\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{x}^{a-1} \left ( y \left ( x \right ) \right ) ^{1-b}f \left ( {\frac {{x}^{a}}{a}}+{\frac { \left ( y \left ( x \right ) \right ) ^{b}}{b}} \right ) =0} \]
Mathematica: cpu = 155.676769 (sec), leaf count = 235 \[ \text {Solve}\left [\int _1^{y(x)} \left (-\int _1^x \left (\frac {K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1}-\frac {K[1]^{a-1} K[2]^{b-1} f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right ) f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{\left (f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1\right )^2}\right ) \, dK[1]-\frac {K[2]^{b-1}}{f\left (\frac {K[2]^b}{b}+\frac {x^a}{a}\right )+1}\right ) \, dK[2]+\int _1^x \frac {K[1]^{a-1} f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right )}{f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right )+1} \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.359 (sec), leaf count = 153 \[ \left \{ y \left ( x \right ) =\sqrt [b]{-{\frac {1}{a} \left ( -{\it RootOf} \left ( \int ^{{\it \_Z}}\! \left ( \left ( \sqrt [a]{a} \right ) ^{a}f \left ( {\frac { \left ( \sqrt [a]{a} \right ) ^{a}b+ \left ( \sqrt [b]{-b+{\it \_a}} \right ) ^{b}a}{ab}} \right ) \left ( \sqrt [b]{-b+{\it \_a}} \right ) ^{-b}{\it \_a}- \left ( \sqrt [a]{a} \right ) ^{a}f \left ( {\frac { \left ( \sqrt [a]{a} \right ) ^{a}b+ \left ( \sqrt [b]{-b+{\it \_a}} \right ) ^{b}a}{ab}} \right ) \left ( \sqrt [b]{-b+{\it \_a}} \right ) ^{-b}b+a \right ) ^{-1}{d{\it \_a}}{a}^ {2}+{\it \_C1}\,ab-{x}^{a}b \right ) a+{x}^{a}b \right ) }} \right \} \]