\(\int x^3 (d+e x^2)^3 (a+b \arctan (c x)) \, dx\) [1136]

Optimal result

Integrand size = 21, antiderivative size = 240 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {b \left (10 c^6 d^3-20 c^4 d^2 e+15 c^2 d e^2-4 e^3\right ) x}{40 c^9}-\frac {b \left (10 c^6 d^3-20 c^4 d^2 e+15 c^2 d e^2-4 e^3\right ) x^3}{120 c^7}-\frac {b e \left (20 c^4 d^2-15 c^2 d e+4 e^2\right ) x^5}{200 c^5}-\frac {b \left (15 c^2 d-4 e\right ) e^2 x^7}{280 c^3}-\frac {b e^3 x^9}{90 c}+\frac {b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right ) \arctan (c x)}{40 c^{10} e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2} \] Output:

1/40*b*(10*c^6*d^3-20*c^4*d^2*e+15*c^2*d*e^2-4*e^3)*x/c^9-1/120*b*(10*c^6* 
d^3-20*c^4*d^2*e+15*c^2*d*e^2-4*e^3)*x^3/c^7-1/200*b*e*(20*c^4*d^2-15*c^2* 
d*e+4*e^2)*x^5/c^5-1/280*b*(15*c^2*d-4*e)*e^2*x^7/c^3-1/90*b*e^3*x^9/c+1/4 
0*b*(c^2*d-e)^4*(c^2*d+4*e)*arctan(c*x)/c^10/e^2-1/8*d*(e*x^2+d)^4*(a+b*ar 
ctan(c*x))/e^2+1/10*(e*x^2+d)^5*(a+b*arctan(c*x))/e^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.02 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b e^3 \left (315 c x-105 c^3 x^3+63 c^5 x^5-45 c^7 x^7+35 c^9 x^9-315 \arctan (c x)\right )}{3150 c^{10}}-\frac {b d^2 e \left (15 c x-5 c^3 x^3+3 c^5 x^5-15 \arctan (c x)\right )}{30 c^6}-\frac {b d^3 \left (-3 c x+c^3 x^3+3 \arctan (c x)\right )}{12 c^4}-\frac {b d e^2 \left (-105 c x+35 c^3 x^3-21 c^5 x^5+15 c^7 x^7+105 \arctan (c x)\right )}{280 c^8}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {1}{2} d^2 e x^6 (a+b \arctan (c x))+\frac {3}{8} d e^2 x^8 (a+b \arctan (c x))+\frac {1}{10} e^3 x^{10} (a+b \arctan (c x)) \] Input:

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

Output:

-1/3150*(b*e^3*(315*c*x - 105*c^3*x^3 + 63*c^5*x^5 - 45*c^7*x^7 + 35*c^9*x 
^9 - 315*ArcTan[c*x]))/c^10 - (b*d^2*e*(15*c*x - 5*c^3*x^3 + 3*c^5*x^5 - 1 
5*ArcTan[c*x]))/(30*c^6) - (b*d^3*(-3*c*x + c^3*x^3 + 3*ArcTan[c*x]))/(12* 
c^4) - (b*d*e^2*(-105*c*x + 35*c^3*x^3 - 21*c^5*x^5 + 15*c^7*x^7 + 105*Arc 
Tan[c*x]))/(280*c^8) + (d^3*x^4*(a + b*ArcTan[c*x]))/4 + (d^2*e*x^6*(a + b 
*ArcTan[c*x]))/2 + (3*d*e^2*x^8*(a + b*ArcTan[c*x]))/8 + (e^3*x^10*(a + b* 
ArcTan[c*x]))/10
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.26, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5511, 27, 403, 403, 27, 403, 403, 299, 216}

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.

\(\displaystyle \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx\)

