# A definite integral in terms of Meijer G-function

I am trying to find some relevant functional identities involving Meijer G-functions in order to prove

$$\int_0^\infty\frac{\log(x+1)}{x}\mathrm{e}^{-zx}\,\mathrm{d}x = G^{3,1}_{2,3}\left(z \middle| \begin{array}{c} 0,1 \\ 0,0,0 \\ \end{array} \right), \quad (z>0).$$

This equality came out of Mathematica, in whose syntax the right-hand side reads MeijerG[{{0}, {1}}, {{0, 0, 0}, {}}, z].

In addition, it would be very nice to further express this Meijer G-function in terms of some simpler functions, such as hypergeometric functions (which could then be expressed as an infinite series). What I really need are proof hints rather than Mathematica magic, as I am not too familiar with complex analysis integration techniques.

NB: This question is related to an earlier question of mine: Closed form of $\int_0^\infty \frac{\log(x)-\log(a)}{x-a}e^{-x} \mathrm{d}x$. to which @Jason has given a valid answer