#### 2.292   ODE No. 292

$y'(x) (a y(x)+b x+c)^2+(\alpha y(x)+\beta x+\gamma )^2=0$ Mathematica : cpu = 36.2462 (sec), leaf count = 1716

$\text {Solve}\left [(\alpha b-a \beta ) \text {RootSum}\left [-c y(x)^2 \alpha ^3-b \text {\#1} y(x)^2 \alpha ^3+a \beta \text {\#1} y(x)^2 \alpha ^2+a \gamma y(x)^2 \alpha ^2-2 b \beta \text {\#1}^2 y(x) \alpha ^2-2 \beta c \text {\#1} y(x) \alpha ^2-2 b \gamma \text {\#1} y(x) \alpha ^2-2 c \gamma y(x) \alpha ^2-b \beta ^2 \text {\#1}^3 \alpha -a^2 b y(x)^3 \alpha -\beta ^2 c \text {\#1}^2 \alpha -2 b \beta \gamma \text {\#1}^2 \alpha -2 a b c y(x)^2 \alpha -2 a b^2 \text {\#1} y(x)^2 \alpha -b \gamma ^2 \text {\#1} \alpha -2 \beta c \gamma \text {\#1} \alpha -b c^2 y(x) \alpha -b^3 \text {\#1}^2 y(x) \alpha +2 a \beta ^2 \text {\#1}^2 y(x) \alpha -2 b^2 c \text {\#1} y(x) \alpha +4 a \beta \gamma \text {\#1} y(x) \alpha +2 a \gamma ^2 y(x) \alpha -c \gamma ^2 \alpha +\beta c^3+a \beta ^3 \text {\#1}^3+a^3 \beta y(x)^3+b^2 \beta c \text {\#1}^2-b^3 \gamma \text {\#1}^2+3 a \beta ^2 \gamma \text {\#1}^2+3 a^2 \beta c y(x)^2+2 a^2 b \beta \text {\#1} y(x)^2-a^2 b \gamma y(x)^2+2 b \beta c^2 \text {\#1}+3 a \beta \gamma ^2 \text {\#1}-2 b^2 c \gamma \text {\#1}+3 a \beta c^2 y(x)+a b^2 \beta \text {\#1}^2 y(x)+4 a b \beta c \text {\#1} y(x)-2 a b^2 \gamma \text {\#1} y(x)-2 a b c \gamma y(x)+a \gamma ^3-b c^2 \gamma \& ,\frac {\beta ^2 \log (x-\text {\#1}) \text {\#1}^2+2 \beta \gamma \log (x-\text {\#1}) \text {\#1}+2 \alpha \beta \log (x-\text {\#1}) y(x) \text {\#1}+\alpha ^2 \log (x-\text {\#1}) y(x)^2+\gamma ^2 \log (x-\text {\#1})+2 \alpha \gamma \log (x-\text {\#1}) y(x)}{b y(x)^2 \alpha ^3-a \beta y(x)^2 \alpha ^2+2 \beta c y(x) \alpha ^2+4 b \beta \text {\#1} y(x) \alpha ^2+2 b \gamma y(x) \alpha ^2+3 b \beta ^2 \text {\#1}^2 \alpha +2 a b^2 y(x)^2 \alpha +2 \beta ^2 c \text {\#1} \alpha +4 b \beta \gamma \text {\#1} \alpha +2 b^2 c y(x) \alpha +2 b^3 \text {\#1} y(x) \alpha -4 a \beta ^2 \text {\#1} y(x) \alpha -4 a \beta \gamma y(x) \alpha +b \gamma ^2 \alpha +2 \beta c \gamma \alpha -2 b \beta c^2-3 a \beta ^3 \text {\#1}^2-2 a^2 b \beta y(x)^2-2 b^2 \beta c \text {\#1}+2 b^3 \gamma \text {\#1}-6 a \beta ^2 \gamma \text {\#1}-4 a b \beta c y(x)-2 a b^2 \beta \text {\#1} y(x)+2 a b^2 \gamma y(x)-3 a \beta \gamma ^2+2 b^2 c \gamma }\& \right ]+\int _1^{y(x)}\left (\frac {-\beta K[1]^2 a^3+\alpha b K[1]^2 a^2-2 \beta c K[1] a^2-2 b \beta x K[1] a^2-\beta c^2 a-b^2 \beta x^2 a-2 b \beta c x a+2 \alpha b c K[1] a+2 \alpha b^2 x K[1] a+\alpha b c^2+\alpha