Integrand size = 21, antiderivative size = 271 \[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {a b d x}{2 c^3}-\frac {a b e x}{3 c^5}+\frac {b^2 d x^2}{12 c^2}-\frac {4 b^2 e x^2}{45 c^4}+\frac {b^2 e x^4}{60 c^2}+\frac {b^2 d x \arctan (c x)}{2 c^3}-\frac {b^2 e x \arctan (c x)}{3 c^5}-\frac {b d x^3 (a+b \arctan (c x))}{6 c}+\frac {b e x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e x^5 (a+b \arctan (c x))}{15 c}-\frac {d (a+b \arctan (c x))^2}{4 c^4}+\frac {e (a+b \arctan (c x))^2}{6 c^6}+\frac {1}{4} d x^4 (a+b \arctan (c x))^2+\frac {1}{6} e x^6 (a+b \arctan (c x))^2-\frac {b^2 d \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {23 b^2 e \log \left (1+c^2 x^2\right )}{90 c^6} \]
1/2*a*b*d*x/c^3-1/3*a*b*e*x/c^5+1/12*b^2*d*x^2/c^2-4/45*b^2*e*x^2/c^4+1/60 *b^2*e*x^4/c^2+1/2*b^2*d*x*arctan(c*x)/c^3-1/3*b^2*e*x*arctan(c*x)/c^5-1/6 *b*d*x^3*(a+b*arctan(c*x))/c+1/9*b*e*x^3*(a+b*arctan(c*x))/c^3-1/15*b*e*x^ 5*(a+b*arctan(c*x))/c-1/4*d*(a+b*arctan(c*x))^2/c^4+1/6*e*(a+b*arctan(c*x) )^2/c^6+1/4*d*x^4*(a+b*arctan(c*x))^2+1/6*e*x^6*(a+b*arctan(c*x))^2-1/3*b^ 2*d*ln(c^2*x^2+1)/c^4+23/90*b^2*e*ln(c^2*x^2+1)/c^6
Time = 0.13 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.89 \[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {c x \left (15 a^2 c^5 x^3 \left (3 d+2 e x^2\right )+b^2 c x \left (-16 e+3 c^2 \left (5 d+e x^2\right )\right )-2 a b \left (30 e-5 c^2 \left (9 d+2 e x^2\right )+3 c^4 \left (5 d x^2+2 e x^4\right )\right )\right )+2 b \left (b c x \left (-30 e+5 c^2 \left (9 d+2 e x^2\right )-3 c^4 \left (5 d x^2+2 e x^4\right )\right )+15 a \left (-3 c^2 d+2 e+c^6 \left (3 d x^4+2 e x^6\right )\right )\right ) \arctan (c x)+15 b^2 \left (-3 c^2 d+2 e+c^6 \left (3 d x^4+2 e x^6\right )\right ) \arctan (c x)^2+2 b^2 \left (-30 c^2 d+23 e\right ) \log \left (1+c^2 x^2\right )}{180 c^6} \]
(c*x*(15*a^2*c^5*x^3*(3*d + 2*e*x^2) + b^2*c*x*(-16*e + 3*c^2*(5*d + e*x^2 )) - 2*a*b*(30*e - 5*c^2*(9*d + 2*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4))) + 2 *b*(b*c*x*(-30*e + 5*c^2*(9*d + 2*e*x^2) - 3*c^4*(5*d*x^2 + 2*e*x^4)) + 15 *a*(-3*c^2*d + 2*e + c^6*(3*d*x^4 + 2*e*x^6)))*ArcTan[c*x] + 15*b^2*(-3*c^ 2*d + 2*e + c^6*(3*d*x^4 + 2*e*x^6))*ArcTan[c*x]^2 + 2*b^2*(-30*c^2*d + 23 *e)*Log[1 + c^2*x^2])/(180*c^6)
