The classic theory of locally optimal designs is developed on the center+error model assuming Gaussianity and homoscedasticity for random error, in which, the Maximum Likelihood Estimator (MLE) turns out to be the most efficient in model parameter estimation. However, these assumptions are typically absent in practice. In this work, we study the locally D-optimal design based on our new oracle Second-order Least Square Estimator (SLSE). We compare asymptotic efficiency of locally D-optimal designs obtained via SLSE, the Maximum quasi-Likelihood Estimator (MqLE) and Maximum Gaussian Likelihood Estimator (MGLE), in the case where the underlying probability distribution of response is non-Gaussian and heteroscedastic. We find that even with less stringent assumptions, asymptotic efficiency of the locally D-optimal designs obtained via MqLE is comparable to oracle SLSE in some cases, albeit lesser in general. As a demonstration of how the locally D-optimal design is numerically found, we apply our feasibility-based particle swarm optimization algorithm to the locally D-optimal design based on the original SLSE.