\(\frac{an}{a-x}+\frac{\left ( a+n \right )\left ( anx+nx^{2}+x^{3} \right )}{x^{3}+nx^{2}-a^{2}x-a^{2}n}=\frac{ax}{n+x}\frac{nx^{2}}{x^{2}-a^{2}}\)\(\Leftrightarrow \frac{an}{a-x}+\frac{\left ( a+n \right )\left ( anx+nx^{2}+x^{3} \right )}{x^{2}\left ( x+n \right )-a^{2}\left ( x+n \right )}=\frac{ax}{n+x}\frac{nx^{2}}{x^{2}-a^{2}}\)\(\Leftrightarrow \frac{-an\left ( x+n \right )\left ( x +a\right )+\left ( a+n \right )+\left ( ax +nx^{2}+x^{3}\right )}{\left ( x+n\right )\left ( x^{2}-a^{2} \right )}=\frac{ax\left ( x^{2}-a^{2}x \right )+nx^{2}\left ( n+x \right )}{\left ( x+n\right )\left ( x^{2}-a^{2} \right )}\)\(\Leftrightarrow -an\left ( x^{2}+nx+ax+an \right )+a^{2}nx+anx^{2}+ax^{3}+n^{2}ax+n^{2}x^{2}+nx^{3}\)\(=ax^{3}-a^{3}x+n^{2}x^{2}+nx^{3} \)\(\Leftrightarrow -anx^{2}-an^{2}x-a^{2}nx-a^{2}n^{2}+a^{2}nx+anx^{2}+ax^{3}+n^{2}ax+n^{2}x^{2}+nx^{3}\)\(=ax^{3}-a^{3}x+n^{2}x^{2}+nx^{3} \)\(\Leftrightarrow -a^{2}n^{2}=-a^{3}x\Leftrightarrow x=\frac{a^{2}n^{2}}{a^{3}} =\frac{n^{2}}{a}\)

\(\frac{an}{a-x}+\frac{\left ( a+n \right )\left ( anx+nx^{2}+x^{3} \right )}{x^{3}+nx^{2}-a^{2}x-a^{2}n}=\frac{ax}{n+x}\frac{nx^{2}}{x^{2}-a^{2}}\)\(\Leftrightarrow \frac{an}{a-x}+\frac{\left ( a+n \right )\left ( anx+nx^{2}+x^{3} \right )}{x^{2}\left ( x+n \right )-a^{2}\left ( x+n \right )}=\frac{ax}{n+x}\frac{nx^{2}}{x^{2}-a^{2}}\)\(\Leftrightarrow \frac{-an\left ( x+n \right )\left ( x +a\right )+\left ( a+n \right )+\left ( ax +nx^{2}+x^{3}\right )}{\left ( x+n\right )\left ( x^{2}-a^{2} \right )}=\frac{ax\left ( x^{2}-a^{2}x \right )+nx^{2}\left ( n+x \right )}{\left ( x+n\right )\left ( x^{2}-a^{2} \right )}\)\(\Leftrightarrow -an\left ( x^{2}+nx+ax+an \right )+a^{2}nx+anx^{2}+ax^{3}+n^{2}ax+n^{2}x^{2}+nx^{3}\)\(=ax^{3}-a^{3}x+n^{2}x^{2}+nx^{3} \)\(\Leftrightarrow -anx^{2}-an^{2}x-a^{2}nx-a^{2}n^{2}+a^{2}nx+anx^{2}+ax^{3}+n^{2}ax+n^{2}x^{2}+nx^{3}\)\(=ax^{3}-a^{3}x+n^{2}x^{2}+nx^{3} \)\(\Leftrightarrow -a^{2}n^{2}=-a^{3}x\Leftrightarrow x=\frac{a^{2}n^{2}}{a^{3}} =\frac{n^{2}}{a}\)