Inspired by recent improved measurements of charm semileptonic decays at BESIII, we study a large set of $D(D_s)$-meson semileptonic decays where the hadron in the final state is one of $D^0$, $rho$, $omega$, $eta^{(prime)}$ in the case of $D^+$ decays, and $D^0$, $phi$, $K^0$, $K^ast(892)^0$, $eta^{(prime)}$ in the case of $D^+_s$ decays. The required hadronic form factors are computed in the full kinematical range of momentum transfer by employing the covariant confined quark model developed by us. A detailed comparison of the form factors with those from other approaches is provided. We calculate the decay branching fractions and their ratios, which show good agreement with available experimental data. We also give predictions for the forward-backward asymmetry and the longitudinal and transverse polarizations of the charged lepton in the final state.
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