\[ \int \frac {(d+e x)^m \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {4 \left (e x +d \right )^{1+m}}{25 e \left (1+m \right )}-\frac {\left (1367 d -293 e +\left (423 d -1367 e \right ) x \right ) \left (e x +d \right )^{1+m}}{700 \left (5 d^{2}-2 d e +3 e^{2}\right ) \left (5 x^{2}+2 x +3\right )}+\frac {\left (e x +d \right )^{1+m} \hypergeom \! \left (\left [1, 1+m \right ], \left [2+m \right ], \frac {5 e x +5 d}{5 d -e \left (1+\mathrm {I} \sqrt {14}\right )}\right ) \left (80360 d^{2}-32144 d e +48216 e^{2}-5922 d e m +19138 e^{2} m -\mathrm {I} \left (6565 d^{2}-2 d e \left (1313-3206 m \right )+e^{2} \left (3939-98 m \right )\right ) \sqrt {14}\right )}{19600 \left (5 d^{2}-2 d e +3 e^{2}\right ) \left (1+m \right ) \left (5 d -e \left (1+\mathrm {I} \sqrt {14}\right )\right )}+\frac {\left (e x +d \right )^{1+m} \hypergeom \! \left (\left [1, 1+m \right ], \left [2+m \right ], \frac {5 e x +5 d}{5 d -e +\mathrm {I} \sqrt {14}\, e}\right ) \left (80360 d^{2}-32144 d e +48216 e^{2}-5922 d e m +19138 e^{2} m +\mathrm {I} \left (6565 d^{2}-2 d e \left (1313-3206 m \right )+e^{2} \left (3939-98 m \right )\right ) \sqrt {14}\right )}{19600 \left (5 d^{2}-2 d e +3 e^{2}\right ) \left (1+m \right ) \left (5 d +\mathrm {I} e \left (\mathrm {I}+\sqrt {14}\right )\right )} \]
command
Integrate[((d + e*x)^m*(2 + x + 3*x^2 - 5*x^3 + 4*x^4))/(3 + 2*x + 5*x^2)^2,x]
Mathematica 13.1 output
\[ \int \frac {(d+e x)^m \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^2} \, dx \]
Mathematica 12.3 output
\[ \frac {(d+e x)^{m+1} \left (-\frac {\sqrt {14} \left (\frac {\left (2115 d^2+d e \left (-846+\left (-6412+423 i \sqrt {14}\right ) m\right )+e^2 \left (1269+\left (98-1367 i \sqrt {14}\right ) m\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac {5 (d+e x)}{5 d+\left (-1-i \sqrt {14}\right ) e}\right )}{5 i d+\left (\sqrt {14}-i\right ) e}-\frac {\left (2115 d^2-d e \left (846+\left (6412+423 i \sqrt {14}\right ) m\right )+e^2 \left (1269+\left (98+1367 i \sqrt {14}\right ) m\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac {5 (d+e x)}{5 d+i \left (i+\sqrt {14}\right ) e}\right )}{5 i d-\left (\sqrt {14}+i\right ) e}\right )}{(m+1) \left (5 d^2-2 d e+3 e^2\right )}-\frac {28 (d (423 x+1367)-e (1367 x+293))}{\left (5 x^2+2 x+3\right ) \left (5 d^2-2 d e+3 e^2\right )}+\frac {56 \left (31 \sqrt {14}+287 i\right ) \, _2F_1\left (1,m+1;m+2;\frac {5 (d+e x)}{5 d+\left (-1-i \sqrt {14}\right ) e}\right )}{(m+1) \left (5 i d+\left (\sqrt {14}-i\right ) e\right )}+\frac {56 \left (31 \sqrt {14}-287 i\right ) \, _2F_1\left (1,m+1;m+2;\frac {5 (d+e x)}{5 d+i \left (i+\sqrt {14}\right ) e}\right )}{(m+1) \left (\left (\sqrt {14}+i\right ) e-5 i d\right )}+\frac {3136}{e m+e}\right )}{19600} \]