b^3 x^2+2 \alpha b^2 c x}{c K[1]^2 \alpha ^3+b x K[1]^2 \alpha ^3-a \beta x K[1]^2 \alpha ^2-a \gamma K[1]^2 \alpha ^2+2 b \beta x^2 K[1] \alpha ^2+2 \beta c x K[1] \alpha ^2+2 b \gamma x K[1] \alpha ^2+2 c \gamma K[1] \alpha ^2+b \beta ^2 x^3 \alpha +a^2 b K[1]^3 \alpha +\beta ^2 c x^2 \alpha +2 b \beta \gamma x^2 \alpha +2 a b c K[1]^2 \alpha +2 a b^2 x K[1]^2 \alpha +b \gamma ^2 x \alpha +2 \beta c \gamma x \alpha +b c^2 K[1] \alpha +b^3 x^2 K[1] \alpha -2 a \beta ^2 x^2 K[1] \alpha +2 b^2 c x K[1] \alpha -4 a \beta \gamma x K[1] \alpha -2 a \gamma ^2 K[1] \alpha +c \gamma ^2 \alpha -\beta c^3-a \beta ^3 x^3-a^3 \beta K[1]^3-b^2 \beta c x^2+b^3 \gamma x^2-3 a \beta ^2 \gamma x^2-3 a^2 \beta c K[1]^2-2 a^2 b \beta x K[1]^2+a^2 b \gamma K[1]^2-2 b \beta c^2 x-3 a \beta \gamma ^2 x+2 b^2 c \gamma x-3 a \beta c^2 K[1]-a b^2 \beta x^2 K[1]-4 a b \beta c x K[1]+2 a b^2 \gamma x K[1]+2 a b c \gamma K[1]-a \gamma ^3+b c^2 \gamma }-\frac {(\alpha b-a \beta ) \left (-c^2-2 b x c-2 a K[1] c-b^2 x^2-a^2 K[1]^2-2 a b x K[1]\right )}{-c K[1]^2 \alpha ^3-b x K[1]^2 \alpha ^3+a \beta x K[1]^2 \alpha ^2+a \gamma K[1]^2 \alpha ^2-2 b \beta x^2 K[1] \alpha ^2-2 \beta c x K[1] \alpha ^2-2 b \gamma x K[1] \alpha ^2-2 c \gamma K[1] \alpha ^2-b \beta ^2 x^3 \alpha -a^2 b K[1]^3 \alpha -\beta ^2 c x^2 \alpha -2 b \beta \gamma x^2 \alpha -2 a b c K[1]^2 \alpha -2 a b^2 x K[1]^2 \alpha -b \gamma ^2 x \alpha -2 \beta c \gamma x \alpha -b c^2 K[1] \alpha -b^3 x^2 K[1] \alpha +2 a \beta ^2 x^2 K[1] \alpha -2 b^2 c x K[1] \alpha +4 a \beta \gamma x K[1] \alpha +2 a \gamma ^2 K[1] \alpha -c \gamma ^2 \alpha +\beta c^3+a \beta ^3 x^3+a^3 \beta K[1]^3+b^2 \beta c x^2-b^3 \gamma x^2+3 a \beta ^2 \gamma x^2+3 a^2 \beta c K[1]^2+2 a^2 b \beta x K[1]^2-a^2 b \gamma K[1]^2+2 b \beta c^2 x+3 a \beta \gamma ^2 x-2 b^2 c \gamma x+3 a \beta c^2 K[1]+a b^2 \beta x^2 K[1]+4 a b \beta c x K[1]-2 a b^2 \gamma x K[1]-2 a b c \gamma K[1]+a \gamma ^3-b c^2 \gamma }\right )dK[1]=c_1,y(x)\right ]$ Maple : cpu = 0.032 (sec), leaf count = 115

$\left \{ y \left ( x \right ) ={\frac {1}{a\beta -b\alpha } \left ( \left ( \left ( bx+c \right ) \alpha -a \left ( \beta \,x+\gamma \right ) \right ) {\it RootOf} \left ( \int ^{{\it \_Z}}\!{\frac { \left ( {\it \_a}\,a-b \right ) ^{2}}{{{\it \_a}}^{3}{a}^{2}-2\,{{\it \_a}}^{2}ab-{{\it \_a}}^{2}{\alpha }^{2}+2\,{\it \_a}\,\alpha \,\beta +{\it \_a}\,{b}^{2}-{\beta }^{2}}}{d{\it \_a}}+\ln \left ( ax\beta -\alpha \,bx+a\gamma -\alpha \,c \right ) +{\it \_C1} \right ) +\gamma \,b-\beta \,c \right ) } \right \}$