Time = 0.80 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5515, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (d x^3 (a+b \arctan (c x))^2+e x^5 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e (a+b \arctan (c x))^2}{6 c^6}-\frac {d (a+b \arctan (c x))^2}{4 c^4}+\frac {b e x^3 (a+b \arctan (c x))}{9 c^3}+\frac {1}{4} d x^4 (a+b \arctan (c x))^2-\frac {b d x^3 (a+b \arctan (c x))}{6 c}+\frac {1}{6} e x^6 (a+b \arctan (c x))^2-\frac {b e x^5 (a+b \arctan (c x))}{15 c}-\frac {a b e x}{3 c^5}+\frac {a b d x}{2 c^3}-\frac {b^2 e x \arctan (c x)}{3 c^5}+\frac {b^2 d x \arctan (c x)}{2 c^3}-\frac {4 b^2 e x^2}{45 c^4}+\frac {b^2 d x^2}{12 c^2}+\frac {b^2 e x^4}{60 c^2}+\frac {23 b^2 e \log \left (c^2 x^2+1\right )}{90 c^6}-\frac {b^2 d \log \left (c^2 x^2+1\right )}{3 c^4}\) |
(a*b*d*x)/(2*c^3) - (a*b*e*x)/(3*c^5) + (b^2*d*x^2)/(12*c^2) - (4*b^2*e*x^ 2)/(45*c^4) + (b^2*e*x^4)/(60*c^2) + (b^2*d*x*ArcTan[c*x])/(2*c^3) - (b^2* e*x*ArcTan[c*x])/(3*c^5) - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) + (b*e*x^3* (a + b*ArcTan[c*x]))/(9*c^3) - (b*e*x^5*(a + b*ArcTan[c*x]))/(15*c) - (d*( a + b*ArcTan[c*x])^2)/(4*c^4) + (e*(a + b*ArcTan[c*x])^2)/(6*c^6) + (d*x^4 *(a + b*ArcTan[c*x])^2)/4 + (e*x^6*(a + b*ArcTan[c*x])^2)/6 - (b^2*d*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*e*Log[1 + c^2*x^2])/(90*c^6)
3.13.47.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*ArcTan[c*x] )^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d , e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])
Time = 0.34 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.11
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{6} e \,x^{6}+\frac {1}{4} d \,x^{4}\right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{4} e \,x^{6}}{6}+\frac {\arctan \left (c x \right )^{2} c^{4} x^{4} d}{4}-\frac {\frac {2 \arctan \left (c x \right ) e \,c^{5} x^{5}}{5}+\arctan \left (c x \right ) d \,c^{5} x^{3}-\frac {2 \arctan \left (c x \right ) e \,c^{3} x^{3}}{3}-3 \arctan \left (c x \right ) c^{3} x d +2 \arctan \left (c x \right ) e c x +3 \arctan \left (c x \right )^{2} c^{2} d -2 \arctan \left (c x \right )^{2} e -\frac {d \,c^{4} x^{2}}{2}-\frac {e \,c^{4} x^{4}}{10}+\frac {8 e \,c^{2} x^{2}}{15}-\frac {\left (-60 c^{2} d +46 e \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {\left (45 c^{2} d -30 e \right ) \arctan \left (c x \right )^{2}}{30}}{6 c^{2}}\right )}{c^{4}}+\frac {2 a b \left (\frac {c^{4} \arctan \left (c x \right ) e \,x^{6}}{6}+\frac {\arctan \left (c x \right ) d \,c^{4} x^{4}}{4}-\frac {\frac {2 e \,c^{5} x^{5}}{5}+d \,c^{5} x^{3}-\frac {2 e \,c^{3} x^{3}}{3}-3 c^{3} x d +2 e c x +\left (3 c^{2} d -2 e \right ) \arctan \left (c x \right )}{12 c^{2}}\right )}{c^{4}}\) | \(302\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (\frac {1}{4} d \,c^{6} x^{4}+\frac {1}{6} e \,c^{6} x^{6}\right )}{c^{2}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d \,c^{6} x^{4}}{4}+\frac {\arctan \left (c x \right )^{2} e \,c^{6} x^{6}}{6}-\frac {\arctan \left (c x \right ) d \,c^{5} x^{3}}{6}-\frac {\arctan \left (c x \right ) e \,c^{5} x^{5}}{15}+\frac {\arctan \left (c x \right ) c^{3} x d}{2}+\frac {\arctan \left (c x \right ) e \,c^{3} x^{3}}{9}-\frac {\arctan \left (c x \right ) e c x}{3}-\frac {\arctan \left (c x \right )^{2} c^{2} d}{2}+\frac {\arctan \left (c x \right )^{2} e}{3}+\frac {d \,c^{4} x^{2}}{12}+\frac {e \,c^{4} x^{4}}{60}-\frac {4 e \,c^{2} x^{2}}{45}+\frac {\left (-60 c^{2} d +46 e \right ) \ln \left (c^{2} x^{2}+1\right )}{180}+\frac {\left (45 c^{2} d -30 e \right ) \arctan \left (c x \right )^{2}}{180}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\arctan \left (c x \right ) e \,c^{6} x^{6}}{6}-\frac {d \,c^{5} x^{3}}{12}-\frac {e \,c^{5} x^{5}}{30}+\frac {c^{3} x d}{4}+\frac {e \,c^{3} x^{3}}{18}-\frac {e c x}{6}-\frac {\left (3 c^{2} d -2 e \right ) \arctan \left (c x \right )}{12}\right )}{c^{2}}}{c^{4}}\) | \(306\) |
| default | \(\frac {\frac {a^{2} \left (\frac {1}{4} d \,c^{6} x^{4}+\frac {1}{6} e \,c^{6} x^{6}\right )}{c^{2}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d \,c^{6} x^{4}}{4}+\frac {\arctan \left (c x \right )^{2} e \,c^{6} x^{6}}{6}-\frac {\arctan \left (c x \right ) d \,c^{5} x^{3}}{6}-\frac {\arctan \left (c x \right ) e \,c^{5} x^{5}}{15}+\frac {\arctan \left (c x \right ) c^{3} x d}{2}+\frac {\arctan \left (c x \right ) e \,c^{3} x^{3}}{9}-\frac {\arctan \left (c x \right ) e c x}{3}-\frac {\arctan \left (c x \right )^{2} c^{2} d}{2}+\frac {\arctan \left (c x \right )^{2} e}{3}+\frac {d \,c^{4} x^{2}}{12}+\frac {e \,c^{4} x^{4}}{60}-\frac {4 e \,c^{2} x^{2}}{45}+\frac {\left (-60 c^{2} d +46 e \right ) \ln \left (c^{2} x^{2}+1\right )}{180}+\frac {\left (45 c^{2} d -30 e \right ) \arctan \left (c x \right )^{2}}{180}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\arctan \left (c x \right ) e \,c^{6} x^{6}}{6}-\frac {d \,c^{5} x^{3}}{12}-\frac {e \,c^{5} x^{5}}{30}+\frac {c^{3} x d}{4}+\frac {e \,c^{3} x^{3}}{18}-\frac {e c x}{6}-\frac {\left (3 c^{2} d -2 e \right ) \arctan \left (c x \right )}{12}\right )}{c^{2}}}{c^{4}}\) | \(306\) |
