$\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$
$\qquad = \displaystyle \int \dfrac{y \, dy}{(1 + 2y)^{1/4}}$
Let
$u = (1 + 2y)^{1/4}$
$u^4 = 1 + 2y$
$y = \frac{1}{2}(u^4 - 1)$
$dy = 2u^3 \, du$
Hence,
$\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$
$\qquad = \displaystyle \int \dfrac{\frac{1}{2}(u^4 - 1)(2u^3 \, du)}{u}$
$\qquad = \displaystyle \int (u^4 - 1)(u^2 \, du)$
$\qquad = \displaystyle \int (u^6 - u^2) \, du$
$\qquad = \frac{1}{7}u^7 - \frac{1}{3}u^3 + C$
$\qquad = \frac{1}{21}u^3(3u^4 - 7) + C$
Revert to the original variable of integration
$\displaystyle \int \dfrac{y \, dy}{\sqrt[4]{1 + 2y}}$
$\qquad = \frac{1}{21}(1 + 2y)^{3/4}[ \, 3(1 + 2y) - 7 \, ] + C$
$\qquad = \frac{1}{21}(1 + 2y)^{3/4}(6y - 4) + C$
$\qquad = \frac{2}{21}(1 + 2y)^{3/4}(3y - 2) + C$ ← answer
