3.1247 \(\int x^3 (d+e x^2) (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=271 \[ \frac{a b d x}{2 c^3}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{b e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{a b e x}{3 c^5}+\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left (c^2 x^2+1\right )}{3 c^4}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 e x^4}{60 c^2}-\frac{4 b^2 e x^2}{45 c^4}+\frac{23 b^2 e \log \left (c^2 x^2+1\right )}{90 c^6}-\frac{b^2 e x \tan ^{-1}(c x)}{3 c^5} \]

[Out]

(a*b*d*x)/(2*c^3) - (a*b*e*x)/(3*c^5) + (b^2*d*x^2)/(12*c^2) - (4*b^2*e*x^2)/(45*c^4) + (b^2*e*x^4)/(60*c^2) +
(b^2*d*x*ArcTan[c*x])/(2*c^3) - (b^2*e*x*ArcTan[c*x])/(3*c^5) - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) + (b*e*x^
3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e*x^5*(a + b*ArcTan[c*x]))/(15*c) - (d*(a + b*ArcTan[c*x])^2)/(4*c^4) + (e
*(a + b*ArcTan[c*x])^2)/(6*c^6) + (d*x^4*(a + b*ArcTan[c*x])^2)/4 + (e*x^6*(a + b*ArcTan[c*x])^2)/6 - (b^2*d*L
og[1 + c^2*x^2])/(3*c^4) + (23*b^2*e*Log[1 + c^2*x^2])/(90*c^6)

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Rubi [A]  time = 0.651159, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {4980, 4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac{a b d x}{2 c^3}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{b e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{a b e x}{3 c^5}+\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left (c^2 x^2+1\right )}{3 c^4}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 e x^4}{60 c^2}-\frac{4 b^2 e x^2}{45 c^4}+\frac{23 b^2 e \log \left (c^2 x^2+1\right )}{90 c^6}-\frac{b^2 e x \tan ^{-1}(c x)}{3 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*d*x)/(2*c^3) - (a*b*e*x)/(3*c^5) + (b^2*d*x^2)/(12*c^2) - (4*b^2*e*x^2)/(45*c^4) + (b^2*e*x^4)/(60*c^2) +
(b^2*d*x*ArcTan[c*x])/(2*c^3) - (b^2*e*x*ArcTan[c*x])/(3*c^5) - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) + (b*e*x^
3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e*x^5*(a + b*ArcTan[c*x]))/(15*c) - (d*(a + b*ArcTan[c*x])^2)/(4*c^4) + (e
*(a + b*ArcTan[c*x])^2)/(6*c^6) + (d*x^4*(a + b*ArcTan[c*x])^2)/4 + (e*x^6*(a + b*ArcTan[c*x])^2)/6 - (b^2*d*L
og[1 + c^2*x^2])/(3*c^4) + (23*b^2*e*Log[1 + c^2*x^2])/(90*c^6)

Rule 4980

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With
[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,
c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4916

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

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 4846

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4884

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

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d x^3 \left (a+b \tan ^{-1}(c x)\right )^2+e x^5 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} (b c d) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{3} (b c e) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{(b d) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac{(b d) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c}-\frac{(b e) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{(b e) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} \left (b^2 d\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{(b d) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac{(b d) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3}+\frac{1}{15} \left (b^2 e\right ) \int \frac{x^5}{1+c^2 x^2} \, dx+\frac{(b e) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{(b e) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3}\\ &=\frac{a b d x}{2 c^3}-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{b e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{12} \left (b^2 d\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{\left (b^2 d\right ) \int \tan ^{-1}(c x) \, dx}{2 c^3}+\frac{1}{30} \left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac{(b e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^5}+\frac{(b e) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac{\left (b^2 e\right ) \int \frac{x^3}{1+c^2 x^2} \, dx}{9 c^2}\\ &=\frac{a b d x}{2 c^3}-\frac{a b e x}{3 c^5}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{b e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{12} \left (b^2 d\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (b^2 d\right ) \int \frac{x}{1+c^2 x^2} \, dx}{2 c^2}+\frac{1}{30} \left (b^2 e\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (b^2 e\right ) \int \tan ^{-1}(c x) \, dx}{3 c^5}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{18 c^2}\\ &=\frac{a b d x}{2 c^3}-\frac{a b e x}{3 c^5}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 e x^2}{30 c^4}+\frac{b^2 e x^4}{60 c^2}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}-\frac{b^2 e x \tan ^{-1}(c x)}{3 c^5}-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{b e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b^2 d \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{b^2 e \log \left (1+c^2 x^2\right )}{30 c^6}+\frac{\left (b^2 e\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 c^4}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{18 c^2}\\ &=\frac{a b d x}{2 c^3}-\frac{a b e x}{3 c^5}+\frac{b^2 d x^2}{12 c^2}-\frac{4 b^2 e x^2}{45 c^4}+\frac{b^2 e x^4}{60 c^2}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}-\frac{b^2 e x \tan ^{-1}(c x)}{3 c^5}-\frac{b d x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac{b e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac{e \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{4} d x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b^2 d \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{23 b^2 e \log \left (1+c^2 x^2\right )}{90 c^6}\\ \end{align*}

