Tough integral with many radicals

Question

I am completed baffled with this integral

$$\int\left[\dfrac{1}{x^{1/3}+x^{1/4}}+\dfrac{\ln(1+x^{1/6})}{x^{1/3}+x^{1/2}}\right]\mathrm dx$$

Any tips?

Answer 1

Hint: In $$ \int\frac{\ln(1+x^{1/6})}{x^{1/3}+x^{1/2}}\;dx , $$ substitution $y=1+x^{1/6}$ does a lot for you.

In $$ \int\frac{1}{x^{1/3}+x^{1/4}}\;dx $$ try $y=x^{1/12}$.