Integrand size = 21, antiderivative size = 380 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=-\frac {a b d^2 x}{c}+\frac {a b d e x}{c^3}-\frac {a b e^2 x}{3 c^5}+\frac {b^2 d e x^2}{6 c^2}-\frac {4 b^2 e^2 x^2}{45 c^4}+\frac {b^2 e^2 x^4}{60 c^2}-\frac {b^2 d^2 x \arctan (c x)}{c}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b^2 e^2 x \arctan (c x)}{3 c^5}-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {e^2 (a+b \arctan (c x))^2}{6 c^6}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2+\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {2 b^2 d e \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {23 b^2 e^2 \log \left (1+c^2 x^2\right )}{90 c^6} \]
-a*b*d^2*x/c+a*b*d*e*x/c^3-1/3*a*b*e^2*x/c^5+1/6*b^2*d*e*x^2/c^2-4/45*b^2* e^2*x^2/c^4+1/60*b^2*e^2*x^4/c^2-b^2*d^2*x*arctan(c*x)/c+b^2*d*e*x*arctan( c*x)/c^3-1/3*b^2*e^2*x*arctan(c*x)/c^5-1/3*b*d*e*x^3*(a+b*arctan(c*x))/c+1 /9*b*e^2*x^3*(a+b*arctan(c*x))/c^3-1/15*b*e^2*x^5*(a+b*arctan(c*x))/c+1/2* d^2*(a+b*arctan(c*x))^2/c^2-1/2*d*e*(a+b*arctan(c*x))^2/c^4+1/6*e^2*(a+b*a rctan(c*x))^2/c^6+1/2*d^2*x^2*(a+b*arctan(c*x))^2+1/2*d*e*x^4*(a+b*arctan( c*x))^2+1/6*e^2*x^6*(a+b*arctan(c*x))^2+1/2*b^2*d^2*ln(c^2*x^2+1)/c^2-2/3* b^2*d*e*ln(c^2*x^2+1)/c^4+23/90*b^2*e^2*ln(c^2*x^2+1)/c^6
Time = 0.17 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.83 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {c x \left (30 a^2 c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )+b^2 c e x \left (-16 e+3 c^2 \left (10 d+e x^2\right )\right )-4 a b \left (15 e^2-5 c^2 e \left (9 d+e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )\right )+4 b \left (-b c x \left (15 e^2-5 c^2 e \left (9 d+e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )+15 a \left (3 c^4 d^2-3 c^2 d e+e^2+c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right )\right ) \arctan (c x)+30 b^2 \left (3 c^4 d^2-3 c^2 d e+e^2+c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )\right ) \arctan (c x)^2+2 b^2 \left (45 c^4 d^2-60 c^2 d e+23 e^2\right ) \log \left (1+c^2 x^2\right )}{180 c^6} \]
(c*x*(30*a^2*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + b^2*c*e*x*(-16*e + 3*c^ 2*(10*d + e*x^2)) - 4*a*b*(15*e^2 - 5*c^2*e*(9*d + e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4))) + 4*b*(-(b*c*x*(15*e^2 - 5*c^2*e*(9*d + e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4))) + 15*a*(3*c^4*d^2 - 3*c^2*d*e + e^2 + c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6)))*ArcTan[c*x] + 30*b^2*(3*c^4*d^2 - 3*c^2*d*e + e^2 + c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6))*ArcTan[c*x]^2 + 2*b^2*(45*c^4*d^2 - 60*c^2*d*e + 23*e^2)*Log[1 + c^2*x^2])/(180*c^6)
Time = 0.95 (sec) , antiderivative size = 380, 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.