Mathematica [A]  time = 0.235419, size = 240, normalized size = 0.89 \[ \frac{c x \left (15 a^2 c^5 x^3 \left (3 d+2 e x^2\right )-2 a b \left (3 c^4 \left (5 d x^2+2 e x^4\right )-5 c^2 \left (9 d+2 e x^2\right )+30 e\right )+b^2 c x \left (3 c^2 \left (5 d+e x^2\right )-16 e\right )\right )+2 b \tan ^{-1}(c x) \left (15 a \left (c^6 \left (3 d x^4+2 e x^6\right )-3 c^2 d+2 e\right )+b c x \left (-3 c^4 \left (5 d x^2+2 e x^4\right )+5 c^2 \left (9 d+2 e x^2\right )-30 e\right )\right )+2 b^2 \left (23 e-30 c^2 d\right ) \log \left (c^2 x^2+1\right )+15 b^2 \tan ^{-1}(c x)^2 \left (c^6 \left (3 d x^4+2 e x^6\right )-3 c^2 d+2 e\right )}{180 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(15*a^2*c^5*x^3*(3*d + 2*e*x^2) + b^2*c*x*(-16*e + 3*c^2*(5*d + e*x^2)) - 2*a*b*(30*e - 5*c^2*(9*d + 2*e*
x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4))) + 2*b*(b*c*x*(-30*e + 5*c^2*(9*d + 2*e*x^2) - 3*c^4*(5*d*x^2 + 2*e*x^4)) +
15*a*(-3*c^2*d + 2*e + c^6*(3*d*x^4 + 2*e*x^6)))*ArcTan[c*x] + 15*b^2*(-3*c^2*d + 2*e + c^6*(3*d*x^4 + 2*e*x^6
))*ArcTan[c*x]^2 + 2*b^2*(-30*c^2*d + 23*e)*Log[1 + c^2*x^2])/(180*c^6)

