Integrand size = 21, antiderivative size = 239 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) x^2}{630 c^7}-\frac {b e \left (189 c^4 d^2-135 c^2 d e+35 e^2\right ) x^4}{1260 c^5}-\frac {b \left (27 c^2 d-7 e\right ) e^2 x^6}{378 c^3}-\frac {b e^3 x^8}{72 c}+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))+\frac {b \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \log \left (1+c^2 x^2\right )}{630 c^9} \] Output:
-1/630*b*(105*c^6*d^3-189*c^4*d^2*e+135*c^2*d*e^2-35*e^3)*x^2/c^7-1/1260*b *e*(189*c^4*d^2-135*c^2*d*e+35*e^2)*x^4/c^5-1/378*b*(27*c^2*d-7*e)*e^2*x^6 /c^3-1/72*b*e^3*x^8/c+1/3*d^3*x^3*(a+b*arctan(c*x))+3/5*d^2*e*x^5*(a+b*arc tan(c*x))+3/7*d*e^2*x^7*(a+b*arctan(c*x))+1/9*e^3*x^9*(a+b*arctan(c*x))+1/ 630*b*(105*c^6*d^3-189*c^4*d^2*e+135*c^2*d*e^2-35*e^3)*ln(c^2*x^2+1)/c^9
Time = 0.14 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.05 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))+\frac {1}{216} b e^3 \left (\frac {12 x^2}{c^7}-\frac {6 x^4}{c^5}+\frac {4 x^6}{c^3}-\frac {3 x^8}{c}-\frac {12 \log \left (1+c^2 x^2\right )}{c^9}\right )-\frac {1}{28} b d e^2 \left (\frac {6 x^2}{c^5}-\frac {3 x^4}{c^3}+\frac {2 x^6}{c}-\frac {6 \log \left (1+c^2 x^2\right )}{c^7}\right )+\frac {3}{20} b d^2 e \left (\frac {2 x^2}{c^3}-\frac {x^4}{c}-\frac {2 \log \left (1+c^2 x^2\right )}{c^5}\right )-\frac {1}{6} b d^3 \left (\frac {x^2}{c}-\frac {\log \left (1+c^2 x^2\right )}{c^3}\right ) \] Input:
Integrate[x^2*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]
Output:
(d^3*x^3*(a + b*ArcTan[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcTan[c*x]))/5 + (3 *d*e^2*x^7*(a + b*ArcTan[c*x]))/7 + (e^3*x^9*(a + b*ArcTan[c*x]))/9 + (b*e ^3*((12*x^2)/c^7 - (6*x^4)/c^5 + (4*x^6)/c^3 - (3*x^8)/c - (12*Log[1 + c^2 *x^2])/c^9))/216 - (b*d*e^2*((6*x^2)/c^5 - (3*x^4)/c^3 + (2*x^6)/c - (6*Lo g[1 + c^2*x^2])/c^7))/28 + (3*b*d^2*e*((2*x^2)/c^3 - x^4/c - (2*Log[1 + c^ 2*x^2])/c^5))/20 - (b*d^3*(x^2/c - Log[1 + c^2*x^2]/c^3))/6
Time = 0.68 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5511, 27, 2331, 2123, 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.
