3.1138 \(\int x^2 (d+e x^2)^3 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=239 \[ \frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e x^4 \left (189 c^4 d^2-135 c^2 d e+35 e^2\right )}{1260 c^5}-\frac{b x^2 \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right )}{630 c^7}+\frac{b \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right ) \log \left (c^2 x^2+1\right )}{630 c^9}-\frac{b e^2 x^6 \left (27 c^2 d-7 e\right )}{378 c^3}-\frac{b e^3 x^8}{72 c} \]

[Out]

-(b*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*x^2)/(630*c^7) - (b*e*(189*c^4*d^2 - 135*c^2*d*e +
35*e^2)*x^4)/(1260*c^5) - (b*(27*c^2*d - 7*e)*e^2*x^6)/(378*c^3) - (b*e^3*x^8)/(72*c) + (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*(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])/(630*c^9)

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Rubi [A]  time = 0.384412, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {270, 4976, 12, 1799, 1620} \[ \frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e x^4 \left (189 c^4 d^2-135 c^2 d e+35 e^2\right )}{1260 c^5}-\frac{b x^2 \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right )}{630 c^7}+\frac{b \left (-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right ) \log \left (c^2 x^2+1\right )}{630 c^9}-\frac{b e^2 x^6 \left (27 c^2 d-7 e\right )}{378 c^3}-\frac{b e^3 x^8}{72 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*x^2)/(630*c^7) - (b*e*(189*c^4*d^2 - 135*c^2*d*e +
35*e^2)*x^4)/(1260*c^5) - (b*(27*c^2*d - 7*e)*e^2*x^6)/(378*c^3) - (b*e^3*x^8)/(72*c) + (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*(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])/(630*c^9)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[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]

Rule 1620

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]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{315 \left (1+c^2 x^2\right )} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \frac{x \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \left (\frac{105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3}{c^8}+\frac{e \left (189 c^4 d^2-135 c^2 d e+35 e^2\right ) x}{c^6}+\frac{5 \left (27 c^2 d-7 e\right ) e^2 x^2}{c^4}+\frac{35 e^3 x^3}{c^2}+\frac{-105 c^6 d^3+189 c^4 d^2 e-135 c^2 d e^2+35 e^3}{c^8 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\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 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )+\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}\\ \end{align*}

Mathematica [A]  time = 0.208617, size = 252, normalized size = 1.05 \[ \frac{3}{5} d^2 e x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{20} b d^2 e \left (\frac{2 x^2}{c^3}-\frac{2 \log \left (c^2 x^2+1\right )}{c^5}-\frac{x^4}{c}\right )-\frac{1}{6} b d^3 \left (\frac{x^2}{c}-\frac{\log \left (c^2 x^2+1\right )}{c^3}\right )-\frac{1}{28} b d e^2 \left (-\frac{3 x^4}{c^3}+\frac{6 x^2}{c^5}-\frac{6 \log \left (c^2 x^2+1\right )}{c^7}+\frac{2 x^6}{c}\right )+\frac{1}{216} b e^3 \left (\frac{4 x^6}{c^3}-\frac{6 x^4}{c^5}+\frac{12 x^2}{c^7}-\frac{12 \log \left (c^2 x^2+1\right )}{c^9}-\frac{3 x^8}{c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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*Log[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

