\[ \int x^m \left (d+e x^2\right )^3 (a+b \text {ArcTan}(c x)) \, dx \]
Optimal antiderivative \[ -\frac {b e \left (e^{2} \left (m^{2}+8 m +15\right )-3 c^{2} d e \left (m^{2}+10 m +21\right )+3 c^{4} d^{2} \left (m^{2}+12 m +35\right )\right ) x^{2+m}}{c^{5} \left (2+m \right ) \left (7+m \right ) \left (m^{2}+8 m +15\right )}+\frac {b \,e^{2} \left (e \left (5+m \right )-3 c^{2} d \left (7+m \right )\right ) x^{4+m}}{c^{3} \left (4+m \right ) \left (5+m \right ) \left (7+m \right )}-\frac {b \,e^{3} x^{6+m}}{c \left (6+m \right ) \left (7+m \right )}+\frac {d^{3} x^{1+m} \left (a +b \arctan \! \left (c x \right )\right )}{1+m}+\frac {3 d^{2} e \,x^{3+m} \left (a +b \arctan \! \left (c x \right )\right )}{3+m}+\frac {3 d \,e^{2} x^{5+m} \left (a +b \arctan \! \left (c x \right )\right )}{5+m}+\frac {e^{3} x^{7+m} \left (a +b \arctan \! \left (c x \right )\right )}{7+m}+\frac {b \left (e^{3} \left (m^{3}+9 m^{2}+23 m +15\right )-3 c^{2} d \,e^{2} \left (m^{3}+11 m^{2}+31 m +21\right )+3 c^{4} d^{2} e \left (m^{3}+13 m^{2}+47 m +35\right )-c^{6} d^{3} \left (m^{3}+15 m^{2}+71 m +105\right )\right ) x^{2+m} \hypergeom \! \left (\left [1, 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], -c^{2} x^{2}\right )}{c^{5} \left (m^{2}+12 m +35\right ) \left (m^{3}+6 m^{2}+11 m +6\right )} \]
command
Integrate[x^m*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]
Mathematica 13.1 output
\[ \int x^m \left (d+e x^2\right )^3 (a+b \text {ArcTan}(c x)) \, dx \]
Mathematica 12.3 output
\[ x^{m+1} \left (\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{m+1}+\frac {3 d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )}{m+3}+\frac {3 d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{m+5}+\frac {e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )}{m+7}-\frac {b c d^3 x \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-c^2 x^2\right )}{m^2+3 m+2}-\frac {3 b c d^2 e x^3 \, _2F_1\left (1,\frac {m+4}{2};\frac {m+6}{2};-c^2 x^2\right )}{m^2+7 m+12}-\frac {3 b c d e^2 x^5 \, _2F_1\left (1,\frac {m+6}{2};\frac {m+8}{2};-c^2 x^2\right )}{(m+5) (m+6)}-\frac {b c e^3 x^7 \, _2F_1\left (1,\frac {m}{2}+4;\frac {m}{2}+5;-c^2 x^2\right )}{(m+7) (m+8)}\right ) \]