\[ y'(x)-\frac {1}{\sqrt {\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4}}=0 \] ✓ Mathematica : cpu = 10.2818 (sec), leaf count = 1117
DSolve[-(1/Sqrt[a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4]) + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1-\frac {2 \operatorname {EllipticF}\left (\arcsin \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 )}\right \}\right \}\] ✓ Maple : cpu = 0.02 (sec), leaf count = 1089
dsolve(diff(y(x),x)-1/(a4*x^4+a3*x^3+a2*x^2+a1*x+a0)^(1/2) = 0,y(x))
\[\text {Expression too large to display}\]
Hand solution
\begin {equation} y^{\prime }-\frac {1}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}=0\tag {1} \end {equation}
To Do.