I don;t see how to do it. Convexity (or concavity) only over a restricted range is often not possible to deal with in CVX. But I’m not saying it definitely can’t be done. So I’ll leave your question at X/(x+1)^2 convex for x>=2 open, so that someone can weigh in more definitively.
Edit: My suspicion is confirmed by mcg’s answer at the linked question.