not exactly sure what you asking, but taking a stab at it, seems to be more of a calculus question than physics
if a = a0 (constant)
velocity = integral of a = a0 * t + v0 (constant)
distance = integral of v = 1/2 a0 * t^2 + v0 * t + s0 (constant) - this should seem familiar to you as the standard equation
if a is a function of time, let's say a = a0 * t - acceleration increases linearly with time
velocity = integral of a = 1/2 a0 * t^2 + v0 - if acceleration is linear, then velocity is ^2
distance = integral of v = 1/6 a0 * t^3 + v0 * t + s0 - if velocity is ^2, then distance is ^3
so no, the expression is different as you integrate a as various functions of t