#### 2.519   ODE No. 519

$y'(x)^3-f(x) \left (a y(x)^2+b y(x)+c\right )^2=0$ Mathematica : cpu = 0.605574 (sec), leaf count = 353

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {\#1} a+b) \left (\frac {a (\text {\#1} (\text {\#1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {(b+2 a \text {\#1})^2}{b^2-4 a c}\right )}{a (\text {\#1} (\text {\#1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x\sqrt [3]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {\#1} a+b) \left (\frac {a (\text {\#1} (\text {\#1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {(b+2 a \text {\#1})^2}{b^2-4 a c}\right )}{a (\text {\#1} (\text {\#1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {\#1} a+b) \left (\frac {a (\text {\#1} (\text {\#1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\frac {(b+2 a \text {\#1})^2}{b^2-4 a c}\right )}{a (\text {\#1} (\text {\#1} a+b)+c)^{2/3}}\& \right ]\left [\int _1^x(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ]\right \}\right \}$ Maple : cpu = 2.348 (sec), leaf count = 197

$\left \{ \int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it \_a}\,b+c \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!-{\sqrt [3]{f \left ( {\it \_a} \right ) \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{2}} \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it \_a}\,b+c \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!-{\frac {-1+i\sqrt {3}}{2}\sqrt [3]{f \left ( {\it \_a} \right ) \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{2}} \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{2}a+{\it \_a}\,b+c \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!{\frac {1+i\sqrt {3}}{2}\sqrt [3]{f \left ( {\it \_a} \right ) \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{2}} \left ( a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0 \right \}$