The Purcell effect is commonly used to increase the spontaneous emission rate by modifying the local environment of a light emitter. Here, we propose a silicon dielectric cuboid nanoantenna for simultaneously enhancing electric dipole (ED), magnetic dipole (MD) and electric quadrupole (EQ) emission. We study the scattering cross section, polarization charge distribution, and electromagnetic field distribution for electromagnetic plane wave illuminating the silicon dielectric cuboid nanoantenna, from which we have identified simultaneous existence of ED, MD and EQ resonance modes in this nanoantenna. We have calculated the Purcell factor of ED, MD and EQ emitters with different moment orientations as a function of radiation wavelength by placing these point radiation source within the nanoantenna, respectively. We find that the resonances wavelengths of the Purcell factor spectrum are matching with the resonance modes in the nanoantenna. Moreover, the maximum Purcell factor of these ED, MD and EQ emitters is 18, 150 and 118 respectively, occurring at the resonance wavelength of 475, 750, and 562 nm, respectively, all within the visible range. The polarization charge distribution features allow us to clarify the excitation and radiation of these resonance modes as the physical origin of large Purcell factor simultaneously occurring in this silicon cuboid nanoantenna. Our theoretical results might help to deeply explore and design the dielectric nanoantenna as an ideal candidate to enhance ED, MD and EQ emission simultaneously with very small loss in the visible range, which is superior than the more popular scheme of plasmonic nanoantenna.