| \(\displaystyle \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5515 |
| \(\displaystyle \int \left (d^2 x (a+b \arctan (c x))^2+2 d e x^3 (a+b \arctan (c x))^2+e^2 x^5 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 (a+b \arctan (c x))^2}{6 c^6}-\frac {d e (a+b \arctan (c x))^2}{2 c^4}+\frac {b e^2 x^3 (a+b \arctan (c x))}{9 c^3}+\frac {d^2 (a+b \arctan (c x))^2}{2 c^2}+\frac {1}{2} d^2 x^2 (a+b \arctan (c x))^2+\frac {1}{2} d e x^4 (a+b \arctan (c x))^2-\frac {b d e x^3 (a+b \arctan (c x))}{3 c}+\frac {1}{6} e^2 x^6 (a+b \arctan (c x))^2-\frac {b e^2 x^5 (a+b \arctan (c x))}{15 c}-\frac {a b e^2 x}{3 c^5}+\frac {a b d e x}{c^3}-\frac {a b d^2 x}{c}-\frac {b^2 e^2 x \arctan (c x)}{3 c^5}+\frac {b^2 d e x \arctan (c x)}{c^3}-\frac {b^2 d^2 x \arctan (c x)}{c}-\frac {4 b^2 e^2 x^2}{45 c^4}+\frac {b^2 d^2 \log \left (c^2 x^2+1\right )}{2 c^2}+\frac {b^2 d e x^2}{6 c^2}+\frac {b^2 e^2 x^4}{60 c^2}+\frac {23 b^2 e^2 \log \left (c^2 x^2+1\right )}{90 c^6}-\frac {2 b^2 d e \log \left (c^2 x^2+1\right )}{3 c^4}\) |
-((a*b*d^2*x)/c) + (a*b*d*e*x)/c^3 - (a*b*e^2*x)/(3*c^5) + (b^2*d*e*x^2)/( 6*c^2) - (4*b^2*e^2*x^2)/(45*c^4) + (b^2*e^2*x^4)/(60*c^2) - (b^2*d^2*x*Ar cTan[c*x])/c + (b^2*d*e*x*ArcTan[c*x])/c^3 - (b^2*e^2*x*ArcTan[c*x])/(3*c^ 5) - (b*d*e*x^3*(a + b*ArcTan[c*x]))/(3*c) + (b*e^2*x^3*(a + b*ArcTan[c*x] ))/(9*c^3) - (b*e^2*x^5*(a + b*ArcTan[c*x]))/(15*c) + (d^2*(a + b*ArcTan[c *x])^2)/(2*c^2) - (d*e*(a + b*ArcTan[c*x])^2)/(2*c^4) + (e^2*(a + b*ArcTan [c*x])^2)/(6*c^6) + (d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + (d*e*x^4*(a + b*Ar cTan[c*x])^2)/2 + (e^2*x^6*(a + b*ArcTan[c*x])^2)/6 + (b^2*d^2*Log[1 + c^2 *x^2])/(2*c^2) - (2*b^2*d*e*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*e^2*Log[1 + c^2*x^2])/(90*c^6)
3.13.56.3.1 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 = 0.43 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.29
| method | result | size |
| parts | \(\frac {a^{2} \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b^{2} \arctan \left (c x \right )^{2} e^{2} x^{6}}{6}+\frac {b^{2} \arctan \left (c x \right )^{2} e \,x^{4} d}{2}+\frac {b^{2} \arctan \left (c x \right )^{2} x^{2} d^{2}}{2}-\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{5}}{15 c}-\frac {b^{2} e \arctan \left (c x \right ) x^{3} d}{3 c}-\frac {b^{2} d^{2} x \arctan \left (c x \right )}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{3}}{9 c^{3}}+\frac {b^{2} d e x \arctan \left (c x \right )}{c^{3}}-\frac {b^{2} e^{2} x \arctan \left (c x \right )}{3 c^{5}}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{2 c^{2}}-\frac {b^{2} e \arctan \left (c x \right )^{2} d}{2 c^{4}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2}}{6 c^{6}}+\frac {b^{2} d e \,x^{2}}{6 c^{2}}+\frac {b^{2} e^{2} x^{4}}{60 c^{2}}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{4}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{6}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{2} e^{2} x^{6}}{6}+\frac {\arctan \left (c x \right ) c^{2} e \,x^{4} d}{2}+\frac {\arctan \left (c x \right ) c^{2} x^{2} d^{2}}{2}+\frac {\arctan \left (c x \right ) c^{2} d^{3}}{6 e}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 c^{4} d^{2} e +3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 c^{4} e}\right )}{c^{2}}\) | \(492\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2} c^{2} x^{2}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e d \,x^{4}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e^{2} x^{6}}{6}-b^{2} c \,d^{2} x \arctan \left (c x \right )-\frac {b^{2} c e \arctan \left (c x \right ) d \,x^{3}}{3}-\frac {b^{2} c \,e^{2} \arctan \left (c x \right ) x^{5}}{15}+\frac {b^{2} d e x \arctan \left (c x \right )}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{3}}{9 c}-\frac {b^{2} e^{2} x \arctan \left (c x \right )}{3 c^{3}}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{2}-\frac {b^{2} e \arctan \left (c x \right )^{2} d}{2 c^{2}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2}}{6 c^{4}}+\frac {b^{2} d e \,x^{2}}{6}+\frac {b^{2} e^{2} x^{4}}{60}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{2}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 c^{4} d^{2} e +3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\) | \(494\) |
| default | \(\frac {\frac {a^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2} c^{2} x^{2}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e d \,x^{4}}{2}+\frac {b^{2} c^{2} \arctan \left (c x \right )^{2} e^{2} x^{6}}{6}-b^{2} c \,d^{2} x \arctan \left (c x \right )-\frac {b^{2} c e \arctan \left (c x \right ) d \,x^{3}}{3}-\frac {b^{2} c \,e^{2} \arctan \left (c x \right ) x^{5}}{15}+\frac {b^{2} d e x \arctan \left (c x \right )}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) x^{3}}{9 c}-\frac {b^{2} e^{2} x \arctan \left (c x \right )}{3 c^{3}}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{2}-\frac {b^{2} e \arctan \left (c x \right )^{2} d}{2 c^{2}}+\frac {b^{2} e^{2} \arctan \left (c x \right )^{2}}{6 c^{4}}+\frac {b^{2} d e \,x^{2}}{6}+\frac {b^{2} e^{2} x^{4}}{60}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{2}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{6} d^{3}}{6 e}+\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{6}-\frac {3 c^{5} d^{2} e x +c^{5} d \,e^{2} x^{3}+\frac {e^{3} c^{5} x^{5}}{5}-3 c^{3} x d \,e^{2}-\frac {e^{3} c^{3} x^{3}}{3}+c x \,e^{3}+\left (c^{6} d^{3}-3 c^{4} d^{2} e +3 e^{2} d \,c^{2}-e^{3}\right ) \arctan \left (c x \right )}{6 e}\right )}{c^{4}}}{c^{2}}\) | \(494\) |
