\[ y'(x)-\text {R1}\left (x,\sqrt {\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4}\right ) \text {R2}\left (y(x),\sqrt {\text {b0}+\text {b1} y(x)+\text {b2} y(x)^2+\text {b3} y(x)^3+\text {b4} y(x)^4}\right )=0 \] ✓ Mathematica : cpu = 0.187597 (sec), leaf count = 89
DSolve[-(R1[x, Sqrt[a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4]]*R2[y[x], Sqrt[b0 + b1*y[x] + b2*y[x]^2 + b3*y[x]^3 + b4*y[x]^4]]) + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {R2}\left (K[1],\sqrt {\text {b4} K[1]^4+\text {b3} K[1]^3+\text {b2} K[1]^2+\text {b1} K[1]+\text {b0}}\right )}dK[1]\& \right ]\left [\int _1^x\text {R1}\left (K[2],\sqrt {\text {a4} K[2]^4+\text {a3} K[2]^3+\text {a2} K[2]^2+\text {a1} K[2]+\text {a0}}\right )dK[2]+c_1\right ]\right \}\right \}\] ✓ Maple : cpu = 0.02 (sec), leaf count = 64
dsolve(diff(y(x),x)-R1(x,(a4*x^4+a3*x^3+a2*x^2+a1*x+a0)^(1/2))*R2(y(x),(b4*y(x)^4+b3*y(x)^3+b2*y(x)^2+b1*y(x)+b0)^(1/2)) = 0,y(x))
\[\int \mathit {R1} \left (x , \sqrt {\mathit {a4} \,x^{4}+\mathit {a3} \,x^{3}+\mathit {a2} \,x^{2}+\mathit {a1} x +\mathit {a0}}\right )d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\mathit {R2} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} \mathit {b4} +\textit {\_a}^{3} \mathit {b3} +\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0}}\right )}d \textit {\_a} \right )+c_{1} = 0\]