$a^3 + b^3 + (\frac{b(2a^3 + b^3)}{a^3 - b^3})^3 = (\frac{a(a^3 + 2b^3)}{a^3 - b^3})^3$
$a^3 + b^3 = (\frac{a(a^3 + 2b^3)}{a^3 - b^3})^3 - (\frac{b(2a^3 + b^3)}{a^3 - b^3})^3$
$a^3 + b^3 = \frac{a^3(a^3 + 2b^3)^3}{(a^3 - b^3)^3} - \frac{b^3(2a^3 + b^3)^3}{(a^3 - b^3)^3}$
$(a^3 + b^3)(a^3 - b^3)^3 = a^3(a^3 + 2b^3)^3 - b^3(2a^3 + b^3)^3$
$a^3(a^3 + 2b^3)^3 - b^3(2a^3 + b^3)^3 = a^3(a^9 + 6a^6b^3 + 6a^3b^6 + 8b^6) - b^3(8a^9 + 6a^6b^3 + 6a^3b^6 + b^9) = a^{12} + 6a^9b^3 + 6a^6b^6 + 8a^3b^9 - 8a^9b^3 - 6a^6b^6 - 6a^3b^9 - b^{12} = a^{12} - 2a^9b^3 + 2a^3b^9 - b^12$
$(a^3 + b^3)(a^3 - b^3)^3 = (a^3 + b^3)(a^9 - 3a^6b^3 + 3a^3b^6 - b^6) = a^{12} + a^9b^3 - 3a^9b^3 - 3a^6b^6 + 3a^6b^6 + 3a^3b^9 - a^3b^9 - a^3b^9 - b^{12} = a^{12} - 2a^9b^3 + 2a^3b^9 - b^{12}$
Значит:
$a^3 + b^3 + (\frac{b(2a^3 + b^3)}{a^3 - b^3})^3 = (\frac{a(a^3 + 2b^3)}{a^3 - b^3})^3$