Full range of blow up exponents for the quintic wave equation in three dimensions

For the critical focusing wave equation \Box u = u^5 on R^{3+1} in the radial case, we prove the existence of type II blow up solutions with scaling parameter \lambda(t) = t^{-1-\nu} for all \nu >0. This extends the previous work by the authors and Tataru where the condition \nu> 1/2 had been imposed, and gives the optimal range of polynomial blow up rates in light of recent work by Duyckaerts, Kenig and Merle.