Hubble-rate-dependent Dark Energy in Brans-Dicke Cosmology

Three general cases of dynamical interacting dark energy models ($\mathcal{D}$-class) are investigated in the context of Brans-Dicke cosmology. Some of important cosmological quantities are calculated for every cases as a function of redshift parameter. The most important part of this paper deals with fitting models with two different expansion history: (SNIa+BAO$_A$+$Omh^{2}$ and SNIa+BAO$_A$+H(z)) and with two different sets of data for Hubble parameter. This provides a remarkable feature to could analytically see the effects of each analyzes and each data sets on final results. The best fitted values of parameters according to these analyzes and data points, $\chi_{tot}^2/dof$, AIC and BIC are reported. By these diagnostic tools we found that some of these models have no chance against $\Lambda$CDM, even without need to study the structure formation, and could be ruled out. While some (e.g. $\mathcal{BD-D}C2$ and $\mathcal{BD-D}A^*$) render the best fit quality,i.e. the value of AIC and BIC and figures show that they fit perfectly with overall data and reveals a strong evidence in favor of these two models against $\Lambda$CDM.