\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {{x}^{3}+3\,a{x}^{2}+3\,{a}^{2}x+{a}^{3}+x \left ( y \left ( x \right ) \right ) ^{2}+a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{3}}{ \left ( x+a \right ) ^{3}}}=0} \]
Mathematica: cpu = 0.156520 (sec), leaf count = 111 \[ \text {Solve}\left [-\frac {19}{3} \text {RootSum}\left [-19 \text {$\#$1}^3+6 \sqrt [3]{38} \text {$\#$1}-19\& ,\frac {\log \left (\frac {\frac {3 y(x)}{(a+x)^3}+\frac {1}{(a+x)^2}}{\sqrt [3]{38} \sqrt [3]{\frac {1}{(a+x)^6}}}-\text {$\#$1}\right )}{2 \sqrt [3]{38}-19 \text {$\#$1}^2}\& \right ]=\frac {1}{9} 38^{2/3} \left (\frac {1}{(a+x)^6}\right )^{2/3} (a+x)^4 \log (a+x)+c_1,y(x)\right ] \]
Maple: cpu = 0.032 (sec), leaf count = 37 \[ \left \{ y \left ( x \right ) =-{\it RootOf} \left ( -\int ^{{\it \_Z}}\! \left ( {{\it \_a}}^{3}-{{\it \_a}}^{2}-{\it \_a}-1 \right ) ^{-1}{d{ \it \_a}}+\ln \left ( x+a \right ) +{\it \_C1} \right ) \left ( x+a \right ) \right \} \]