| parallelrisch | \(\frac {46 e^{2} b^{2} \ln \left (c^{2} x^{2}+1\right )+30 \arctan \left (c x \right )^{2} b^{2} e^{2}+180 x^{4} \arctan \left (c x \right ) a b \,c^{6} d e +16 e^{2} b^{2}+180 a b \,c^{4} d^{2} \arctan \left (c x \right )-180 a b \,c^{2} d e \arctan \left (c x \right )+3 b^{2} c^{4} e^{2} x^{4}-16 b^{2} c^{2} e^{2} x^{2}-30 b^{2} c^{2} d e -60 x^{3} \arctan \left (c x \right ) b^{2} c^{5} d e +180 x \arctan \left (c x \right ) b^{2} c^{3} d e -120 b^{2} c^{2} d e \ln \left (c^{2} x^{2}+1\right )-12 a b \,c^{5} e^{2} x^{5}-180 a b \,c^{5} d^{2} x +20 a b \,c^{3} e^{2} x^{3}-60 a b c \,e^{2} x +30 b^{2} c^{4} d e \,x^{2}+30 c^{6} a^{2} e^{2} x^{6}+90 c^{6} a^{2} d^{2} x^{2}+60 x^{6} \arctan \left (c x \right ) a b \,c^{6} e^{2}+180 x^{2} \arctan \left (c x \right ) a b \,c^{6} d^{2}+90 x^{4} \arctan \left (c x \right )^{2} b^{2} c^{6} d e +30 x^{6} \arctan \left (c x \right )^{2} b^{2} c^{6} e^{2}+90 x^{2} \arctan \left (c x \right )^{2} b^{2} c^{6} d^{2}+90 \arctan \left (c x \right )^{2} b^{2} c^{4} d^{2}-90 c^{4} a^{2} d^{2}-60 a b \,c^{5} d e \,x^{3}+180 a b \,c^{3} d e x +60 a b \,e^{2} \arctan \left (c x \right )+90 b^{2} c^{4} d^{2} \ln \left (c^{2} x^{2}+1\right )+90 c^{6} a^{2} d e \,x^{4}-90 \arctan \left (c x \right )^{2} b^{2} c^{2} d e +20 x^{3} \arctan \left (c x \right ) b^{2} c^{3} e^{2}-12 x^{5} \arctan \left (c x \right ) b^{2} c^{5} e^{2}-180 x \arctan \left (c x \right ) b^{2} c^{5} d^{2}-60 x \arctan \left (c x \right ) b^{2} c \,e^{2}}{180 c^{6}}\) | \(537\) |
| risch | \(\frac {x^{6} e^{2} a^{2}}{6}+\frac {x^{2} d^{2} a^{2}}{2}-\frac {4 b^{2} e^{2} x^{2}}{45 c^{4}}+\frac {b^{2} e^{2} x^{4}}{60 c^{2}}+\frac {23 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{6}}-\frac {a b \,e^{2} x}{3 c^{5}}+\frac {b^{2} d e \,x^{2}}{6 c^{2}}-\frac {2 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {a b \,d^{2} x}{c}+\frac {a b d e x}{c^{3}}-\frac {b^{2} e^{2} x^{6} \ln \left (-i c x +1\right )^{2}}{24}-\frac {b^{2} d^{2} x^{2} \ln \left (-i c x +1\right )^{2}}{8}-\frac {b^{2} d^{2} \ln \left (-i c x +1\right )^{2}}{8 c^{2}}-\frac {b^{2} e^{2} \ln \left (-i c x +1\right )^{2}}{24 c^{6}}-\frac {i b \left (30 x^{6} a \,c^{6} e^{2}+15 i b \,c^{6} e^{2} x^{6} \ln \left (-i c x +1\right )+90 x^{4} a \,c^{6} d e -6 b \,c^{5} e^{2} x^{5}+45 i b \,c^{6} d e \,x^{4} \ln \left (-i c x +1\right )+90 x^{2} a \,c^{6} d^{2}-30 b \,c^{5} d e \,x^{3}+45 i b \,c^{6} d^{2} x^{2} \ln \left (-i c x +1\right )-90 b \,c^{5} d^{2} x +10 b \,c^{3} e^{2} x^{3}+90 b \,c^{3} d e x +45 i \ln \left (-i c x +1\right ) b \,c^{4} d^{2}-30 b c \,e^{2} x -45 i \ln \left (-i c x +1\right ) b \,c^{2} d e +15 i \ln \left (-i c x +1\right ) b \,e^{2}\right ) \ln \left (i c x +1\right )}{180 c^{6}}-\frac {b^{2} d e \,x^{4} \ln \left (-i c x +1\right )^{2}}{8}+\frac {b^{2} d e \ln \left (-i c x +1\right )^{2}}{8 c^{4}}-\frac {i b^{2} e^{2} x^{5} \ln \left (-i c x +1\right )}{30 c}-\frac {i b^{2} d^{2} x \ln \left (-i c x +1\right )}{2 c}+\frac {i b^{2} e^{2} x^{3} \ln \left (-i c x +1\right )}{18 c^{3}}-\frac {i b^{2} e^{2} x \ln \left (-i c x +1\right )}{6 c^{5}}+\frac {i a b \,e^{2} x^{6} \ln \left (-i c x +1\right )}{6}-\frac {b^{2} \left (e^{2} c^{6} x^{6}+3 d \,c^{6} e \,x^{4}+3 d^{2} c^{6} x^{2}+3 c^{4} d^{2}-3 c^{2} d e +e^{2}\right ) \ln \left (i c x +1\right )^{2}}{24 c^{6}}+\frac {i a b d e \,x^{4} \ln \left (-i c x +1\right )}{2}-\frac {i b^{2} d e \,x^{3} \ln \left (-i c x +1\right )}{6 c}+\frac {i b^{2} d e x \ln \left (-i c x +1\right )}{2 c^{3}}-\frac {a b d e \,x^{3}}{3 c}-\frac {a b d e \arctan \left (c x \right )}{c^{4}}-\frac {a b \,e^{2} x^{5}}{15 c}+\frac {a b \,e^{2} x^{3}}{9 c^{3}}+\frac {a b \,d^{2} \arctan \left (c x \right )}{c^{2}}+\frac {a b \,e^{2} \arctan \left (c x \right )}{3 c^{6}}+\frac {x^{4} e d \,a^{2}}{2}+\frac {i a b \,d^{2} x^{2} \ln \left (-i c x +1\right )}{2}\) | \(849\) |
1/6*a^2*(e*x^2+d)^3/e+1/6*b^2*arctan(c*x)^2*e^2*x^6+1/2*b^2*arctan(c*x)^2* e*x^4*d+1/2*b^2*arctan(c*x)^2*x^2*d^2-1/15*b^2/c*e^2*arctan(c*x)*x^5-1/3*b ^2/c*e*arctan(c*x)*x^3*d-b^2*d^2*x*arctan(c*x)/c+1/9*b^2/c^3*e^2*arctan(c* x)*x^3+b^2*d*e*x*arctan(c*x)/c^3-1/3*b^2*e^2*x*arctan(c*x)/c^5+1/2*b^2/c^2 *arctan(c*x)^2*d^2-1/2*b^2/c^4*e*arctan(c*x)^2*d+1/6*b^2/c^6*e^2*arctan(c* x)^2+1/6*b^2*d*e*x^2/c^2+1/60*b^2*e^2*x^4/c^2-4/45*b^2*e^2*x^2/c^4+1/2*b^2 *d^2*ln(c^2*x^2+1)/c^2-2/3*b^2*d*e*ln(c^2*x^2+1)/c^4+23/90*b^2*e^2*ln(c^2* x^2+1)/c^6+2*a*b/c^2*(1/6*arctan(c*x)*c^2*e^2*x^6+1/2*arctan(c*x)*c^2*e*x^ 4*d+1/2*arctan(c*x)*c^2*x^2*d^2+1/6*arctan(c*x)*c^2/e*d^3-1/6/c^4/e*(3*c^5 *d^2*e*x+c^5*d*e^2*x^3+1/5*e^3*c^5*x^5-3*c^3*x*d*e^2-1/3*e^3*c^3*x^3+c*x*e ^3+(c^6*d^3-3*c^4*d^2*e+3*c^2*d*e^2-e^3)*arctan(c*x)))
Time = 0.28 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.09 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {30 \, a^{2} c^{6} e^{2} x^{6} - 12 \, a b c^{5} e^{2} x^{5} + 3 \, {\left (30 \, a^{2} c^{6} d e + b^{2} c^{4} e^{2}\right )} x^{4} - 20 \, {\left (3 \, a b c^{5} d e - a b c^{3} e^{2}\right )} x^{3} + 2 \, {\left (45 \, a^{2} c^{6} d^{2} + 15 \, b^{2} c^{4} d e - 8 \, b^{2} c^{2} e^{2}\right )} x^{2} + 30 \, {\left (b^{2} c^{6} e^{2} x^{6} + 3 \, b^{2} c^{6} d e x^{4} + 3 \, b^{2} c^{6} d^{2} x^{2} + 3 \, b^{2} c^{4} d^{2} - 3 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} \arctan \left (c x\right )^{2} - 60 \, {\left (3 \, a b c^{5} d^{2} - 3 \, a b c^{3} d e + a b c e^{2}\right )} x + 4 \, {\left (15 \, a b c^{6} e^{2} x^{6} + 45 \, a b c^{6} d e x^{4} - 3 \, b^{2} c^{5} e^{2} x^{5} + 45 \, a b c^{6} d^{2} x^{2} + 45 \, a b c^{4} d^{2} - 45 \, a b c^{2} d e + 15 \, a b e^{2} - 5 \, {\left (3 \, b^{2} c^{5} d e - b^{2} c^{3} e^{2}\right )} x^{3} - 15 \, {\left (3 \, b^{2} c^{5} d^{2} - 3 \, b^{2} c^{3} d e + b^{2} c e^{2}\right )} x\right )} \arctan \left (c x\right ) + 2 \, {\left (45 \, b^{2} c^{4} d^{2} - 60 \, b^{2} c^{2} d e + 23 \, b^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \]
1/180*(30*a^2*c^6*e^2*x^6 - 12*a*b*c^5*e^2*x^5 + 3*(30*a^2*c^6*d*e + b^2*c ^4*e^2)*x^4 - 20*(3*a*b*c^5*d*e - a*b*c^3*e^2)*x^3 + 2*(45*a^2*c^6*d^2 + 1 5*b^2*c^4*d*e - 8*b^2*c^2*e^2)*x^2 + 30*(b^2*c^6*e^2*x^6 + 3*b^2*c^6*d*e*x ^4 + 3*b^2*c^6*d^2*x^2 + 3*b^2*c^4*d^2 - 3*b^2*c^2*d*e + b^2*e^2)*arctan(c *x)^2 - 60*(3*a*b*c^5*d^2 - 3*a*b*c^3*d*e + a*b*c*e^2)*x + 4*(15*a*b*c^6*e ^2*x^6 + 45*a*b*c^6*d*e*x^4 - 3*b^2*c^5*e^2*x^5 + 45*a*b*c^6*d^2*x^2 + 45* a*b*c^4*d^2 - 45*a*b*c^2*d*e + 15*a*b*e^2 - 5*(3*b^2*c^5*d*e - b^2*c^3*e^2 )*x^3 - 15*(3*b^2*c^5*d^2 - 3*b^2*c^3*d*e + b^2*c*e^2)*x)*arctan(c*x) + 2* (45*b^2*c^4*d^2 - 60*b^2*c^2*d*e + 23*b^2*e^2)*log(c^2*x^2 + 1))/c^6
