3.53   ODE No. 53

$$\boxed{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)-\frac{{\left(f\left(x\right)\right)}^{1-n}\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){\left(y\left(x\right)\right)}^n}{{(ag\left(x\right)+b)}^n}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)y\left(x\right)}{f\left(x\right)}-f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)=0}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 57.744833 (sec), leaf count = 95 $$\text{Solve}[{\int }_1^{y(x){(f(x{)}^{-n}(ag(x)+b{)}^{-n})}^{\frac{1}{n}}}\frac{1}{-{\left(a^n\right)}^{\frac{1}{n}}K[1]+K[1{]}^n+1}dK[1]=\frac{f(x)(ag(x)+b)\log (ag(x)+b){(f(x{)}^{-n}(ag(x)+b{)}^{-n})}^{\frac{1}{n}}}{a}+c_1,y(x)]$$

Maple: cpu = 0.062 (sec), leaf count = 281 $$\{y\left(x\right)=\frac{(ag\left(x\right)+b)f\left(x\right)}{a}\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-{\int }^{\mathit{\_}\mathit{Z}}\frac{{\left(\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){(ag\left(x\right)+b)}^{-n}{\left(f\left(x\right)\right)}^{1-n}\right)}^{-n-1}{\left(f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{-2n+1}{\left({\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^3{(ag\left(x\right)+b)}^{-n-1}{\left(f\left(x\right)\right)}^{2-n}an\right)}^n{n}^{-n}}{\mathit{\_}\mathit{a}{\left(\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){(ag\left(x\right)+b)}^{-n}{\left(f\left(x\right)\right)}^{1-n}\right)}^{-n-1}{\left(f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{-2n+1}{\left({\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^3{(ag\left(x\right)+b)}^{-n-1}{\left(f\left(x\right)\right)}^{2-n}an\right)}^n{n}^{-n}-{\left(\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){(ag\left(x\right)+b)}^{-n}{\left(f\left(x\right)\right)}^{1-n}\right)}^{-n-1}{\left(f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{-2n+1}{\left({\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^3{(ag\left(x\right)+b)}^{-n-1}{\left(f\left(x\right)\right)}^{2-n}an\right)}^n{n}^{-n}-{\mathit{\_}\mathit{a}}^n}d\mathit{\_}\mathit{a}-\ln (ag\left(x\right)+b)+\mathit{\_}\mathit{C}\mathit{1})\}$$

Sage: cpu = 69.46 (sec), leaf count = 0 $$[[\int -\frac{{(ag\left(x\right)+b)}^{n}f{\left(x\right)}^{n}y\left(x\right)D[0]\left(f\right)\left(x\right)+y{\left(x\right)}^{n}f{\left(x\right)}^{2}D[0]\left(g\right)\left(x\right)+{(ag\left(x\right)+b)}^{n}f{\left(x\right)}^{n+2}D[0]\left(g\right)\left(x\right)}{{(ag\left(x\right)+b)}^{n}af{\left(x\right)}^{n+1}y\left(x\right)-(af{\left(x\right)}^{n+2}g\left(x\right)+bf{\left(x\right)}^{n+2}){(ag\left(x\right)+b)}^n-(af{\left(x\right)}^{2}g\left(x\right)+bf{\left(x\right)}^2)y{\left(x\right)}^n}dx+\int \frac{{(ag\left(x\right)+b)}^{n}f{\left(x\right)}^n-({(ag\left(x\right)+b)}^{n}af{\left(x\right)}^{n}y\left(x\right)-(af{\left(x\right)}^{n+1}g\left(x\right)+bf{\left(x\right)}^{n+1}){(ag\left(x\right)+b)}^n-(af\left(x\right)g\left(x\right)+bf\left(x\right))y{\left(x\right)}^n)\int \frac{{(ag\left(x\right)+b)}^{2n}af{\left(x\right)}^{2n+1}D[0]\left(g\right)\left(x\right)+(af{\left(x\right)}^{2n}g\left(x\right)+bf{\left(x\right)}^{2n}){(ag\left(x\right)+b)}^{2n}D[0]\left(f\right)\left(x\right)-((an-a){(ag\left(x\right)+b)}^{n}f{\left(x\right)}^{n+1}D[0]\left(g\right)\left(x\right)+((an-a)f{\left(x\right)}^{n}g\left(x\right)+(bn-b)f{\left(x\right)}^n){(ag\left(x\right)+b)}^{n}D[0]\left(f\right)\left(x\right))y{\left(x\right)}^n}{{(ag\left(x\right)+b)}^{2n}{a}^{2}f{\left(x\right)}^{2n}y{\left(x\right)}^2-2(a^{2}f{\left(x\right)}^{2n+1}g\left(x\right)+abf{\left(x\right)}^{2n+1}){(ag\left(x\right)+b)}^{2n}y\left(x\right)+(a^{2}f{\left(x\right)}^{2n+2}g{\left(x\right)}^2+2abf{\left(x\right)}^{2n+2}g\left(x\right)+b^{2}f{\left(x\right)}^{2n+2}){(ag\left(x\right)+b)}^{2n}+(a^{2}f{\left(x\right)}^{2}g{\left(x\right)}^2+2abf{\left(x\right)}^{2}g\left(x\right)+b^{2}f{\left(x\right)}^2)y{\left(x\right)}^{2n}-2((a^{2}f{\left(x\right)}^{n+1}g\left(x\right)+abf{\left(x\right)}^{n+1}){(ag\left(x\right)+b)}^{n}y\left(x\right)-(a^{2}f{\left(x\right)}^{n+2}g{\left(x\right)}^2+2abf{\left(x\right)}^{n+2}g\left(x\right)+b^{2}f{\left(x\right)}^{n+2}){(ag\left(x\right)+b)}^n)y{\left(x\right)}^n}dx}{{(ag\left(x\right)+b)}^{n}af{\left(x\right)}^{n}y\left(x\right)-(af{\left(x\right)}^{n+1}g\left(x\right)+bf{\left(x\right)}^{n+1}){(ag\left(x\right)+b)}^n-(af\left(x\right)g\left(x\right)+bf\left(x\right))y{\left(x\right)}^n}d\left(y\left(x\right)\right)=c],\mathtt{\text{lie}}]$$