\[ \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {8 c \left (d \left (2 c x +b \right )\right )^{1+m} \hypergeom \! \left (\left [2, \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], \frac {\left (2 c x +b \right )^{2}}{-4 a c +b^{2}}\right )}{\left (-4 a c +b^{2}\right )^{2} d \left (1+m \right )} \]
command
Integrate[(b*d + 2*c*d*x)^m/(a + b*x + c*x^2)^2,x]
Mathematica 13.1 output
\[ \int \frac {(b d+2 c d x)^m}{\left (a+b x+c x^2\right )^2} \, dx \]
Mathematica 12.3 output
\[ \frac {8 c (b+2 c x) (d (b+2 c x))^m \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{(m+1) \left (b^2-4 a c\right )^2} \]