#### 2.290   ODE No. 290

$y'(x) \left (a y(x)^2+2 b x y(x)+c x^2\right )+b y(x)^2+2 c x y(x)+d x^2=0$ Mathematica : cpu = 0.452494 (sec), leaf count = 831

$\left \{\left \{y(x)\to -\frac {b x}{a}+\frac {\sqrt [3]{-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}+\sqrt {4 \left (9 a c x^2-9 b^2 x^2\right )^3+\left (-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}\right ){}^2}}}{3 \sqrt [3]{2} a}-\frac {\sqrt [3]{2} \left (9 a c x^2-9 b^2 x^2\right )}{3 a \sqrt [3]{-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}+\sqrt {4 \left (9 a c x^2-9 b^2 x^2\right )^3+\left (-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}\right ){}^2}}}\right \},\left \{y(x)\to -\frac {b x}{a}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}+\sqrt {4 \left (9 a c x^2-9 b^2 x^2\right )^3+\left (-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}\right ){}^2}}}{6 \sqrt [3]{2} a}+\frac {\left (1+i \sqrt {3}\right ) \left (9 a c x^2-9 b^2 x^2\right )}{3\ 2^{2/3} a \sqrt [3]{-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}+\sqrt {4 \left (9 a c x^2-9 b^2 x^2\right )^3+\left (-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}\right ){}^2}}}\right \},\left \{y(x)\to -\frac {b x}{a}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}+\sqrt {4 \left (9 a c x^2-9 b^2 x^2\right )^3+\left (-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}\right ){}^2}}}{6 \sqrt [3]{2} a}+\frac {\left (1-i \sqrt {3}\right ) \left (9 a c x^2-9 b^2 x^2\right )}{3\ 2^{2/3} a \sqrt [3]{-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}+\sqrt {4 \left (9 a c x^2-9 b^2 x^2\right )^3+\left (-54 b^3 x^3+81 a b c x^3-27 a^2 d x^3+27 a^2 e^{3 c_1}\right ){}^2}}}\right \}\right \}$ Maple : cpu = 0.099 (sec), leaf count = 1388

$\left \{ y \left ( x \right ) ={\frac {1}{{\it \_C1}} \left ( {\frac {1}{2\,a}\sqrt [3]{-4\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+4\,\sqrt {{{\it \_C1}}^{6}{a}^{2}{d}^{2}{x}^{6}-6\,{{\it \_C1}}^{6}abcd{x}^{6}+4\,{{\it \_C1}}^{6}a{c}^{3}{x}^{6}+4\,{{\it \_C1}}^{6}{b}^{3}d{x}^{6}-3\,{{\it \_C1}}^{6}{b}^{2}{c}^{2}{x}^{6}-2\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+6\,c{x}^{3}{{\it \_C1}}^{3}ba-4\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+{a}^{2}}a+4\,{a}^{2}}}-2\,{\frac {{{\it \_C1}}^{2}{x}^{2} \left ( ac-{b}^{2} \right ) }{a\sqrt [3]{-4\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+4\,\sqrt {{{\it \_C1}}^{6}{a}^{2}{d}^{2}{x}^{6}-6\,{{\it \_C1}}^{6}abcd{x}^{6}+4\,{{\it \_C1}}^{6}a{c}^{3}{x}^{6}+4\,{{\it \_C1}}^{6}{b}^{3}d{x}^{6}-3\,{{\it \_C1}}^{6}{b}^{2}{c}^{2}{x}^{6}-2\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+6\,c{x}^{3}{{\it \_C1}}^{3}ba-4\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+{a}^{2}}a+4\,{a}^{2}}}}-{\frac {bx{\it \_C1}}{a}} \right ) },y \left ( x \right ) =-{\frac {1}{4\,{\it \_C1}\,a} \left ( 4\,bx{\it \_C1}\,\sqrt [3]{4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+ \left ( -4\,{{\it \_C1}}^{3}d{x}^{3}+4 \right ) {a}^{2}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}}+ \left ( 4\,i{x}^{2} \left ( ac-{b}^{2} \right ) {{\it \_C1}}^{2}+i \left ( -4\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+4\,{a}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}-4\,{{\it \_C1}}^{2}{x}^{2} \left ( ac-{b}^{2} \right ) + \left ( -4\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+4\,{a}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+ \left ( -4\,{{\it \_C1}}^{3}d{x}^{3}+4 \right ) {a}^{2}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}}}}},y \left ( x \right ) ={\frac {1}{4\,{\it \_C1}\,a} \left ( -4\,bx{\it \_C1}\,\sqrt [3]{4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+ \left ( -4\,{{\it \_C1}}^{3}d{x}^{3}+4 \right ) {a}^{2}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}}+ \left ( 4\,i{x}^{2} \left ( ac-{b}^{2} \right ) {{\it \_C1}}^{2}+i \left ( -4\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+4\,{a}^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+4\,{{\it \_C1}}^{2}{x}^{2} \left ( ac-{b}^{2} \right ) - \left ( -4\,{{\it \_C1}}^{3}{a}^{2}d{x}^{3}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}+4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+4\,{a}^{2} \right ) ^{{\frac {2}{3}}} \right ) {\frac {1}{\sqrt [3]{4\,\sqrt { \left ( {a}^{2}{d}^{2}+ \left ( -6\,bcd+4\,{c}^{3} \right ) a+4\,{b}^{3}d-3\,{b}^{2}{c}^{2} \right ) {x}^{6}{{\it \_C1}}^{6}-2\,{x}^{3} \left ( {a}^{2}d-3\,abc+2\,{b}^{3} \right ) {{\it \_C1}}^{3}+{a}^{2}}a+ \left ( -4\,{{\it \_C1}}^{3}d{x}^{3}+4 \right ) {a}^{2}+12\,c{x}^{3}{{\it \_C1}}^{3}ba-8\,{b}^{3}{x}^{3}{{\it \_C1}}^{3}}}}} \right \}$