| parallelrisch | \(-\frac {-60 x^{6} \arctan \left (c x \right ) a b \,c^{6} e -90 a d b \arctan \left (c x \right ) x^{4} c^{6}-30 c^{6} a^{2} e \,x^{6}-45 c^{6} a^{2} d \,x^{4}-30 \arctan \left (c x \right )^{2} b^{2} e -16 b^{2} e +60 b^{2} c^{2} d \ln \left (c^{2} x^{2}+1\right )-3 b^{2} c^{4} e \,x^{4}-15 b^{2} c^{4} d \,x^{2}+16 b^{2} c^{2} e \,x^{2}-46 b^{2} e \ln \left (c^{2} x^{2}+1\right )+12 a b \,c^{5} e \,x^{5}+30 a b \,c^{5} d \,x^{3}+90 a b \,c^{2} d \arctan \left (c x \right )-20 a b \,c^{3} e \,x^{3}-90 a b \,c^{3} d x +60 a b c e x +45 \arctan \left (c x \right )^{2} b^{2} c^{2} d -45 b^{2} d \arctan \left (c x \right )^{2} x^{4} c^{6}+30 x^{3} \arctan \left (c x \right ) b^{2} c^{5} d -20 x^{3} \arctan \left (c x \right ) b^{2} c^{3} e +12 x^{5} \arctan \left (c x \right ) b^{2} c^{5} e -90 x \arctan \left (c x \right ) b^{2} c^{3} d +60 x \arctan \left (c x \right ) b^{2} c e -30 x^{6} \arctan \left (c x \right )^{2} b^{2} c^{6} e -60 a b e \arctan \left (c x \right )+15 b^{2} c^{2} d}{180 c^{6}}\) | \(354\) |
| risch | \(-\frac {4 b^{2} e \,x^{2}}{45 c^{4}}+\frac {b^{2} e \,x^{4}}{60 c^{2}}+\frac {23 b^{2} e \ln \left (c^{2} x^{2}+1\right )}{90 c^{6}}-\frac {a b e x}{3 c^{5}}+\frac {x^{4} d \,a^{2}}{4}+\frac {x^{6} e \,a^{2}}{6}+\frac {b^{2} d \,x^{2}}{12 c^{2}}-\frac {b^{2} d \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {a b d x}{2 c^{3}}-\frac {i b^{2} e x \ln \left (-i c x +1\right )}{6 c^{5}}+\frac {i a b e \,x^{6} \ln \left (-i c x +1\right )}{6}+\frac {i a b d \,x^{4} \ln \left (-i c x +1\right )}{4}-\frac {b^{2} \left (2 e \,c^{6} x^{6}+3 d \,c^{6} x^{4}-3 c^{2} d +2 e \right ) \ln \left (i c x +1\right )^{2}}{48 c^{6}}-\frac {i b^{2} e \,x^{5} \ln \left (-i c x +1\right )}{30 c}-\frac {i b^{2} d \,x^{3} \ln \left (-i c x +1\right )}{12 c}+\frac {i b^{2} e \,x^{3} \ln \left (-i c x +1\right )}{18 c^{3}}+\frac {i b^{2} d x \ln \left (-i c x +1\right )}{4 c^{3}}-\frac {a b e \,x^{5}}{15 c}-\frac {a b d \,x^{3}}{6 c}+\frac {a b e \,x^{3}}{9 c^{3}}-\frac {a b d \arctan \left (c x \right )}{2 c^{4}}+\frac {a b e \arctan \left (c x \right )}{3 c^{6}}-\frac {b^{2} e \,x^{6} \ln \left (-i c x +1\right )^{2}}{24}-\frac {b^{2} d \,x^{4} \ln \left (-i c x +1\right )^{2}}{16}+\frac {b^{2} d \ln \left (-i c x +1\right )^{2}}{16 c^{4}}-\frac {b^{2} e \ln \left (-i c x +1\right )^{2}}{24 c^{6}}-\frac {i b \left (60 a \,c^{6} e \,x^{6}+30 i b \,c^{6} e \,x^{6} \ln \left (-i c x +1\right )+90 a \,c^{6} d \,x^{4}-12 b \,c^{5} e \,x^{5}+45 i b \,c^{6} d \,x^{4} \ln \left (-i c x +1\right )-30 b \,c^{5} d \,x^{3}+20 b \,c^{3} e \,x^{3}+90 b \,c^{3} d x -60 b c e x -45 i b \,c^{2} d \ln \left (-i c x +1\right )+30 i b e \ln \left (-i c x +1\right )\right ) \ln \left (i c x +1\right )}{360 c^{6}}\) | \(573\) |
