3.1137 \(\int x^3 (d+e x^2)^3 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=240 \[ \frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac{d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}-\frac{b e x^5 \left (20 c^4 d^2-15 c^2 d e+4 e^2\right )}{200 c^5}-\frac{b x^3 \left (-20 c^4 d^2 e+10 c^6 d^3+15 c^2 d e^2-4 e^3\right )}{120 c^7}+\frac{b x \left (-20 c^4 d^2 e+10 c^6 d^3+15 c^2 d e^2-4 e^3\right )}{40 c^9}-\frac{b e^2 x^7 \left (15 c^2 d-4 e\right )}{280 c^3}+\frac{b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right ) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac{b e^3 x^9}{90 c} \]

[Out]

(b*(10*c^6*d^3 - 20*c^4*d^2*e + 15*c^2*d*e^2 - 4*e^3)*x)/(40*c^9) - (b*(10*c^6*d^3 - 20*c^4*d^2*e + 15*c^2*d*e
^2 - 4*e^3)*x^3)/(120*c^7) - (b*e*(20*c^4*d^2 - 15*c^2*d*e + 4*e^2)*x^5)/(200*c^5) - (b*(15*c^2*d - 4*e)*e^2*x
^7)/(280*c^3) - (b*e^3*x^9)/(90*c) + (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)

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Rubi [A]  time = 0.458801, antiderivative size = 285, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 43, 4976, 12, 528, 388, 203} \[ \frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}-\frac{d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}-\frac{b x \left (25 c^4 d^2-135 c^2 d e+84 e^2\right ) \left (d+e x^2\right )^2}{4200 c^5 e}+\frac{b x \left (750 c^4 d^2 e+5 c^6 d^3-1071 c^2 d e^2+420 e^3\right ) \left (d+e x^2\right )}{12600 c^7 e}+\frac{b x \left (-4977 c^4 d^2 e^2+1815 c^6 d^3 e+325 c^8 d^4+4305 c^2 d e^3-1260 e^4\right )}{12600 c^9 e}+\frac{b \left (c^2 d-e\right )^4 \left (c^2 d+4 e\right ) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac{b x \left (23 c^2 d-36 e\right ) \left (d+e x^2\right )^3}{2520 c^3 e}-\frac{b x \left (d+e x^2\right )^4}{90 c e} \]

Antiderivative was successfully verified.

[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 266

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 43

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 4976

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]))

Rule 12

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

Rule 528

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 388

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 203

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

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{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 ) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac{d \left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 e^2}+\frac{\left (d+e x^2\right )^5 \left (a+b \tan ^{-1}(c x)\right )}{10 e^2}\\ \end{align*}

Mathematica [A]  time = 0.226935, size = 262, normalized size = 1.09 \[ \frac{c x \left (315 a c^9 x^3 \left (20 d^2 e x^2+10 d^3+15 d e^2 x^4+4 e^3 x^6\right )-b \left (5 c^8 \left (252 d^2 e x^4+210 d^3 x^2+135 d e^2 x^6+28 e^3 x^8\right )-15 c^6 \left (140 d^2 e x^2+210 d^3+63 d e^2 x^4+12 e^3 x^6\right )+63 c^4 e \left (100 d^2+25 d e x^2+4 e^2 x^4\right )-105 c^2 e^2 \left (45 d+4 e x^2\right )+1260 e^3\right )\right )+315 b \tan ^{-1}(c x) \left (c^{10} x^4 \left (20 d^2 e x^2+10 d^3+15 d e^2 x^4+4 e^3 x^6\right )+20 c^4 d^2 e-10 c^6 d^3-15 c^2 d e^2+4 e^3\right )}{12600 c^{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(315*a*c^9*x^3*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) - b*(1260*e^3 - 105*c^2*e^2*(45*d + 4*e
*x^2) + 63*c^4*e*(100*d^2 + 25*d*e*x^2 + 4*e^2*x^4) - 15*c^6*(210*d^3 + 140*d^2*e*x^2 + 63*d*e^2*x^4 + 12*e^3*
x^6) + 5*c^8*(210*d^3*x^2 + 252*d^2*e*x^4 + 135*d*e^2*x^6 + 28*e^3*x^8))) + 315*b*(-10*c^6*d^3 + 20*c^4*d^2*e
- 15*c^2*d*e^2 + 4*e^3 + c^10*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6))*ArcTan[c*x])/(12600*c^10
)

