality:

$$\sum_{n=1}^\infty (n+k)!\left(\frac{a}{n}\right)^n=a(k+1)!\int_{0}^{1}\frac{dx}{(1+ax\ln x)^{k+2}}$$

when $|a|<e$

then, does this series have a closed form?