#### 2.274   ODE No. 274

$\left (a+x^2+y(x)^2\right ) y'(x)+b+x^2+2 x y(x)=0$ Mathematica : cpu = 0.1738 (sec), leaf count = 411

$\left \{\left \{y(x)\to \frac {\sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+\left (-81 b x+81 c_1-27 x^3\right ){}^2}-81 b x+81 c_1-27 x^3}}{3 \sqrt [3]{2}}-\frac {3 \sqrt [3]{2} \left (a+x^2\right )}{\sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+\left (-81 b x+81 c_1-27 x^3\right ){}^2}-81 b x+81 c_1-27 x^3}}\right \},\left \{y(x)\to \frac {3 \left (1+i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+\left (-81 b x+81 c_1-27 x^3\right ){}^2}-81 b x+81 c_1-27 x^3}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+\left (-81 b x+81 c_1-27 x^3\right ){}^2}-81 b x+81 c_1-27 x^3}}{6 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {3 \left (1-i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+\left (-81 b x+81 c_1-27 x^3\right ){}^2}-81 b x+81 c_1-27 x^3}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {2916 \left (a+x^2\right )^3+\left (-81 b x+81 c_1-27 x^3\right ){}^2}-81 b x+81 c_1-27 x^3}}{6 \sqrt [3]{2}}\right \}\right \}$ Maple : cpu = 0.029 (sec), leaf count = 657

$\left \{ y \left ( x \right ) ={\frac {1}{2} \left ( \left ( -4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}-4\,{x}^{2}-4\,a \right ) {\frac {1}{\sqrt [3]{-4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}}}}}},y \left ( x \right ) ={\frac {1}{4} \left ( \left ( i \left ( -4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+4\,i{x}^{2}+4\,ia \right ) \sqrt {3}- \left ( -4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+4\,{x}^{2}+4\,a \right ) {\frac {1}{\sqrt [3]{-4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}}}}}},y \left ( x \right ) =-{\frac {1}{4} \left ( \left ( i \left ( -4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}+4\,i{x}^{2}+4\,ia \right ) \sqrt {3}+ \left ( -4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}} \right ) ^{{\frac {2}{3}}}-4\,{x}^{2}-4\,a \right ) {\frac {1}{\sqrt [3]{-4\,{x}^{3}-12\,bx-12\,{\it \_C1}+4\,\sqrt {5\,{x}^{6}+ \left ( 12\,a+6\,b \right ) {x}^{4}+6\,{x}^{3}{\it \_C1}+ \left ( 12\,{a}^{2}+9\,{b}^{2} \right ) {x}^{2}+18\,bx{\it \_C1}+4\,{a}^{3}+9\,{{\it \_C1}}^{2}}}}}} \right \}$