##### 4.13.39 $$y'(x) \left (x^2 (a+2 b)+2 x (2 b+c) y(x)+3 c y(x)^2\right )+2 x (a+2 b) y(x)+3 a x^2+(2 b+c) y(x)^2=0$$

ODE
$y'(x) \left (x^2 (a+2 b)+2 x (2 b+c) y(x)+3 c y(x)^2\right )+2 x (a+2 b) y(x)+3 a x^2+(2 b+c) y(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class A], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.763602 (sec), leaf count = 1105

$\left \{\left \{y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (4 b^2-2 c b+c (c-3 a)\right ) x^2}{\sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}-2 (2 b+c) x+2^{2/3} \sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}{6 c}\right \},\left \{y(x)\to \frac {-\frac {2 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) \left (4 b^2-2 c b+c (c-3 a)\right ) x^2}{\sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}-4 (2 b+c) x+i 2^{2/3} \left (i+\sqrt {3}\right ) \sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}{12 c}\right \},\left \{y(x)\to -\frac {\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (4 b^2-2 c b+c (c-3 a)\right ) x^2}{\sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}+4 (2 b+c) x+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}{12 c}\right \}\right \}$

Maple
cpu = 0.297 (sec), leaf count = 3250

$\text {Expression too large to display}$ Mathematica raw input

DSolve[3*a*x^2 + 2*(a + 2*b)*x*y[x] + (2*b + c)*y[x]^2 + ((a + 2*b)*x^2 + 2*(2*b + c)*x*y[x] + 3*c*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-2*(2*b + c)*x + (2*2^(1/3)*(4*b^2 - 2*b*c + c*(-3*a + c))*x^2)/(27*c
^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^
3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(
3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))^2])^(1/3) +
2^(2/3)*(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3
 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^
3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))
^2])^(1/3))/(6*c)}, {y[x] -> (-4*(2*b + c)*x - ((2*I)*2^(1/3)*(-I + Sqrt[3])*(4*
b^2 - 2*b*c + c*(-3*a + c))*x^2)/(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12
*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2
*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^
C[1] + 2*(-3*a + b)*x^3))^2])^(1/3) + I*2^(2/3)*(I + Sqrt[3])*(27*c^2*E^C[1] - 1
6*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^3 - 2*c^3*x^3
 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(3*a + 2*b)*c*
x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))^2])^(1/3))/(12*c)}, {y[x]
 -> -1/12*(4*(2*b + c)*x + (2*2^(1/3)*(1 - I*Sqrt[3])*(4*b^2 - 2*b*c + c*(-3*a +
 c))*x^2)/(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x
^3 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*
b^3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3
))^2])^(1/3) + 2^(2/3)*(1 + I*Sqrt[3])*(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^
3 + 12*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*
c - (2*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2
*(9*E^C[1] + 2*(-3*a + b)*x^3))^2])^(1/3))/c}}

Maple raw input

dsolve(((a+2*b)*x^2+2*(c+2*b)*x*y(x)+3*c*y(x)^2)*diff(y(x),x)+3*a*x^2+2*(a+2*b)*x*y(x)+(c+2*b)*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = (1/6/c*(72*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*
b^2*c*x^3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1
^6*a^2*b^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1
^6*a*b^2*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*
b^3*c*x^6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3
*x^3+24*_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c
^2)^(1/3)-2/3*_C1^2*x^2*(3*a*c-4*b^2+2*b*c-c^2)/c/(72*_C1^3*a*b*c*x^3-72*_C1^3*a
*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+
12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a
^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a
*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*
x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*
_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)-1/3*_C1*x*(c+2*b)/c)/_C1, y(x) = (-
1/12/c*(72*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^
3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b
^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2
*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^
6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*
_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3
)+1/3*_C1^2*x^2*(3*a*c-4*b^2+2*b*c-c^2)/c/(72*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3
-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/
2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x
^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6
-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_
C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^
3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)-1/3*_C1*x*(c+2*b)/c-1/2*I*3^(1/2)*(1/6/c*(7
2*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^
3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16
*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16
*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6
*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2
*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)+2/3*_C1
^2*x^2*(3*a*c-4*b^2+2*b*c-c^2)/c/(72*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3-64*_C1^3
*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/2)*(4*_C1
^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x^6+16*_C1
^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6-16*_C1^6
*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_C1^3*a*c^
2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^3*x^3+27*
c^2)^(1/2)*c+108*c^2)^(1/3)))/_C1, y(x) = (-1/12/c*(72*_C1^3*a*b*c*x^3-72*_C1^3*
a*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3
+12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*
a^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*
a*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c
*x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4
*_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)+1/3*_C1^2*x^2*(3*a*c-4*b^2+2*b*c-c
^2)/c/(72*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3
+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^
2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*
c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6
-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_
C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)
-1/3*_C1*x*(c+2*b)/c+1/2*I*3^(1/2)*(1/6/c*(72*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3
-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^3*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/
2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16*_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x
^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16*_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6
-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6*b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_
C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2*c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^
3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)+2/3*_C1^2*x^2*(3*a*c-4*b^2+2*b*c-c^2)/c/(72
*_C1^3*a*b*c*x^3-72*_C1^3*a*c^2*x^3-64*_C1^3*b^3*x^3+48*_C1^3*b^2*c*x^3+24*_C1^3
*b*c^2*x^3-8*_C1^3*c^3*x^3+12*3^(1/2)*(4*_C1^6*a^3*c*x^6-4*_C1^6*a^2*b^2*x^6-16*
_C1^6*a^2*b*c*x^6+8*_C1^6*a^2*c^2*x^6+16*_C1^6*a*b^3*x^6+8*_C1^6*a*b^2*c*x^6-16*
_C1^6*a*b*c^2*x^6+4*_C1^6*a*c^3*x^6-16*_C1^6*b^4*x^6+16*_C1^6*b^3*c*x^6-4*_C1^6*
b^2*c^2*x^6+36*_C1^3*a*b*c*x^3-36*_C1^3*a*c^2*x^3-32*_C1^3*b^3*x^3+24*_C1^3*b^2*
c*x^3+12*_C1^3*b*c^2*x^3-4*_C1^3*c^3*x^3+27*c^2)^(1/2)*c+108*c^2)^(1/3)))/_C1]