\(\int x^3 (d+e x^2)^3 (a+b \arctan (c x)) \, dx\) [1137]

Optimal result

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

[Out]

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
/40*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*arctan(c*x))/e^2+1/10*(e*x^2+d)^5*(a
+b*arctan(c*x))/e^2

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 45, 5096, 12, 542, 396, 209} \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {\left (d+e x^2\right )^5 (a+b \arctan (c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \arctan (c x))}{8 e^2}+\frac {b \arctan (c x) \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right )}{40 c^{10} e^2}-\frac {b x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{4200 c^5 e}+\frac {b x \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) \left (d+e x^2\right )}{12600 c^7 e}+\frac {b x \left (325 c^8 d^4+1815 c^6 d^3 e-4977 c^4 d^2 e^2+4305 c^2 d e^3-1260 e^4\right )}{12600 c^9 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e} \]

[In]

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

[Out]

(b*(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)/(12600*c^9*e) + (b*(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))/(12600*c^7*e) - (b*(25*c^4*d^2 - 135*c^2*d*e +
84*e^2)*x*(d + e*x^2)^2)/(4200*c^5*e) - (b*(23*c^2*d - 36*e)*x*(d + e*x^2)^3)/(2520*c^3*e) - (b*x*(d + e*x^2)^
4)/(90*c*e) + (b*(c^2*d - e)^4*(c^2*d + 4*e)*ArcTan[c*x])/(40*c^10*e^2) - (d*(d + e*x^2)^4*(a + b*ArcTan[c*x])
)/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcTan[c*x]))/(10*e^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 209

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 396

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

Rule 542

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

Rule 5096

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]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[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] &&
!ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = -\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}-(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{40 e^2 \left (1+c^2 x^2\right )} \, dx \\ & = -\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}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{1+c^2 x^2} \, dx}{40 e^2} \\ & = -\frac {b x \left (d+e x^2\right )^4}{90 c e}-\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}-\frac {b \int \frac {\left (d+e x^2\right )^3 \left (-d \left (9 c^2 d+4 e\right )+\left (23 c^2 d-36 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{360 c e^2} \\ & = -\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\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}-\frac {b \int \frac {\left (d+e x^2\right )^2 \left (-3 d \left (21 c^4 d^2+17 c^2 d e-12 e^2\right )+3 e \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x^2\right )}{1+c^2 x^2} \, dx}{2520 c^3 e^2} \\ & = -\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\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}-\frac {b \int \frac {\left (d+e x^2\right ) \left (-3 d \left (105 c^6 d^3+110 c^4 d^2 e-195 c^2 d e^2+84 e^3\right )-3 e \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x^2\right )}{1+c^2 x^2} \, dx}{12600 c^5 e^2} \\ & = \frac {b \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x \left (d+e x^2\right )}{12600 c^7 e}-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\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}-\frac {b \int \frac {-3 d \left (315 c^8 d^4+325 c^6 d^3 e-1335 c^4 d^2 e^2+1323 c^2 d e^3-420 e^4\right )-3 e \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 ) x^2}{1+c^2 x^2} \, dx}{37800 c^7 e^2} \\ & = \frac {b \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 ) x}{12600 c^9 e}+\frac {b \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x \left (d+e x^2\right )}{12600 c^7 e}-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}-\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}+\frac {\left (b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx}{40 c^9 e^2} \\ & = \frac {b \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 ) x}{12600 c^9 e}+\frac {b \left (5 c^6 d^3+750 c^4 d^2 e-1071 c^2 d e^2+420 e^3\right ) x \left (d+e x^2\right )}{12600 c^7 e}-\frac {b \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) x \left (d+e x^2\right )^2}{4200 c^5 e}-\frac {b \left (23 c^2 d-36 e\right ) x \left (d+e x^2\right )^3}{2520 c^3 e}-\frac {b x \left (d+e x^2\right )^4}{90 c e}+\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.02 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=-\frac {b e^3 \left (315 c x-105 c^3 x^3+63 c^5 x^5-45 c^7 x^7+35 c^9 x^9-315 \arctan (c x)\right )}{3150 c^{10}}-\frac {b d^2 e \left (15 c x-5 c^3 x^3+3 c^5 x^5-15 \arctan (c x)\right )}{30 c^6}-\frac {b d^3 \left (-3 c x+c^3 x^3+3 \arctan (c x)\right )}{12 c^4}-\frac {b d e^2 \left (-105 c x+35 c^3 x^3-21 c^5 x^5+15 c^7 x^7+105 \arctan (c x)\right )}{280 c^8}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {1}{2} d^2 e x^6 (a+b \arctan (c x))+\frac {3}{8} d e^2 x^8 (a+b \arctan (c x))+\frac {1}{10} e^3 x^{10} (a+b \arctan (c x)) \]

[In]

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

[Out]

-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 - 15*ArcTan[c*x]))/(30*c^6) - (b*d^3*(-3*c*x + c^3*x^3 + 3*ArcTan[c*x]))/(1
2*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*ArcTan[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

Maple [A] (verified)

