$I=\int \frac{\left(x+x^{\frac{2}{3}}+x^{\frac{1}{6}}\right)}{x\left(1+x^{\frac{1}{3}}\right)} d x$ is equal to

$\frac{3}{2} x^{\frac{2}{3}}+6 \tan ^{-1}\left(x^{\frac{1}{6}}\right)+c$

Substituting $x=p^6, d x=6 p^5 d p$

$=\int \frac{6 p^5\left(p^6+p^4+p\right)}{p^6\left(1+p^2\right)} d p=\int \frac{6\left(p^5+p^3+1\right)}{\left(p^2+1\right)} d p=\int 6 p^3 d p+\int\left(\frac{6}{p^2+1}\right) d p$

$=\frac{6 p^4}{4}+6 \tan ^{-1} p=\frac{3}{2} x^{\frac{2}{3}}+6 \tan ^{-1}\left(x^{\frac{1}{6}}\right)+c$

Hence (1) is the correct answer.