Assuming a Gaussian-shaped input signal pulse, we deduced the analytical expression of the output-pulse amplitude envelope in the Brillouin slow light system utilizing broadband pump with a rectangular spectrum under small-signal gain approximation. The expression is applicable in case that the pump spectrum has a sharp rising or trailing edge and a flat top. The influence of the fiber dispersion, the stimulated Brilouin scattering (SBS) gain non-uniformity and the SBS-gain-induced dispersion on the pulse distortion has also been quantitatively analyzed. The calculated results show that the analytical solution agrees well with the numerical one if the spectrum of the signal pulse lies in the pump spectrum. SBS-gain-induced third-order dispersion is the main physical factor responsible for the pulse distortion, which also limits the retardation increase of the short pulse in the cascaded Brillouin slow light system.