12.8 Problem number 1228

$$\int x^m{(d+e{x}^2)}^3(a+b\text{ArcTan}(cx))dx$$

Optimal antiderivative $$-\frac{be(e^2(m^2+8m+15)-3c^{2}de(m^2+10m+21)+3c^4{d}^2(m^2+12m+35))x^{2+m}}{c^5(2+m)(7+m)(m^2+8m+15)}+\frac{b{e}^2(e(5+m)-3c^{2}d(7+m))x^{4+m}}{c^3(4+m)(5+m)(7+m)}-\frac{b{e}^3{x}^{6+m}}{c(6+m)(7+m)}+\frac{d^3{x}^{1+m}(a+b\arctan \left(cx\right))}{1+m}+\frac{3d^{2}e{x}^{3+m}(a+b\arctan \left(cx\right))}{3+m}+\frac{3d{e}^2{x}^{5+m}(a+b\arctan \left(cx\right))}{5+m}+\frac{e^3{x}^{7+m}(a+b\arctan \left(cx\right))}{7+m}+\frac{b(e^3(m^3+9m^2+23m+15)-3c^{2}d{e}^2(m^3+11m^2+31m+21)+3c^4{d}^{2}e(m^3+13m^2+47m+35)-c^6{d}^3(m^3+15m^2+71m+105))x^{2+m}\mathrm{hypergeom}([1,1+\frac{m}{2}],[2+\frac{m}{2}],-c^2{x}^2)}{c^5(m^2+12m+35)(m^3+6m^2+11m+6)}$$

command

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

Mathematica 13.1 output

$$\int x^m{(d+e{x}^2)}^3(a+b\text{ArcTan}(cx))dx$$

Mathematica 12.3 output

$$x^{m+1}(\frac{d^3(a+b{\tan }^{-1}(cx))}{m+1}+\frac{3d^{2}e{x}^2(a+b{\tan }^{-1}(cx))}{m+3}+\frac{3d{e}^2{x}^4(a+b{\tan }^{-1}(cx))}{m+5}+\frac{e^3{x}^6(a+b{\tan }^{-1}(cx))}{m+7}-\frac{bc{d}^{3}x{}_2{F}_1(1,\frac{m+2}{2};\frac{m+4}{2};-c^2{x}^2)}{m^2+3m+2}-\frac{3bc{d}^{2}e{x}^3{}_2{F}_1(1,\frac{m+4}{2};\frac{m+6}{2};-c^2{x}^2)}{m^2+7m+12}-\frac{3bcd{e}^2{x}^5{}_2{F}_1(1,\frac{m+6}{2};\frac{m+8}{2};-c^2{x}^2)}{(m+5)(m+6)}-\frac{bc{e}^3{x}^7{}_2{F}_1(1,\frac{m}{2}+4;\frac{m}{2}+5;-c^2{x}^2)}{(m+7)(m+8)})$$