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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.

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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]

Mathematica graphics

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