Using a differential transmission pump-probe experiment in heterodyne detection, the ultrafast gain and refractive-index dynamics of the ground-state transition in InAs/GaAs quantum-dot amplifiers emitting near 1.3 mu m at working condition, that is at room temperature and under electrical injection were measured. An ultrafast gain recovery on a subpicosecond time scale is observed at high electrical injection indicating fast carrier relaxation into the dot ground state, which is appealing for high-speed applications with these devices. Comparing p-doped and undoped devices of otherwise identical structure and operating at the same gain, a faster absorption recovery but a slower gain dynamics in p-doped amplifiers was observed. This finding should help in elucidating the role of p-doping in the design of QD-based devices with high-speed performances.