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The second order linear differential equation $${\left(p\left(x\right)y^{\text{'}}\right)}^{\text{'}}+q\left(x\right)y=0,x\in (0,x_0]$$ is considered, where $p$, $q\in C^1(0,x_0]$, $p\left(x\right)>0$, $q\left(x\right)>0$ for $x\in (0,x_0]$. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near $x=0$ without the Hartman–Wintner condition.