Maximum-sized matroids with no minors isomorphic to U2,5, F7, F-7 or P7

Let M be the class of simple matroids which do not contain the 5-point line U-2,U-5, the Fano plane F-7, the non-Fano plane F-7(-) or the matroid P-7 as minors. Let h(n) be the maximum number of points in a rank-n matroid in M. We show that h(2) = 4, h(3) = 7, and h(n) = ((n+1)(2)) for n greater than or equal to 4, and we also find all the maximum-sized matroids for each rank.