Integrate[] does not work when the function has an antiderivative function

Question

This function

(1 + (1 + 1/(2*Sqrt[x]))/(2*Sqrt[Sqrt[x] + x]))/(2*Sqrt[x + Sqrt[Sqrt[x] + x]])

has the antiderivative function, since

D[Sqrt[x + Sqrt[x + Sqrt[x]]], x]== (1 + (1 + 1/(2*Sqrt[x]))/(2*Sqrt[Sqrt[x] + x]))/(2*Sqrt[x + Sqrt[Sqrt[x] + x]])

However, when I integrate it in Mathematica, I can't get the result as expected:

Integrate[(1 + (1 + 1/(2*Sqrt[x]))/(2*Sqrt[Sqrt[x] + x]))/(2*
Sqrt[x + Sqrt[Sqrt[x] + x]]), x, Assumptions :> x > 0]

Why didn't Integrate[] function work? I already tried many functions like Apart[], FullSimplify[], ExpandAll[], they didn't work either.

Answer 1

This is not an answer to your question (hence the community tag) since I do not know why Integrate does not solve this, but to point out that the command Int solves this instantly with no problem. This is using Albert Rich Rubi package:

ShowSteps = False;
Int[(1 + (1 + 1/(2*Sqrt[x]))/(2*Sqrt[Sqrt[x] + x]))/(2*Sqrt[x + Sqrt[Sqrt[x] + x]]), x]