We present an analysis of deep multiwavelength data of z = 0.3-3 starburst galaxies selected by their 70 μm emission in the Extended-Chandra Deep Field-South (E-CDF-S) and Extended Groth Strip (EGS). We identify active galactic nuclei (AGNs) in these infrared sources through their X-ray emission and quantify the fraction of sources hosting an AGN. We find that the fraction depends strongly on the star formation rate (SFR, inferred from the 70 μm luminosity after subtracting the estimated AGN contribution), particularly for more luminous AGNs. At the highest SFRs (∼ 1000 M⊙ yr-1), the fraction of galaxies that host an X-ray detected AGN with L 0.5-8.0 keV ≳ 1043M erg s-1 rises to ∼ 40%, a value roughly consistent with the AGN fraction found for high-redshift submillimeter galaxies. The strong dependence of the AGN fraction on SFR implies that the duty cycle of luminous AGN activity relates to the growth rate of the host galaxy.