The complete kinetic theory model for concentrated polymer solutions and melts proposed by Curtiss and Bird is solved for shear flow: (a) analytically by providing a solution for the single-link (or configurational) distribution function as a real basis spherical harmonics expansion and then calculating the materials functions in shear flow up to second order in the dimensionless shear rate and, (b) numerically via the execution of Brownian dynamics simulations. These two methods are actually complementary to each other as the former is accurate only for small dimensionless shear rates where the latter produces results with increasingly large uncertainties. The analytical expansions of the material functions with respect to the dimensionless shear rate reduce to those of the extensively studied, simplified Curtiss-Bird model when ε' = 0, and to the rigid rod when ε' = 1. It is known that the power-law behavior at high shear rates is very different for these two extremal cases. We employ Brownian dynamics simulation to not only recover the limiting cases but to find a gradual variation of the power-law behaviors at large dimensionless shear rates upon varying ε'. The fact that experimental data are usually located between these two extremes strongly advocates the significance of studying the solution of the Curtiss-Bird model. This is exemplified in this work by comparing the solution of this model with available rheological data for semiflexible biological systems that are clearly not captured by the original Doi-Edwards or simplified Curtiss-Bird models.