\(\Big \downarrow \) 5511

\(\displaystyle -b c \int -\frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{40 e^2 \left (c^2 x^2+1\right )}dx+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \int \frac {\left (d-4 e x^2\right ) \left (e x^2+d\right )^4}{c^2 x^2+1}dx}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\int \frac {\left (e x^2+d\right )^3 \left (d \left (9 d c^2+4 e\right )-\left (23 c^2 d-36 e\right ) e x^2\right )}{c^2 x^2+1}dx}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\frac {\int \frac {3 \left (e x^2+d\right )^2 \left (d \left (21 d^2 c^4+17 d e c^2-12 e^2\right )-e \left (25 d^2 c^4-135 d e c^2+84 e^2\right ) x^2\right )}{c^2 x^2+1}dx}{7 c^2}-\frac {e x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{7 c^2}}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \left (\frac {\frac {3 \int \frac {\left (e x^2+d\right )^2 \left (d \left (21 d^2 c^4+17 d e c^2-12 e^2\right )-e \left (25 d^2 c^4-135 d e c^2+84 e^2\right ) x^2\right )}{c^2 x^2+1}dx}{7 c^2}-\frac {e x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{7 c^2}}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\frac {3 \left (\frac {\int \frac {\left (e x^2+d\right ) \left (e \left (5 d^3 c^6+750 d^2 e c^4-1071 d e^2 c^2+420 e^3\right ) x^2+d \left (105 d^3 c^6+110 d^2 e c^4-195 d e^2 c^2+84 e^3\right )\right )}{c^2 x^2+1}dx}{5 c^2}-\frac {e x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{5 c^2}\right )}{7 c^2}-\frac {e x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{7 c^2}}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b c \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {e \left (325 d^4 c^8+1815 d^3 e c^6-4977 d^2 e^2 c^4+4305 d e^3 c^2-1260 e^4\right ) x^2+d \left (315 d^4 c^8+325 d^3 e c^6-1335 d^2 e^2 c^4+1323 d e^3 c^2-420 e^4\right )}{c^2 x^2+1}dx}{3 c^2}+\frac {e x \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) \left (d+e x^2\right )}{3 c^2}}{5 c^2}-\frac {e x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{5 c^2}\right )}{7 c^2}-\frac {e x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{7 c^2}}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b c \left (\frac {\frac {3 \left (\frac {\frac {\frac {315 \left (c^2 d+4 e\right ) \left (c^2 d-e\right )^4 \int \frac {1}{c^2 x^2+1}dx}{c^2}+\frac {e x \left (325 c^8 d^4+1815 c^6 d^3 e-4977 c^4 d^2 e^2+4305 c^2 d e^3-1260 e^4\right )}{c^2}}{3 c^2}+\frac {e x \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) \left (d+e x^2\right )}{3 c^2}}{5 c^2}-\frac {e x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{5 c^2}\right )}{7 c^2}-\frac {e x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{7 c^2}}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}+\frac {b c \left (\frac {\frac {3 \left (\frac {\frac {\frac {315 \arctan (c x) \left (c^2 d+4 e\right ) \left (c^2 d-e\right )^4}{c^3}+\frac {e x \left (325 c^8 d^4+1815 c^6 d^3 e-4977 c^4 d^2 e^2+4305 c^2 d e^3-1260 e^4\right )}{c^2}}{3 c^2}+\frac {e x \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) \left (d+e x^2\right )}{3 c^2}}{5 c^2}-\frac {e x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{5 c^2}\right )}{7 c^2}-\frac {e x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{7 c^2}}{9 c^2}-\frac {4 e x \left (d+e x^2\right )^4}{9 c^2}\right )}{40 e^2}\)

Input:

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

Output:

-1/8*(d*(d + e*x^2)^4*(a + b*ArcTan[c*x]))/e^2 + ((d + e*x^2)^5*(a + b*Arc 
Tan[c*x]))/(10*e^2) + (b*c*((-4*e*x*(d + e*x^2)^4)/(9*c^2) + (-1/7*((23*c^ 
2*d - 36*e)*e*x*(d + e*x^2)^3)/c^2 + (3*(-1/5*(e*(25*c^4*d^2 - 135*c^2*d*e 
 + 84*e^2)*x*(d + e*x^2)^2)/c^2 + ((e*(5*c^6*d^3 + 750*c^4*d^2*e - 1071*c^ 
2*d*e^2 + 420*e^3)*x*(d + e*x^2))/(3*c^2) + ((e*(325*c^8*d^4 + 1815*c^6*d^ 
3*e - 4977*c^4*d^2*e^2 + 4305*c^2*d*e^3 - 1260*e^4)*x)/c^2 + (315*(c^2*d - 
 e)^4*(c^2*d + 4*e)*ArcTan[c*x])/c^3)/(3*c^2))/(5*c^2)))/(7*c^2))/(9*c^2)) 
)/(40*e^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 5511

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]))
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.25

