This paper addresses a hypothesis testing problem for comparing exponentially distributed data against a new class termed “New Better than Renewal Used in Moment Generating Function” (NBRUmg f ). A measure of departure from exponentiality is constructed using the Laplace transform, followed by the development of a U-statistic-based test for the hypothesis. Additionally, a test based on the goodness of fit approach is examined as a special case. The asymptotic normality of the proposed statistic is introduced, and Pitman’s asymptotic efficiency of the two tests is computed and compared with other tests. Percentiles of the test statistics are computed for certain sample sizes in the case of complete data, and the powers of the tests are computed for popular reliability distributions. Finally, practical applications of the proposed tests are demonstrated in multiple cases.