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

[transgalactic]

[tex]
\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}})}
[/tex]
i got read of one root but instead i got another one
??

 

[quantumdude]

Notice that the denominator can be written as follows:

[tex]x\left(1+2x^{1/2}+x^{1/3}\right)[/tex]

[tex]x\left(1+2x^{3/6}+x^{2/6}\right)[/tex]

Let [itex]u=x^{1/6}[/itex].