Why does $$\frac{n!n^x}{(x+1)_n}=\left(\frac{n}{n+1}\right)^x\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x$$ where the subscript n is the rising factorial in the left denominator
my attempt: the index n in the product indicates the indicates the term $\left(1+\frac{x}{j}\right)^{-1}$ may be some series of infinite geometric sums from the rising factorial but how? This is page 2 in Andrews, Askey, Roy Special Functions.
$$\prod_{j=1}^n\left(\frac{j+1}j\right)^x$$ telescopes and equals $(n+1)^x$. Multiplying by $$\left(\frac n{n+1}\right)^x$$ gives $n^x$. $$\prod_{j=1}^n\left(1+\frac xj\right)^{-1}=\frac{n!}{(x+1)(x+2)\cdots(x+n)} =\frac{n!}{(x+1)^{(n)}}$$ etc. (I prefer $x^{(n)}$ and $x_{(n)}$ for the rising and falling factorials resp.)