We investigate the evolution of the H$beta$+[OIII] and [OII] luminosity functions from $z sim 0.8$ to $sim5$ in four redshift slices per emission line using data from the High-{it z} Emission Line Survey (HiZELS). This is the first time that the H$beta$+[OIII] and [OII] luminosity functions have been studied at these redshifts in a self-consistent analysis. This is also the largest sample of [OII] and H$beta$+[OIII] emitters (3475 and 3298 emitters, respectively) in this redshift range, with large co-moving volumes $sim 1 times 10^6$ Mpc$^{-3}$ in two independent volumes (COSMOS and UDS), greatly reducing the effects of cosmic variance. The emitters were selected by a combination of photometric redshift and color-color selections, as well as spectroscopic follow-up, including recent spectroscopic observations using DEIMOS and MOSFIRE on the Keck Telescopes and FMOS on Subaru. We find a strong increase in $L_star$ and a decrease in $phi_star$ for both H$beta$+[OIII] and [OII] emitters. We derive the [OII] star-formation history of the Universe since $zsim5$ and find that the cosmic SFRD rises from $z sim 5$ to $sim 3$ and then drops towards $z sim 0$. We also find that our star-formation history is able to reproduce the evolution of the stellar mass density up to $zsim 5$ based only on a single tracer of star-formation. When comparing the H$beta$+[OIII] SFRDs to the [OII] and H$alpha$ SFRD measurements in the literature, we find that there is a remarkable agreement, suggesting that the H$beta$+[OIII] sample is dominated by star-forming galaxies at high-$z$ rather than AGNs.
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