Answer
$$\frac{4}{9} (x+1)^{9 / 4}+\frac{4}{5} (x+1)^{5 / 4}+C $$
Work Step by Step
Given
$$ \int(x+2)(x+1)^{1 / 4} d x $$
Let
$$u=x+1\ \ \ \Rightarrow \ \ \ du=dx $$
Then
\begin{aligned} \int(x+2)(x+1)^{1 / 4} d x &=\int(u+1) u^{1 / 4} d u \\ &=\int\left(u^{5 / 4}+u^{1 / 4}\right) d u \\ &=\frac{4}{9} u^{9 / 4}+\frac{4}{5} u^{5 / 4}+C\\
&= \frac{4}{9} (x+1)^{9 / 4}+\frac{4}{5} (x+1)^{5 / 4}+C \end{aligned}
