The duration and fidelity of qubit readout are critical factors for applications in quantum-information processing as they limit the fidelity of algorithms which reuse qubits after measurement or apply feedback based on the measurement result. Here we present fast multiplexed readout of five qubits using a single 1.2-GHz-wide readout channel. Using a readout pulse length of 80 ns and populating readout resonators for less than 250 ns, we find an average probability of correct assignment for the five measured qubits to be 97%. The differences between the individual readout errors and those found when measuring the qubits simultaneously are within 1%. We employ individual Purcell filters for each readout resonator to suppress off-resonant driving, which we characterize by measuring the dephasing imposed on unintentionally measured qubits. We expect the readout scheme presented here to become particularly useful for the selective readout of individual qubits in multiqubit quantum processors.