\[ \boxed { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) +a \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2}+d \left ( y \left ( x \right ) \right ) ^{1-a}=0} \]
Mathematica: cpu = 1.508191 (sec), leaf count = 744 \[ \left \{\left \{y(x)\to \left (-\frac {a d \exp \left (\frac {1}{2} x \left (\sqrt {-4 a c+b^2-4 c}+b\right )-\frac {x \left (b \sqrt {-4 a c+b^2-4 c}-4 (a+1) c+b^2\right )}{\sqrt {-4 a c+b^2-4 c}+b}-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}\right )}{(a+1) c}-\frac {d \exp \left (\frac {1}{2} x \left (\sqrt {-4 a c+b^2-4 c}+b\right )-\frac {x \left (b \sqrt {-4 a c+b^2-4 c}-4 (a+1) c+b^2\right )}{\sqrt {-4 a c+b^2-4 c}+b}-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}\right )}{(a+1) c}+\frac {a b c_1 \exp \left (-\frac {x \left (b \sqrt {-4 a c+b^2-4 c}-4 (a+1) c+b^2\right )}{\sqrt {-4 a c+b^2-4 c}+b}-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}\right )}{b \sqrt {-4 a c+b^2-4 c}-4 a c+b^2-4 c}+\frac {b c_1 \exp \left (-\frac {x \left (b \sqrt {-4 a c+b^2-4 c}-4 (a+1) c+b^2\right )}{\sqrt {-4 a c+b^2-4 c}+b}-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}\right )}{b \sqrt {-4 a c+b^2-4 c}-4 a c+b^2-4 c}+\frac {a c_1 \sqrt {-4 a c+b^2-4 c} \exp \left (-\frac {x \left (b \sqrt {-4 a c+b^2-4 c}-4 (a+1) c+b^2\right )}{\sqrt {-4 a c+b^2-4 c}+b}-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}\right )}{b \sqrt {-4 a c+b^2-4 c}-4 a c+b^2-4 c}+\frac {c_1 \sqrt {-4 a c+b^2-4 c} \exp \left (-\frac {x \left (b \sqrt {-4 a c+b^2-4 c}-4 (a+1) c+b^2\right )}{\sqrt {-4 a c+b^2-4 c}+b}-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}\right )}{b \sqrt {-4 a c+b^2-4 c}-4 a c+b^2-4 c}+c_2 e^{-\frac {2 (a+1) c x}{\sqrt {-4 a c+b^2-4 c}+b}}\right ){}^{\frac {1}{a+1}}\right \}\right \} \]
Maple: cpu = 0.187 (sec), leaf count = 158 \[ \left \{ y \left ( x \right ) ={1 \left ( {{c}^{2} \left ( -4\,ac+{b}^{2}- 4\,c \right ) \left ( {\it \_C2}\,c{{\rm e}^{-{\frac {x}{2} \left ( -b+ \sqrt {-4\,ac+{b}^{2}-4\,c} \right ) }}}a-{{\rm e}^{{\frac {x}{2} \left ( b+\sqrt {-4\,ac+{b}^{2}-4\,c} \right ) }}}{\it \_C1}\,ac+{\it \_C2}\,c{{\rm e}^{-{\frac {x}{2} \left ( -b+\sqrt {-4\,ac+{b}^{2}-4\,c} \right ) }}}-d{{\rm e}^{bx}}\sqrt {-4\,ac+{b}^{2}-4\,c}-{{\rm e}^{{ \frac {x}{2} \left ( b+\sqrt {-4\,ac+{b}^{2}-4\,c} \right ) }}}{\it \_C1 }\,c \right ) ^{-2}} \right ) ^{-{\frac {1}{2\,a+2}}} \left ( {{\rm e}^{{ \frac {bx}{a+1}}}} \right ) ^{-1}} \right \} \]