\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -{\it R1} \left ( x,\sqrt {{\it a4}\,{x}^{4}+{\it a3}\,{x}^{3}+{\it a2}\,{x}^{2}+{\it a1}\,x+{\it a0}} \right ) {\it R2} \left ( y \left ( x \right ) ,\sqrt {{\it b4}\, \left ( y \left ( x \right ) \right ) ^{4}+{\it b3}\, \left ( y \left ( x \right ) \right ) ^{3}+{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{\it b1}\,y \left ( x \right ) +{\it b0}} \right ) =0} \]
Mathematica: cpu = 0.818104 (sec), leaf count = 87 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{\text {R2}\left (K[1],\sqrt {\text {b1} K[1]+\text {b2} K[1]^2+\text {b3} K[1]^3+\text {b4} K[1]^4+\text {b0}}\right )} \, dK[1]\& \right ]\left [\int _1^x \text {R1}\left (K[2],\sqrt {\text {a1} K[2]+\text {a2} K[2]^2+\text {a3} K[2]^3+\text {a4} K[2]^4+\text {a0}}\right ) \, dK[2]+c_1\right ]\right \}\right \} \]
Maple: cpu = 0.016 (sec), leaf count = 64 \[ \left \{ \int \!{\it R1} \left ( x,\sqrt {{\it a4}\,{x}^{4}+{\it a3}\,{ x}^{3}+{\it a2}\,{x}^{2}+{\it a1}\,x+{\it a0}} \right ) \,{\rm d}x- \int ^{y \left ( x \right ) }\! \left ( {\it R2} \left ( {\it \_a},\sqrt { {{\it \_a}}^{4}{\it b4}+{{\it \_a}}^{3}{\it b3}+{{\it \_a}}^{2}{\it b2 }+{\it \_a}\,{\it b1}+{\it b0}} \right ) \right ) ^{-1}{d{\it \_a}}+{ \it \_C1}=0 \right \} \]