\[ \int_{0}^{1} \frac{d x}{(1+x)(2+x) \sqrt{x(1-x)}}=\frac{\pi A}{\sqrt{6}(\sqrt{3}+1)} \] then ....

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\[
\int_{0}^{1} \frac{d x}{(1+x)(2+x) \sqrt{x(1-x)}}=\frac{\pi A}{\sqrt{6}(\sqrt{3}+1)}
\]
\( \mathrm{P} \)
then \( A \) is equal to



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