\[ y'(x)-\sqrt {\frac {\text {b0}+\text {b1} y(x)+\text {b2} y(x)^2+\text {b3} y(x)^3+\text {b4} y(x)^4}{\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4}}=0 \] ✓ Mathematica : cpu = 3.40963 (sec), leaf count = 2237
DSolve[-Sqrt[(b0 + b1*y[x] + b2*y[x]^2 + b3*y[x]^3 + b4*y[x]^4)/(a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4)] + Derivative[1][y][x] == 0,y[x],x]
\[\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-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )}{\left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )}{\left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )}\right ) \left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right )^2 \sqrt {\frac {\left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]\right )}{\left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,3\right ]\right )}} \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right ) \sqrt {\frac {\left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right ) \left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )}{\left (x-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]\right )^2 \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )^2}}}{\sqrt {\text {a0}+x (\text {a1}+x (\text {a2}+x (\text {a3}+\text {a4} x)))} \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {$\#$1}^4+\text {a3} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {a0}\& ,4\right ]\right )},y(x)\right ]\] ✓ Maple : cpu = 0.121 (sec), leaf count = 113
dsolve(diff(y(x),x)-((b4*y(x)^4+b3*y(x)^3+b2*y(x)^2+b1*y(x)+b0)/(a4*x^4+a3*x^3+a2*x^2+a1*x+a0))^(1/2) = 0,y(x))
\[\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4} \mathit {b4} +\textit {\_a}^{3} \mathit {b3} +\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0}}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {\frac {\mathit {b4} y \left (x \right )^{4}+\mathit {b3} y \left (x \right )^{3}+\mathit {b2} y \left (x \right )^{2}+\mathit {b1} y \left (x \right )+\mathit {b0}}{\textit {\_a}^{4} \mathit {a4} +\textit {\_a}^{3} \mathit {a3} +\textit {\_a}^{2} \mathit {a2} +\textit {\_a} \mathit {a1} +\mathit {a0}}}}{\sqrt {\mathit {b4} y \left (x \right )^{4}+\mathit {b3} y \left (x \right )^{3}+\mathit {b2} y \left (x \right )^{2}+\mathit {b1} y \left (x \right )+\mathit {b0}}}d \textit {\_a} +c_{1} = 0\]