An efficient and reliable method is presented for calculations the generalized exponential (GE) integrals. The basic series expressions of the generalized GE integrals are established. Evaluation of GE integrals for different values of the parameters, show the efficiencies of the new approach. The numerical results illustrate clearly a further reduction in calculation times. The relationships obtained are valid for the arbitrary values parameters and the computation results obtained are in good agreement with the literature. Numerical results obtained and comparisons with numerical results from the literature are listed.