Evaluate the following integral: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(4e^{2x}-\frac{2}{3}{x}^{-\frac{4}{3}})\sqrt[\phantom{2}]{e^{2x}+x^{-\frac{1}{3}}}dx.$

$(A)\frac{4}{3}(e^{2x}+x^{-\frac{1}{3}}{)}^{\frac{3}{2}}+C$

$(B)\frac{2}{3}(e^{2x}+x^{-\frac{1}{3}}{)}^{\frac{3}{2}}+C$

$(C)\frac{2}{3}\sqrt[\phantom{2}]{e^{2x}+x^{-\frac{1}{3}}}+C$

$(D)\frac{2}{6}(e^{2x}+x^{-\frac{1}{3}}{)}^{\frac{1}{2}}+C$