  Consider the Kirchhoff equation $$ \partial_{tt} u - \Delta u \Big( 1 +
\int_{\mathbb{T}^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus
$\mathbb{T}^d$. In a previous paper we proved that, after a first step of
quasilinear normal form, the resonant cubic terms show an integrable behavior,
namely they give no contribution to the energy estimates. This leads to the
question whether the same structure also emerges at the next steps of normal
form. In this paper, we perform the second step and give a negative answer to
the previous question: the quintic resonant terms give a nonzero contribution
to the energy estimates. This is not only a formal calculation, as we prove
that the normal form transformation is bounded between Sobolev spaces.
