Does transforming the variable of integration from z to w-x work?

IntLap[s_, x_, \[Alpha]_, Lt1_, Lt2_] := 
 Integrate[(1 - 1/(1 + s*(w)^-\[Alpha]))*(w - x), {w, Lt1 + x, 
   Lt2 + x}]
Assuming[s \[Element] Reals && 
  x \[Element] Reals && \[Alpha] \[Element] Reals && 
  Lt1 \[Element] Reals && Lt2 \[Element] Reals && s > 0 && 
  x > 0 && \[Alpha] > 2 && Lt1 > -x && Lt2 > Lt1, 
 Simplify[IntLap[s, x, \[Alpha], Lt1, Lt2]]]

This gives me

-x (-(Lt1 + x) Hypergeometric2F1[1, 1/\[Alpha], 
      1 + 1/\[Alpha], -((Lt1 + x)^\[Alpha]/s)] + (Lt2 + 
       x) Hypergeometric2F1[1, 1/\[Alpha], 
      1 + 1/\[Alpha], -((Lt2 + x)^\[Alpha]/s)]) + (
 s ((Lt1 + x)^(2 - \[Alpha])
      Hypergeometric2F1[1, (-2 + \[Alpha])/\[Alpha], 
      2 - 2/\[Alpha], -s (Lt1 + x)^-\[Alpha]] - (Lt2 + x)^(
     2 - \[Alpha])
      Hypergeometric2F1[1, (-2 + \[Alpha])/\[Alpha], 
      2 - 2/\[Alpha], -s (Lt2 + x)^-\[Alpha]]))/(-2 + \[Alpha])

for output.