∫(x+1)/√(x^2+x+1)dx

$$ \begin{aligned} & \int \frac{x+1}{\sqrt{x^{2}+x+1}} d x \\=& \int \frac{x+1}{\sqrt{\left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}}} d x \\=& \frac{2}{\sqrt{3}} \int \frac{x+1}{\sqrt{\frac{(2 x+1)^{2}}{3}+1}} d x \\=& \frac{2}{\sqrt{3}} \int \frac{\frac{\sqrt{3} u-1}{2}+1}{\sqrt{u^{2}+1}} d \frac{\sqrt{3} u-1}{2} \cdot\left(u=\frac{2 x+1}{\sqrt{3}}, x=\frac{\sqrt{3} u-1}{2}\right) \\=& \frac{1}{2} \int \frac{\sqrt3 u+1}{\sqrt{u^{2}+1}} d u \\=& \frac{\sqrt 3}{2} \int \frac{u}{\sqrt{u^{2}+1}} d u+\frac{1}{2} \int \frac{1}{\sqrt{u^{2}+1}} d u \\=&\sqrt{x^{2}+x+1} +\frac{1}{2} arcsinh \frac{\sqrt{3}}{3}(2 x+1)+C \end{aligned} $$

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