4.2 Problem number 811

$$\int \frac{(f+gx{)}^n(a+2cdx+ce{x}^2)}{(d+ex{)}^3}dx$$

Optimal antiderivative $$-\frac{(a-\frac{c{d}^2}{e}){(gx+f)}^{1+n}}{2(-dg+ef){(ex+d)}^2}-\frac{(c{d}^2-ae)g(1-n){(gx+f)}^{1+n}}{2e{(-dg+ef)}^2(ex+d)}+\frac{(ae{g}^2(1-n)n-c(2e^2{f}^2-4defg+d^2{g}^2(-n^2+n+2))){(gx+f)}^{1+n}\mathrm{hypergeom}([1,1+n],[2+n],\frac{e(gx+f)}{-dg+ef})}{2e{(-dg+ef)}^3(1+n)}$$

command

Integrate[((f + g*x)^n*(a + 2*c*d*x + c*e*x^2))/(d + e*x)^3,x]

Mathematica 13.1 output

$$\int \frac{(f+gx{)}^n(a+2cdx+ce{x}^2)}{(d+ex{)}^3}dx$$

Mathematica 12.3 output

$$-\frac{(f+gx{)}^{n+1}(g^2(ae-c{d}^2){}_2{F}_1(3,n+1;n+2;\frac{e(f+gx)}{ef-dg})+c(ef-dg{)}^2{}_2{F}_1(1,n+1;n+2;\frac{e(f+gx)}{ef-dg}))}{e(n+1)(ef-dg{)}^3}$$