Chemists are frequently interested in rate equations, which are first-order differential equations. Numerical integration of these equations allows the researcher to accurately predict the concentrations of chemical species at any time given the initial conditions. Explicit Runge-Kutta (RK) integration is widely used for solving the rate equations. In this article, Adomian decomposition methods (ADM) are used to obtain the solutions of chemical rate equations. The Adomian method outlined here outperforms high-order RK routines in the arenas of accuracy and truncation error. Additionally, four modifications are introduced that place the Adomian integration on par with RK in terms of speed (a primary reason for which Adomian decomposition methods are currently underemployed). The inclusion of up to the fifth term in the Adomian expansion gives a truncation error of order O(h(10)). The method as presented yields solutions which are step-size independent in the nonstiff regime. The problem of rapid polynomial divergence is addressed through discretizing the time axis. Performance of the ADM method against an implicit algorithm is also given.