\(A=\dfrac{1}{a^3+b^3}+\dfrac{1}{a^2b}+\dfrac{1}{ab^2}\)
\(=\dfrac{1}{\left(a+b\right)\left(a^2-ab+b^2\right)}+\dfrac{1}{ab}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\)
\(\ge\dfrac{1}{a^2-ab+b^2}+\dfrac{4}{ab\left(a+b\right)}\)
\(\ge\dfrac{1}{a^2-ab+b^2}+\dfrac{4}{ab}\)
\(=\dfrac{1}{a^2-ab+b^2}+\dfrac{1}{ab}+\dfrac{1}{ab}+\dfrac{1}{ab}+\dfrac{1}{ab}\)
\(\ge\dfrac{25}{a^2+3ab+b^2}=\dfrac{25}{\left(a+b\right)^2+ab}\ge\dfrac{25}{1+\dfrac{\left(a+b\right)^2}{4}}\ge\dfrac{25}{1+\dfrac{1}{4}}=20\)
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