3.2 Problem number 1428

$$\int \frac{(bd+2cdx{)}^m}{{(a+bx+c{x}^2)}^3}dx$$

Optimal antiderivative $$-\frac{32c^2{\left(d(2cx+b)\right)}^{1+m}\mathrm{hypergeom}([3,\frac{1}{2}+\frac{m}{2}],[\frac{3}{2}+\frac{m}{2}],\frac{{(2cx+b)}^2}{-4ac+b^2})}{{(-4ac+b^2)}^{3}d(1+m)}$$

command

Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^3,x]

Mathematica 13.1 output

$$\int \frac{(bd+2cdx{)}^m}{{(a+bx+c{x}^2)}^3}dx$$

Mathematica 12.3 output

$$-\frac{32c^2(b+2cx)(d(b+2cx){)}^m{}_2{F}_1(3,\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2cx{)}^2}{b^2-4ac})}{(m+1){(b^2-4ac)}^3}$$