Accurately and efficiently predicting the fundamental-frequency temporal shape of broadband long-pulsed lasers is very important in research on the properties of high-power laser amplifiers. In this study, we first propose that analytic electric polarization in the temporal domain is applied to broadband long-pulsed pulse amplification. We first verify the accuracy of this algorithm in the dozens of picoseconds range and the results are consistent with Miro software. Then we simulate the broadband long-pulsed amplification. The simulation results indicate that the front edge of the output pulse is more enlarged than the end edge owing to saturation and that the gain narrowing induces severe amplitude modulation. Analytic electric polarization in the temporal domain is effective and precise for investigating the broadband pulse amplification in the time scale from dozens of picoseconds to nanoseconds, and the computation time can be decreased by at least 4 orders of magnitude.