\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +a \left ( {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) \right ) \left ( y \left ( x \right ) \right ) ^{3}+6\,a\phi \left ( x \right ) \left ( y \left ( x \right ) \right ) ^{2}+{\frac { \left ( 2\,a+1 \right ) y \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) }}+2\,a+2=0} \]
Mathematica: cpu = 28.631636 (sec), leaf count = 52 \[ \text {DSolve}\left [a y(x)^3 \phi '(x)+\frac {(2 a+1) y(x) \phi ''(x)}{\phi '(x)}+6 a \phi (x) y(x)^2+2 a+y'(x)+2=0,y(x),x\right ] \]
Maple: cpu = 0.702 (sec), leaf count = 0 \[ \text {could not solve} \]
Sage: cpu = 0 (sec), leaf count = 0 \[ \text {Maxima was unable to solve this ODE} \]