method	result	size
parts	\(a \left (\frac {1}{10} e^{3} x^{10}+\frac {3}{8} e^{2} d \,x^{8}+\frac {1}{2} e \,d^{2} x^{6}+\frac {1}{4} d^{3} x^{4}\right )+\frac {b \left (\frac {\arctan \left (c x \right ) c^{4} e^{3} x^{10}}{10}+\frac {3 \arctan \left (c x \right ) c^{4} e^{2} d \,x^{8}}{8}+\frac {\arctan \left (c x \right ) c^{4} d^{2} e \,x^{6}}{2}+\frac {\arctan \left (c x \right ) d^{3} c^{4} x^{4}}{4}-\frac {\frac {4 e^{3} c^{9} x^{9}}{9}+\frac {15 d \,c^{9} e^{2} x^{7}}{7}+4 d^{2} c^{9} e \,x^{5}+\frac {10 d^{3} c^{9} x^{3}}{3}-\frac {4 e^{3} c^{7} x^{7}}{7}-3 d \,c^{7} e^{2} x^{5}-\frac {20 d^{2} c^{7} e \,x^{3}}{3}-10 c^{7} x \,d^{3}+\frac {4 e^{3} c^{5} x^{5}}{5}+5 c^{5} d \,e^{2} x^{3}+20 c^{5} d^{2} e x -\frac {4 e^{3} c^{3} x^{3}}{3}-15 c^{3} x d \,e^{2}+4 c x \,e^{3}+\left (10 c^{6} d^{3}-20 c^{4} d^{2} e +15 e^{2} d \,c^{2}-4 e^{3}\right ) \arctan \left (c x \right )}{40 c^{6}}\right )}{c^{4}}\)	\(301\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} d^{3} c^{10} x^{4}+\frac {1}{2} d^{2} c^{10} e \,x^{6}+\frac {3}{8} d \,c^{10} e^{2} x^{8}+\frac {1}{10} e^{3} c^{10} x^{10}\right )}{c^{6}}+\frac {b \left (\frac {\arctan \left (c x \right ) d^{3} c^{10} x^{4}}{4}+\frac {\arctan \left (c x \right ) d^{2} c^{10} e \,x^{6}}{2}+\frac {3 \arctan \left (c x \right ) d \,c^{10} e^{2} x^{8}}{8}+\frac {\arctan \left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {d^{3} c^{9} x^{3}}{12}-\frac {d^{2} c^{9} e \,x^{5}}{10}-\frac {3 d \,c^{9} e^{2} x^{7}}{56}-\frac {e^{3} c^{9} x^{9}}{90}+\frac {c^{7} x \,d^{3}}{4}+\frac {d^{2} c^{7} e \,x^{3}}{6}+\frac {3 d \,c^{7} e^{2} x^{5}}{40}+\frac {e^{3} c^{7} x^{7}}{70}-\frac {c^{5} d^{2} e x}{2}-\frac {c^{5} d \,e^{2} x^{3}}{8}-\frac {e^{3} c^{5} x^{5}}{50}+\frac {3 c^{3} x d \,e^{2}}{8}+\frac {e^{3} c^{3} x^{3}}{30}-\frac {c x \,e^{3}}{10}-\frac {\left (10 c^{6} d^{3}-20 c^{4} d^{2} e +15 e^{2} d \,c^{2}-4 e^{3}\right ) \arctan \left (c x \right )}{40}\right )}{c^{6}}}{c^{4}}\)	\(315\)
default	\(\frac {\frac {a \left (\frac {1}{4} d^{3} c^{10} x^{4}+\frac {1}{2} d^{2} c^{10} e \,x^{6}+\frac {3}{8} d \,c^{10} e^{2} x^{8}+\frac {1}{10} e^{3} c^{10} x^{10}\right )}{c^{6}}+\frac {b \left (\frac {\arctan \left (c x \right ) d^{3} c^{10} x^{4}}{4}+\frac {\arctan \left (c x \right ) d^{2} c^{10} e \,x^{6}}{2}+\frac {3 \arctan \left (c x \right ) d \,c^{10} e^{2} x^{8}}{8}+\frac {\arctan \left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {d^{3} c^{9} x^{3}}{12}-\frac {d^{2} c^{9} e \,x^{5}}{10}-\frac {3 d \,c^{9} e^{2} x^{7}}{56}-\frac {e^{3} c^{9} x^{9}}{90}+\frac {c^{7} x \,d^{3}}{4}+\frac {d^{2} c^{7} e \,x^{3}}{6}+\frac {3 d \,c^{7} e^{2} x^{5}}{40}+\frac {e^{3} c^{7} x^{7}}{70}-\frac {c^{5} d^{2} e x}{2}-\frac {c^{5} d \,e^{2} x^{3}}{8}-\frac {e^{3} c^{5} x^{5}}{50}+\frac {3 c^{3} x d \,e^{2}}{8}+\frac {e^{3} c^{3} x^{3}}{30}-\frac {c x \,e^{3}}{10}-\frac {\left (10 c^{6} d^{3}-20 c^{4} d^{2} e +15 e^{2} d \,c^{2}-4 e^{3}\right ) \arctan \left (c x \right )}{40}\right )}{c^{6}}}{c^{4}}\)	\(315\)
