I have tried evaluating this series

$$\sum_{n=1}^{\infty}\frac{H_{n}^3}{(n+1)2^n} $$

using some methods but it's seems to me that it is very hard. However, I noticed that the series converges faster than the Riemann series.

My question here is:

Is there some mathematical technique for evaluating the above series?

Note1: Here, $H_n$ denotes the harmonic numbers.

Edit : I have a wrong type I meant in the denomenator $2^n$

Thank you for any help.