<p>The modified Korteweg-de Vries-Burgers equation with a nonlinear source is considered. The Cauchy problem for the equation is not solved by the inverse scattering transform because the equation does not pass the Weiss-Tabor-Carnevale test with the Kruskal variable. However, the equation admits a translation group in the independent variables, so we can search for solutions by applying a traveling wave reduction. The Painlevé test for the resulting nonlinear ordinary differential equation is used to determine its integrability. Constraints on the parameters of the mathematical model are found as necessary conditions for integrability. Guided by the resulting Fuchs indices, we construct the first integral of the nonlinear ordinary differential equation. Under the identified constraints, the general solution with three arbitrary constants is obtained in terms of complex functions. A version of the simplest equation method is then applied to find exact solutions. Exact solutions containing one and two arbitrary constants are obtained.</p>