##### 4.15.22 $$y'(x) (a (y(x)+x)+1)^n+a (y(x)+x)^n=0$$

ODE
$y'(x) (a (y(x)+x)+1)^n+a (y(x)+x)^n=0$ ODE Classiﬁcation

[[_homogeneous, class C], _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 2.40113 (sec), leaf count = 169

$\text {Solve}\left [c_1=\int _1^x-\frac {a (K[1]+y(x))^n}{a (K[1]+y(x))^n-(a (K[1]+y(x))+1)^n}dK[1]+\int _1^{y(x)}\frac {(a (x+K[2])+1)^n+\left (a (x+K[2])^n-(a (x+K[2])+1)^n\right ) \int _1^x\frac {a n (K[1]+K[2])^{n-1} (a (K[1]+K[2])+1)^{n-1}}{\left ((a (K[1]+K[2])+1)^n-a (K[1]+K[2])^n\right )^2}dK[1]}{(a (x+K[2])+1)^n-a (x+K[2])^n}dK[2],y(x)\right ]$

Maple
cpu = 0.14 (sec), leaf count = 45

$\left [y \left (x \right ) = -x +\RootOf \left (-x -\left (\int _{}^{\textit {\_Z}}-\frac {\left (a \textit {\_a} +1\right )^{n}}{-a \,\textit {\_a}^{n}+\left (a \textit {\_a} +1\right )^{n}}d \textit {\_a} \right )+\textit {\_C1} \right )\right ]$ Mathematica raw input

DSolve[a*(x + y[x])^n + (1 + a*(x + y[x]))^n*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == Inactive[Integrate][-((a*(K[1] + y[x])^n)/(a*(K[1] + y[x])^n - (1
+ a*(K[1] + y[x]))^n)), {K[1], 1, x}] + Inactive[Integrate][((1 + a*(x + K[2]))^
n + (a*(x + K[2])^n - (1 + a*(x + K[2]))^n)*Inactive[Integrate][(a*n*(K[1] + K[2
])^(-1 + n)*(1 + a*(K[1] + K[2]))^(-1 + n))/(-(a*(K[1] + K[2])^n) + (1 + a*(K[1]
 + K[2]))^n)^2, {K[1], 1, x}])/(-(a*(x + K[2])^n) + (1 + a*(x + K[2]))^n), {K[2]
, 1, y[x]}], y[x]]

Maple raw input

dsolve((1+a*(x+y(x)))^n*diff(y(x),x)+a*(x+y(x))^n = 0, y(x))

Maple raw output

[y(x) = -x+RootOf(-x-Intat(-(_a*a+1)^n/(-a*_a^n+(_a*a+1)^n),_a = _Z)+_C1)]