| \(\displaystyle \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx\) |
| \(\Big \downarrow \) 5511 |
| \(\displaystyle -b c \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{315 \left (c^2 x^2+1\right )}dx+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{c^2 x^2+1}dx+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {1}{630} b c \int \frac {x^2 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{c^2 x^2+1}dx^2+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {1}{630} b c \int \left (\frac {35 e^3 x^6}{c^2}+\frac {5 \left (27 c^2 d-7 e\right ) e^2 x^4}{c^4}+\frac {e \left (189 d^2 c^4-135 d e c^2+35 e^2\right ) x^2}{c^6}+\frac {105 d^3 c^6-189 d^2 e c^4+135 d e^2 c^2-35 e^3}{c^8}+\frac {-105 d^3 c^6+189 d^2 e c^4-135 d e^2 c^2+35 e^3}{c^8 \left (c^2 x^2+1\right )}\right )dx^2+\frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} d^3 x^3 (a+b \arctan (c x))+\frac {3}{5} d^2 e x^5 (a+b \arctan (c x))+\frac {3}{7} d e^2 x^7 (a+b \arctan (c x))+\frac {1}{9} e^3 x^9 (a+b \arctan (c x))-\frac {1}{630} b c \left (\frac {35 e^3 x^8}{4 c^2}+\frac {5 e^2 x^6 \left (27 c^2 d-7 e\right )}{3 c^4}+\frac {e x^4 \left (189 c^4 d^2-135 c^2 d e+35 e^2\right )}{2 c^6}-\frac {\left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right ) \log \left (c^2 x^2+1\right )}{c^{10}}+\frac {x^2 \left (105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right )}{c^8}\right )\) |
Input:
Int[x^2*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]
Output:
(d^3*x^3*(a + b*ArcTan[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcTan[c*x]))/5 + (3 *d*e^2*x^7*(a + b*ArcTan[c*x]))/7 + (e^3*x^9*(a + b*ArcTan[c*x]))/9 - (b*c *(((105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*x^2)/c^8 + (e*(1 89*c^4*d^2 - 135*c^2*d*e + 35*e^2)*x^4)/(2*c^6) + (5*(27*c^2*d - 7*e)*e^2* x^6)/(3*c^4) + (35*e^3*x^8)/(4*c^2) - ((105*c^6*d^3 - 189*c^4*d^2*e + 135* c^2*d*e^2 - 35*e^3)*Log[1 + c^2*x^2])/c^10))/630
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Sim p[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 + c^2 *x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] && !ILt Q[(m - 1)/2, 0]))
Time = 0.35 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.14
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{3} x^{9}+\frac {3}{7} e^{2} d \,x^{7}+\frac {3}{5} d^{2} e \,x^{5}+\frac {1}{3} d^{3} x^{3}\right )+\frac {b \left (\frac {\arctan \left (c x \right ) c^{3} e^{3} x^{9}}{9}+\frac {3 \arctan \left (c x \right ) c^{3} e^{2} d \,x^{7}}{7}+\frac {3 \arctan \left (c x \right ) c^{3} d^{2} e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) d^{3} c^{3} x^{3}}{3}-\frac {\frac {105 c^{8} d^{3} x^{2}}{2}+\frac {189 c^{8} d^{2} e \,x^{4}}{4}+\frac {45 c^{8} d \,e^{2} x^{6}}{2}-\frac {189 c^{6} d^{2} e \,x^{2}}{2}+\frac {35 c^{8} e^{3} x^{8}}{8}-\frac {135 c^{6} d \,e^{2} x^{4}}{4}-\frac {35 c^{6} e^{3} x^{6}}{6}+\frac {135 c^{4} d \,e^{2} x^{2}}{2}+\frac {35 c^{4} e^{3} x^{4}}{4}-\frac {35 c^{2} e^{3} x^{2}}{2}+\frac {\left (-105 c^{6} d^{3}+189 c^{4} d^{2} e -135 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{315 c^{6}}\right )}{c^{3}}\) | \(272\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arctan \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arctan \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arctan \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arctan \left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {c^{8} d^{3} x^{2}}{6}-\frac {3 c^{8} d^{2} e \,x^{4}}{20}+\frac {3 c^{6} d^{2} e \,x^{2}}{10}-\frac {c^{8} d \,e^{2} x^{6}}{14}+\frac {3 c^{6} d \,e^{2} x^{4}}{28}-\frac {c^{8} e^{3} x^{8}}{72}-\frac {3 c^{4} d \,e^{2} x^{2}}{14}+\frac {c^{6} e^{3} x^{6}}{54}-\frac {c^{4} e^{3} x^{4}}{36}+\frac {c^{2} e^{3} x^{2}}{18}-\frac {\left (-105 c^{6} d^{3}+189 c^{4} d^{2} e -135 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{630}\right )}{c^{6}}}{c^{3}}\) | \(285\) |
| default | \(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arctan \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arctan \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arctan \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arctan \left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {c^{8} d^{3} x^{2}}{6}-\frac {3 c^{8} d^{2} e \,x^{4}}{20}+\frac {3 c^{6} d^{2} e \,x^{2}}{10}-\frac {c^{8} d \,e^{2} x^{6}}{14}+\frac {3 c^{6} d \,e^{2} x^{4}}{28}-\frac {c^{8} e^{3} x^{8}}{72}-\frac {3 c^{4} d \,e^{2} x^{2}}{14}+\frac {c^{6} e^{3} x^{6}}{54}-\frac {c^{4} e^{3} x^{4}}{36}+\frac {c^{2} e^{3} x^{2}}{18}-\frac {\left (-105 c^{6} d^{3}+189 c^{4} d^{2} e -135 e^{2} d \,c^{2}+35 e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{630}\right )}{c^{6}}}{c^{3}}\) | \(285\) |
| parallelrisch | \(\frac {840 x^{9} \arctan \left (c x \right ) b \,c^{9} e^{3}+840 a \,c^{9} e^{3} x^{9}+3240 x^{7} \arctan \left (c x \right ) b \,c^{9} d \,e^{2}-105 b \,c^{8} e^{3} x^{8}+3240 a \,c^{9} d \,e^{2} x^{7}+4536 x^{5} \arctan \left (c x \right ) b \,c^{9} d^{2} e -540 b \,c^{8} d \,e^{2} x^{6}+4536 a \,c^{9} d^{2} e \,x^{5}+2520 x^{3} \arctan \left (c x \right ) b \,c^{9} d^{3}+140 b \,c^{6} e^{3} x^{6}-1134 b \,c^{8} d^{2} e \,x^{4}+2520 a \,c^{9} d^{3} x^{3}+810 b \,c^{6} d \,e^{2} x^{4}-1260 b \,c^{8} d^{3} x^{2}-210 b \,c^{4} e^{3} x^{4}+2268 b \,c^{6} d^{2} e \,x^{2}+1260 \ln \left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3}-1620 b \,c^{4} d \,e^{2} x^{2}-2268 \ln \left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e +420 b \,c^{2} e^{3} x^{2}+1620 \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2}-420 \ln \left (c^{2} x^{2}+1\right ) b \,e^{3}}{7560 c^{9}}\) | \(323\) |