a^2*(1/6*e*x^6+1/4*d*x^4)+b^2/c^4*(1/6*arctan(c*x)^2*c^4*e*x^6+1/4*arctan( c*x)^2*c^4*x^4*d-1/6/c^2*(2/5*arctan(c*x)*e*c^5*x^5+arctan(c*x)*d*c^5*x^3- 2/3*arctan(c*x)*e*c^3*x^3-3*arctan(c*x)*c^3*x*d+2*arctan(c*x)*e*c*x+3*arct an(c*x)^2*c^2*d-2*arctan(c*x)^2*e-1/2*d*c^4*x^2-1/10*e*c^4*x^4+8/15*e*c^2* x^2-1/30*(-60*c^2*d+46*e)*ln(c^2*x^2+1)-1/30*(45*c^2*d-30*e)*arctan(c*x)^2 ))+2*a*b/c^4*(1/6*c^4*arctan(c*x)*e*x^6+1/4*arctan(c*x)*d*c^4*x^4-1/12/c^2 *(2/5*e*c^5*x^5+d*c^5*x^3-2/3*e*c^3*x^3-3*c^3*x*d+2*e*c*x+(3*c^2*d-2*e)*ar ctan(c*x)))
Time = 0.30 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07 \[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {30 \, a^{2} c^{6} e x^{6} - 12 \, a b c^{5} e x^{5} + 3 \, {\left (15 \, a^{2} c^{6} d + b^{2} c^{4} e\right )} x^{4} - 10 \, {\left (3 \, a b c^{5} d - 2 \, a b c^{3} e\right )} x^{3} + {\left (15 \, b^{2} c^{4} d - 16 \, b^{2} c^{2} e\right )} x^{2} + 15 \, {\left (2 \, b^{2} c^{6} e x^{6} + 3 \, b^{2} c^{6} d x^{4} - 3 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \arctan \left (c x\right )^{2} + 30 \, {\left (3 \, a b c^{3} d - 2 \, a b c e\right )} x + 2 \, {\left (30 \, a b c^{6} e x^{6} + 45 \, a b c^{6} d x^{4} - 6 \, b^{2} c^{5} e x^{5} - 45 \, a b c^{2} d - 5 \, {\left (3 \, b^{2} c^{5} d - 2 \, b^{2} c^{3} e\right )} x^{3} + 30 \, a b e + 15 \, {\left (3 \, b^{2} c^{3} d - 2 \, b^{2} c e\right )} x\right )} \arctan \left (c x\right ) - 2 \, {\left (30 \, b^{2} c^{2} d - 23 \, b^{2} e\right )} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \]
1/180*(30*a^2*c^6*e*x^6 - 12*a*b*c^5*e*x^5 + 3*(15*a^2*c^6*d + b^2*c^4*e)* x^4 - 10*(3*a*b*c^5*d - 2*a*b*c^3*e)*x^3 + (15*b^2*c^4*d - 16*b^2*c^2*e)*x ^2 + 15*(2*b^2*c^6*e*x^6 + 3*b^2*c^6*d*x^4 - 3*b^2*c^2*d + 2*b^2*e)*arctan (c*x)^2 + 30*(3*a*b*c^3*d - 2*a*b*c*e)*x + 2*(30*a*b*c^6*e*x^6 + 45*a*b*c^ 6*d*x^4 - 6*b^2*c^5*e*x^5 - 45*a*b*c^2*d - 5*(3*b^2*c^5*d - 2*b^2*c^3*e)*x ^3 + 30*a*b*e + 15*(3*b^2*c^3*d - 2*b^2*c*e)*x)*arctan(c*x) - 2*(30*b^2*c^ 2*d - 23*b^2*e)*log(c^2*x^2 + 1))/c^6
