We report the results of a study of the exclusive charmless semileptonic decays, B^0 --> pi^- l^+ nu, B^+ --> pi^0 l^+ nu, B^+ --> omega l^+ nu, B^+ --> eta l^+ nu and B^+ --> eta^ l^+ nu, (l = e or mu) undertaken with approximately 462x10^6 Bbar{B} pairs collected at the Upsilon(4S) resonance with the BABAR detector. The analysis uses events in which the signal B decays are reconstructed with a loose neutrino reconstruction technique. We obtain partial branching fractions in several bins of q^2, the square of the momentum transferred to the lepton-neutrino pair, for B^0 --> pi^- l^+ nu, B^+ --> pi^0 l^+ nu, B^+ --> omega l^+ nu and B^+ --> eta l^+ nu. From these distributions, we extract the form-factor shapes f_+(q^2) and the total branching fractions BF(B^0 --> pi^- l^+ nu) = (1.45 +/- 0.04_{stat} +/- 0.06_{syst})x10^-4 (combined pi^- and pi^0 decay channels assuming isospin symmetry), BF(B^+ --> omega l^+ nu) = (1.19 +/- 0.16_{stat} +/- 0.09_{syst})x10^-4 and BF(B^+ --> eta l^+ nu) = (0.38 +/- 0.05_{stat} +/- 0.05_{syst})x10^-4. We also measure BF(B^+ --> eta^ l^+ nu) = (0.24 +/- 0.08_{stat} +/- 0.03_{syst})x10^-4. We obtain values for the magnitude of the CKM matrix element V_{ub} by direct comparison with three different QCD calculations in restricted q^2 ranges of B --> pi l^+ nu decays. From a simultaneous fit to the experimental data over the full q^2 range and the FNAL/MILC lattice QCD predictions, we obtain |V_{ub}| = (3.25 +/- 0.31)x10^-3, where the error is the combined experimental and theoretical uncertainty.
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