What am I doing wrong in the derivation of Bass diffusion model?
You are missing an integration constant

$$
\log\left(\frac{p + qF(t)}{1 - F(t)}\right) = (p + q)t + \color{red}{\tilde{C}}
$$

This constant you can name it whatever you want, I'm going to name it as

$$
\color{red}{\tilde{C}} = \color{blue}{C}(p + q) + \ln q
$$

where $C$ is just another constant. So I basically changed one constant for  another one (completely allowed). Now the problem becomes

\begin{eqnarray}
\ln\left(\frac{p + qF(t)}{1 - F(t)}\right) &amp;=&amp; (p + q)t + \color{blue}{C}(p + q) + \ln q = (p + q)(t + C) + \ln q\\
p + qF(t) &amp;=&amp; e^{(p + q)(t + C) + \ln q}(1 - F(t)) = qe^{(p + q)(t + C)}(1 - F(t)) \\
[q + qe^{(p + q)(t + C)}]F(t)&amp;=&amp; qe^{(p + q)(t + C)} - p \\
F(t) &amp;=&amp; \frac{1}{q} \frac{qe^{(p + q)(t + C)} - p}{1 + e^{(p + q)(t + C)}}
\end{eqnarray}

Rearranging a bit the terms

$$\bbox[5px,border:2px solid blue]
{
F(t) = \frac{1}{q}\frac{q - pe^{-(p + q)(t + C)}}{1 + e^{-(p + q)(t + C)}}
}
$$