The topic of this paper is solving the burnup equations using dedicated matrix exponential methods that are based on two different types of rational approximation near the negative real axis. The previously introduced Chebyshev Rational Approximation Method (CRAM) is now analyzed in detail for its accuracy and convergence, and correct partial fraction coefficients for approximation orders 14 and 16 are given to facilitate its implementation and improve the accuracy. As a new approach, rational approximation based on quadrature formulas derived from complex contour integrals is proposed, which forms an attractive alternative to CRAM, as its coefficients are easy to compute for any order of approximation. This gives the user the option to routinely choose between computational efficiency and accuracy all the way up to the level permitted by the available arithmetic precision. The presented results for two test cases are validated against reference solutions computed using high-precision arithmetics. The observed behavior of the methods confirms the previous conclusions of CRAM's excellent suitability for burnup calculations and establishes the quadrature-based approximation as a viable and flexible alternative that, like CRAM, has its foundation in the specific eigenvalue properties of burnup matrices.