##### 4.10.45 $$(6 y(x)-x+5) y'(x)=4 y(x)-x+3$$

ODE
$(6 y(x)-x+5) y'(x)=4 y(x)-x+3$ 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 = 0.335516 (sec), leaf count = 1177

$\left \{\left \{y(x)\to \frac {1}{6} \left (x+\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}+1}-5\right )\right \},\left \{y(x)\to \frac {1}{6} \left (x-\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}-1}-5\right )\right \},\left \{y(x)\to \frac {1}{6} \left (x+\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}+1}-5\right )\right \},\left \{y(x)\to \frac {1}{6} \left (x-\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}-1}-5\right )\right \}\right \}$

Maple
cpu = 0.059 (sec), leaf count = 29

$\left [y \left (x \right ) = -1-\frac {-4 \textit {\_C1} \left (x +1\right )+3+\sqrt {-8 \textit {\_C1} \left (x +1\right )+9}}{8 \textit {\_C1}}\right ]$ Mathematica raw input

DSolve[(5 - x + 6*y[x])*y'[x] == 3 - x + 4*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (-5 + x + (2*(1 + x))/(1 + Sqrt[3/(1 + x)^2 - Sqrt[-((Cosh[(4*C[1])/9]
 + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh
[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1
])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))
] + x*Sqrt[3/(1 + x)^2 - Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2
*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1
+ x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^
2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))]))/6}, {y[x] -> (-5 + x - (2*(
1 + x))/(-1 + Sqrt[3/(1 + x)^2 - Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((
1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2
 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 +
(1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))] + x*Sqrt[3/(1 + x)^2
- Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4
*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9]
 + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1
 + x)^2*Sinh[(4*C[1])/9]))]))/6}, {y[x] -> (-5 + x + (2*(1 + x))/(1 + Sqrt[3/(1
+ x)^2 + Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*
Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C
[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/
9] + (1 + x)^2*Sinh[(4*C[1])/9]))] + x*Sqrt[3/(1 + x)^2 + Sqrt[-((Cosh[(4*C[1])/
9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Si
nh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C
[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]
))]))/6}, {y[x] -> (-5 + x - (2*(1 + x))/(-1 + Sqrt[3/(1 + x)^2 + Sqrt[-((Cosh[(
4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 +
 x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*S
inh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*
C[1])/9]))] + x*Sqrt[3/(1 + x)^2 + Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/
((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] -
(2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1
+ (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))]))/6}}

Maple raw input

dsolve((5-x+6*y(x))*diff(y(x),x) = 3-x+4*y(x), y(x))

Maple raw output

[y(x) = -1-1/8/_C1*(-4*_C1*(x+1)+3+(-8*_C1*(x+1)+9)^(1/2))]