\[ a y(x)^3+b y(x)^2+c y(x)+4 y(x) y''(x)-3 y'(x)^2=0 \] ✓ Mathematica : cpu = 3.81569 (sec), leaf count = 2281
DSolve[c*y[x] + b*y[x]^2 + a*y[x]^3 - 3*Derivative[1][y][x]^2 + 4*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\text {Solve}\left [-\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}}\right )|\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}} \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ){}^2 \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2}} \sqrt {y(x)}}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \sqrt {-\frac {1}{3} a y(x)^3-b y(x)^2+c_1 y(x)^{3/2}+c y(x)}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}}\right )|\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,3\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right )}} \sqrt {\frac {\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right )}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ){}^2 \left (\sqrt {y(x)}-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]\right ){}^2}} \sqrt {y(x)}}{\left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,1\right ]\right ) \left (\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,2\right ]-\text {Root}\left [a \text {$\#$1}^4+3 b \text {$\#$1}^2-3 c_1 \text {$\#$1}-3 c\& ,4\right ]\right ) \sqrt {-\frac {1}{3} a y(x)^3-b y(x)^2+c_1 y(x)^{3/2}+c y(x)}}=x+c_2,y(x)\right ]\right \}\] ✓ Maple : cpu = 1.436 (sec), leaf count = 87
dsolve(4*diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2+a*y(x)^3+b*y(x)^2+c*y(x)=0,y(x))
\[y \left (x \right ) = 0\]