We present a lattice determination of the hadronic susceptibilities that, thanks to unitarity and analyticity, constrain the form factors entering the semileptonic $b\rightarrow c$ transitions. We evaluate the transverse and longitudinal susceptibilities of the vector and axial polarization functions at zero momentum transfer from the moments of appropriate two-point correlation functions. The latter are obtained on the lattice employing gauge ensembles of the Extended Twisted Mass Collaboration (ETMC) with $N_f$=2+1+1 flavors of Wilson-clover twisted-mass quarks with masses of all the dynamical quark flavors tuned close to their physical values. The simulations are carried out at four values of the lattice spacing, $a \simeq (0.057,0.068, 0.080, 0.091)$~fm, with spatial lattice sizes up to $L \simeq 7.6$~fm. Heavy-quark masses up to $\approx 3.5$ times the physical charm mass are employed, allowing for a smooth extrapolation to the physical b-quark mass.