Sullivan minimal models of classifying spaces for non-formal spaces of small rank

We observe certain rational homotopical conditions of a simply connected two stage CW complex X such that the rational cohomology of the classifying space Baut1X for fibrations with fiber X is (not) free and we illustrate their examples. First, we consider when is Baut1X a rational factor of Baut1(X×Sn) for a non-formal elliptic space X of rank 3 and an odd-integer n. Second, we compute the Sullivan minimal models of Baut1X when X are certain non-formal pure spaces of rank 5.