Velocity changing with instantenous + constant acceleration

suchara

A velocity at time t, V2, is dependant on
2 accelerations...
a = constant acceleration
a2 = acceleration dependant on velocity (instantaneous)

So basically...the whole thing goes like this

V2 = V1 + at + (Sqrt(xV1 - y) + z)
a,x,y,z are known constants

V3 = V2 + at2 + (Sqrt(xV2 - y) + z)

V4 = V3 + at3 + (Sqrt(xV3 - y) + z)

How would I come up with an equation for the overall velocity equation where lets say..given a starting velocity V1, I can come up with the instantaneous velocity after a certain time?
Like... how can I calc V4 given V1 and total time (t1+t2+t3)

I wanna come up with a d(t) equation by integrating a v(t) equation but Im stuck at this part..