4.2.45 $y^{\prime }(x)=a{x}^{\frac{n}{1-n}}+by(x{)}^n$

ODE
$$y^{\prime }(x)=a{x}^{\frac{n}{1-n}}+by(x{)}^n$$ ODE Classification

[[_homogeneous, `class G`], _Chini]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 172.366 (sec), leaf count = 109

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

Maple ✓
cpu = 0.305 (sec), leaf count = 61

$$\{-{\int }_{\mathit{\_}\mathit{b}}^{y\left(x\right)}1x^{\frac{n}{n-1}}{((bx(n-1){\mathit{\_}\mathit{a}}^n+\mathit{\_}\mathit{a})x^{\frac{n}{n-1}}+a(n-1)x)}^{-1}\mathrm{d}\mathit{\_}\mathit{a}(n-1)+\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

dsolve(diff(y(x),x) = a*x^(n/(1-n))+b*y(x)^n, y(x),'implicit')

Maple raw output

-Int(1/((b*x*(n-1)*_a^n+_a)*x^(1/(n-1)*n)+a*(n-1)*x)*x^(1/(n-1)*n),_a = _b .. y(
x))*(n-1)+ln(x)-_C1 = 0