parallelrisch	\(\frac {4725 x^{8} \arctan \left (c x \right ) b \,c^{10} d \,e^{2}+6300 b \,c^{4} d^{2} e \arctan \left (c x \right )-4725 b \,c^{2} d \,e^{2} \arctan \left (c x \right )-3150 b \,c^{6} d^{3} \arctan \left (c x \right )+6300 x^{6} \arctan \left (c x \right ) b \,c^{10} d^{2} e +1260 a \,c^{10} e^{3} x^{10}-140 b \,c^{9} e^{3} x^{9}+3150 a \,c^{10} d^{3} x^{4}+180 b \,c^{7} e^{3} x^{7}-1050 b \,c^{9} d^{3} x^{3}-252 b \,c^{5} e^{3} x^{5}+3150 b \,c^{7} d^{3} x +420 b \,c^{3} e^{3} x^{3}-1260 b c \,e^{3} x +4725 a \,c^{10} d \,e^{2} x^{8}+6300 a \,c^{10} d^{2} e \,x^{6}-675 b \,c^{9} d \,e^{2} x^{7}-1260 b \,c^{9} d^{2} e \,x^{5}+945 b \,c^{7} d \,e^{2} x^{5}+2100 b \,c^{7} d^{2} e \,x^{3}-6300 b \,c^{5} d^{2} e x +4725 b \,c^{3} d \,e^{2} x -1575 b \,c^{5} d \,e^{2} x^{3}+3150 d^{3} b \arctan \left (c x \right ) x^{4} c^{10}+1260 x^{10} \arctan \left (c x \right ) b \,c^{10} e^{3}+1260 b \,e^{3} \arctan \left (c x \right )}{12600 c^{10}}\)	\(339\)
risch	\(-\frac {3 b d \,e^{2} x^{7}}{56 c}-\frac {b \,d^{2} e \,x^{5}}{10 c}+\frac {3 b d \,e^{2} x^{5}}{40 c^{3}}+\frac {b \,d^{2} e \,x^{3}}{6 c^{3}}-\frac {b d \,e^{2} x^{3}}{8 c^{5}}-\frac {b \,d^{2} e x}{2 c^{5}}+\frac {3 b d \,e^{2} x}{8 c^{7}}+\frac {b \,d^{2} e \arctan \left (c x \right )}{2 c^{6}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{8 c^{8}}-\frac {b \,e^{3} x^{9}}{90 c}+\frac {b \,d^{3} x}{4 c^{3}}-\frac {b \,d^{3} x^{3}}{12 c}-\frac {b \,d^{3} \arctan \left (c x \right )}{4 c^{4}}+\frac {i b \,d^{2} e \,x^{6} \ln \left (-i c x +1\right )}{4}+\frac {b \,e^{3} x^{3}}{30 c^{7}}-\frac {b \,e^{3} x}{10 c^{9}}+\frac {b \,e^{3} \arctan \left (c x \right )}{10 c^{10}}+\frac {i b \,e^{3} x^{10} \ln \left (-i c x +1\right )}{20}+\frac {i b \,d^{3} x^{4} \ln \left (-i c x +1\right )}{8}+\frac {3 i b d \,e^{2} x^{8} \ln \left (-i c x +1\right )}{16}+\frac {b \,e^{3} x^{7}}{70 c^{3}}-\frac {b \,e^{3} x^{5}}{50 c^{5}}+\frac {3 x^{8} e^{2} d a}{8}+\frac {x^{6} e \,d^{2} a}{2}-\frac {i b \left (4 e^{3} x^{10}+15 e^{2} d \,x^{8}+20 e \,d^{2} x^{6}+10 d^{3} x^{4}\right ) \ln \left (i c x +1\right )}{80}+\frac {x^{10} e^{3} a}{10}+\frac {x^{4} d^{3} a}{4}\)	\(382\)
orering	\(\frac {\left (420 c^{10} e^{4} x^{12}+2080 c^{10} d \,e^{3} x^{10}+4065 c^{10} d^{2} e^{2} x^{8}-60 c^{8} e^{4} x^{10}+3780 c^{10} d^{3} e \,x^{6}-425 c^{8} d \,e^{3} x^{8}+1050 c^{10} d^{4} x^{4}-1395 c^{8} d^{2} e^{2} x^{6}+108 c^{6} e^{4} x^{8}-3570 c^{8} d^{3} e \,x^{4}+981 c^{6} d \,e^{3} x^{6}-1050 c^{8} d^{4} x^{2}+6615 c^{6} d^{2} e^{2} x^{4}-252 c^{4} e^{4} x^{6}-3150 c^{6} d^{3} e \,x^{2}-4809 c^{4} d \,e^{3} x^{4}-2100 c^{6} d^{4}+8925 c^{4} d^{2} e^{2} x^{2}+1260 c^{2} e^{4} x^{4}+4200 c^{4} d^{3} e -7455 c^{2} d \,e^{3} x^{2}-3150 c^{2} d^{2} e^{2}+2100 e^{4} x^{2}+840 d \,e^{3}\right ) \left (a +b \arctan \left (c x \right )\right )}{2100 c^{10} \left (e \,x^{2}+d \right )}-\frac {\left (140 c^{8} e^{3} x^{8}+675 c^{8} d \,e^{2} x^{6}+1260 c^{8} d^{2} e \,x^{4}-180 c^{6} e^{3} x^{6}+1050 c^{8} d^{3} x^{2}-945 c^{6} d \,e^{2} x^{4}-2100 c^{6} d^{2} e \,x^{2}+252 c^{4} e^{3} x^{4}-3150 c^{6} d^{3}+1575 c^{4} d \,e^{2} x^{2}+6300 c^{4} d^{2} e -420 c^{2} e^{3} x^{2}-4725 e^{2} d \,c^{2}+1260 e^{3}\right ) \left (c^{2} x^{2}+1\right ) \left (3 x^{2} \left (e \,x^{2}+d \right )^{3} \left (a +b \arctan \left (c x \right )\right )+6 x^{4} \left (e \,x^{2}+d \right )^{2} \left (a +b \arctan \left (c x \right )\right ) e +\frac {x^{3} \left (e \,x^{2}+d \right )^{3} b c}{c^{2} x^{2}+1}\right )}{12600 x^{2} c^{10} \left (e \,x^{2}+d \right )^{3}}\)	\(541\)

