In[]:=
ant=DSolve[{x'[t]==a+vx[t]/(l+vt),x[0]==0},x[t],t][[1]]//FullSimplify
Out[]=
x[t]-
a(l+tv)(Log[l]-Log[l+tv])
v

In[]:=
time=Solve[(x[t]/.ant)==l+vt,t][[1]]
Out[]=
t
-1+
v
a

l
v

In[]:=
t/.time/.{a->1,l->100,v->2}//N
Out[]=
319.453
In[]:=
t/.time/.{a->1,l->100,v->1000}//N
Out[]=
1.970071114017047×
433
10