A search is presented for production of a heavy up-type quark ($t^\prime$) together with its antiparticle, assuming a significant branching ratio for subsequent $t^\prime$ decay into a Standard Model Higgs boson and a top quark, as predicted by vector-like quark models. The search is based on 14.3 fb$^{-1}$ of $pp$ collisions at $\sqrt{s}=8$ TeV recorded in 2012 with the ATLAS detector at the CERN Large Hadron Collider. Data are analysed in the lepton+jets final state, characterised by an isolated electron or muon with moderately high transverse momentum, significant missing transverse momentum, and at least six jets. The search exploits the high total transverse momenta of all final state objects and the high multiplicity of $b$ jets characteristic of signal events with at least one Higgs boson decaying into $b\bar{b}$, to discriminate against the dominant background from top quark pair production. No significant excess of events above the Standard Model expectation is observed, and upper limits are derived for vector-like quarks of various masses in the two-dimensional plane of $BR(t^\prime \to Wb)$ versus $BR(t^\prime \to Ht)$, where $H$ is the Standard Model Higgs boson, assumed to have a mass of $125$ GeV. Under the branching ratio assumptions corresponding to a weak-isospin doublet (singlet) scenario, a $t^\prime$ quark with mass lower than 790 (640) GeV is excluded at the 95\% confidence level.

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