Brain teaser:  What are we doing wrong?

Beginning with Newton's second law,

F = dp/dt

Shouldn't we be able to say that if the mass of the system at time t is m = M +  bt, that

F = m(dv/dt)

= (M+bt) (dv/dt) ?

Can't we then separate the variables and integrate:

(F / (M + bt)) dt = dv

∫ (F / (M + bt)) dt = ∫ dv

(F / b) ln(M + bt) + C = v

where C would be some constant which sets the initial velocity? Here, since v(t=0) = 0, we

would have C = - (F/b)  ln (M)

Is there a difference between the textbook solution and this new one:

v = (F/b) (ln(M+bt) - ln(M)) ?

Yes, they must be different... there are no natural logs in the right solution. To confirm that the new solution doesn't agree with the earlier Textbook solution, the red curve

on the v(t)  graph shows this new one. While at short times the solutions agree, they don't at long times.

This new solution is wrong. Why?  Where did we go wrong in deriving it?