##### 4.10.31 $$(x-3 y(x)) y'(x)-y(x)+3 x+4=0$$

ODE
$(x-3 y(x)) y'(x)-y(x)+3 x+4=0$ ODE Classiﬁcation

[[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]

Book solution method
Equation linear in the variables, $$y'(x)=f\left ( \frac {X_1}{X_2} \right )$$

Mathematica
cpu = 0.221946 (sec), leaf count = 781

$\left \{\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {\#1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\right )+\text {\#1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {\#1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {\#1}^2 \left (36 x^2+108 x+81\right )+\text {\#1} (-12 x-18)+1\& ,1\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {\#1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\right )+\text {\#1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {\#1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {\#1}^2 \left (36 x^2+108 x+81\right )+\text {\#1} (-12 x-18)+1\& ,2\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {\#1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\right )+\text {\#1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {\#1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {\#1}^2 \left (36 x^2+108 x+81\right )+\text {\#1} (-12 x-18)+1\& ,3\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {\#1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\right )+\text {\#1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {\#1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {\#1}^2 \left (36 x^2+108 x+81\right )+\text {\#1} (-12 x-18)+1\& ,4\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {\#1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\right )+\text {\#1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {\#1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {\#1}^2 \left (36 x^2+108 x+81\right )+\text {\#1} (-12 x-18)+1\& ,5\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {\#1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\right )+\text {\#1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {\#1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {\#1}^2 \left (36 x^2+108 x+81\right )+\text {\#1} (-12 x-18)+1\& ,6\right ]}\right )\right \}\right \}$

Maple
cpu = 0.249 (sec), leaf count = 242

$\left [y \left (x \right ) = -\frac {1}{2}-\frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (-144 \left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}-\frac {3456 \left (3+2 x \right )^{3} \textit {\_C1}}{\left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+864 i \sqrt {3}\, \left (\frac {\left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {4 \left (3+2 x \right )^{3} \textit {\_C1}}{\left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{2985984 \textit {\_C1}}-\left (3+2 x \right )^{3}}{2 \left (3+2 x \right )^{2}}\right ]$ Mathematica raw input

DSolve[4 + 3*x - y[x] + (x - 3*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x - Root[1 + (-18 - 12*x)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 43
2*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*
#1^4 + (11664 + 16*E^(12*C[1]) + 46656*x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9
216*x^5 + 1024*x^6)*#1^6 & , 1]^(-1))/3}, {y[x] -> (x - Root[1 + (-18 - 12*x)*#1
 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 -
 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 46656*
x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9216*x^5 + 1024*x^6)*#1^6 & , 2]^(-1))/3
}, {y[x] -> (x - Root[1 + (-18 - 12*x)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 +
432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4
)*#1^4 + (11664 + 16*E^(12*C[1]) + 46656*x + 77760*x^2 + 69120*x^3 + 34560*x^4 +
 9216*x^5 + 1024*x^6)*#1^6 & , 3]^(-1))/3}, {y[x] -> (x - Root[1 + (-18 - 12*x)*
#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944
 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 4665
6*x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9216*x^5 + 1024*x^6)*#1^6 & , 4]^(-1))
/3}, {y[x] -> (x - Root[1 + (-18 - 12*x)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216
+ 432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x
^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 46656*x + 77760*x^2 + 69120*x^3 + 34560*x^4
 + 9216*x^5 + 1024*x^6)*#1^6 & , 5]^(-1))/3}, {y[x] -> (x - Root[1 + (-18 - 12*x
)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 432*x + 288*x^2 + 64*x^3)*#1^3 + (-19
44 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 46
656*x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9216*x^5 + 1024*x^6)*#1^6 & , 6]^(-1
))/3}}

Maple raw input

dsolve((x-3*y(x))*diff(y(x),x)+4+3*x-y(x) = 0, y(x))

Maple raw output

[y(x) = -1/2-1/2*(1/2985984/_C1*(-1/2+1/2*I*3^(1/2))^6*(-144*(108*(3+2*x)^3*_C1+
12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))^(1/3)-3456*(3+2*x)^3*_C1/(108
*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))^(1/3)+864*I*3^
(1/2)*(1/6*(108*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))
^(1/3)-4*(3+2*x)^3*_C1/(108*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_
C1^2)^(1/2))^(1/3)))^2-(3+2*x)^3)/(3+2*x)^2]