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We prove the $L^p$ bound for the Hilbert transform along variable non-flatcurves $(t,u(x)[t]^\alpha+v(x)[t]^\beta)$, where $\alpha$ and $\beta$ satisfy  $\alpha\neq \beta,\ \alpha\neq 1,\ \beta\neq 1.$  Comparing with the associated theorem in \cite{GHLJ} investigating the case$\alpha=\beta\neq 1$, our result is more general while the proof is moreinvolved. To achieve our goal, we divide the frequency of the objectivefunction into three cases and take different strategies to control these cases.Furthermore, we need to introduce a "short" shift maximal function$\mathfrak{M}^{[n]}$ to establish some pointwise estimate.

$L^p$ bound for the Hilbert transform along variable non-flat curves