Time = 0.54 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.51 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\begin {cases} \frac {a^{2} d^{2} x^{2}}{2} + \frac {a^{2} d e x^{4}}{2} + \frac {a^{2} e^{2} x^{6}}{6} + a b d^{2} x^{2} \operatorname {atan}{\left (c x \right )} + a b d e x^{4} \operatorname {atan}{\left (c x \right )} + \frac {a b e^{2} x^{6} \operatorname {atan}{\left (c x \right )}}{3} - \frac {a b d^{2} x}{c} - \frac {a b d e x^{3}}{3 c} - \frac {a b e^{2} x^{5}}{15 c} + \frac {a b d^{2} \operatorname {atan}{\left (c x \right )}}{c^{2}} + \frac {a b d e x}{c^{3}} + \frac {a b e^{2} x^{3}}{9 c^{3}} - \frac {a b d e \operatorname {atan}{\left (c x \right )}}{c^{4}} - \frac {a b e^{2} x}{3 c^{5}} + \frac {a b e^{2} \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {b^{2} d^{2} x^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2} + \frac {b^{2} d e x^{4} \operatorname {atan}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e^{2} x^{6} \operatorname {atan}^{2}{\left (c x \right )}}{6} - \frac {b^{2} d^{2} x \operatorname {atan}{\left (c x \right )}}{c} - \frac {b^{2} d e x^{3} \operatorname {atan}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{15 c} + \frac {b^{2} d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{2}} + \frac {b^{2} d^{2} \operatorname {atan}^{2}{\left (c x \right )}}{2 c^{2}} + \frac {b^{2} d e x^{2}}{6 c^{2}} + \frac {b^{2} e^{2} x^{4}}{60 c^{2}} + \frac {b^{2} d e x \operatorname {atan}{\left (c x \right )}}{c^{3}} + \frac {b^{2} e^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{9 c^{3}} - \frac {2 b^{2} d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{4}} - \frac {b^{2} d e \operatorname {atan}^{2}{\left (c x \right )}}{2 c^{4}} - \frac {4 b^{2} e^{2} x^{2}}{45 c^{4}} - \frac {b^{2} e^{2} x \operatorname {atan}{\left (c x \right )}}{3 c^{5}} + \frac {23 b^{2} e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{90 c^{6}} + \frac {b^{2} e^{2} \operatorname {atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\a^{2} \left (\frac {d^{2} x^{2}}{2} + \frac {d e x^{4}}{2} + \frac {e^{2} x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**2*x**2/2 + a**2*d*e*x**4/2 + a**2*e**2*x**6/6 + a*b*d** 2*x**2*atan(c*x) + a*b*d*e*x**4*atan(c*x) + a*b*e**2*x**6*atan(c*x)/3 - a* b*d**2*x/c - a*b*d*e*x**3/(3*c) - a*b*e**2*x**5/(15*c) + a*b*d**2*atan(c*x )/c**2 + a*b*d*e*x/c**3 + a*b*e**2*x**3/(9*c**3) - a*b*d*e*atan(c*x)/c**4 - a*b*e**2*x/(3*c**5) + a*b*e**2*atan(c*x)/(3*c**6) + b**2*d**2*x**2*atan( c*x)**2/2 + b**2*d*e*x**4*atan(c*x)**2/2 + b**2*e**2*x**6*atan(c*x)**2/6 - b**2*d**2*x*atan(c*x)/c - b**2*d*e*x**3*atan(c*x)/(3*c) - b**2*e**2*x**5* atan(c*x)/(15*c) + b**2*d**2*log(x**2 + c**(-2))/(2*c**2) + b**2*d**2*atan (c*x)**2/(2*c**2) + b**2*d*e*x**2/(6*c**2) + b**2*e**2*x**4/(60*c**2) + b* *2*d*e*x*atan(c*x)/c**3 + b**2*e**2*x**3*atan(c*x)/(9*c**3) - 2*b**2*d*e*l og(x**2 + c**(-2))/(3*c**4) - b**2*d*e*atan(c*x)**2/(2*c**4) - 4*b**2*e**2 *x**2/(45*c**4) - b**2*e**2*x*atan(c*x)/(3*c**5) + 23*b**2*e**2*log(x**2 + c**(-2))/(90*c**6) + b**2*e**2*atan(c*x)**2/(6*c**6), Ne(c, 0)), (a**2*(d **2*x**2/2 + d*e*x**4/2 + e**2*x**6/6), True))
