Answer
$\frac{a^2+b^2}{a^2+ab+b^2}$.
Work Step by Step
The given expression is
$\Rightarrow \frac{ab}{a^2+ab+b^2}+\left ( \frac{ac-ad-bc+bd}{ac-ad+bc-bd} \div \frac{a^3-b^3}{a^3+b^3} \right )$
Factor each term in the bracket as shown below.
$\Rightarrow ac-ad-bc+bd$
Group each term.
$\Rightarrow (ac-ad)+(-bc+bd)$
Factor each group.
$\Rightarrow a(c-d)-b(c-d)$
Factor out $(c-d)$.
$\Rightarrow (c-d)(a-b)$
$\Rightarrow ac-ad+bc-bd$
Group each term.
$\Rightarrow (ac-ad)+(bc-bd)$
Factor each group.
$\Rightarrow a(c-d)+b(c-d)$
Factor out $(c-d)$.
$\Rightarrow (c-d)(a+b)$
$\Rightarrow a^3-b^3$
Use the special formula $A^3-B^3=(A-B)(A^2+AB+B^2)$
$\Rightarrow (a-b)(a^2+ab+b^2)$
$\Rightarrow a^3+b^3$
Use the special formula $A^3+B^3=(A+B)(A^2-AB+B^2)$
$\Rightarrow (a+b)(a^2-ab+b^2)$
Back substitute all the factors into the given expression.
$\Rightarrow \frac{ab}{a^2+ab+b^2}+\left ( \frac{(c-d)(a-b)}{(c-d)(a+b)} \div \frac{(a-b)(a^2+ab+b^2)}{(a+b)(a^2-ab+b^2)} \right )$
Invert the divisor and multiply in the bracket.
$\Rightarrow \frac{ab}{a^2+ab+b^2}+\left ( \frac{(c-d)(a-b)}{(c-d)(a+b)} \cdot \frac{(a+b)(a^2-ab+b^2)}{(a-b)(a^2+ab+b^2)} \right )$
Cancel common terms.
$\Rightarrow \frac{ab}{a^2+ab+b^2}+\left ( \frac{a^2-ab+b^2}{a^2+ab+b^2} \right )$
$\Rightarrow \frac{ab+a^2-ab+b^2}{a^2+ab+b^2}$
Simplify.
$\Rightarrow \frac{a^2+b^2}{a^2+ab+b^2}$.
