  We study the decays B0 to J/psi pi+pi- and B+ to J/psi pi+pi0, including
intermediate resonances, using a sample of 382 million BBbar pairs recorded by
the BaBar detector at the PEP-II e+e- B factory. We measure the branching
fractions B(B0 ->J/psi rho0) = (2.7 +/- 0.3 +/- 0.17) x 10-5 and B(B+ ->J/psi
rho+) = (5.0 +/- 0.7 +/-0.31) x 10-5.
  We also set the following upper limits at the 90% confidence level: B(B0 ->
J/psi pi+ \pi- non-resonant) < 1.2 x 10-5, B(B0 -> J/psi f_2(1270)) < 4.6 x
10-6, and B(B+ -> J/psi pi+ pi0 non-resonant) < 4.4 x 10-6. We measure the
charge asymmetry in charged B decays to J/psi rho to be -0.11 +/- 0.12 +/-
0.08.
