Verified answer

∫[√x/(1+ ³√x)]dx = [Cambio x= t^6 --> dx= 6·t^5· dt] =

∫[√t^6 / (1+ ³√t^6] · 6·t^5· dt =

∫6·t^8 /(1+t²) dt =

∫ 6·t^6 - 6·t^4 + 6·t² - 6 + 6/(t²+1) dt=

6/7·t^7 - 6/5·t^5 + 2·t³ - 6t + 6 ·arctg(t) + C =

6/7·x^(7/6) - 6/5·x^(5/6) + 2· √x- 6x^(1/6) + 6 ·arctg[x^(1/6)]+ C

Saludos.