Two new, simple, general and rigorous closed-form integral solutions to the base current density and to the base transit time for electron transport in NPN bipolar transistors with heavily doped base and non-uniform band structure are presented. The expressions include the effects of non-uniform band structure, the influence of Fermi-Dirac statistics, the position-dependent mobility and the velocity saturation effect at the base-collector depletion-layer edge on the base side. An application of this new simplified base current density of equation to Early voltage in a narrow-base bipolar transistor is illustrated. A simple example for the effects of the minority-carrier density and of the base built-in field on the minority-carrier base transit time is also qualitatively discussed.