Time = 0.31 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.14 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {1}{6} \, b^{2} e^{2} x^{6} \arctan \left (c x\right )^{2} + \frac {1}{6} \, a^{2} e^{2} x^{6} + \frac {1}{2} \, b^{2} d e x^{4} \arctan \left (c x\right )^{2} + \frac {1}{2} \, a^{2} d e x^{4} + \frac {1}{2} \, b^{2} d^{2} x^{2} \arctan \left (c x\right )^{2} + \frac {1}{2} \, a^{2} d^{2} x^{2} + {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} - \frac {1}{2} \, {\left (2 \, c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac {\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d^{2} + \frac {1}{3} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b d e - \frac {1}{6} \, {\left (2 \, c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d e + \frac {1}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b e^{2} - \frac {1}{180} \, {\left (4 \, c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac {3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} e^{2} \]
1/6*b^2*e^2*x^6*arctan(c*x)^2 + 1/6*a^2*e^2*x^6 + 1/2*b^2*d*e*x^4*arctan(c *x)^2 + 1/2*a^2*d*e*x^4 + 1/2*b^2*d^2*x^2*arctan(c*x)^2 + 1/2*a^2*d^2*x^2 + (x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d^2 - 1/2*(2*c*(x/c^ 2 - arctan(c*x)/c^3)*arctan(c*x) + (arctan(c*x)^2 - log(c^2*x^2 + 1))/c^2) *b^2*d^2 + 1/3*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x) /c^5))*a*b*d*e - 1/6*(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*e + 1/ 45*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan (c*x)/c^7))*a*b*e^2 - 1/180*(4*c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15* 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^2
\[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
Time = 5.64 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.05 \[ \int x \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {a^2\,d^2\,x^2}{2}+\frac {a^2\,e^2\,x^6}{6}+\frac {b^2\,d^2\,\ln \left (c^2\,x^2+1\right )}{2\,c^2}+\frac {23\,b^2\,e^2\,\ln \left (c^2\,x^2+1\right )}{90\,c^6}+\frac {b^2\,e^2\,x^4}{60\,c^2}-\frac {4\,b^2\,e^2\,x^2}{45\,c^4}+\frac {b^2\,d^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2\,c^2}+\frac {b^2\,e^2\,{\mathrm {atan}\left (c\,x\right )}^2}{6\,c^6}+\frac {b^2\,d^2\,x^2\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+\frac {b^2\,e^2\,x^6\,{\mathrm {atan}\left (c\,x\right )}^2}{6}+\frac {a^2\,d\,e\,x^4}{2}-\frac {b^2\,e^2\,x^5\,\mathrm {atan}\left (c\,x\right )}{15\,c}+\frac {b^2\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )}{9\,c^3}-\frac {a\,b\,d^2\,x}{c}-\frac {a\,b\,e^2\,x}{3\,c^5}+a\,b\,d^2\,x^2\,\mathrm {atan}\left (c\,x\right )+\frac {a\,b\,e^2\,x^6\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {2\,b^2\,d\,e\,\ln \left (c^2\,x^2+1\right )}{3\,c^4}-\frac {a\,b\,e^2\,x^5}{15\,c}+\frac {a\,b\,e^2\,x^3}{9\,c^3}+\frac {b^2\,d\,e\,x^2}{6\,c^2}-\frac {b^2\,d\,e\,{\mathrm {atan}\left (c\,x\right )}^2}{2\,c^4}-\frac {b^2\,d^2\,x\,\mathrm {atan}\left (c\,x\right )}{c}-\frac {b^2\,e^2\,x\,\mathrm {atan}\left (c\,x\right )}{3\,c^5}+\frac {b^2\,d\,e\,x^4\,{\mathrm {atan}\left (c\,x\right )}^2}{2}+\frac {a\,b\,d^2\,\mathrm {atan}\left (\frac {b\,c\,e^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}+\frac {3\,b\,c^5\,d^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}-\frac {3\,b\,c^3\,d\,e\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}\right )}{c^2}+\frac {a\,b\,e^2\,\mathrm {atan}\left (\frac {b\,c\,e^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}+\frac {3\,b\,c^5\,d^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}-\frac {3\,b\,c^3\,d\,e\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}\right )}{3\,c^6}-\frac {b^2\,d\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{3\,c}+\frac {a\,b\,d\,e\,x}{c^3}+a\,b\,d\,e\,x^4\,\mathrm {atan}\left (c\,x\right )-\frac {a\,b\,d\,e\,x^3}{3\,c}+\frac {b^2\,d\,e\,x\,\mathrm {atan}\left (c\,x\right )}{c^3}-\frac {a\,b\,d\,e\,\mathrm {atan}\left (\frac {b\,c\,e^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}+\frac {3\,b\,c^5\,d^2\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}-\frac {3\,b\,c^3\,d\,e\,x}{3\,b\,c^4\,d^2-3\,b\,c^2\,d\,e+b\,e^2}\right )}{c^4} \]
(a^2*d^2*x^2)/2 + (a^2*e^2*x^6)/6 + (b^2*d^2*log(c^2*x^2 + 1))/(2*c^2) + ( 23*b^2*e^2*log(c^2*x^2 + 1))/(90*c^6) + (b^2*e^2*x^4)/(60*c^2) - (4*b^2*e^ 2*x^2)/(45*c^4) + (b^2*d^2*atan(c*x)^2)/(2*c^2) + (b^2*e^2*atan(c*x)^2)/(6 *c^6) + (b^2*d^2*x^2*atan(c*x)^2)/2 + (b^2*e^2*x^6*atan(c*x)^2)/6 + (a^2*d *e*x^4)/2 - (b^2*e^2*x^5*atan(c*x))/(15*c) + (b^2*e^2*x^3*atan(c*x))/(9*c^ 3) - (a*b*d^2*x)/c - (a*b*e^2*x)/(3*c^5) + a*b*d^2*x^2*atan(c*x) + (a*b*e^ 2*x^6*atan(c*x))/3 - (2*b^2*d*e*log(c^2*x^2 + 1))/(3*c^4) - (a*b*e^2*x^5)/ (15*c) + (a*b*e^2*x^3)/(9*c^3) + (b^2*d*e*x^2)/(6*c^2) - (b^2*d*e*atan(c*x )^2)/(2*c^4) - (b^2*d^2*x*atan(c*x))/c - (b^2*e^2*x*atan(c*x))/(3*c^5) + ( b^2*d*e*x^4*atan(c*x)^2)/2 + (a*b*d^2*atan((b*c*e^2*x)/(b*e^2 + 3*b*c^4*d^ 2 - 3*b*c^2*d*e) + (3*b*c^5*d^2*x)/(b*e^2 + 3*b*c^4*d^2 - 3*b*c^2*d*e) - ( 3*b*c^3*d*e*x)/(b*e^2 + 3*b*c^4*d^2 - 3*b*c^2*d*e)))/c^2 + (a*b*e^2*atan(( b*c*e^2*x)/(b*e^2 + 3*b*c^4*d^2 - 3*b*c^2*d*e) + (3*b*c^5*d^2*x)/(b*e^2 + 3*b*c^4*d^2 - 3*b*c^2*d*e) - (3*b*c^3*d*e*x)/(b*e^2 + 3*b*c^4*d^2 - 3*b*c^ 2*d*e)))/(3*c^6) - (b^2*d*e*x^3*atan(c*x))/(3*c) + (a*b*d*e*x)/c^3 + a*b*d *e*x^4*atan(c*x) - (a*b*d*e*x^3)/(3*c) + (b^2*d*e*x*atan(c*x))/c^3 - (a*b* d*e*atan((b*c*e^2*x)/(b*e^2 + 3*b*c^4*d^2 - 3*b*c^2*d*e) + (3*b*c^5*d^2*x) /(b*e^2 + 3*b*c^4*d^2 - 3*b*c^2*d*e) - (3*b*c^3*d*e*x)/(b*e^2 + 3*b*c^4*d^ 2 - 3*b*c^2*d*e)))/c^4