$\int \frac{(\mathrm{x}+{\mathrm{x}}^{\frac{2}{3}}+{\mathrm{x}}^{\frac{1}{6}})}{\mathrm{x}(1+{\mathrm{x}}^{\frac{1}{3}})}\mathrm{dx}$ is equal to 

Substituting ${\text{x=p}}^6,\mathrm{dx}=6\hspace{0.17em}\hspace{0.17em}{\mathrm{p}}^5\hspace{0.17em}\hspace{0.17em}\mathrm{dp}$
$\mathrm{I}=\int \frac{6{\mathrm{p}}^5({\mathrm{p}}^6+{\mathrm{p}}^4+\mathrm{p})}{{\mathrm{p}}^6(1+{\mathrm{p}}^2)}\mathrm{dp}=\int \frac{6\text{\hspace{0.17em}}({\mathrm{p}}^5+{\mathrm{p}}^3+1)}{({\mathrm{p}}^2+1)}\mathrm{dp}=\int 6{\mathrm{p}}^3\mathrm{d}\mathrm{p}+\int \left(\frac{6}{{\mathrm{p}}^2+1}\right)\mathrm{dp}$
$=\frac{6{\mathrm{p}}^4}{4}+6{\tan }^{-1}\mathrm{p}=\frac{3}{2}{\mathrm{x}}^{\frac{2}{3}}+6{\tan }^{-1}\left({\mathrm{x}}^{\frac{1}{6}}\right)+\mathrm{c}$