log(1+x^1.5) is convex for x <= (1/2)^(2/3), and concave for x >= (1/2)^(2/3)
Even if your problem were constrained to be on only one of these sides, log(1+x^1.5) is not concave or convex over the entirety of its natural domain (x >= 0), so it’s unlikely there would be a CVX representation valid on even that side.
You’re not the first one on this forum to make an invalid claim of convexity (or concavity). Unfortunately, you are not likely to be the last.