Evaluate: `int(x^2+x+1)/((x+1)^2(x+2))\ dx`

Explanation

Let `I=int frac{x^{2}+x+1}{(x^{2}+1)(x+2)} d x`
We express

`frac{x^{2}+x+1}{(x^{2}+1)(x+2)}=frac{A}{x+2}+frac{B x+C}{x^{2}+1} `

`Rightarrow x^{2}+x+1=A(x^{2}+1)+(B x+C)(x+2)`

Equating the coefficients of `x^{wedge} 2, x` and constants, we get

`1=A+B `and `1=2 B+C `and `1=A+2 C` or `A=frac{3}{5}` and `B=frac{2}{5}` and `C=frac{1}{5}`

`therefore I=int(frac{frac{3}{5}}{x+2}+frac{frac{2}{5} x+frac{1}{5}}{x^{2}+1}) d x`

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