  The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in
the open subset of
$\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing
all cubic 4-folds and conjecture that the cubics corresponding to these
divisors are precisely the rational ones. Rationality has been known
classically for the first family $C_{14}$. We use congruences of 5-secant
conics to prove rationality for the first three of the families $C_d$,
corresponding to $d=14, 26, 38$ in Hassett's notation.