| risch | \(-\frac {i b \left (35 e^{3} x^{9}+135 e^{2} d \,x^{7}+189 d^{2} e \,x^{5}+105 d^{3} x^{3}\right ) \ln \left (i c x +1\right )}{630}+\frac {3 i b \,d^{2} e \,x^{5} \ln \left (-i c x +1\right )}{10}+\frac {3 i b d \,e^{2} x^{7} \ln \left (-i c x +1\right )}{14}+\frac {x^{9} e^{3} a}{9}+\frac {i b \,e^{3} x^{9} \ln \left (-i c x +1\right )}{18}+\frac {3 x^{7} e^{2} d a}{7}-\frac {b \,e^{3} x^{8}}{72 c}+\frac {i b \,d^{3} x^{3} \ln \left (-i c x +1\right )}{6}+\frac {3 x^{5} e \,d^{2} a}{5}-\frac {b d \,e^{2} x^{6}}{14 c}+\frac {x^{3} d^{3} a}{3}-\frac {3 b \,d^{2} e \,x^{4}}{20 c}+\frac {b \,e^{3} x^{6}}{54 c^{3}}-\frac {b \,d^{3} x^{2}}{6 c}+\frac {3 b d \,e^{2} x^{4}}{28 c^{3}}+\frac {3 b \,d^{2} e \,x^{2}}{10 c^{3}}-\frac {b \,e^{3} x^{4}}{36 c^{5}}+\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,d^{3}}{6 c^{3}}-\frac {3 b d \,e^{2} x^{2}}{14 c^{5}}-\frac {3 \ln \left (-c^{2} x^{2}-1\right ) b \,d^{2} e}{10 c^{5}}+\frac {b \,e^{3} x^{2}}{18 c^{7}}+\frac {3 \ln \left (-c^{2} x^{2}-1\right ) b d \,e^{2}}{14 c^{7}}-\frac {\ln \left (-c^{2} x^{2}-1\right ) b \,e^{3}}{18 c^{9}}\) | \(368\) |
Input:
int(x^2*(e*x^2+d)^3*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/9*e^3*x^9+3/7*e^2*d*x^7+3/5*d^2*e*x^5+1/3*d^3*x^3)+b/c^3*(1/9*arctan( c*x)*c^3*e^3*x^9+3/7*arctan(c*x)*c^3*e^2*d*x^7+3/5*arctan(c*x)*c^3*d^2*e*x ^5+1/3*arctan(c*x)*d^3*c^3*x^3-1/315/c^6*(105/2*c^8*d^3*x^2+189/4*c^8*d^2* e*x^4+45/2*c^8*d*e^2*x^6-189/2*c^6*d^2*e*x^2+35/8*c^8*e^3*x^8-135/4*c^6*d* e^2*x^4-35/6*c^6*e^3*x^6+135/2*c^4*d*e^2*x^2+35/4*c^4*e^3*x^4-35/2*c^2*e^3 *x^2+1/2*(-105*c^6*d^3+189*c^4*d^2*e-135*c^2*d*e^2+35*e^3)*ln(c^2*x^2+1)))
Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.16 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {840 \, a c^{9} e^{3} x^{9} + 3240 \, a c^{9} d e^{2} x^{7} - 105 \, b c^{8} e^{3} x^{8} + 4536 \, a c^{9} d^{2} e x^{5} + 2520 \, a c^{9} d^{3} x^{3} - 20 \, {\left (27 \, b c^{8} d e^{2} - 7 \, b c^{6} e^{3}\right )} x^{6} - 6 \, {\left (189 \, b c^{8} d^{2} e - 135 \, b c^{6} d e^{2} + 35 \, b c^{4} e^{3}\right )} x^{4} - 12 \, {\left (105 \, b c^{8} d^{3} - 189 \, b c^{6} d^{2} e + 135 \, b c^{4} d e^{2} - 35 \, b c^{2} e^{3}\right )} x^{2} + 24 \, {\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \arctan \left (c x\right ) + 12 \, {\left (105 \, b c^{6} d^{3} - 189 \, b c^{4} d^{2} e + 135 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \log \left (c^{2} x^{2} + 1\right )}{7560 \, c^{9}} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="fricas")
Output:
1/7560*(840*a*c^9*e^3*x^9 + 3240*a*c^9*d*e^2*x^7 - 105*b*c^8*e^3*x^8 + 453 6*a*c^9*d^2*e*x^5 + 2520*a*c^9*d^3*x^3 - 20*(27*b*c^8*d*e^2 - 7*b*c^6*e^3) *x^6 - 6*(189*b*c^8*d^2*e - 135*b*c^6*d*e^2 + 35*b*c^4*e^3)*x^4 - 12*(105* b*c^8*d^3 - 189*b*c^6*d^2*e + 135*b*c^4*d*e^2 - 35*b*c^2*e^3)*x^2 + 24*(35 *b*c^9*e^3*x^9 + 135*b*c^9*d*e^2*x^7 + 189*b*c^9*d^2*e*x^5 + 105*b*c^9*d^3 *x^3)*arctan(c*x) + 12*(105*b*c^6*d^3 - 189*b*c^4*d^2*e + 135*b*c^2*d*e^2 - 35*b*e^3)*log(c^2*x^2 + 1))/c^9
