velocity = acceleration * t
final velocity = u + a t^{3/2) t
= u + a t^(3/2 + 1) b
= u + a t^(5/2)b
To caluclate b, use the acceleration by
d final velocity/dt = 5/2 a b t^(3/2)
Compare this with acceleration
You will get b = 2/5
The answer will be
velocity final = u + a 2/5 t^(5/2)