<s>[INST] what method should I use? The power formula? or The logarithmic formula? (indefinite integral) Should I use the power formula or logarithmic to evaluate this integral:
$$\int \frac {x+1} {\sqrt[3]{x^2 + 2x + 1}}~dx$$
 [/INST] I assume the integral is
\begin{align}
\int\frac{x+1}{\sqrt[3]{x^2+2x+1}}\,dx
\end{align}
Then
\begin{align}
\require{cancel}
\int\frac{x+1}{\sqrt[3]{x^2+2x+1}}\,dx&=\int\frac{x+1}{\sqrt[3]{(x+1)^2}}\,dx\\
&=\int\frac{x+1}{(x+1)^{\frac{2}{3}}}\,dx\\
&=\int\frac{(x+1)^{\frac{2}{3}+\frac{1}{3}}}{(x+1)^{\frac{2}{3}}}\,dx\\
&=\int\frac{\cancel{(x+1)^{\frac{2}{3}}}(x+1)^{\frac{1}{3}}}{\cancel{(x+1)^{\frac{2}{3}}}}\,dx\\
&=\int(x+1)^{\frac{1}{3}}\,dx\qquad;\;\text{let}\;u=x+1\;\text{then}\;dx=du\\
&=\int u^{\frac{1}{3}}\,du\\
&=\frac{1}{\frac{1}{3}+1}u^{\frac{1}{3}+1}+C\\
&=\frac{3}{4}u^{\frac{4}{3}}+C\\
&=\frac{3}{4}(x+1)^{\frac{4}{3}}+C
\end{align}
</s>