Induction motors (IM) play essential tasks in distinct production sectors because of their low cost and robustness. Considering that most of the energy demand in industry is allocated for powering up IM, recent research has focused on detecting and predicting faults to avoid severe disturbances. Broken rotor bars (BRB) in IM cause a significant deficit of energy, above all in those applications where constant changes in speed are required, increasing the probability of a catastrophic failure. Variable speed drives (VSD) introduce harmonic components to the power supply current controlling the IM rotating speed, which make it difficult to identify BRB. Therefore, in this work, an innovative methodology is proposed for detecting BRB in VSD-fed IM with a wide rotating-speed bandwidth during their start-up transient. The introduced procedure performs a statistical analysis for computing the mean, median, mode, variance, skewness, and kurtosis, to identify slight changes on the acquired current signal. These values are fed into an artificial neural network (ANN), which carries out the IM operational condition classification as healthy (HLT) or with BRB. Experimentally obtained results corroborate the effectiveness of the proposed approach to detecting BRB even for dynamically varying rotating speed, reaching a high accuracy of 99%, similar to recently reported techniques.