4.883   (a + x)y′(x)2 + y′(x)(a1 + b1x+ c1y(x))+ a2+ b2x+ c2y(x) = 0

ODE

(a+ x)y′(x)2 + y′(x)(a1 +b1x + c1y(x ))+ a2 + b2x+ c2y(x) = 0

ODE Classification

[_dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 242.704 (sec), leaf count = 286

    ⌊ (                                                                                                                                (                                                                                                        (                (                   ) )           )                                                                        )                 ⌋
      ||                                                                                                                                                                                                                                           (b1c1− (c1+2)c2)tan−1 ∘b1+2(c1+1)K$1535584+c2--                                                                                      ||
    || ||||                                                                             (                       (                      ) ) ||       (K$1535584(b1 + c1K$1535584+ c2+ K$1535584)+ b2)−2cc11+2(aK$1535584(c1K$1535584+ 2c2)+ a1c2− a2c1)exp|(                   −b12−2b1
    || ||{                                                                                (b1c1 − (c1 + 2)c2)tan−1  ∘b1+22(c1+1)K$1535584+c22-   ||    ∫                                                                                                          (c1+1) −b1 −2b1c2+4b2(c1+1)−c22                 |
Solve||  x = (c1K$1535584+ c2)(K$1535584(b1 +c1K$1535584 + c2+ K$1535584)+ b2)− c21c1++22 exp (−----------∘------------−b1-−2b1c2+4b2(c1+1)−c2--) ||c1 −   ---------------------------------------------------------------------------------------------------------------------------
    || ||||                                                                                    (c1+ 1)  − b12 − 2b1c2 + 4b2(c1+ 1) − c22    ||                                                                   (c1K$1535584+ c2)                                                                       ||                                 c1K$1535584+ c2                        |
    ⌈ ||||                                                                                                                                (                                                                                                                                                           )                                                                        ||||                 ⌉
      (                                                                                                                                                                                                                                                                                                                                                                     )

Maple
cpu = 0.131 (sec), leaf count = 2085

(
|{                                                                              c1b1    (              ---------1---------)---------1--------- c1c2    (              ---------1---------)---------1---------(  b1-    (              ---------1---------)---------1--------) 2(  -c2     (              --------1---------)---------1---------) 2(∫    (  2                       &
 [x(-T) = -------------T-c1+-c2-------------((c1+ 1)-T2 + (b1+ c2)-T +b2)−2cc11+2 ec1+1arctan (2(c1+1) T+b1+c2)∘−b12−2b1c2+(4c1+4)b2−c22 ∘−b12−2b1c2+(4c1+4)b2−c22ec1+1arctan (2(c1+1)
|(         ((c1 +1) T 2 + (b1 + c2)-T + b2)(c1+1)−1                                                                                                                                                                                                                                                                                                  (                 (-Tc1+ c2)2 &

Mathematica raw input

DSolve[a2 + b2*x + c2*y[x] + (a1 + b1*x + c1*y[x])*y'[x] + (a + x)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

Solve[{x == ((c2 + c1*K$1535584)*(C[1] - Integrate[(E^(((b1*c1 - (2 + c1)*c2)*Ar
cTan[(b1 + c2 + 2*(1 + c1)*K$1535584)/Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^
2]])/((1 + c1)*Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^2]))*(-(a2*c1) + a1*c2 
+ a*K$1535584*(2*c2 + c1*K$1535584)))/((c2 + c1*K$1535584)^2*(b2 + K$1535584*(b1
 + c2 + K$1535584 + c1*K$1535584))^(c1/(2 + 2*c1))), K$1535584]))/(E^(((b1*c1 - 
(2 + c1)*c2)*ArcTan[(b1 + c2 + 2*(1 + c1)*K$1535584)/Sqrt[-b1^2 + 4*b2*(1 + c1) 
- 2*b1*c2 - c2^2]])/((1 + c1)*Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^2]))*(b2
 + K$1535584*(b1 + c2 + K$1535584 + c1*K$1535584))^((2 + c1)/(2 + 2*c1))), y[x] 
== -((a2 + a1*K$1535584 + a*K$1535584^2 + b2*x + b1*K$1535584*x + K$1535584^2*x)
/(c2 + c1*K$1535584))}, {y[x], K$1535584}]

Maple raw input

dsolve((a+x)*diff(y(x),x)^2+(a1+b1*x+c1*y(x))*diff(y(x),x)+a2+b2*x+c2*y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = ((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4
)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/
2))/(c1+1)*c1*b1)*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*
_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2)*exp(1/(-b1^2-2*b
1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*
b2-c2^2)^(1/2))/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan
((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*(_T*c1
+c2)*(Int(-1/(_T*c1+c2)^2*(_T^2*a*c1+2*_T*a*c2+a1*c2-a2*c1)*((c1+1)*_T^2+(b1+c2)
*_T+b2)^(c1/(2*c1+2))*((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-
c2^2)^(1/2))*b1)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)
*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*b1)/((c1+1)*_T^2+(b
1+c2)*_T+b2)/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1
+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*b1)^2/exp(1/(-b1^2-2*b1*c2+(
4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^
2)^(1/2))/(c1+1)*c2)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c
1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2),_T)+_C1)/((
(c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1)))/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1
/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2, y(
_T) = (-((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4)
*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2
))/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T
+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*exp(1/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-
c2^2)^(1/2))/(c1+1)*c1*b1)*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((
2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2)*(_T*c1+
c2)*(_T^2+_T*b1+b2)*Int(-1/(_T*c1+c2)^2*(_T^2*a*c1+2*_T*a*c2+a1*c2-a2*c1)*((c1+1
)*_T^2+(b1+c2)*_T+b2)^(c1/(2*c1+2))*((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1
/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*ar
ctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*b1)/(
(c1+1)*_T^2+(b1+c2)*_T+b2)/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((
2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*b1)^2/exp(1/(-
b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(
4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2
)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c
2),_T)-((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*
b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)
)/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+
b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*exp(1/(-b1^2-2*b1*c2
+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c
2^2)^(1/2))/(c1+1)*c1*c2)*_C1*(_T*c1+c2)*(_T^2+_T*b1+b2)*exp(1/(-b1^2-2*b1*c2+(4
*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2
)^(1/2))/(c1+1)*c1*b1)-((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1/(-b1^2-2*b1*
c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2
-c2^2)^(1/2))*b1)^2*(_T^2*a+_T*a1+a2))/(_T*c1+c2)/(((c1+1)*_T^2+(b1+c2)*_T+b2)^(
1/(c1+1)))/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c
2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2]