Answer
$$ \displaystyle\int \dfrac{{x}^{2}+2x+3}{\sqrt{4x-{x}^{2}}}\, \mathrm d x = -{{x\,\sqrt{-\left(x-4\right)\,x}+10\,\sqrt{-\left(x-4\right)\,x}- 26\,\arcsin \left({{x-2}\over{2}}\right)}\over{2}} $$
Explanation
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