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Maple [A]  time = 0.039, size = 297, normalized size = 1.2 \begin{align*}{\frac{a{e}^{3}{x}^{9}}{9}}+{\frac{3\,ad{e}^{2}{x}^{7}}{7}}+{\frac{3\,a{d}^{2}e{x}^{5}}{5}}+{\frac{a{d}^{3}{x}^{3}}{3}}+{\frac{b\arctan \left ( cx \right ){e}^{3}{x}^{9}}{9}}+{\frac{3\,b\arctan \left ( cx \right ) d{e}^{2}{x}^{7}}{7}}+{\frac{3\,b\arctan \left ( cx \right ){d}^{2}e{x}^{5}}{5}}+{\frac{b\arctan \left ( cx \right ){d}^{3}{x}^{3}}{3}}-{\frac{b{d}^{3}{x}^{2}}{6\,c}}-{\frac{3\,b{d}^{2}e{x}^{4}}{20\,c}}-{\frac{bd{e}^{2}{x}^{6}}{14\,c}}+{\frac{3\,b{x}^{2}{d}^{2}e}{10\,{c}^{3}}}-{\frac{b{e}^{3}{x}^{8}}{72\,c}}+{\frac{3\,b{x}^{4}d{e}^{2}}{28\,{c}^{3}}}+{\frac{b{x}^{6}{e}^{3}}{54\,{c}^{3}}}-{\frac{3\,b{x}^{2}d{e}^{2}}{14\,{c}^{5}}}-{\frac{b{e}^{3}{x}^{4}}{36\,{c}^{5}}}+{\frac{b{e}^{3}{x}^{2}}{18\,{c}^{7}}}+{\frac{b{d}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{3}}}-{\frac{3\,b\ln \left ({c}^{2}{x}^{2}+1 \right ){d}^{2}e}{10\,{c}^{5}}}+{\frac{3\,b\ln \left ({c}^{2}{x}^{2}+1 \right ) d{e}^{2}}{14\,{c}^{7}}}-{\frac{b\ln \left ({c}^{2}{x}^{2}+1 \right ){e}^{3}}{18\,{c}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+1/9*b*arctan(c*x)*e^3*x^9+3/7*b*arctan(c*x)*d*e^2*
x^7+3/5*b*arctan(c*x)*d^2*e*x^5+1/3*b*arctan(c*x)*d^3*x^3-1/6*b*d^3*x^2/c-3/20/c*b*d^2*e*x^4-1/14/c*b*d*e^2*x^
6+3/10/c^3*b*x^2*d^2*e-1/72*b*e^3*x^8/c+3/28/c^3*b*x^4*d*e^2+1/54/c^3*b*x^6*e^3-3/14/c^5*b*x^2*d*e^2-1/36/c^5*
b*e^3*x^4+1/18/c^7*b*e^3*x^2+1/6*b*d^3*ln(c^2*x^2+1)/c^3-3/10/c^5*b*ln(c^2*x^2+1)*d^2*e+3/14/c^7*b*ln(c^2*x^2+
1)*d*e^2-1/18/c^9*b*ln(c^2*x^2+1)*e^3

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Maxima [A]  time = 1.04122, size = 358, normalized size = 1.5 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*log(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

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Fricas [A]  time = 1.35348, size = 647, normalized size = 2.71 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 4536*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*(10
5*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

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Sympy [A]  time = 6.92135, size = 389, normalized size = 1.63 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Giac [A]  time = 1.10018, size = 424, normalized size = 1.77 \begin{align*} \frac{840 \, b c^{9} x^{9} \arctan \left (c x\right ) e^{3} + 840 \, a c^{9} x^{9} e^{3} + 3240 \, b c^{9} d x^{7} \arctan \left (c x\right ) e^{2} + 3240 \, a c^{9} d x^{7} e^{2} + 4536 \, b c^{9} d^{2} x^{5} \arctan \left (c x\right ) e - 105 \, b c^{8} x^{8} e^{3} + 4536 \, a c^{9} d^{2} x^{5} e + 2520 \, b c^{9} d^{3} x^{3} \arctan \left (c x\right ) - 540 \, b c^{8} d x^{6} e^{2} + 2520 \, a c^{9} d^{3} x^{3} - 1134 \, b c^{8} d^{2} x^{4} e - 1260 \, b c^{8} d^{3} x^{2} + 140 \, b c^{6} x^{6} e^{3} + 810 \, b c^{6} d x^{4} e^{2} + 2268 \, b c^{6} d^{2} x^{2} e + 1260 \, b c^{6} d^{3} \log \left (c^{2} x^{2} + 1\right ) - 210 \, b c^{4} x^{4} e^{3} - 1620 \, b c^{4} d x^{2} e^{2} - 2268 \, b c^{4} d^{2} e \log \left (c^{2} x^{2} + 1\right ) + 420 \, b c^{2} x^{2} e^{3} + 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7560*(840*b*c^9*x^9*arctan(c*x)*e^3 + 840*a*c^9*x^9*e^3 + 3240*b*c^9*d*x^7*arctan(c*x)*e^2 + 3240*a*c^9*d*x^
7*e^2 + 4536*b*c^9*d^2*x^5*arctan(c*x)*e - 105*b*c^8*x^8*e^3 + 4536*a*c^9*d^2*x^5*e + 2520*b*c^9*d^3*x^3*arcta
n(c*x) - 540*b*c^8*d*x^6*e^2 + 2520*a*c^9*d^3*x^3 - 1134*b*c^8*d^2*x^4*e - 1260*b*c^8*d^3*x^2 + 140*b*c^6*x^6*
e^3 + 810*b*c^6*d*x^4*e^2 + 2268*b*c^6*d^2*x^2*e + 1260*b*c^6*d^3*log(c^2*x^2 + 1) - 210*b*c^4*x^4*e^3 - 1620*
b*c^4*d*x^2*e^2 - 2268*b*c^4*d^2*e*log(c^2*x^2 + 1) + 420*b*c^2*x^2*e^3 + 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