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.