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Maple [A]  time = 0.038, size = 315, normalized size = 1.3 \begin{align*} -{\frac{b{d}^{2}e{x}^{5}}{10\,c}}+{\frac{b{x}^{3}{d}^{2}e}{6\,{c}^{3}}}-{\frac{b{x}^{3}d{e}^{2}}{8\,{c}^{5}}}-{\frac{b{d}^{2}ex}{2\,{c}^{5}}}-{\frac{3\,\arctan \left ( cx \right ) bd{e}^{2}}{8\,{c}^{8}}}+{\frac{b{d}^{2}\arctan \left ( cx \right ) e}{2\,{c}^{6}}}+{\frac{3\,b\arctan \left ( cx \right ) d{e}^{2}{x}^{8}}{8}}+{\frac{b\arctan \left ( cx \right ){d}^{2}e{x}^{6}}{2}}-{\frac{3\,bd{e}^{2}{x}^{7}}{56\,c}}+{\frac{3\,bd{e}^{2}x}{8\,{c}^{7}}}+{\frac{3\,b{x}^{5}d{e}^{2}}{40\,{c}^{3}}}+{\frac{a{e}^{3}{x}^{10}}{10}}+{\frac{a{x}^{4}{d}^{3}}{4}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{4\,{c}^{4}}}+{\frac{b{x}^{7}{e}^{3}}{70\,{c}^{3}}}+{\frac{b\arctan \left ( cx \right ){x}^{4}{d}^{3}}{4}}+{\frac{b\arctan \left ( cx \right ){e}^{3}{x}^{10}}{10}}-{\frac{b{e}^{3}x}{10\,{c}^{9}}}+{\frac{b{e}^{3}{x}^{3}}{30\,{c}^{7}}}-{\frac{b{x}^{5}{e}^{3}}{50\,{c}^{5}}}+{\frac{a{d}^{2}e{x}^{6}}{2}}+{\frac{3\,ad{e}^{2}{x}^{8}}{8}}+{\frac{b\arctan \left ( cx \right ){e}^{3}}{10\,{c}^{10}}}+{\frac{b{d}^{3}x}{4\,{c}^{3}}}-{\frac{b{d}^{3}{x}^{3}}{12\,c}}-{\frac{b{e}^{3}{x}^{9}}{90\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10/c*b*d^2*e*x^5+1/6/c^3*b*x^3*d^2*e-1/8/c^5*b*x^3*d*e^2-1/2/c^5*b*d^2*e*x-3/8/c^8*b*arctan(c*x)*d*e^2+1/2/
c^6*b*arctan(c*x)*d^2*e+3/8*b*arctan(c*x)*d*e^2*x^8+1/2*b*arctan(c*x)*d^2*e*x^6-3/56/c*b*d*e^2*x^7+3/8/c^7*b*d
*e^2*x+3/40/c^3*b*x^5*d*e^2+1/10*a*e^3*x^10+1/4*a*x^4*d^3-1/4*b*d^3*arctan(c*x)/c^4+1/70/c^3*b*x^7*e^3+1/4*b*a
rctan(c*x)*x^4*d^3+1/10*b*arctan(c*x)*e^3*x^10-1/10/c^9*b*e^3*x+1/30/c^7*b*e^3*x^3-1/50/c^5*b*x^5*e^3+1/2*a*d^
2*e*x^6+3/8*a*d*e^2*x^8+1/10/c^10*b*arctan(c*x)*e^3+1/4*b*d^3*x/c^3-1/12*b*d^3*x^3/c-1/90*b*e^3*x^9/c

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Maxima [A]  time = 1.48413, size = 362, normalized size = 1.51 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.42384, size = 711, normalized size = 2.96 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 9.67284, size = 411, normalized size = 1.71 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.64996, size = 479, normalized size = 2. \begin{align*} \frac{1260 \, b c^{10} x^{10} \arctan \left (c x\right ) e^{3} + 1260 \, a c^{10} x^{10} e^{3} + 4725 \, b c^{10} d x^{8} \arctan \left (c x\right ) e^{2} + 4725 \, a c^{10} d x^{8} e^{2} + 6300 \, b c^{10} d^{2} x^{6} \arctan \left (c x\right ) e - 140 \, b c^{9} x^{9} e^{3} + 6300 \, a c^{10} d^{2} x^{6} e + 3150 \, b c^{10} d^{3} x^{4} \arctan \left (c x\right ) - 675 \, b c^{9} d x^{7} e^{2} + 3150 \, a c^{10} d^{3} x^{4} - 1260 \, b c^{9} d^{2} x^{5} e - 1050 \, b c^{9} d^{3} x^{3} + 180 \, b c^{7} x^{7} e^{3} + 945 \, b c^{7} d x^{5} e^{2} + 2100 \, b c^{7} d^{2} x^{3} e + 3150 \, b c^{7} d^{3} x - 252 \, b c^{5} x^{5} e^{3} - 3150 \, b c^{6} d^{3} \arctan \left (c x\right ) - 1575 \, b c^{5} d x^{3} e^{2} - 6300 \, \pi b c^{4} d^{2} e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 6300 \, b c^{5} d^{2} x e + 6300 \, b c^{4} d^{2} \arctan \left (c x\right ) e + 420 \, b c^{3} x^{3} e^{3} + 4725 \, b c^{3} d x e^{2} - 4725 \, b c^{2} d \arctan \left (c x\right ) e^{2} - 1260 \, \pi b e^{3} \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 1260 \, b c x e^{3} + 1260 \, b \arctan \left (c x\right ) e^{3}}{12600 \, c^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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