The electrodynamic properties of Ba(Fe$_{0.92}$Co$_{0.08})_2$As$_{2}$ and Ba(Fe$_{0.95}$Ni$_{0.05})_As$_{2}$ single crystals have been investigated by reflectivity measurements in a wide frequency range. In the metallic state, the optical conductivity consists of a broad incoherent background and a narrow Drude-like component which determines the transport properties; only the latter contribution strongly depends on the composition and temperature. This subsystem reveals a $T^2$ behavior in the dc resistivity and scattering rate disclosing a hidden Fermi-liquid behavior in the 122 iron-pnictide family. An extended Drude analysis yields the frequency dependence of the effective mass (with $m^*/m_bapprox 5$ in the static limit) and scattering rate that does not disclose a simple power law. The spectral weight shifts to lower energies upon cooling; a significant fraction is not recovered within the infrared range of frequencies.
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