  We present an analysis of deep multiwavelength data for z ~ 0.3-3 starburst
galaxies selected by their 70 um emission in the Extended-Chandra Deep
Field-South and Extended Groth Strip. We identify active galactic nuclei (AGNs)
in these infrared sources through their X-ray emission and quantify the
fraction that host an AGN. We find that the fraction depends strongly on both
the mid-infrared color and rest-frame mid-infrared luminosity of the source,
rising to ~ 50-70% at the warmest colors and highest mid-infrared luminosities
(corresponding to ultraluminous infrared galaxies), similar to the trends found
locally. Additionally, we find that the AGN fraction depends strongly on the
star formation rate of the host galaxy (inferred from the observed 70 um
luminosity after subtracting the estimated AGN contribution), particularly for
more luminous AGNs (L_X > 10^43 erg/s). At the highest star formation rates (~
1000 M_sun/yr), the fraction of galaxies with an X-ray detected AGN rises to ~
30%, roughly consistent with that found in high-redshift submillimeter
galaxies. Assuming that the AGN fraction is driven by the star formation rate
(rather than stellar mass or redshift, for which our sample is largely
degenerate), this result implies that the duty cycle of luminous AGN activity
increases with the star formation rate of the host galaxy: specifically, we
find that luminous X-ray detected AGNs are at least ~ 5-10 times more common in
systems with high star formation rates (> 300 M_sun/yr) than in systems with
lower star formation rates (< 30 M_sun/yr). Lastly, we investigate the ratio
between the supermassive black hole accretion rate and the growth rate of the
host galaxy and find that, for sources with detected AGNs and star formation,
this ratio in distant starbursts agrees well with that expected from the local
scaling relation assuming the black holes and bulges grew at the same epoch.