Time = 0.32 (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	\(\frac {-3150 b \,c^{6} d^{3} \arctan \left (c x \right )-4725 b \,c^{2} d \,e^{2} \arctan \left (c x \right )+6300 b \,c^{4} d^{2} e \arctan \left (c x \right )+1260 a \,c^{10} e^{3} x^{10}-140 b \,c^{9} e^{3} x^{9}+3150 a \,c^{10} d^{3} x^{4}+180 b \,c^{7} e^{3} x^{7}-1050 b \,c^{9} d^{3} x^{3}-252 b \,c^{5} e^{3} x^{5}+3150 b \,c^{7} d^{3} x +420 b \,c^{3} e^{3} x^{3}-1260 b c \,e^{3} x +1260 b \,e^{3} \arctan \left (c x \right )+4725 a \,c^{10} d \,e^{2} x^{8}+6300 a \,c^{10} d^{2} e \,x^{6}-675 b \,c^{9} d \,e^{2} x^{7}-1260 b \,c^{9} d^{2} e \,x^{5}+945 b \,c^{7} d \,e^{2} x^{5}+2100 b \,c^{7} d^{2} e \,x^{3}-1575 b \,c^{5} d \,e^{2} x^{3}-6300 b \,c^{5} d^{2} e x +4725 b \,c^{3} d \,e^{2} x +1260 x^{10} \arctan \left (c x \right ) b \,c^{10} e^{3}+3150 d^{3} b \arctan \left (c x \right ) x^{4} c^{10}+4725 x^{8} \arctan \left (c x \right ) b \,c^{10} d \,e^{2}+6300 x^{6} \arctan \left (c x \right ) b \,c^{10} d^{2} e}{12600 c^{10}}\)	\(339\)
risch	\(\frac {3 i b d \,e^{2} x^{8} \ln \left (-i c x +1\right )}{16}+\frac {i b \,d^{2} e \,x^{6} \ln \left (-i c x +1\right )}{4}-\frac {b \,e^{3} x^{9}}{90 c}-\frac {3 b d \,e^{2} x^{7}}{56 c}-\frac {b \,d^{2} e \,x^{5}}{10 c}+\frac {3 b d \,e^{2} x^{5}}{40 c^{3}}+\frac {b \,d^{2} e \,x^{3}}{6 c^{3}}-\frac {b d \,e^{2} x^{3}}{8 c^{5}}-\frac {b \,d^{2} e x}{2 c^{5}}+\frac {3 b d \,e^{2} x}{8 c^{7}}+\frac {b \,d^{2} e \arctan \left (c x \right )}{2 c^{6}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{8 c^{8}}+\frac {b \,e^{3} x^{7}}{70 c^{3}}-\frac {b \,e^{3} x^{5}}{50 c^{5}}+\frac {b \,e^{3} x^{3}}{30 c^{7}}-\frac {b \,e^{3} x}{10 c^{9}}+\frac {b \,e^{3} \arctan \left (c x \right )}{10 c^{10}}+\frac {3 x^{8} e^{2} d a}{8}+\frac {x^{6} e \,d^{2} a}{2}-\frac {b \,d^{3} \arctan \left (c x \right )}{4 c^{4}}+\frac {b \,d^{3} x}{4 c^{3}}-\frac {b \,d^{3} x^{3}}{12 c}+\frac {x^{10} e^{3} a}{10}+\frac {x^{4} d^{3} a}{4}-\frac {i b \left (4 e^{3} x^{10}+15 e^{2} d \,x^{8}+20 e \,d^{2} x^{6}+10 d^{3} x^{4}\right ) \ln \left (i c x +1\right )}{80}+\frac {i b \,d^{3} x^{4} \ln \left (-i c x +1\right )}{8}+\frac {i b \,e^{3} x^{10} \ln \left (-i c x +1\right )}{20}\)	\(382\)

[In]

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

[Out]

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*arctan(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)

none

Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.27 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1260 \, a c^{10} e^{3} x^{10} + 4725 \, a c^{10} d e^{2} x^{8} - 140 \, b c^{9} e^{3} x^{9} + 6300 \, a c^{10} d^{2} e x^{6} + 3150 \, a c^{10} d^{3} x^{4} - 45 \, {\left (15 \, b c^{9} d e^{2} - 4 \, b c^{7} e^{3}\right )} x^{7} - 63 \, {\left (20 \, b c^{9} d^{2} e - 15 \, b c^{7} d e^{2} + 4 \, b c^{5} e^{3}\right )} x^{5} - 105 \, {\left (10 \, b c^{9} d^{3} - 20 \, b c^{7} d^{2} e + 15 \, b c^{5} d e^{2} - 4 \, b c^{3} e^{3}\right )} x^{3} + 315 \, {\left (10 \, b c^{7} d^{3} - 20 \, b c^{5} d^{2} e + 15 \, b c^{3} d e^{2} - 4 \, b c e^{3}\right )} x + 315 \, {\left (4 \, b c^{10} e^{3} x^{10} + 15 \, b c^{10} d e^{2} x^{8} + 20 \, b c^{10} d^{2} e x^{6} + 10 \, b c^{10} d^{3} x^{4} - 10 \, b c^{6} d^{3} + 20 \, b c^{4} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b e^{3}\right )} \arctan \left (c x\right )}{12600 \, c^{10}} \]