Time = 0.63 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.63 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{3}}{3} + \frac {3 a d^{2} e x^{5}}{5} + \frac {3 a d e^{2} x^{7}}{7} + \frac {a e^{3} x^{9}}{9} + \frac {b d^{3} x^{3} \operatorname {atan}{\left (c x \right )}}{3} + \frac {3 b d^{2} e x^{5} \operatorname {atan}{\left (c x \right )}}{5} + \frac {3 b d e^{2} x^{7} \operatorname {atan}{\left (c x \right )}}{7} + \frac {b e^{3} x^{9} \operatorname {atan}{\left (c x \right )}}{9} - \frac {b d^{3} x^{2}}{6 c} - \frac {3 b d^{2} e x^{4}}{20 c} - \frac {b d e^{2} x^{6}}{14 c} - \frac {b e^{3} x^{8}}{72 c} + \frac {b d^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6 c^{3}} + \frac {3 b d^{2} e x^{2}}{10 c^{3}} + \frac {3 b d e^{2} x^{4}}{28 c^{3}} + \frac {b e^{3} x^{6}}{54 c^{3}} - \frac {3 b d^{2} e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10 c^{5}} - \frac {3 b d e^{2} x^{2}}{14 c^{5}} - \frac {b e^{3} x^{4}}{36 c^{5}} + \frac {3 b d e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{14 c^{7}} + \frac {b e^{3} x^{2}}{18 c^{7}} - \frac {b e^{3} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{18 c^{9}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{3}}{3} + \frac {3 d^{2} e x^{5}}{5} + \frac {3 d e^{2} x^{7}}{7} + \frac {e^{3} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(e*x**2+d)**3*(a+b*atan(c*x)),x)
Output:
Piecewise((a*d**3*x**3/3 + 3*a*d**2*e*x**5/5 + 3*a*d*e**2*x**7/7 + a*e**3* x**9/9 + b*d**3*x**3*atan(c*x)/3 + 3*b*d**2*e*x**5*atan(c*x)/5 + 3*b*d*e** 2*x**7*atan(c*x)/7 + b*e**3*x**9*atan(c*x)/9 - b*d**3*x**2/(6*c) - 3*b*d** 2*e*x**4/(20*c) - b*d*e**2*x**6/(14*c) - b*e**3*x**8/(72*c) + b*d**3*log(x **2 + c**(-2))/(6*c**3) + 3*b*d**2*e*x**2/(10*c**3) + 3*b*d*e**2*x**4/(28* c**3) + b*e**3*x**6/(54*c**3) - 3*b*d**2*e*log(x**2 + c**(-2))/(10*c**5) - 3*b*d*e**2*x**2/(14*c**5) - b*e**3*x**4/(36*c**5) + 3*b*d*e**2*log(x**2 + c**(-2))/(14*c**7) + b*e**3*x**2/(18*c**7) - b*e**3*log(x**2 + c**(-2))/( 18*c**9), Ne(c, 0)), (a*(d**3*x**3/3 + 3*d**2*e*x**5/5 + 3*d*e**2*x**7/7 + e**3*x**9/9), True))
Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.11 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{9} \, a e^{3} x^{9} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{3} + \frac {3}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b d^{2} e + \frac {1}{28} \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b d e^{2} + \frac {1}{216} \, {\left (24 \, x^{9} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{6} x^{8} - 4 \, c^{4} x^{6} + 6 \, c^{2} x^{4} - 12 \, x^{2}}{c^{8}} + \frac {12 \, \log \left (c^{2} x^{2} + 1\right )}{c^{10}}\right )}\right )} b e^{3} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="maxima")
Output:
1/9*a*e^3*x^9 + 3/7*a*d*e^2*x^7 + 3/5*a*d^2*e*x^5 + 1/3*a*d^3*x^3 + 1/6*(2 *x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*b*d^3 + 3/20*(4*x^5 *arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c^2*x^2 + 1)/c^6))*b*d^2*e + 1/28*(12*x^7*arctan(c*x) - c*((2*c^4*x^6 - 3*c^2*x^4 + 6*x^2)/c^6 - 6*l og(c^2*x^2 + 1)/c^8))*b*d*e^2 + 1/216*(24*x^9*arctan(c*x) - c*((3*c^6*x^8 - 4*c^4*x^6 + 6*c^2*x^4 - 12*x^2)/c^8 + 12*log(c^2*x^2 + 1)/c^10))*b*e^3
Time = 0.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.35 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {840 \, b c^{9} e^{3} x^{9} \arctan \left (c x\right ) + 840 \, a c^{9} e^{3} x^{9} + 3240 \, b c^{9} d e^{2} x^{7} \arctan \left (c x\right ) + 3240 \, a c^{9} d e^{2} x^{7} - 105 \, b c^{8} e^{3} x^{8} + 4536 \, b c^{9} d^{2} e x^{5} \arctan \left (c x\right ) + 4536 \, a c^{9} d^{2} e x^{5} - 540 \, b c^{8} d e^{2} x^{6} + 2520 \, b c^{9} d^{3} x^{3} \arctan \left (c x\right ) + 2520 \, a c^{9} d^{3} x^{3} - 1134 \, b c^{8} d^{2} e x^{4} + 140 \, b c^{6} e^{3} x^{6} - 1260 \, b c^{8} d^{3} x^{2} + 810 \, b c^{6} d e^{2} x^{4} + 2268 \, b c^{6} d^{2} e x^{2} - 210 \, b c^{4} e^{3} x^{4} + 1260 \, b c^{6} d^{3} \log \left (c^{2} x^{2} + 1\right ) - 1620 \, b c^{4} d e^{2} x^{2} - 2268 \, b c^{4} d^{2} e \log \left (c^{2} x^{2} + 1\right ) + 420 \, b c^{2} e^{3} x^{2} + 1620 \, b c^{2} d e^{2} \log \left (c^{2} x^{2} + 1\right ) - 420 \, b e^{3} \log \left (c^{2} x^{2} + 1\right )}{7560 \, c^{9}} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="giac")
Output:
1/7560*(840*b*c^9*e^3*x^9*arctan(c*x) + 840*a*c^9*e^3*x^9 + 3240*b*c^9*d*e ^2*x^7*arctan(c*x) + 3240*a*c^9*d*e^2*x^7 - 105*b*c^8*e^3*x^8 + 4536*b*c^9 *d^2*e*x^5*arctan(c*x) + 4536*a*c^9*d^2*e*x^5 - 540*b*c^8*d*e^2*x^6 + 2520 *b*c^9*d^3*x^3*arctan(c*x) + 2520*a*c^9*d^3*x^3 - 1134*b*c^8*d^2*e*x^4 + 1 40*b*c^6*e^3*x^6 - 1260*b*c^8*d^3*x^2 + 810*b*c^6*d*e^2*x^4 + 2268*b*c^6*d ^2*e*x^2 - 210*b*c^4*e^3*x^4 + 1260*b*c^6*d^3*log(c^2*x^2 + 1) - 1620*b*c^ 4*d*e^2*x^2 - 2268*b*c^4*d^2*e*log(c^2*x^2 + 1) + 420*b*c^2*e^3*x^2 + 1620 *b*c^2*d*e^2*log(c^2*x^2 + 1) - 420*b*e^3*log(c^2*x^2 + 1))/c^9
Time = 1.31 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.24 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {a\,d^3\,x^3}{3}+\frac {a\,e^3\,x^9}{9}+\frac {b\,d^3\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {b\,e^3\,\ln \left (c^2\,x^2+1\right )}{18\,c^9}-\frac {b\,d^3\,x^2}{6\,c}-\frac {b\,e^3\,x^8}{72\,c}+\frac {b\,e^3\,x^6}{54\,c^3}-\frac {b\,e^3\,x^4}{36\,c^5}+\frac {b\,e^3\,x^2}{18\,c^7}+\frac {3\,a\,d^2\,e\,x^5}{5}+\frac {3\,a\,d\,e^2\,x^7}{7}+\frac {b\,d^3\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}+\frac {b\,e^3\,x^9\,\mathrm {atan}\left (c\,x\right )}{9}+\frac {3\,b\,d^2\,e\,x^5\,\mathrm {atan}\left (c\,x\right )}{5}+\frac {3\,b\,d\,e^2\,x^7\,\mathrm {atan}\left (c\,x\right )}{7}-\frac {3\,b\,d^2\,e\,\ln \left (c^2\,x^2+1\right )}{10\,c^5}+\frac {3\,b\,d\,e^2\,\ln \left (c^2\,x^2+1\right )}{14\,c^7}-\frac {3\,b\,d^2\,e\,x^4}{20\,c}+\frac {3\,b\,d^2\,e\,x^2}{10\,c^3}-\frac {b\,d\,e^2\,x^6}{14\,c}+\frac {3\,b\,d\,e^2\,x^4}{28\,c^3}-\frac {3\,b\,d\,e^2\,x^2}{14\,c^5} \] Input:
int(x^2*(a + b*atan(c*x))*(d + e*x^2)^3,x)
Output:
(a*d^3*x^3)/3 + (a*e^3*x^9)/9 + (b*d^3*log(c^2*x^2 + 1))/(6*c^3) - (b*e^3* log(c^2*x^2 + 1))/(18*c^9) - (b*d^3*x^2)/(6*c) - (b*e^3*x^8)/(72*c) + (b*e ^3*x^6)/(54*c^3) - (b*e^3*x^4)/(36*c^5) + (b*e^3*x^2)/(18*c^7) + (3*a*d^2* e*x^5)/5 + (3*a*d*e^2*x^7)/7 + (b*d^3*x^3*atan(c*x))/3 + (b*e^3*x^9*atan(c *x))/9 + (3*b*d^2*e*x^5*atan(c*x))/5 + (3*b*d*e^2*x^7*atan(c*x))/7 - (3*b* d^2*e*log(c^2*x^2 + 1))/(10*c^5) + (3*b*d*e^2*log(c^2*x^2 + 1))/(14*c^7) - (3*b*d^2*e*x^4)/(20*c) + (3*b*d^2*e*x^2)/(10*c^3) - (b*d*e^2*x^6)/(14*c) + (3*b*d*e^2*x^4)/(28*c^3) - (3*b*d*e^2*x^2)/(14*c^5)
Time = 0.21 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.35 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {2520 \mathit {atan} \left (c x \right ) b \,c^{9} d^{3} x^{3}+4536 \mathit {atan} \left (c x \right ) b \,c^{9} d^{2} e \,x^{5}+3240 \mathit {atan} \left (c x \right ) b \,c^{9} d \,e^{2} x^{7}+840 \mathit {atan} \left (c x \right ) b \,c^{9} e^{3} x^{9}+1260 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{6} d^{3}-2268 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{4} d^{2} e +1620 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,c^{2} d \,e^{2}-420 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b \,e^{3}+2520 a \,c^{9} d^{3} x^{3}+4536 a \,c^{9} d^{2} e \,x^{5}+3240 a \,c^{9} d \,e^{2} x^{7}+840 a \,c^{9} e^{3} x^{9}-1260 b \,c^{8} d^{3} x^{2}-1134 b \,c^{8} d^{2} e \,x^{4}-540 b \,c^{8} d \,e^{2} x^{6}-105 b \,c^{8} e^{3} x^{8}+2268 b \,c^{6} d^{2} e \,x^{2}+810 b \,c^{6} d \,e^{2} x^{4}+140 b \,c^{6} e^{3} x^{6}-1620 b \,c^{4} d \,e^{2} x^{2}-210 b \,c^{4} e^{3} x^{4}+420 b \,c^{2} e^{3} x^{2}}{7560 c^{9}} \] Input:
int(x^2*(e*x^2+d)^3*(a+b*atan(c*x)),x)
Output:
(2520*atan(c*x)*b*c**9*d**3*x**3 + 4536*atan(c*x)*b*c**9*d**2*e*x**5 + 324 0*atan(c*x)*b*c**9*d*e**2*x**7 + 840*atan(c*x)*b*c**9*e**3*x**9 + 1260*log (c**2*x**2 + 1)*b*c**6*d**3 - 2268*log(c**2*x**2 + 1)*b*c**4*d**2*e + 1620 *log(c**2*x**2 + 1)*b*c**2*d*e**2 - 420*log(c**2*x**2 + 1)*b*e**3 + 2520*a *c**9*d**3*x**3 + 4536*a*c**9*d**2*e*x**5 + 3240*a*c**9*d*e**2*x**7 + 840* a*c**9*e**3*x**9 - 1260*b*c**8*d**3*x**2 - 1134*b*c**8*d**2*e*x**4 - 540*b *c**8*d*e**2*x**6 - 105*b*c**8*e**3*x**8 + 2268*b*c**6*d**2*e*x**2 + 810*b *c**6*d*e**2*x**4 + 140*b*c**6*e**3*x**6 - 1620*b*c**4*d*e**2*x**2 - 210*b *c**4*e**3*x**4 + 420*b*c**2*e**3*x**2)/(7560*c**9)