#### 2.404   ODE No. 404

$a y'(x)^2+b x^2 y'(x)+c x y(x)=0$ Mathematica : cpu = 2.70508 (sec), leaf count = 795

$\left \{\text {Solve}\left [\int _1^x\left (\frac {(3 b+2 c) K[1]^2}{2 \left (3 b K[1]^3+c K[1]^3+9 a y(x)\right )}+\frac {3 \sqrt {K[1] \left (b^2 K[1]^3-4 a c y(x)\right )}}{2 \left (3 b K[1]^3+c K[1]^3+9 a y(x)\right )}\right )dK[1]+\int _1^{y(x)}\left (\frac {9 \sqrt {x \left (b^2 x^3-4 a c K[2]\right )} a}{2 (3 b+c) x^2 \left (3 b x^3+c x^3+9 a K[2]\right )}+\frac {3 (3 b+2 c) a}{2 (3 b+c) \left (3 b x^3+c x^3+9 a K[2]\right )}-\int _1^x\left (-\frac {9 a (3 b+2 c) K[1]^2}{2 \left (3 b K[1]^3+c K[1]^3+9 a K[2]\right )^2}-\frac {3 a c K[1]}{\left (3 b K[1]^3+c K[1]^3+9 a K[2]\right ) \sqrt {K[1] \left (b^2 K[1]^3-4 a c K[2]\right )}}-\frac {27 a \sqrt {K[1] \left (b^2 K[1]^3-4 a c K[2]\right )}}{2 \left (3 b K[1]^3+c K[1]^3+9 a K[2]\right )^2}\right )dK[1]-\frac {\sqrt {x \left (b^2 x^3-4 a c K[2]\right )}}{2 (3 b+c) x^2 K[2]}+\frac {b}{2 (3 b+c) K[2]}\right )dK[2]=c_1,y(x)\right ],\text {Solve}\left [\int _1^x\left (\frac {(3 b+2 c) K[3]^2}{2 \left (3 b K[3]^3+c K[3]^3+9 a y(x)\right )}-\frac {3 \sqrt {K[3] \left (b^2 K[3]^3-4 a c y(x)\right )}}{2 \left (3 b K[3]^3+c K[3]^3+9 a y(x)\right )}\right )dK[3]+\int _1^{y(x)}\left (-\frac {9 \sqrt {x \left (b^2 x^3-4 a c K[4]\right )} a}{2 (3 b+c) x^2 \left (3 b x^3+c x^3+9 a K[4]\right )}+\frac {3 (3 b+2 c) a}{2 (3 b+c) \left (3 b x^3+c x^3+9 a K[4]\right )}-\int _1^x\left (-\frac {9 a (3 b+2 c) K[3]^2}{2 \left (3 b K[3]^3+c K[3]^3+9 a K[4]\right )^2}+\frac {3 a c K[3]}{\left (3 b K[3]^3+c K[3]^3+9 a K[4]\right ) \sqrt {K[3] \left (b^2 K[3]^3-4 a c K[4]\right )}}+\frac {27 a \sqrt {K[3] \left (b^2 K[3]^3-4 a c K[4]\right )}}{2 \left (3 b K[3]^3+c K[3]^3+9 a K[4]\right )^2}\right )dK[3]+\frac {\sqrt {x \left (b^2 x^3-4 a c K[4]\right )}}{2 (3 b+c) x^2 K[4]}+\frac {b}{2 (3 b+c) K[4]}\right )dK[4]=c_1,y(x)\right ]\right \}$ Maple : cpu = 0.503 (sec), leaf count = 389

$\left \{ \int _{{\it \_b}}^{x}\!{1 \left ( -b{{\it \_a}}^{2}-\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,acy \left ( x \right ) } \right ) \left ( b{{\it \_a}}^{3}+\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,acy \left ( x \right ) }{\it \_a}+6\,ay \left ( x \right ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-2\,{\frac {a}{b{x}^{3}+\sqrt {{b}^{2}{x}^{4}-4\,{\it \_f}\,acx}x+6\,{\it \_f}\,a}}-\int _{{\it \_b}}^{x}\!6\,{\frac {a}{ \left ( b{{\it \_a}}^{3}+\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,{\it \_f}\,ac}{\it \_a}+6\,{\it \_f}\,a \right ) ^{2}} \left ( 2\,{\frac {{\it \_a}\,{\it \_f}\,ac}{\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,{\it \_f}\,ac}}}+b{{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,{\it \_f}\,ac} \right ) }\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{1 \left ( -b{{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,acy \left ( x \right ) } \right ) \left ( b{{\it \_a}}^{3}+6\,ay \left ( x \right ) -\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,acy \left ( x \right ) }{\it \_a} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!2\,{\frac {a}{-b{x}^{3}+\sqrt {{b}^{2}{x}^{4}-4\,{\it \_f}\,acx}x-6\,{\it \_f}\,a}}-\int _{{\it \_b}}^{x}\!-6\,{\frac {a}{ \left ( b{{\it \_a}}^{3}+6\,{\it \_f}\,a-\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,{\it \_f}\,ac}{\it \_a} \right ) ^{2}} \left ( 2\,{\frac {{\it \_a}\,{\it \_f}\,ac}{\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,{\it \_f}\,ac}}}-b{{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{4}{b}^{2}-4\,{\it \_a}\,{\it \_f}\,ac} \right ) }\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \}$