\({\log _2}x + 6{\log _4}x-3{\log _8}x \le 6\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _2}x + 3{\log _2}x-{\log _2}x \le 6\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,3{\log _2}x \le 6\left| {:3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,} \right.{\log _2}x \le 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\log _2}x \le {\log _2}4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \le 4,}\\{x > 0\,}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {0;4} \right].} \right.\)
Целые решения: \(1,\,\,2,\,\,3,\,\,4.\) Их сумма равна: \(1 + 2 + 3 + 4 = 10.\)
Ответ: 10.