 Solution

Differentiating $X(z)X(w)\sim -({\alpha }^{\prime }/2)\log (z-w)$ gives the contractions needed for Wick theorem. The free-boson part of $T$ supplies $1/[2(z-w{)}^4]$, corresponding to central charge one. The cross-contractions between $:\partial X\partial X:$ and ${\partial }^{2}X$ produce the required lower poles but no fourth-order scalar term. Finally,so the product of the two improvement terms contributes $3{\alpha }^{\prime }{q}^2/(z-w{)}^4$. Matching this with $c/[2(z-w{)}^4]$ in the stress-tensor operator-product expansion gives the linear dilaton conformal field theory