$\int \frac{2x+1}{x^4+2x^3+3x^2+2x+5}dx=$

$\begin{array}{l}I=\int \frac{2x+1}{x^4+2x^3+3x^2+2x+5}dx\\ =\int \frac{2x+1}{4+x^4+x^2+1+2x^{2}x+2x\cdot 1+2\cdot x^2\cdot 1}dx\\ =\int \frac{2x+1}{4+{\left(x^2+x+1\right)}^2}dx\\ \int \frac{(2x+1)}{(2{)}^2+{\left(x^2+x+1\right)}^2}dx\\ \text{ Let }{x}^2+x+1=t\Rightarrow (2x+1)dx=dt\\ =\int \frac{1}{2^2+t^2}dt=\frac{1}{2}{\tan }^{-1}\left(\frac{x^2+x+1}{2}\right)+c\end{array}$