This does appear to be concave for x > 0, although I haven’t proven it. It is not concave for all x < 0 (for instance, its second derivative at x = -1 is 0.69717488…).
You will have to wait to see if someone can figure out a CVX formulation for this. Perhaps you can be inspired by section 5.2.9 Lambert W-function of Mosek Modeling Cookbook and How to handle log(x*exp(x))? .
Ok, thank you for your reply.