  The present work studies one of Einstein's alternative formulations based on
the non-metricity scalar $Q$ generalized as $f(Q)$ theory. More specifically,
we consider the power-law form of $f(Q)$ gravity i.e. $f(Q)=Q+\alpha\, Q^n$.
Here, we analyze the behavior of the cosmological model at the background and
perturbation level. At the background level, we find the effective evolution of
the model is the same as that of the $\Lambda$CDM for $|n|<1$. Interestingly,
the geometric component of the theory solely determined the late-time
acceleration of the Universe. We also examine the integrability of the model by
employing the method of singularity analysis. In particular, we find the
conditions under which field equations pass the Painlev\'{e} test and hence
possess the Painlev\'{e} property. While the equations pass the Painlev\'{e}
test in the presence of dust for any value of $n$, the test is valid after the
addition of radiation fluid only for $n<1$. Finally, at the perturbation level,
the behavior of matter growth index signifies a deviation of the model from the
$\Lambda$CDM even for $|n|<1$.
