<s>[INST] Calculation of integral with different roots I am supposed to calculate integral:$$\int \frac{4x+5\sqrt{x+2}}{\sqrt[3]{\left ( x+2 \right )^{2}}}dx$$ but I do not know, how to use the substitution.
Can anyone help me?
 [/INST] Hint. One may write
\begin{align}
\frac{4x+5\sqrt{x+2}}{\sqrt[3]{\left ( x+2 \right )^{2}}}&=\frac{4x+8-8}{\sqrt[3]{\left ( x+2 \right )^{2}}}+\frac{5\sqrt{x+2}}{\sqrt[3]{\left ( x+2 \right )^{2}}}
\\\\&=(x+2)^{1/3}-\frac{8}{(x+2)^{2/3}}+\frac{5}{(x+2)^{1/6}}
\end{align} then just perform the change of variable $t=x+2$.
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