Let's first number the equations for convenience:

\begin{align} x&=x_0+vt+\frac 1 2 at^2\tag{1}\\ &\qquad\text{and}\\ x&=x_0+vt+\frac 1 2 at^2+\frac 1 6 j t^3+\frac 1 {24} st^4 +\frac 1 {120} ct^5\dots\tag{2}\\ &=\frac 1 {0!} x_0+\frac 1 {1!} vt +\frac 1 {2!}at^2+\frac 1 {3!} jt^3+\frac 1 {4!} st^4 +\frac 1 {5!} ct^5\dots \tag{3} \end{align}

Now the equation $(2)$ and $(3)$ are the most general equations for any particle's motion, whereas the equation $(1)$ is the equation which holds true only in the special case where the acceleration is constant. This implies that the higher derivatives of position are zero (i.e. $a=\text{constant}\implies j=s=c=\dots=0$). Thus the eauations $(2)$ and $(3)$ reduce to equation $(1)$.

In essence, the equation $(2)$ is just the Taylor expansion of displacement $x$ and therefore it's the most general form of representing the displacement $x$.

Let's first number the equations for convenience:

\begin{align} x&=x_0+vt+\frac 1 2 at^2\tag{1}\\ &\qquad\text{and}\\ x&=x_0+vt+\frac 1 2 at^2+\frac 1 6 j t^3+\frac 1 {24} st^4 +\frac 1 {120} ct^5\dots\tag{2}\\ &=\frac 1 {0!} x_0+\frac 1 {1!} vt +\frac 1 {2!}at^2+\frac 1 {3!} jt^3+\frac 1 {4!} st^4 +\frac 1 {5!} ct^5\dots \tag{3} \end{align}

Now the equations $(2)$ and $(3)$ are the most general equations for any particle's motion, whereas the equation $(1)$ is the equation which holds true only in the special case where the acceleration is constant. This implies that the higher derivatives of position are zero (i.e. $a=\text{constant}\implies j=s=c=\dots=0$). Thus the equations $(2)$ and $(3)$ reduce to equation $(1)$.

In essence, the equation $(2)$ is just the Taylor expansion of displacement $x$ and therefore it's the most general form of representing the displacement $x$ and it holds true in all cases.