[In]

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

[Out]

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^1
0*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 - 1
05*(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 + 1
5*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 = 0.76 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.71 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{4}}{4} + \frac {a d^{2} e x^{6}}{2} + \frac {3 a d e^{2} x^{8}}{8} + \frac {a e^{3} x^{10}}{10} + \frac {b d^{3} x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d^{2} e x^{6} \operatorname {atan}{\left (c x \right )}}{2} + \frac {3 b d e^{2} x^{8} \operatorname {atan}{\left (c x \right )}}{8} + \frac {b e^{3} x^{10} \operatorname {atan}{\left (c x \right )}}{10} - \frac {b d^{3} x^{3}}{12 c} - \frac {b d^{2} e x^{5}}{10 c} - \frac {3 b d e^{2} x^{7}}{56 c} - \frac {b e^{3} x^{9}}{90 c} + \frac {b d^{3} x}{4 c^{3}} + \frac {b d^{2} e x^{3}}{6 c^{3}} + \frac {3 b d e^{2} x^{5}}{40 c^{3}} + \frac {b e^{3} x^{7}}{70 c^{3}} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d^{2} e x}{2 c^{5}} - \frac {b d e^{2} x^{3}}{8 c^{5}} - \frac {b e^{3} x^{5}}{50 c^{5}} + \frac {b d^{2} e \operatorname {atan}{\left (c x \right )}}{2 c^{6}} + \frac {3 b d e^{2} x}{8 c^{7}} + \frac {b e^{3} x^{3}}{30 c^{7}} - \frac {3 b d e^{2} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} - \frac {b e^{3} x}{10 c^{9}} + \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{10 c^{10}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{4}}{4} + \frac {d^{2} e x^{6}}{2} + \frac {3 d e^{2} x^{8}}{8} + \frac {e^{3} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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)

none

Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.12 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{3} + \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d^{2} e + \frac {1}{280} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b d e^{2} + \frac {1}{3150} \, {\left (315 \, x^{10} \arctan \left (c x\right ) - c {\left (\frac {35 \, c^{8} x^{9} - 45 \, c^{6} x^{7} + 63 \, c^{4} x^{5} - 105 \, c^{2} x^{3} + 315 \, x}{c^{10}} - \frac {315 \, \arctan \left (c x\right )}{c^{11}}\right )}\right )} b e^{3} \]

[In]

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

[Out]

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*a
rctan(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 - 10
5*c^2*x^3 + 315*x)/c^10 - 315*arctan(c*x)/c^11))*b*e^3

Giac [F]

\[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.50 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \arctan (c x)) \, dx=x^3\,\left (\frac {\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{c^2}+\frac {b\,d^2\,e}{2\,c}}{3\,c^2}-\frac {b\,d^3}{12\,c}\right )-x^8\,\left (\frac {a\,e^3}{8\,c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{8\,c^2}\right )+x^6\,\left (\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{6\,c^2}+\frac {a\,d\,e\,\left (d\,c^2+e\right )}{2\,c^2}\right )+x^7\,\left (\frac {b\,e^3}{70\,c^3}-\frac {3\,b\,d\,e^2}{56\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3\,x^4}{4}+\frac {b\,d^2\,e\,x^6}{2}+\frac {3\,b\,d\,e^2\,x^8}{8}+\frac {b\,e^3\,x^{10}}{10}\right )-x^5\,\left (\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{5\,c^2}+\frac {b\,d^2\,e}{10\,c}\right )+x^2\,\left (\frac {\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{c^2}}{2\,c^2}+\frac {a\,d^3}{2\,c^2}\right )-x^4\,\left (\frac {\frac {\frac {a\,e^3}{c^2}-\frac {a\,e^2\,\left (3\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {3\,a\,d\,e\,\left (d\,c^2+e\right )}{c^2}}{4\,c^2}-\frac {a\,d^2\,\left (d\,c^2+3\,e\right )}{4\,c^2}\right )+\frac {a\,e^3\,x^{10}}{10}-\frac {x\,\left (\frac {\frac {\frac {b\,e^3}{10\,c^3}-\frac {3\,b\,d\,e^2}{8\,c}}{c^2}+\frac {b\,d^2\,e}{2\,c}}{c^2}-\frac {b\,d^3}{4\,c}\right )}{c^2}-\frac {b\,e^3\,x^9}{90\,c}+\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (-10\,c^6\,d^3+20\,c^4\,d^2\,e-15\,c^2\,d\,e^2+4\,e^3\right )}{-10\,b\,c^6\,d^3+20\,b\,c^4\,d^2\,e-15\,b\,c^2\,d\,e^2+4\,b\,e^3}\right )\,\left (-10\,c^6\,d^3+20\,c^4\,d^2\,e-15\,c^2\,d\,e^2+4\,e^3\right )}{40\,c^{10}} \]

[In]

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

[Out]

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)