\[ y'(x)^3-f(x) \left (a y(x)^2+b y(x)+c\right )^2=0 \] ✓ Mathematica : cpu = 3.35254 (sec), leaf count = 473
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \left (2 \text {$\#$1} a-\sqrt {b^2-4 a c}+b\right ) \left (\frac {2 \text {$\#$1} a+\sqrt {b^2-4 a c}+b}{\sqrt {b^2-4 a c}}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {-b-2 a \text {$\#$1}+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{2\ 2^{2/3} a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x \sqrt [3]{f(K[1])} \, dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \left (2 \text {$\#$1} a-\sqrt {b^2-4 a c}+b\right ) \left (\frac {2 \text {$\#$1} a+\sqrt {b^2-4 a c}+b}{\sqrt {b^2-4 a c}}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {-b-2 a \text {$\#$1}+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{2\ 2^{2/3} a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x -\sqrt [3]{-1} \sqrt [3]{f(K[2])} \, dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \left (2 \text {$\#$1} a-\sqrt {b^2-4 a c}+b\right ) \left (\frac {2 \text {$\#$1} a+\sqrt {b^2-4 a c}+b}{\sqrt {b^2-4 a c}}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {-b-2 a \text {$\#$1}+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{2\ 2^{2/3} a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x (-1)^{2/3} \sqrt [3]{f(K[3])} \, dK[3]+c_1\right ]\right \}\right \}\] ✓ Maple : cpu = 0.511 (sec), leaf count = 197
\[ \left \{ \int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it \_a}\,b+c \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [3]{f \left ( {\it \_a} \right ) \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{2}} \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it \_a}\,b+c \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}-1}{2}\sqrt [3]{f \left ( {\it \_a} \right ) \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{2}} \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it \_a}\,b+c \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}+1}{2}\sqrt [3]{f \left ( {\it \_a} \right ) \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{2}} \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \]