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The branching fractions of the decays $B^{+} \to \eta \ell^{+} \nu_{\ell}$ and $B^{+} \to \eta^{\prime} \ell^{+} \nu_{\ell}$ are measured, where $\ell$ is either an electron or a muon, using a data sample of $711\,{\rm fb}^{-1}$ containing $772 \times 10^6 B\bar{B}$ pairs collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$ collider. To reduce the dependence of the result on the form factor model, the measurement is performed over the entire $q^2$ range. The resulting branching fractions are ${\cal B}(B^{+} \rightarrow \eta \ell^{+} \nu_{\ell}) = (2.83 \pm 0.55_{\rm (stat.)} \pm 0.34_{\rm (syst.)}) \times 10^{-5}$ and ${\cal B}(B^{+} \rightarrow \eta' \ell^{+} \nu_{\ell}) = (2.79 \pm 1.29_{\rm (stat.)} \pm 0.30_{\rm (syst.)}) \times 10^{-5}$.