Semiconductor quantum boxes (QB's) are well suited to cavity quantum electrodynamic experiments in the solid state because of their sharp emission. We study by time-resolved photoluminescence InAs QB's placed in the core of small-volume and high-finesse GaAs/AlAs pillar microresonators. A spontaneous emission rate enhancement by a factor of up to 5 is selectively observed for the QB's which are on resonance with one-cavity mode. We explain its magnitude by considering the Purcell figure of merit of the micropillars and the effect of the random spatial and spectral distributions of the QB's.