The tagged quasi-free np→npπ+π- reaction has been studied experimentally with the High Acceptance Di-Electron Spectrometer (HADES) at GSI at a deuteron incident beam energy of 1.25 GeV/nucleon (s~2.42 GeV/c for the quasi-free collision). For the first time, differential distributions of solid statistics for π+π- production in np collisions have been collected in the region corresponding to the large transverse momenta of the secondary particles. The invariant mass and angular distributions for the np→npπ+π- reaction are compared with different models. This comparison confirms the dominance of the t-channel with δδ contribution. It also validates the changes previously introduced in the Valencia model to describe two-pion production data in other isospin channels, although some deviations are observed, especially for the π+π- invariant mass spectrum. The extracted total cross section is also in much better agreement with this model. Our new measurement puts useful constraints for the existence of the conjectured dibaryon resonance at mass M~2.38 GeV and with width Γ~70 MeV.