Input:

int(x^3*(e*x^2+d)^3*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)
 

Output:

a*(1/10*e^3*x^10+3/8*e^2*d*x^8+1/2*e*d^2*x^6+1/4*d^3*x^4)+b/c^4*(1/10*arct 
an(c*x)*c^4*e^3*x^10+3/8*arctan(c*x)*c^4*e^2*d*x^8+1/2*arctan(c*x)*c^4*d^2 
*e*x^6+1/4*arctan(c*x)*d^3*c^4*x^4-1/40/c^6*(4/9*e^3*c^9*x^9+15/7*d*c^9*e^ 
2*x^7+4*d^2*c^9*e*x^5+10/3*d^3*c^9*x^3-4/7*e^3*c^7*x^7-3*d*c^7*e^2*x^5-20/ 
3*d^2*c^7*e*x^3-10*c^7*x*d^3+4/5*e^3*c^5*x^5+5*c^5*d*e^2*x^3+20*c^5*d^2*e* 
x-4/3*e^3*c^3*x^3-15*c^3*x*d*e^2+4*c*x*e^3+(10*c^6*d^3-20*c^4*d^2*e+15*c^2 
*d*e^2-4*e^3)*arctan(c*x)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.27 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1260 \, a c^{10} e^{3} x^{10} + 4725 \, a c^{10} d e^{2} x^{8} - 140 \, b c^{9} e^{3} x^{9} + 6300 \, a c^{10} d^{2} e x^{6} + 3150 \, a c^{10} d^{3} x^{4} - 45 \, {\left (15 \, b c^{9} d e^{2} - 4 \, b c^{7} e^{3}\right )} x^{7} - 63 \, {\left (20 \, b c^{9} d^{2} e - 15 \, b c^{7} d e^{2} + 4 \, b c^{5} e^{3}\right )} x^{5} - 105 \, {\left (10 \, b c^{9} d^{3} - 20 \, b c^{7} d^{2} e + 15 \, b c^{5} d e^{2} - 4 \, b c^{3} e^{3}\right )} x^{3} + 315 \, {\left (10 \, b c^{7} d^{3} - 20 \, b c^{5} d^{2} e + 15 \, b c^{3} d e^{2} - 4 \, b c e^{3}\right )} x + 315 \, {\left (4 \, b c^{10} e^{3} x^{10} + 15 \, b c^{10} d e^{2} x^{8} + 20 \, b c^{10} d^{2} e x^{6} + 10 \, b c^{10} d^{3} x^{4} - 10 \, b c^{6} d^{3} + 20 \, b c^{4} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b e^{3}\right )} \arctan \left (c x\right )}{12600 \, c^{10}} \] Input:

integrate(x^3*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="fricas")
 

Output:

1/12600*(1260*a*c^10*e^3*x^10 + 4725*a*c^10*d*e^2*x^8 - 140*b*c^9*e^3*x^9 
+ 6300*a*c^10*d^2*e*x^6 + 3150*a*c^10*d^3*x^4 - 45*(15*b*c^9*d*e^2 - 4*b*c 
^7*e^3)*x^7 - 63*(20*b*c^9*d^2*e - 15*b*c^7*d*e^2 + 4*b*c^5*e^3)*x^5 - 105 
*(10*b*c^9*d^3 - 20*b*c^7*d^2*e + 15*b*c^5*d*e^2 - 4*b*c^3*e^3)*x^3 + 315* 
(10*b*c^7*d^3 - 20*b*c^5*d^2*e + 15*b*c^3*d*e^2 - 4*b*c*e^3)*x + 315*(4*b* 
c^10*e^3*x^10 + 15*b*c^10*d*e^2*x^8 + 20*b*c^10*d^2*e*x^6 + 10*b*c^10*d^3* 
x^4 - 10*b*c^6*d^3 + 20*b*c^4*d^2*e - 15*b*c^2*d*e^2 + 4*b*e^3)*arctan(c*x 
))/c^10
 

