The branching fraction and direct $C!P$ asymmetry of the decay $B^{+}rightarrow J/psi rho^{+}$ are measured using proton-proton collision data collected with the LHCb detector at centre-of-mass energies of 7 and 8 TeV, corresponding to a total integrated luminosity of 3$mbox{fb}^{-1}$. The following results are obtained: begin{align} mathcal{B}(B^{+}rightarrow J/psi rho^{+}) &= (3.81 ^{+0.25}_{-0.24} pm 0.35) times 10^{-5}, onumber mathcal{A}^{C!P} (B^{+}rightarrow J/psi rho^{+}) &= -0.045^{+0.056}_{-0.057} pm 0.008, onumber end{align} where the first uncertainties are statistical and the second systematic. Both measurements are the most precise to date.