\[ a x y'(x)+b y(x)+c x^2+y'(x)^2=0 \] ✓ Mathematica : cpu = 0.894982 (sec), leaf count = 1085
DSolve[c*x^2 + b*y[x] + a*x*Derivative[1][y][x] + Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\text {Solve}\left [2 \log (x)+\frac {1}{2} \log (y(x))-\log \left (2 b y(x)+\sqrt {-b y(x)} \sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right )+\text {RootSum}\left [a^4-2 \text {$\#$1}^2 a^2-8 c a^2-2 b \text {$\#$1} a^2-4 b \text {$\#$1}^2 a+\text {$\#$1}^4-2 b \text {$\#$1}^3+16 c^2+8 c \text {$\#$1}^2+8 b c \text {$\#$1}\& ,\frac {\log (x) \text {$\#$1}^3-\log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}^3-b \log (x) \text {$\#$1}^2+b \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}^2-a^2 \log (x) \text {$\#$1}-2 a b \log (x) \text {$\#$1}+4 c \log (x) \text {$\#$1}+a^2 \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}+2 a b \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}-4 c \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}-a^2 b \log (x)+4 b c \log (x)+a^2 b \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right )-4 b c \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right )}{-2 \text {$\#$1}^3+3 b \text {$\#$1}^2+2 a^2 \text {$\#$1}+4 a b \text {$\#$1}-8 c \text {$\#$1}+a^2 b-4 b c}\& \right ]=c_1,y(x)\right ],\text {Solve}\left [2 \log (x)+\frac {1}{2} \log (y(x))-\log \left (2 b y(x)+\sqrt {-b y(x)} \sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right )+\text {RootSum}\left [a^4-2 \text {$\#$1}^2 a^2-8 c a^2+2 b \text {$\#$1} a^2-4 b \text {$\#$1}^2 a+\text {$\#$1}^4+2 b \text {$\#$1}^3+16 c^2+8 c \text {$\#$1}^2-8 b c \text {$\#$1}\& ,\frac {-\log (x) \text {$\#$1}^3+\log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}^3-b \log (x) \text {$\#$1}^2+b \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}^2+a^2 \log (x) \text {$\#$1}+2 a b \log (x) \text {$\#$1}-4 c \log (x) \text {$\#$1}-a^2 \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}-2 a b \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}+4 c \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right ) \text {$\#$1}-a^2 b \log (x)+4 b c \log (x)+a^2 b \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right )-4 b c \log \left (x \text {$\#$1}+2 \sqrt {-b y(x)}-\sqrt {\left (a^2-4 c\right ) x^2-4 b y(x)}\right )}{2 \text {$\#$1}^3+3 b \text {$\#$1}^2-2 a^2 \text {$\#$1}-4 a b \text {$\#$1}+8 c \text {$\#$1}+a^2 b-4 b c}\& \right ]=c_1,y(x)\right ]\right \}\] ✗ Maple : cpu = 0. (sec), leaf count = 0
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x)+b*y(x)+c*x^2 = 0,y(x))
, exception
time expired