Time = 0.52 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.47 \[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\begin {cases} \frac {a^{2} d x^{4}}{4} + \frac {a^{2} e x^{6}}{6} + \frac {a b d x^{4} \operatorname {atan}{\left (c x \right )}}{2} + \frac {a b e x^{6} \operatorname {atan}{\left (c x \right )}}{3} - \frac {a b d x^{3}}{6 c} - \frac {a b e x^{5}}{15 c} + \frac {a b d x}{2 c^{3}} + \frac {a b e x^{3}}{9 c^{3}} - \frac {a b d \operatorname {atan}{\left (c x \right )}}{2 c^{4}} - \frac {a b e x}{3 c^{5}} + \frac {a b e \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {b^{2} d x^{4} \operatorname {atan}^{2}{\left (c x \right )}}{4} + \frac {b^{2} e x^{6} \operatorname {atan}^{2}{\left (c x \right )}}{6} - \frac {b^{2} d x^{3} \operatorname {atan}{\left (c x \right )}}{6 c} - \frac {b^{2} e x^{5} \operatorname {atan}{\left (c x \right )}}{15 c} + \frac {b^{2} d x^{2}}{12 c^{2}} + \frac {b^{2} e x^{4}}{60 c^{2}} + \frac {b^{2} d x \operatorname {atan}{\left (c x \right )}}{2 c^{3}} + \frac {b^{2} e x^{3} \operatorname {atan}{\left (c x \right )}}{9 c^{3}} - \frac {b^{2} d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{4}} - \frac {b^{2} d \operatorname {atan}^{2}{\left (c x \right )}}{4 c^{4}} - \frac {4 b^{2} e x^{2}}{45 c^{4}} - \frac {b^{2} e x \operatorname {atan}{\left (c x \right )}}{3 c^{5}} + \frac {23 b^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{90 c^{6}} + \frac {b^{2} e \operatorname {atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\a^{2} \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d*x**4/4 + a**2*e*x**6/6 + a*b*d*x**4*atan(c*x)/2 + a*b*e* x**6*atan(c*x)/3 - a*b*d*x**3/(6*c) - a*b*e*x**5/(15*c) + a*b*d*x/(2*c**3) + a*b*e*x**3/(9*c**3) - a*b*d*atan(c*x)/(2*c**4) - a*b*e*x/(3*c**5) + a*b *e*atan(c*x)/(3*c**6) + b**2*d*x**4*atan(c*x)**2/4 + b**2*e*x**6*atan(c*x) **2/6 - b**2*d*x**3*atan(c*x)/(6*c) - b**2*e*x**5*atan(c*x)/(15*c) + b**2* d*x**2/(12*c**2) + b**2*e*x**4/(60*c**2) + b**2*d*x*atan(c*x)/(2*c**3) + b **2*e*x**3*atan(c*x)/(9*c**3) - b**2*d*log(x**2 + c**(-2))/(3*c**4) - b**2 *d*atan(c*x)**2/(4*c**4) - 4*b**2*e*x**2/(45*c**4) - b**2*e*x*atan(c*x)/(3 *c**5) + 23*b**2*e*log(x**2 + c**(-2))/(90*c**6) + b**2*e*atan(c*x)**2/(6* c**6), Ne(c, 0)), (a**2*(d*x**4/4 + e*x**6/6), True))
Time = 0.31 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.13 \[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {1}{6} \, b^{2} e x^{6} \arctan \left (c x\right )^{2} + \frac {1}{6} \, a^{2} e x^{6} + \frac {1}{4} \, b^{2} d x^{4} \arctan \left (c x\right )^{2} + \frac {1}{4} \, a^{2} d x^{4} + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b d - \frac {1}{12} \, {\left (2 \, c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d + \frac {1}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b e - \frac {1}{180} \, {\left (4 \, c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac {3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} e \]