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.71 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{4}}{4} + \frac {a d^{2} e x^{6}}{2} + \frac {3 a d e^{2} x^{8}}{8} + \frac {a e^{3} x^{10}}{10} + \frac {b d^{3} x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d^{2} e x^{6} \operatorname {atan}{\left (c x \right )}}{2} + \frac {3 b d e^{2} x^{8} \operatorname {atan}{\left (c x \right )}}{8} + \frac {b e^{3} x^{10} \operatorname {atan}{\left (c x \right )}}{10} - \frac {b d^{3} x^{3}}{12 c} - \frac {b d^{2} e x^{5}}{10 c} - \frac {3 b d e^{2} x^{7}}{56 c} - \frac {b e^{3} x^{9}}{90 c} + \frac {b d^{3} x}{4 c^{3}} + \frac {b d^{2} e x^{3}}{6 c^{3}} + \frac {3 b d e^{2} x^{5}}{40 c^{3}} + \frac {b e^{3} x^{7}}{70 c^{3}} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d^{2} e x}{2 c^{5}} - \frac {b d e^{2} x^{3}}{8 c^{5}} - \frac {b e^{3} x^{5}}{50 c^{5}} + \frac {b d^{2} e \operatorname {atan}{\left (c x \right )}}{2 c^{6}} + \frac {3 b d e^{2} x}{8 c^{7}} + \frac {b e^{3} x^{3}}{30 c^{7}} - \frac {3 b d e^{2} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} - \frac {b e^{3} x}{10 c^{9}} + \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{10 c^{10}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{4}}{4} + \frac {d^{2} e x^{6}}{2} + \frac {3 d e^{2} x^{8}}{8} + \frac {e^{3} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(x**3*(e*x**2+d)**3*(a+b*atan(c*x)),x)
 

Output:

Piecewise((a*d**3*x**4/4 + a*d**2*e*x**6/2 + 3*a*d*e**2*x**8/8 + a*e**3*x* 
*10/10 + b*d**3*x**4*atan(c*x)/4 + b*d**2*e*x**6*atan(c*x)/2 + 3*b*d*e**2* 
x**8*atan(c*x)/8 + b*e**3*x**10*atan(c*x)/10 - b*d**3*x**3/(12*c) - b*d**2 
*e*x**5/(10*c) - 3*b*d*e**2*x**7/(56*c) - b*e**3*x**9/(90*c) + b*d**3*x/(4 
*c**3) + b*d**2*e*x**3/(6*c**3) + 3*b*d*e**2*x**5/(40*c**3) + b*e**3*x**7/ 
(70*c**3) - b*d**3*atan(c*x)/(4*c**4) - b*d**2*e*x/(2*c**5) - b*d*e**2*x** 
3/(8*c**5) - b*e**3*x**5/(50*c**5) + b*d**2*e*atan(c*x)/(2*c**6) + 3*b*d*e 
**2*x/(8*c**7) + b*e**3*x**3/(30*c**7) - 3*b*d*e**2*atan(c*x)/(8*c**8) - b 
*e**3*x/(10*c**9) + b*e**3*atan(c*x)/(10*c**10), Ne(c, 0)), (a*(d**3*x**4/ 
4 + d**2*e*x**6/2 + 3*d*e**2*x**8/8 + e**3*x**10/10), True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.12 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{3} + \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d^{2} e + \frac {1}{280} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b d e^{2} + \frac {1}{3150} \, {\left (315 \, x^{10} \arctan \left (c x\right ) - c {\left (\frac {35 \, c^{8} x^{9} - 45 \, c^{6} x^{7} + 63 \, c^{4} x^{5} - 105 \, c^{2} x^{3} + 315 \, x}{c^{10}} - \frac {315 \, \arctan \left (c x\right )}{c^{11}}\right )}\right )} b e^{3} \] Input:

integrate(x^3*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="maxima")
 

Output:

1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 + 1/12 
*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*b*d^3 + 
 1/30*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arc 
tan(c*x)/c^7))*b*d^2*e + 1/280*(105*x^8*arctan(c*x) - c*((15*c^6*x^7 - 21* 
c^4*x^5 + 35*c^2*x^3 - 105*x)/c^8 + 105*arctan(c*x)/c^9))*b*d*e^2 + 1/3150 
*(315*x^10*arctan(c*x) - c*((35*c^8*x^9 - 45*c^6*x^7 + 63*c^4*x^5 - 105*c^ 
2*x^3 + 315*x)/c^10 - 315*arctan(c*x)/c^11))*b*e^3
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.64 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1260 \, b c^{10} e^{3} x^{10} \arctan \left (c x\right ) + 1260 \, a c^{10} e^{3} x^{10} + 4725 \, b c^{10} d e^{2} x^{8} \arctan \left (c x\right ) + 4725 \, a c^{10} d e^{2} x^{8} - 140 \, b c^{9} e^{3} x^{9} + 6300 \, b c^{10} d^{2} e x^{6} \arctan \left (c x\right ) + 6300 \, a c^{10} d^{2} e x^{6} - 675 \, b c^{9} d e^{2} x^{7} + 3150 \, b c^{10} d^{3} x^{4} \arctan \left (c x\right ) + 3150 \, a c^{10} d^{3} x^{4} - 1260 \, b c^{9} d^{2} e x^{5} + 180 \, b c^{7} e^{3} x^{7} - 1050 \, b c^{9} d^{3} x^{3} + 945 \, b c^{7} d e^{2} x^{5} + 2100 \, b c^{7} d^{2} e x^{3} - 252 \, b c^{5} e^{3} x^{5} + 3150 \, \pi b c^{6} d^{3} \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) + 3150 \, b c^{7} d^{3} x - 1575 \, b c^{5} d e^{2} x^{3} - 3150 \, b c^{6} d^{3} \arctan \left (c x\right ) - 6300 \, \pi b c^{4} d^{2} e \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) - 6300 \, b c^{5} d^{2} e x + 420 \, b c^{3} e^{3} x^{3} + 6300 \, b c^{4} d^{2} e \arctan \left (c x\right ) + 4725 \, \pi b c^{2} d e^{2} \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) + 4725 \, b c^{3} d e^{2} x - 4725 \, b c^{2} d e^{2} \arctan \left (c x\right ) - 1260 \, \pi b e^{3} \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) - 1260 \, b c e^{3} x + 1260 \, b e^{3} \arctan \left (c x\right )}{12600 \, c^{10}} \] Input:

integrate(x^3*(e*x^2+d)^3*(a+b*arctan(c*x)),x, algorithm="giac")
 

