  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 $z\sim5$ 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 $z\sim 5$
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.
