We estimate the neutrino emission from the decay chain of the $pi$-meson and $mu$-lepton, produced by proton-proton inelastic scattering in energetic ($E_{rm iso}gtrsim 10^{52}$~erg) long gamma-ray bursts (GRBs), within the type I binary-driven hypernova (BdHN) model. The BdHN I progenitor is textcolor{red}{a} binary system composed of a carbon-oxygen star (CO$_{rm core}$) and a neutron star (NS) companion. The CO$_{rm core}$ explosion as supernova (SN) triggers a massive accretion process onto the NS. For short orbital periods of few minutes, the NS reaches the critical mass, hence forming a black hole (BH). Recent numerical simulations of the above scenario show that the SN ejecta becomes highly asymmetric, creating a textit{cavity} around the newborn BH site, due to the NS accretion and gravitational collapse. Therefore, the electron-positron ($e^{pm}$) plasma created in the BH formation, during its isotropic and self-accelerating expansion, engulfs different amounts of ejecta baryons along different directions, leading to a direction-dependent Lorentz factor. The protons engulfed inside the high-density ($sim 10^{23}$~particle/cm$^3$) ejecta reach energies in the range $1.24lesssim E_plesssim 6.14$ GeV and interact with the unshocked protons in the ejecta. The protons engulfed from the low density region around the BH reach energies $sim 1$ TeV and interact with the low-density ($sim1$~particle/cm$^3$) protons of the interstellar medium (ISM). The above interactions give rise, respectively, to neutrino energies $E_{ u}leq 2$ GeV and $10leq E_{ u}leq 10^3$ GeV, and for both cases we calculate the spectra and luminosity.
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