Output:

1/12600*(1260*b*c^10*e^3*x^10*arctan(c*x) + 1260*a*c^10*e^3*x^10 + 4725*b* 
c^10*d*e^2*x^8*arctan(c*x) + 4725*a*c^10*d*e^2*x^8 - 140*b*c^9*e^3*x^9 + 6 
300*b*c^10*d^2*e*x^6*arctan(c*x) + 6300*a*c^10*d^2*e*x^6 - 675*b*c^9*d*e^2 
*x^7 + 3150*b*c^10*d^3*x^4*arctan(c*x) + 3150*a*c^10*d^3*x^4 - 1260*b*c^9* 
d^2*e*x^5 + 180*b*c^7*e^3*x^7 - 1050*b*c^9*d^3*x^3 + 945*b*c^7*d*e^2*x^5 + 
 2100*b*c^7*d^2*e*x^3 - 252*b*c^5*e^3*x^5 + 3150*pi*b*c^6*d^3*sgn(c)*sgn(x 
) + 3150*b*c^7*d^3*x - 1575*b*c^5*d*e^2*x^3 - 3150*b*c^6*d^3*arctan(c*x) - 
 6300*pi*b*c^4*d^2*e*sgn(c)*sgn(x) - 6300*b*c^5*d^2*e*x + 420*b*c^3*e^3*x^ 
3 + 6300*b*c^4*d^2*e*arctan(c*x) + 4725*pi*b*c^2*d*e^2*sgn(c)*sgn(x) + 472 
5*b*c^3*d*e^2*x - 4725*b*c^2*d*e^2*arctan(c*x) - 1260*pi*b*e^3*sgn(c)*sgn( 
x) - 1260*b*c*e^3*x + 1260*b*e^3*arctan(c*x))/c^10
 

Mupad [B] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.50 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=x^3\,\left (\frac {\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{c^2}+\frac {b\,d^2\,e}{2\,c}}{3\,c^2}-\frac {b\,d^3}{12\,c}\right )-x^8\,\left (\frac {a\,e^3}{8\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{8\,c^2}\right )+x^6\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{6\,c^2}+\frac {a\,d\,e\,\left (d\,c^2+e\right )}{2\,c^2}\right )+x^7\,\left (\frac {b\,e^3}{70\,c^3}-\frac {3\,b\,d\,e^2}{56\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3\,x^4}{4}+\frac {b\,d^2\,e\,x^6}{2}+\frac {3\,b\,d\,e^2\,x^8}{8}+\frac {b\,e^3\,x^{10}}{10}\right )-x^5\,\left (\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{5\,c^2}+\frac {b\,d^2\,e}{10\,c}\right )+x^2\,\left (\frac {\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{c^2}}{2\,c^2}+\frac {a\,d^3}{2\,c^2}\right )-x^4\,\left (\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{4\,c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{4\,c^2}\right )+\frac {a\,e^3\,x^{10}}{10}-\frac {x\,\left (\frac {\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{c^2}+\frac {b\,d^2\,e}{2\,c}}{c^2}-\frac {b\,d^3}{4\,c}\right )}{c^2}-\frac {b\,e^3\,x^9}{90\,c}+\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (-10\,c^6\,d^3+20\,c^4\,d^2\,e-15\,c^2\,d\,e^2+4\,e^3\right )}{-10\,b\,c^6\,d^3+20\,b\,c^4\,d^2\,e-15\,b\,c^2\,d\,e^2+4\,b\,e^3}\right )\,\left (-10\,c^6\,d^3+20\,c^4\,d^2\,e-15\,c^2\,d\,e^2+4\,e^3\right )}{40\,c^{10}} \] Input:

int(x^3*(a + b*atan(c*x))*(d + e*x^2)^3,x)
 

