\[ \boxed { x \left ( 1+x \right ) ^{2}y \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -x \left ( 1+x \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\, \left ( 1+x \right ) ^{2}y \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -a \left ( x+2 \right ) \left ( y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 0.135017 (sec), leaf count = 29 \[ \left \{\left \{y(x)\to c_2 e^{\frac {-a-c_1}{x}+a \log (x+1)}\right \}\right \} \]
Maple: cpu = 1.544 (sec), leaf count = 31 \[ \left \{ y \left ( x \right ) ={\frac { \left ( 1+x \right ) ^{a}}{{\it \_C1}\,{{\rm e}^{a}}}{{\rm e}^{{\frac {{\it \_C2}}{x}}}} \left ( { {\rm e}^{{\frac {a}{x}}}} \right ) ^{-1}} \right \} \]