I'm not seeing the same issue. Most likely this is being caused by something else that has been defined at some point.
Save any work you have and then restart Mathematica so that it has a fresh kernel. After that try evaluating the following code:
Assuming[x > 0 && Alpha > 0 && k > 0 && Lambda > 0,
FullSimplify@
Integrate[(k (1 + Alpha) (1 - Alpha^k)^(1/Alpha) (t/Lambda)^k)/t, {t, 0, x}]]
This is essentially the same syntax above but without any special characters. Does this work for you? It should give:
(1 + Alpha) (1 - Alpha^k)^(1/Alpha) (x/Lambda)^k
Concerning this piece of code:
Assuming[x > 0 && Alpha > 0 && k > 0 && Lambda > 0, Reduce[x Alpha^(1/k) <= Lambda, Alpha]]
Assuming[...., SomeFunction[....]] is syntactic sugar for SomeFunction[.....,Assumptions->....]. Reduce in this case doesn't have an assumptions option, so Assuming effectively does nothing here.
The operation might be best done an equality instead and done with Solve. (x Alpha^(1/k) == Lambda)
Assuming[x > 0 && Alpha > 0 && k > 0 && Lambda > 0,
Solve[x Alpha^(1/k) == Lambda, Alpha]]
Once you have where they're equal, you can label which regions satisfy the original inequality.