From point of view of this work, we will establish a novelty impressive behavior to the traveling wave solutions to the Benjamin-Bona-Mahony-Burgers equation (BBMBE) with dual power-law nonlinearity which is stretch to the Korteweg-de Varies equation but has more advantages compared with it. The traveling wave solutions of this equation have been achieved for the first time in the framework of two different techniques namely the Paul- Painleve approach method (PPAM) and the Riccati-Bernoulli Sub-ODE method (RBSODM). Furthermore, the corresponding numerical solutions for all achieved exact solutions via the above two methods will be documented in the framework of the variational iteration method.