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We present a theoretical study on inelastic transient electrical currents and the effects of phonon heating in a single-level quantum dot system weakly coupled to a localized vibration degree of freedom, using the nonequilibrium Green's function method under the wide-band-limit and the lowest-order-expansion approximations. The energy transfer between electron and phonon systems is evaluated using both approximations, which separately are exact in the limits of the equilibrium state (t < 0) and steady state (t→∞). The time-dependent phonon number, which determines the system temperature and heating effects on the inelastic current, is calculated using a phenomenological method employing the time-dependent power transfer. The two approximations are shown to provide qualitatively similar dynamical behaviors for the system temperature, which can be grouped under two responses: if the energy corresponding to the applied bias voltage is smaller than or equal to the phonon energy, the temperature first increases because of phonon emission, and then decreases because of phonon absorption; alternatively, if the energy corresponding to the bias voltage is larger than the phonon energy, the temperature increases monotonically until a steady state is reached. The total electrical current is suppressed by phonon heating, while heat transferring between dot and environment mitigates the effects of such heating. Furthermore, the relaxation time of the current is extended by phonon scattering and heating.