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Answer
$$ \displaystyle\int \dfrac{\sqrt{x}}{{x}^{3}+x}\, \mathrm d x = {{\ln \left(x+\sqrt{2}\,\sqrt{x}+1\right)-\ln \left(x-\sqrt{2}\, \sqrt{x}+1\right)+2\,\arctan \left({{2\,\sqrt{x}+\sqrt{2}}\over{ \sqrt{2}}}\right)+2\,\arctan \left({{2\,\sqrt{x}-\sqrt{2}}\over{ \sqrt{2}}}\right)}\over{2\sqrt{2}}} $$
Explanation
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