Methods for solving the stiff system of ordinary differential equations that constitute nuclear reaction networks are surveyed. Three semi-implicit time integration algorithms are examined; a traditional first-order-accurate Euler method, a fourth-order-accurate Kaps-Rentrop method, and a variable-order Bader-Deuflhard method. These three integration methods are coupled to eight different linear algebra packages. Four of the linear algebra packages operate on dense matrices (LAPACK, LUDCMP, LEQS, GIFT), three of them are designed for the direct solution of sparse matrices (MA28, UMFPACK, Y12M), and one uses an iterative method for sparse matrices (BiCG). The scaling properties and behavior of the 24 combinations (3 time integration methods times 8 linear algebra packages) are analyzed by running each combination on seven different nuclear reaction networks. These reaction networks range from a hardwired 13 isotope α-chain and heavy-ion reaction network, which is suitable for most multidimensional simulations of stellar phenomena, to a 489 isotope reaction network, which is suitable for determining the yields of isotopes lighter than technetium in spherically symmetric models of Type II supernovae. Each of the time integration methods and linear algebra packages are capable of generating accurate results, but the efficiency of the various methods—evaluated across several different machine architectures and compiler options—differ dramatically. If the execution speed of reaction networks that contain less than about 50 isotopes is an overriding concern, then the variable-order Bader-Deuflhard time integration method coupled with routines generated from the GIFT matrix package or LAPACK with vendor-optimized BLAS routines is a good choice. If the amount of storage needed for any reaction network is a concern, then any of the sparse matrix packages will reduce the storage costs by 70%-90%. When a balance between accuracy, overall efficiency, and ease of use is desirable, then the variable-order Bader-Deuflhard time integration method coupled with the MA28 sparse matrix package is a strong choice.