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Using 535 x 10^6 B-meson pairs collected by the Belle detector at the KEKB e+e- collider, we measure branching fractions of (7.16 +/- 0.10(stat) +/-0.60(syst)) x 10^-4 for B+ -> J/psi K+ pi+ pi- and (4.31 +/- 0.20(stat) +/-0.50(syst)) x 10^-4 for B+ -> psi-prime K+ pi+ pi-. We perform amplitudeanalyses to determine the resonant structure of the K+ pi+ pi- final state inB+ -> J/psi K+ pi+ pi- and B+ -> psi-prime K+ pi+ pi- and find that theK_1(1270) is a prominent component of both decay modes. There is significantinterference among the different intermediate states, which leads, inparticular, to a striking distortion of the rho line shape due to the omega.Based on the results of the fit to the B+ -> J/psi K+ pi+ pi- data, therelative decay fractions of the K_1(1270) to K rho, K omega, and K^*(892) piare consistent with previous measurements, but the decay fraction toK_0^*(1430) is significantly smaller. Finally, by floating the mass and widthof the K_1(1270) in an additional fit of the B+ -> J/psi K+ pi+ pi- data, wemeasure a mass of (1248.1 +/- 3.3(stat) +/- 1.4(syst)) MeV/c^2 and a width of(119.5 +/- 5.2(stat) +/- 6.7(syst)) MeV/c^2 for the K_1(1270).

Study of the K+ pi+ pi- Final State in B+ -> J/psi K+ pi+ pi- and B+ -> psi-prime K+ pi+ pi-