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

[transgalactic]

<br /> \int \frac{dx}{x(1+2\sqrt{x}+\sqrt[3]{x})}=\int \frac{dx}{x(\sqrt{x}+1+\sqrt{x}+\sqrt[3]{x})}=<br /> \int \frac{dx}{x(\sqrt{x}+\frac{(1-x)}{1-\sqrt[3]{x}})}<br />
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}.