1/6*b^2*e*x^6*arctan(c*x)^2 + 1/6*a^2*e*x^6 + 1/4*b^2*d*x^4*arctan(c*x)^2 + 1/4*a^2*d*x^4 + 1/6*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arct an(c*x)/c^5))*a*b*d - 1/12*(2*c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5)* arctan(c*x) - (c^2*x^2 + 3*arctan(c*x)^2 - 4*log(c^2*x^2 + 1))/c^4)*b^2*d + 1/45*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*ar ctan(c*x)/c^7))*a*b*e - 1/180*(4*c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 1 5*arctan(c*x)/c^7)*arctan(c*x) - (3*c^4*x^4 - 16*c^2*x^2 - 30*arctan(c*x)^ 2 + 46*log(c^2*x^2 + 1))/c^6)*b^2*e
\[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]
Time = 1.73 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.25 \[ \int x^3 \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {46\,b^2\,e\,\ln \left (c^2\,x^2+1\right )+30\,b^2\,e\,{\mathrm {atan}\left (c\,x\right )}^2-60\,b^2\,c^2\,d\,\ln \left (c^2\,x^2+1\right )+45\,a^2\,c^6\,d\,x^4+15\,b^2\,c^4\,d\,x^2+30\,a^2\,c^6\,e\,x^6-16\,b^2\,c^2\,e\,x^2+3\,b^2\,c^4\,e\,x^4+60\,a\,b\,e\,\mathrm {atan}\left (c\,x\right )-45\,b^2\,c^2\,d\,{\mathrm {atan}\left (c\,x\right )}^2+45\,b^2\,c^6\,d\,x^4\,{\mathrm {atan}\left (c\,x\right )}^2+30\,b^2\,c^6\,e\,x^6\,{\mathrm {atan}\left (c\,x\right )}^2-30\,a\,b\,c^5\,d\,x^3+20\,a\,b\,c^3\,e\,x^3-12\,a\,b\,c^5\,e\,x^5+90\,b^2\,c^3\,d\,x\,\mathrm {atan}\left (c\,x\right )-60\,a\,b\,c\,e\,x-30\,b^2\,c^5\,d\,x^3\,\mathrm {atan}\left (c\,x\right )+20\,b^2\,c^3\,e\,x^3\,\mathrm {atan}\left (c\,x\right )-12\,b^2\,c^5\,e\,x^5\,\mathrm {atan}\left (c\,x\right )+90\,a\,b\,c^3\,d\,x-90\,a\,b\,c^2\,d\,\mathrm {atan}\left (c\,x\right )-60\,b^2\,c\,e\,x\,\mathrm {atan}\left (c\,x\right )+90\,a\,b\,c^6\,d\,x^4\,\mathrm {atan}\left (c\,x\right )+60\,a\,b\,c^6\,e\,x^6\,\mathrm {atan}\left (c\,x\right )}{180\,c^6} \]
(46*b^2*e*log(c^2*x^2 + 1) + 30*b^2*e*atan(c*x)^2 - 60*b^2*c^2*d*log(c^2*x ^2 + 1) + 45*a^2*c^6*d*x^4 + 15*b^2*c^4*d*x^2 + 30*a^2*c^6*e*x^6 - 16*b^2* c^2*e*x^2 + 3*b^2*c^4*e*x^4 + 60*a*b*e*atan(c*x) - 45*b^2*c^2*d*atan(c*x)^ 2 + 45*b^2*c^6*d*x^4*atan(c*x)^2 + 30*b^2*c^6*e*x^6*atan(c*x)^2 - 30*a*b*c ^5*d*x^3 + 20*a*b*c^3*e*x^3 - 12*a*b*c^5*e*x^5 + 90*b^2*c^3*d*x*atan(c*x) - 60*a*b*c*e*x - 30*b^2*c^5*d*x^3*atan(c*x) + 20*b^2*c^3*e*x^3*atan(c*x) - 12*b^2*c^5*e*x^5*atan(c*x) + 90*a*b*c^3*d*x - 90*a*b*c^2*d*atan(c*x) - 60 *b^2*c*e*x*atan(c*x) + 90*a*b*c^6*d*x^4*atan(c*x) + 60*a*b*c^6*e*x^6*atan( c*x))/(180*c^6)