Output:

x^3*((((b*e^3)/(10*c^3) - (3*b*d*e^2)/(8*c))/c^2 + (b*d^2*e)/(2*c))/(3*c^2 
) - (b*d^3)/(12*c)) - x^8*((a*e^3)/(8*c^2) - (a*e^2*(e + 3*c^2*d))/(8*c^2) 
) + x^6*(((a*e^3)/c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/(6*c^2) + (a*d*e*(e + c 
^2*d))/(2*c^2)) + x^7*((b*e^3)/(70*c^3) - (3*b*d*e^2)/(56*c)) + atan(c*x)* 
((b*d^3*x^4)/4 + (b*e^3*x^10)/10 + (b*d^2*e*x^6)/2 + (3*b*d*e^2*x^8)/8) - 
x^5*(((b*e^3)/(10*c^3) - (3*b*d*e^2)/(8*c))/(5*c^2) + (b*d^2*e)/(10*c)) + 
x^2*(((((a*e^3)/c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/c^2 + (3*a*d*e*(e + c^2*d 
))/c^2)/c^2 - (a*d^2*(3*e + c^2*d))/c^2)/(2*c^2) + (a*d^3)/(2*c^2)) - x^4* 
((((a*e^3)/c^2 - (a*e^2*(e + 3*c^2*d))/c^2)/c^2 + (3*a*d*e*(e + c^2*d))/c^ 
2)/(4*c^2) - (a*d^2*(3*e + c^2*d))/(4*c^2)) + (a*e^3*x^10)/10 - (x*((((b*e 
^3)/(10*c^3) - (3*b*d*e^2)/(8*c))/c^2 + (b*d^2*e)/(2*c))/c^2 - (b*d^3)/(4* 
c)))/c^2 - (b*e^3*x^9)/(90*c) + (b*atan((b*c*x*(4*e^3 - 10*c^6*d^3 - 15*c^ 
2*d*e^2 + 20*c^4*d^2*e))/(4*b*e^3 - 10*b*c^6*d^3 - 15*b*c^2*d*e^2 + 20*b*c 
^4*d^2*e))*(4*e^3 - 10*c^6*d^3 - 15*c^2*d*e^2 + 20*c^4*d^2*e))/(40*c^10)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.41 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1260 \mathit {atan} \left (c x \right ) b \,e^{3}-3150 \mathit {atan} \left (c x \right ) b \,c^{6} d^{3}+3150 a \,c^{10} d^{3} x^{4}+1260 a \,c^{10} e^{3} x^{10}-1050 b \,c^{9} d^{3} x^{3}-140 b \,c^{9} e^{3} x^{9}+3150 b \,c^{7} d^{3} x +180 b \,c^{7} e^{3} x^{7}-252 b \,c^{5} e^{3} x^{5}+420 b \,c^{3} e^{3} x^{3}-1260 b c \,e^{3} x +2100 b \,c^{7} d^{2} e \,x^{3}+945 b \,c^{7} d \,e^{2} x^{5}-6300 b \,c^{5} d^{2} e x -1575 b \,c^{5} d \,e^{2} x^{3}+4725 b \,c^{3} d \,e^{2} x +3150 \mathit {atan} \left (c x \right ) b \,c^{10} d^{3} x^{4}+1260 \mathit {atan} \left (c x \right ) b \,c^{10} e^{3} x^{10}+6300 \mathit {atan} \left (c x \right ) b \,c^{4} d^{2} e -4725 \mathit {atan} \left (c x \right ) b \,c^{2} d \,e^{2}+6300 a \,c^{10} d^{2} e \,x^{6}+4725 a \,c^{10} d \,e^{2} x^{8}-1260 b \,c^{9} d^{2} e \,x^{5}-675 b \,c^{9} d \,e^{2} x^{7}+6300 \mathit {atan} \left (c x \right ) b \,c^{10} d^{2} e \,x^{6}+4725 \mathit {atan} \left (c x \right ) b \,c^{10} d \,e^{2} x^{8}}{12600 c^{10}} \] Input:

int(x^3*(e*x^2+d)^3*(a+b*atan(c*x)),x)
 

Output:

(3150*atan(c*x)*b*c**10*d**3*x**4 + 6300*atan(c*x)*b*c**10*d**2*e*x**6 + 4 
725*atan(c*x)*b*c**10*d*e**2*x**8 + 1260*atan(c*x)*b*c**10*e**3*x**10 - 31 
50*atan(c*x)*b*c**6*d**3 + 6300*atan(c*x)*b*c**4*d**2*e - 4725*atan(c*x)*b 
*c**2*d*e**2 + 1260*atan(c*x)*b*e**3 + 3150*a*c**10*d**3*x**4 + 6300*a*c** 
10*d**2*e*x**6 + 4725*a*c**10*d*e**2*x**8 + 1260*a*c**10*e**3*x**10 - 1050 
*b*c**9*d**3*x**3 - 1260*b*c**9*d**2*e*x**5 - 675*b*c**9*d*e**2*x**7 - 140 
*b*c**9*e**3*x**9 + 3150*b*c**7*d**3*x + 2100*b*c**7*d**2*e*x**3 + 945*b*c 
**7*d*e**2*x**5 + 180*b*c**7*e**3*x**7 - 6300*b*c**5*d**2*e*x - 1575*b*c** 
5*d*e**2*x**3 - 252*b*c**5*e**3*x**5 + 4725*b*c**3*d*e**2*x + 420*b*c**3*e 
**3*x**3 - 1260*b*c*e**3*x)/(12600*c**10)