##### 4.10.8 $$(-y(x)+5 x+1) y'(x)-5 y(x)+x+5=0$$

ODE
$(-y(x)+5 x+1) y'(x)-5 y(x)+x+5=0$ 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 = 0.207128 (sec), leaf count = 925

$\left \{\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {\#1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {\#1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {\#1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {\#1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {\#1}^2-144 e^{\frac {12 c_1}{25}} x \text {\#1}+4 e^{\frac {12 c_1}{25}}\& ,1\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {\#1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {\#1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {\#1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {\#1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {\#1}^2-144 e^{\frac {12 c_1}{25}} x \text {\#1}+4 e^{\frac {12 c_1}{25}}\& ,2\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {\#1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {\#1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {\#1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {\#1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {\#1}^2-144 e^{\frac {12 c_1}{25}} x \text {\#1}+4 e^{\frac {12 c_1}{25}}\& ,3\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {\#1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {\#1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {\#1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {\#1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {\#1}^2-144 e^{\frac {12 c_1}{25}} x \text {\#1}+4 e^{\frac {12 c_1}{25}}\& ,4\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {\#1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {\#1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {\#1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {\#1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {\#1}^2-144 e^{\frac {12 c_1}{25}} x \text {\#1}+4 e^{\frac {12 c_1}{25}}\& ,5\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {\#1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {\#1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {\#1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {\#1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {\#1}^2-144 e^{\frac {12 c_1}{25}} x \text {\#1}+4 e^{\frac {12 c_1}{25}}\& ,6\right ]}+1\right \}\right \}$

Maple
cpu = 0.066 (sec), leaf count = 208

$\left [y \left (x \right ) = 1+\frac {\left (6 \sqrt {3}\, x \sqrt {\frac {x \left (27 x \textit {\_C1} +2\right )}{\textit {\_C1}}}\, \textit {\_C1}^{2}+54 \textit {\_C1}^{2} x^{2}+18 x \textit {\_C1} +1\right )^{\frac {1}{3}}}{6 \textit {\_C1}}+\frac {12 x \textit {\_C1} +1}{6 \textit {\_C1} \left (6 \sqrt {3}\, x \sqrt {\frac {x \left (27 x \textit {\_C1} +2\right )}{\textit {\_C1}}}\, \textit {\_C1}^{2}+54 \textit {\_C1}^{2} x^{2}+18 x \textit {\_C1} +1\right )^{\frac {1}{3}}}-\frac {3 x \textit {\_C1} +1}{3 \textit {\_C1}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 \sqrt {3}\, x \sqrt {\frac {x \left (27 x \textit {\_C1} +2\right )}{\textit {\_C1}}}\, \textit {\_C1}^{2}+54 \textit {\_C1}^{2} x^{2}+18 x \textit {\_C1} +1\right )^{\frac {1}{3}}}{3 \textit {\_C1}}-\frac {12 x \textit {\_C1} +1}{3 \textit {\_C1} \left (6 \sqrt {3}\, x \sqrt {\frac {x \left (27 x \textit {\_C1} +2\right )}{\textit {\_C1}}}\, \textit {\_C1}^{2}+54 \textit {\_C1}^{2} x^{2}+18 x \textit {\_C1} +1\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

DSolve[5 + x - 5*y[x] + (1 + 5*x - y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + (729 +
 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)*#1^3
+ (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^((12*C
[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 1]^(-1)
}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + (729
 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)*#1^
3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^((12
*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 2]^(-
1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + (7
29 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)*#
1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^((
12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 3]^
(-1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 +
(729 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)
*#1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^
((12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 4
]^(-1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1
+ (729 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^
3)*#1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*
E^((12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & ,
 5]^(-1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#
1 + (729 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*
x^3)*#1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 18662
4*E^((12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 &
 , 6]^(-1)}}

Maple raw input

dsolve((1+5*x-y(x))*diff(y(x),x)+5+x-5*y(x) = 0, y(x))

Maple raw output

[y(x) = 1+1/6/_C1*(6*3^(1/2)*x*(1/_C1*x*(27*_C1*x+2))^(1/2)*_C1^2+54*_C1^2*x^2+1
8*x*_C1+1)^(1/3)+1/6*(12*_C1*x+1)/_C1/(6*3^(1/2)*x*(1/_C1*x*(27*_C1*x+2))^(1/2)*
_C1^2+54*_C1^2*x^2+18*x*_C1+1)^(1/3)-1/3*(3*_C1*x+1)/_C1-1/2*I*3^(1/2)*(1/3/_C1*
(6*3^(1/2)*x*(1/_C1*x*(27*_C1*x+2))^(1/2)*_C1^2+54*_C1^2*x^2+18*x*_C1+1)^(1/3)-1
/3*(12*_C1*x+1)/_C1/(6*3^(1/2)*x*(1/_C1*x*(27*_C1*x+2))^(1/2)*_C1^2+54*_C1^2*x^2
+18*x*_C1+1)^(1/3))]