\[ y'(x)-\sqrt {\left (\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4\right ) \left (\text {b0}+\text {b1} y(x)+\text {b2} y(x)^2+\text {b3} y(x)^3+\text {b4} y(x)^4\right )}=0 \] ✓ Mathematica : cpu = 8.16462 (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 = 0.177 (sec), leaf count = 111
\[\int _{}^{y \relax (x )}\frac {1}{\sqrt {\textit {\_a}^{4} \mathit {b4} +\textit {\_a}^{3} \mathit {b3} +\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0}}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {\left (\mathit {b4} y \relax (x )^{4}+\mathit {b3} y \relax (x )^{3}+\mathit {b2} y \relax (x )^{2}+\mathit {b1} y \relax (x )+\mathit {b0} \right ) \left (\textit {\_a}^{4} \mathit {a4} +\textit {\_a}^{3} \mathit {a3} +\textit {\_a}^{2} \mathit {a2} +\textit {\_a} \mathit {a1} +\mathit {a0} \right )}}{\sqrt {\mathit {b4} y \relax (x )^{4}+\mathit {b3} y \relax (x )^{3}+\mathit {b2} y \relax (x )^{2}+\mathit {b1} y \relax (x )+\mathit {b0}}}d \textit {\_a} +c_{1} = 0\]