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Series with power of generalized harmonic number $\displaystyle\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k$
It's possible to generalize these series?
$$\sum_{k=1}^{\infty}H_k^{(s)}x^k=\frac{\operatorname{Li}_s(x)}{1-x}$$
$$\sum_{k=1}^{\infty}H_k^2 x^k=\frac{\ln(1-x)^2+\operatorname{Li}_2(x)}{1-x}$$
Where:
$$...
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Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $
Prove the following
$$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$
where
$$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$
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