$\displaystyle \int\frac{dx}{(1+x^{\frac{1}{4}})x^{\frac{1}{3}}} $
I am stumped on what to substitute here.
Try to sub. $\displaystyle x = u^{-12} $
$\displaystyle dx = -12u^{-13} du$
the integral goes to
$\displaystyle \int \frac{-12u^{-13} du}{ ( 1+ u^{-12/4}) u^{-12/3} } $
$\displaystyle = \int \frac{-12u^{-13} du}{ ( 1+ u^{-3}) u^{-4} }$
$\displaystyle = -12 \int \frac{du}{u^6(1 + u^3)} $
$\displaystyle = - 12 \int \left[ \frac{1}{1+u^3} - \frac{u^3- 1}{u^6} \right] ~du $
$\displaystyle = -12 \int \left[ \frac{1}{1 + u^3 } - \frac{1}{u^3} + \frac{1}{u^6} \right] ~du $
and so on