How can the root integral be simplified to a more manageable form?

[transgalactic]

$$\int \frac{dx}{x(1+2\sqrt{x}+\sqrt[3]{x})}=\int \frac{dx}{x(\sqrt{x}+1+\sqrt{x}+\sqrt[3]{x})}= \int \frac{dx}{x(\sqrt{x}+\frac{(1-x)}{1-\sqrt[3]{x}})}$$
i got read of one root but instead i got another one
??

 

[quantumdude]

Notice that the denominator can be written as follows:

$$x\left(1+2x^{1/2}+x^{1/3}\right)$$

$$x\left(1+2x^{3/6}+x^{2/6}\right)$$

Let $u=x^{1/6}$.