The given condition is equivalent to $(a+b)^2+(a-b)^2=6(a^2-b^2)\iff a^2+b^2=3(a^2-b^2)$, so $a^2=2b^2$. Note that $|a|\neq|b|$, so it follows that $a=b\neq 0$. Thus, $a^6=8b^6\iff a^3=2b^3\sqrt{2}\iff \frac{a^3}{b^3}=2\sqrt{2}$. Thus, the given expression can be simplified as $\frac{2\sqrt{2}+1}{2\sqrt{2}-1}+\frac{2\sqrt{2}-1}{2\sqrt{2}+1}$, which is equal to $\frac{2(8+1)}{8-1}=\boxed{\frac{18}{7}}$.