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Answer
$$ \displaystyle\int {\mathrm{e}}^{4x}{\cdot}\sqrt{1+{\mathrm{e}}^{2x}}\, \mathrm d x = {{\left(e^{2\,x}+1\right)^{{{3}\over{2}}}\,\left(3\,e^{2\,x}-2 \right)}\over{15}} $$
Explanation
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