This paper introduces a novel cosmological scale factor, $a(t)=e^{H(t)} { (1-e^{-k(t)t}) }^{b(t)}$, as a strong candidate for effectively modeling the universe in a unified and continuous description across all major epochs of cosmic evolution. We provide a detailed analysis, supported by graphical representations of the universe's evolution with smooth transition from the early inflation to the present acceleration expansion, proposing a coherent and comprehensive picture of cosmic history, consistent with current observational constraints. Furthermore, we explore the theoretical foundations of this model within extra-dimensional physics, demonstrating its compatibility with the brane-world paradigm. Through analytical approximations, the model describes a natural transition from a high-energy, brane-dominated early universe to the low-energy regime governed by general relativity, providing a compelling narrative of cosmic evolution across disparate energy scales. Additionally, the explicit presence of quantum statistical corrections to the Hubble parameter of this scale factor, as a prominent feature of the model, offers a phenomenological approach to incorporating quantum statistic into the classical evolution of the universe.