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(x^2-x+1)=x^2-x+1/4+3/4
=(x-1/2)^2 + 3/4
integral(x+1)/sqrt(x^2-x+1)=x+1/sqrt{(x-1/2)^2 + 3/4}
now put x-1/2=t
dx=dt
integral(t+3/2)/sqrt(t^2+3/4)
= int{ t/sqrt(t^2+3/4) + 3/4sqrt(t^2+3/4) }
= sqrt(t^2 +3/4) + 3/4log{sqrt(t^2+3/4) +t} +c
=sqrt{ (x-1/2)^2 +3/4} + 3/4 log{sqrt(x-1/2)^2 +x +1/4 } +c
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