let
y=x^(1/6)
dy = (1/6)x^(-5/6) dx
dx =6y^5 dy
∫ dx/[√x + x^(2/3)]
=∫ [6y^5 /(y^3 + y^4 ) ] dy
=∫ [6y^2 /(1 + y ) ] dy
=∫ [ 6y -6+ 6/(y+1) ] dy
=3y^2 -6y +6|y+1| + C
=3x^(1/3) -6x^(1/6) +6|x^(1/6)+1| + C
where
6y^2
=6y(y+1) -6y
= 6y(y+1) -6(y+1) +6
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