Answer
$$\int 3(\ln 2) 2^{x}\left(1+2^{x}\right)^{3} d x =\frac{3}{4} \left(1+2^{x}\right)^4+c$$
Work Step by Step
Given $$
\int 3(\ln 2) 2^{x}\left(1+2^{x}\right)^{3} d x
$$
Let $ u=1+2^{x}\ \to\ \ du=(\ln 2) 2^{x}dx$, so:
\begin{align*}
\int 3(\ln 2) 2^{x}\left(1+2^{x}\right)^{3} d x&=\int 3 u^{3} du\\
&=\frac{3}{4} u^4+c\\
&=\frac{3}{4} \left(1+2^{x}\right)^4+c
\end{align*}