calculus - Compute $sum_{n=1}^inftyfrac{H_n^2H_n^{(2)}}{n^3}$

How to prove

$$\sum_{n=1}^\infty\frac{H_n^2H_n^{(2)}}{n^3}=\frac{19}{2}\zeta(3)\zeta(4)-2\zeta(2)\zeta(5)-7\zeta(7)\ ?$$
where $H_n^{(p)}=1+\frac1{2^p}+\cdots+\frac1{n^p}$ is the $n$th generalized harmonic number of order $p$.

This series is very advanced and can be found evaluated in the book (Almost) Impossible Integrals, Sums and Series page 300 using only series manipulations, but luckily I was able to evaluate it using only integration, some harmonic identities and results of easy Euler sums.

Can we prove the equality above in different methods besides series manipulation and the idea of my solution below? All approaches are highly appreciated.

Solution is posted in the answer section.

Thanks