  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, 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 $|n|<1$.
