#### 2.69   ODE No. 69

$y'(x)-\sqrt {\left (\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4\right ) \left (\text {b0}+\text {b1} y(x)+\text {b2} y(x)^2+\text {b3} y(x)^3+\text {b4} y(x)^4\right )}=0$ Mathematica : cpu = 7.7341 (sec), leaf count = 1163

$\text {Solve}\left [-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]\right )}{\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right )}{\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right )}\right ) \left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]\right )^2 \sqrt {\frac {\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,3\right ]\right )}{\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,3\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]\right )}} \sqrt {\frac {\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]\right ) \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right )}{\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right )^2 \left (y(x)-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]\right )^2}}}{\left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,2\right ]-\text {Root}\left [\text {b4} \text {\#1}^4+\text {b3} \text {\#1}^3+\text {b2} \text {\#1}^2+\text {b1} \text {\#1}+\text {b0}\& ,4\right ]\right ) \sqrt {\text {b0}+y(x) (\text {b1}+y(x) (\text {b2}+y(x) (\text {b3}+\text {b4} y(x))))}}=c_1+\int _1^x\sqrt {\text {a4} K[1]^4+\text {a3} K[1]^3+\text {a2} K[1]^2+\text {a1} K[1]+\text {a0}}dK[1],y(x)\right ]$ Maple : cpu = 0.116 (sec), leaf count = 111

$\left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {{{\it \_a}}^{4}{\it b4}+{{\it \_a}}^{3}{\it b3}+{{\it \_a}}^{2}{\it b2}+{\it \_a}\,{\it b1}+{\it b0}}}}{d{\it \_a}}+\int ^{x}\!-{\sqrt { \left ( {\it b4}\, \left ( y \left ( x \right ) \right ) ^{4}+{\it b3}\, \left ( y \left ( x \right ) \right ) ^{3}+{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{\it b1}\,y \left ( x \right ) +{\it b0} \right ) \left ( {{\it \_a}}^{4}{\it a4}+{{\it \_a}}^{3}{\it a3}+{{\it \_a}}^{2}{\it a2}+{\it \_a}\,{\it a1}+{\it a0} \right ) }{\frac {1}{\sqrt {{\it b4}\, \left ( y \left ( x \right ) \right ) ^{4}+{\it b3}\, \left ( y \left ( x \right ) \right ) ^{3}+{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{\it b1}\,y \left ( x \right ) +{\it b0}}}}}{d{\it \_a}}+{\it \_C1}=0 \right \}$