The results of an amplitude analysis of the charmless three-body decay B+→π+π+π-, in which CP-violation effects are taken into account, are reported. The analysis is based on a data sample corresponding to an integrated luminosity of 3 fb-1 of pp collisions recorded with the LHCb detector. The most challenging aspect of the analysis is the description of the behavior of the π+π- S-wave contribution, which is achieved by using three complementary approaches based on the isobar model, the K-matrix formalism, and a quasi-model-independent procedure. Additional resonant contributions for all three methods are described using a common isobar model, and include the ρ(770)0, ω(782) and ρ(1450)0 resonances in the π+π- P-wave, the f2(1270) resonance in the π+π- D-wave, and the ρ3(1690)0 resonance in the π+π- F-wave. Significant CP-violation effects are observed in both S- and D-waves, as well as in the interference between the S- and P-waves. The results from all three approaches agree and provide new insight into the dynamics and the origin of CP-violation effects in B+→π+π+π- decays.