## Update ##

With the corrected integration equations

    Assuming[Element[a | b | x1 | x2, Reals],
     Integrate[Sqrt[1 + (2 a x + b)^2], {x, x1, x2}]
    ]

    (* ConditionalExpression[(1/(
     4 a))(-b Sqrt[1 + (b + 2 a x1)^2] - 2 a x1 Sqrt[1 + (b + 2 a x1)^2] +
        b Sqrt[1 + (b + 2 a x2)^2] + 2 a x2 Sqrt[1 + (b + 2 a x2)^2] - 
       ArcSinh[b + 2 a x1] + ArcSinh[b + 2 a x2]), x1 < x2] *)

Define a function to create the forward model for `l` given the other inputs:

    lfun[a_, b_, x1_, x2_] := 
     1/(4 a) (-b Sqrt[1 + (b + 2 a x1)^2] - 
        2 a x1 Sqrt[1 + (b + 2 a x1)^2] + b Sqrt[1 + (b + 2 a x2)^2] + 
        2 a x2 Sqrt[1 + (b + 2 a x2)^2] - ArcSinh[b + 2 a x1] + 
        ArcSinh[b + 2 a x2])

Compute a test example

    lfun[1, 2, 1, 2] // N

    (* 5.1003 *)

Use `FindMinimum` to compute `x2` assuming the other inputs are known:

    With[
     {
      a = 1,
      b = 2,
      x1 = 1,
      l = 5.1003
      },
     FindMinimum[(l - lfun[a, b, x1, x2])^2, x2]
     ]

    (* {1.76173*10^-20, {x2 -> 2.}} *)