Answer
$ \displaystyle \frac{(1+2e^{x})^{3/2}}{3}+C$
Work Step by Step
$\displaystyle \int e^{x}\sqrt{1+2e^{x}}dx=\int e^{x}(1+2e^{x})dx$
Shortcut to apply:
$\displaystyle \quad\int g\cdot u^{n}dx=\frac{g}{u'}\cdot\frac{u^{n+1}}{n+1}+C \quad ($if $\quad n\neq-1)$
$\left[\begin{array}{ll}
g(x)=e^{x}, & u(x)=1+2e^{x},\\
& u'(x)=2e^{x}
\end{array}\right]$
$=\displaystyle \frac{e^{x}}{2e^{x}}\cdot\frac{(1+2e^{x})^{3/2}}{3/2}+C$
$=\displaystyle \frac{(1+2e^{x})^{3/2}}{3}+C$
