$\frac{a^3b^3}{a+b}+\frac{b^3c^3}{b+c}+\frac{c^3a^3}{c+a} =\frac{a^2b^2}{ac+bc}+\frac{b^2c^2}{ab+ac}+\frac{c^2a^2}{bc+ab} \geq \frac{(ab+bc+ca)^2}{2(ab+bc+ca)}=\frac{ab+bc+ca}{2}= \frac12 \left(\frac{1}{a}+ \frac{1}{b}+\frac{1}{c}\right)$