Answer
$$ \displaystyle\int \dfrac{1}{{x}^{\frac{3}{2}}+8}\, \mathrm d x = {{\sqrt{3}\,\ln \left(x-2\,\sqrt{x}+4\right)-2\,\sqrt{3}\,\ln \left(\sqrt{x}+2\right)+6\,\arctan \left({{\sqrt{x}-1}\over{\sqrt{3}}}\right)}\over{6\sqrt{3}}} $$
Explanation
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