  We study well-posedness for a non-integrable generalization of the fifth
order KdV, the second member in the KdV heirarchy. In particular, we use
differentiation-by-parts to establish well-posedness for $s> 35/64$ in low
modulation restricted norm spaces, as well as non-linear smoothing of order
$\varepsilon < \min(2(s-35/64), 1)$. As corollaries, we obtain unconditional
well-posedness for the non-integrable fifth order KdV for $s > 1$ and global
well-posedness for the integrable fifth order KdV for $s\geq 1$. We also show
local well-posedness for the non-integrable fifth order KdV for $s > 1/2$,
contingent upon the conjectured $L^8$ Strichartz estimate. As an application of
the nonlinear smoothing we obtain non-trivial upper bounds on the upper
Minkowski dimension of the solution to the non-integrable fifth order KdV.
