Answer
$$\frac{1}{7}e^{7x}+\frac{3}{5}e^{5x}+e^{3x}+e^{x}+c$$
Work Step by Step
Let $u=e^x$, then $du=e^x dx$. Now, we have
$$\int e^x(e^{2x}+1)^3dx=\int (u^2+1)^3du\\
=\int (u^6+3u^4+3u^3+1 )du\\
=\frac{1}{7}u^7+\frac{3}{5}u^5+\frac{3}{3}u^3+u+c\\
=\frac{1}{7}e^{7x}+\frac{3}{5}e^{5x}+e^{3x}+e^{x}+c$$
