Flavour-changing neutral current (FCNC) transitions such as b -> s mu+ mu- are forbidden at tree level in the Standard Model and proceed only through loop diagrams. Their consequently low rates make them sensitive probes of physics beyond the Standard Model at mass scales well above those directly accessible at colliders. This thesis presents a measurement of the differential branching fraction of the FCNC decay B+ -> K+ pi+ pi- mu+ mu-, in bins of the invariant mass squared of the dimuon system, q^2. The measurement uses proton-proton collision data recorded by the LHCb experiment during Run 1 and Run 2 of the LHC, corresponding to an integrated luminosity of 9 fb^-1. Signal candidates are isolated using a combination of cut-based requirements, dedicated vetoes against misidentified backgrounds, and boosted decision tree classifiers. Efficiencies are determined from simulation, corrected using data-driven techniques derived from the B+ -> K+ pi+ pi- J/psi(-> mu+ mu-) control mode, and modelled as a function of the decay phase space using a machine-learning approach that accommodates the high dimensionality of the five-body final state. Signal and normalisation yields are extracted from fits to the B+ invariant-mass distribution, and the branching fraction is measured relative to that of the B+ -> psi(2S)(-> mu+ mu-) K+ decay. The results present the most precise measurement of this decay channel to date, and are reported in intervals of q^2 between 0.1 and 19.0 GeV^2/c^4, including two alternative definitions of the region close to the kinematic limit. In this high-q^2 region, the signal significances are measured to be 2.8 sigma in q^2 in [15, 19] GeV^2/c^4, and 5.1 sigma in the wider q^2 in [14.3, 19] GeV^2/c^4 bin. The branching fraction results are found to agree with the previous LHCb measurement of this decay. As a cross-check, the branching fraction of the resonant B+ -> K+ pi+ pi- J/psi decay is also measured, improving substantially on the precision of the current world average. For the rare B+ -> K+ pi+ pi- mu+ mu- decay, the experimental precision remains limited by the size of the dataset, which motivates a future measurement with the recently completed LHCb Run 3 dataset; preliminary work in this direction is also discussed at the end of this thesis and shows promising results.