Efficient implementations of the symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles (OCCD) method with the density-fitting approach, denoted by DF-OCCD(T) and DF-OCCD(T)(?), are presented. The computational cost of the DF-OCCD(T) method is compared with that of the conventional OCCD(T). In the conventional OCCD(T) and OCCD(T)(?) methods, one needs to perform four-index integral transformations at each coupled-cluster doubles iterations, which limits its applications to large chemical systems. Our results demonstrate that DF-OCCD(T) provides dramatically lower computational costs compared to OCCD(T), and there are more than 68-fold reductions in the computational time for the C5H12 molecule with the cc-pVTZ basis set. Our results show that the DF-OCCD(T) and DF-OCCD(T)(?) methods are very helpful for the study of single bond-breaking problems. Performances of the DF-OCCD(T) and DF-OCCD(T)(?) methods are noticeably better than that of the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for the potential energy surfaces of the molecules considered. Specifically, the DF-OCCD(T)(?) method provides dramatic improvements upon CCSD(T), and there are 8-14-fold reductions in nonparallelity errors. Overall, we conclude that the DF-OCCD(T)(?) method is very promising for the study of challenging chemical systems, where the CCSD(T) fails.