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Maple [A]  time = 0.053, size = 329, normalized size = 1.2 \begin{align*} -{\frac{{b}^{2}d\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,{c}^{4}}}-{\frac{abe{x}^{5}}{15\,c}}+{\frac{ab\arctan \left ( cx \right ) e}{3\,{c}^{6}}}+{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{3}e}{9\,{c}^{3}}}-{\frac{{b}^{2}\arctan \left ( cx \right ) e{x}^{5}}{15\,c}}-{\frac{abd{x}^{3}}{6\,c}}+{\frac{ab{x}^{3}e}{9\,{c}^{3}}}+{\frac{ab\arctan \left ( cx \right ) e{x}^{6}}{3}}+{\frac{ab\arctan \left ( cx \right ){x}^{4}d}{2}}-{\frac{4\,{b}^{2}e{x}^{2}}{45\,{c}^{4}}}+{\frac{{b}^{2}e{x}^{4}}{60\,{c}^{2}}}+{\frac{23\,{b}^{2}e\ln \left ({c}^{2}{x}^{2}+1 \right ) }{90\,{c}^{6}}}-{\frac{{b}^{2}\arctan \left ( cx \right ) d{x}^{3}}{6\,c}}+{\frac{{a}^{2}e{x}^{6}}{6}}+{\frac{{a}^{2}{x}^{4}d}{4}}-{\frac{ab\arctan \left ( cx \right ) d}{2\,{c}^{4}}}-{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}d}{4\,{c}^{4}}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{4}d}{4}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}e{x}^{6}}{6}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}e}{6\,{c}^{6}}}-{\frac{abex}{3\,{c}^{5}}}-{\frac{{b}^{2}ex\arctan \left ( cx \right ) }{3\,{c}^{5}}}+{\frac{abdx}{2\,{c}^{3}}}+{\frac{{b}^{2}dx\arctan \left ( cx \right ) }{2\,{c}^{3}}}+{\frac{{b}^{2}d{x}^{2}}{12\,{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2*d*ln(c^2*x^2+1)/c^4-1/15/c*a*b*e*x^5+1/3/c^6*a*b*arctan(c*x)*e+1/9/c^3*b^2*arctan(c*x)*x^3*e-1/15/c*b
^2*arctan(c*x)*e*x^5-1/6/c*a*b*d*x^3+1/9/c^3*a*b*x^3*e+1/3*a*b*arctan(c*x)*e*x^6+1/2*a*b*arctan(c*x)*x^4*d-4/4
5*b^2*e*x^2/c^4+1/60*b^2*e*x^4/c^2+23/90*b^2*e*ln(c^2*x^2+1)/c^6-1/6/c*b^2*arctan(c*x)*d*x^3+1/6*a^2*e*x^6+1/4
*a^2*x^4*d-1/2/c^4*a*b*arctan(c*x)*d-1/4/c^4*b^2*arctan(c*x)^2*d+1/4*b^2*arctan(c*x)^2*x^4*d+1/6*b^2*arctan(c*
x)^2*e*x^6+1/6/c^6*b^2*arctan(c*x)^2*e-1/3*a*b*e*x/c^5-1/3*b^2*e*x*arctan(c*x)/c^5+1/2*a*b*d*x/c^3+1/2*b^2*d*x
*arctan(c*x)/c^3+1/12*b^2*d*x^2/c^2

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Maxima [A]  time = 1.59891, size = 413, normalized size = 1.52 \begin{align*} \frac{1}{6} \, b^{2} e x^{6} \arctan \left (c x\right )^{2} + \frac{1}{6} \, a^{2} e x^{6} + \frac{1}{4} \, b^{2} d x^{4} \arctan \left (c x\right )^{2} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{6} \,{\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 )} a b d - \frac{1}{12} \,{\left (2 \, c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac{c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d + \frac{1}{45} \,{\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 )} a b e - \frac{1}{180} \,{\left (4 \, 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 )} \arctan \left (c x\right ) - \frac{3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*e*x^6*arctan(c*x)^2 + 1/6*a^2*e*x^6 + 1/4*b^2*d*x^4*arctan(c*x)^2 + 1/4*a^2*d*x^4 + 1/6*(3*x^4*arctan(
c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*d - 1/12*(2*c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5
)*arctan(c*x) - (c^2*x^2 + 3*arctan(c*x)^2 - 4*log(c^2*x^2 + 1))/c^4)*b^2*d + 1/45*(15*x^6*arctan(c*x) - c*((3
*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*a*b*e - 1/180*(4*c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6
- 15*arctan(c*x)/c^7)*arctan(c*x) - (3*c^4*x^4 - 16*c^2*x^2 - 30*arctan(c*x)^2 + 46*log(c^2*x^2 + 1))/c^6)*b^
2*e

