Answer
$\frac{a^{2}+b^{2}}{a^{2}+ab+b^{2}}$
Work Step by Step
$\frac{ab}{a^{2}+ab+b^{2}} + (\frac{ac-ad-bc+bd}{ac-ad+bc-bd} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}})$
$=\frac{ab}{a^{2}+ab+b^{2}} + (\frac{a(c-d)-b(c-d)}{a(c-d)+b(c-d)} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}})$
$=\frac{ab}{a^{2}+ab+b^{2}} + (\frac{(a-b)(c-d)}{(a+b)(c-d)} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}})$
$=\frac{ab}{a^{2}+ab+b^{2}} + (\frac{(a-b)}{(a+b)} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}})$
Using the formulas $a^{3}-b^{3} =(a-b)(a^{2}+ab+b^{2})$ and $a^{3}+b^{3} =(a+b)(a^{2}-ab+b^{2})$
$=\frac{ab}{a^{2}+ab+b^{2}} + (\frac{(a-b)}{(a+b)} \div \frac{(a-b)(a^{2}+ab+b^{2})}{(a+b)(a^{2}-ab+b^{2})})$
$=\frac{ab}{a^{2}+ab+b^{2}} + (\frac{(a-b)}{(a+b)} \times \frac{(a+b)(a^{2}-ab+b^{2})}{(a-b)(a^{2}+ab+b^{2})})$
$=\frac{ab}{a^{2}+ab+b^{2}} + \frac{(a^{2}-ab+b^{2})}{(a^{2}+ab+b^{2})}$
$= \frac{ab+a^{2}-ab+b^{2}}{(a^{2}+ab+b^{2})}$
$= \frac{a^{2}+b^{2}}{a^{2}+ab+b^{2}}$
