Is there a closed form expression for $$ \sum_{k=n}^\infty\frac{k!^2}{(k+x)(k-n)!(k+n+1)!} $$ where $0<x<1$ ?

(For $n=0$, I know that $$\sum_{k=0}^\infty\frac{1}{(k+x)(k+1)}=\frac{\psi(x)+\gamma}{x-1}$$ where $\psi(x)$ is the digamma function and $\gamma$ is the Euler-Mascheroni constant).