We investigate the power-law, intermediate, and logamediate inflationary models in the framework of DBI noncanonical scalar field with constant sound speed. In the DBI setting, we first represent the power spectrum of both scalar density and tensor gravitational perturbations. Then, we derive different inflationary observables including the scalar spectral index ns, the running of the scalar spectral index dns d ln k, and the tensor-to-scalar ratio r. We show that the 95% CL constraint of the Planck 2015 T+E data on the non-Gaussianity parameter fNL DBI leads to the sound speed bound cs 0.087 in the DBI inflation. Moreover, our results imply that, although the predictions of the power-law, intermediate, and logamediate inflations in the standard canonical framework (cs = 1) are not consistent with the Planck 2015 data, in the DBI scenario with constant sound speed cs < 1, the result of the r - ns diagram for these models can lie inside the 68% CL region favored by Planck 2015 TT,TE,EE+lowP data. We also specify the parameter space of the power-law, intermediate, and logamediate inflations for which our models are compatible with the 68% or 95% CL regions of the Planck 2015 TT,TE,EE+lowP data. Using the allowed ranges of the parameter space of the intermediate and logamediate inflationary models, we estimate the running of the scalar spectral index and find that it is compatible with the 95% CL constraint from the Planck 2015 TT,TE,EE +lowP data.
