Answer
$$ \displaystyle\int {\left(16{x}^{2}+4\right)}^{\frac{3}{2}}\, \mathrm d x = {{3\,{\rm asinh}\; \left(2\,x\right)+16\,x^3\,\sqrt{4\,x^2+1}+10\,x \,\sqrt{4\,x^2+1}}\over{2}} $$
Explanation
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