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Fricas [A]  time = 1.8157, size = 652, normalized size = 2.41 \begin{align*} \frac{30 \, a^{2} c^{6} e x^{6} - 12 \, a b c^{5} e x^{5} + 3 \,{\left (15 \, a^{2} c^{6} d + b^{2} c^{4} e\right )} x^{4} - 10 \,{\left (3 \, a b c^{5} d - 2 \, a b c^{3} e\right )} x^{3} +{\left (15 \, b^{2} c^{4} d - 16 \, b^{2} c^{2} e\right )} x^{2} + 15 \,{\left (2 \, b^{2} c^{6} e x^{6} + 3 \, b^{2} c^{6} d x^{4} - 3 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \arctan \left (c x\right )^{2} + 30 \,{\left (3 \, a b c^{3} d - 2 \, a b c e\right )} x + 2 \,{\left (30 \, a b c^{6} e x^{6} + 45 \, a b c^{6} d x^{4} - 6 \, b^{2} c^{5} e x^{5} - 45 \, a b c^{2} d - 5 \,{\left (3 \, b^{2} c^{5} d - 2 \, b^{2} c^{3} e\right )} x^{3} + 30 \, a b e + 15 \,{\left (3 \, b^{2} c^{3} d - 2 \, b^{2} c e\right )} x\right )} \arctan \left (c x\right ) - 2 \,{\left (30 \, b^{2} c^{2} d - 23 \, b^{2} e\right )} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(30*a^2*c^6*e*x^6 - 12*a*b*c^5*e*x^5 + 3*(15*a^2*c^6*d + b^2*c^4*e)*x^4 - 10*(3*a*b*c^5*d - 2*a*b*c^3*e)
*x^3 + (15*b^2*c^4*d - 16*b^2*c^2*e)*x^2 + 15*(2*b^2*c^6*e*x^6 + 3*b^2*c^6*d*x^4 - 3*b^2*c^2*d + 2*b^2*e)*arct
an(c*x)^2 + 30*(3*a*b*c^3*d - 2*a*b*c*e)*x + 2*(30*a*b*c^6*e*x^6 + 45*a*b*c^6*d*x^4 - 6*b^2*c^5*e*x^5 - 45*a*b
*c^2*d - 5*(3*b^2*c^5*d - 2*b^2*c^3*e)*x^3 + 30*a*b*e + 15*(3*b^2*c^3*d - 2*b^2*c*e)*x)*arctan(c*x) - 2*(30*b^
2*c^2*d - 23*b^2*e)*log(c^2*x^2 + 1))/c^6

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Sympy [A]  time = 6.14251, size = 398, normalized size = 1.47 \begin{align*} \begin{cases} \frac{a^{2} d x^{4}}{4} + \frac{a^{2} e x^{6}}{6} + \frac{a b d x^{4} \operatorname{atan}{\left (c x \right )}}{2} + \frac{a b e x^{6} \operatorname{atan}{\left (c x \right )}}{3} - \frac{a b d x^{3}}{6 c} - \frac{a b e x^{5}}{15 c} + \frac{a b d x}{2 c^{3}} + \frac{a b e x^{3}}{9 c^{3}} - \frac{a b d \operatorname{atan}{\left (c x \right )}}{2 c^{4}} - \frac{a b e x}{3 c^{5}} + \frac{a b e \operatorname{atan}{\left (c x \right )}}{3 c^{6}} + \frac{b^{2} d x^{4} \operatorname{atan}^{2}{\left (c x \right )}}{4} + \frac{b^{2} e x^{6} \operatorname{atan}^{2}{\left (c x \right )}}{6} - \frac{b^{2} d x^{3} \operatorname{atan}{\left (c x \right )}}{6 c} - \frac{b^{2} e x^{5} \operatorname{atan}{\left (c x \right )}}{15 c} + \frac{b^{2} d x^{2}}{12 c^{2}} + \frac{b^{2} e x^{4}}{60 c^{2}} + \frac{b^{2} d x \operatorname{atan}{\left (c x \right )}}{2 c^{3}} + \frac{b^{2} e x^{3} \operatorname{atan}{\left (c x \right )}}{9 c^{3}} - \frac{b^{2} d \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{3 c^{4}} - \frac{b^{2} d \operatorname{atan}^{2}{\left (c x \right )}}{4 c^{4}} - \frac{4 b^{2} e x^{2}}{45 c^{4}} - \frac{b^{2} e x \operatorname{atan}{\left (c x \right )}}{3 c^{5}} + \frac{23 b^{2} e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{90 c^{6}} + \frac{b^{2} e \operatorname{atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text{for}\: c \neq 0 \\a^{2} \left (\frac{d x^{4}}{4} + \frac{e x^{6}}{6}\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)*(a+b*atan(c*x))**2,x)

[Out]

Piecewise((a**2*d*x**4/4 + a**2*e*x**6/6 + a*b*d*x**4*atan(c*x)/2 + a*b*e*x**6*atan(c*x)/3 - a*b*d*x**3/(6*c)
- a*b*e*x**5/(15*c) + a*b*d*x/(2*c**3) + a*b*e*x**3/(9*c**3) - a*b*d*atan(c*x)/(2*c**4) - a*b*e*x/(3*c**5) + a
*b*e*atan(c*x)/(3*c**6) + b**2*d*x**4*atan(c*x)**2/4 + b**2*e*x**6*atan(c*x)**2/6 - b**2*d*x**3*atan(c*x)/(6*c
) - b**2*e*x**5*atan(c*x)/(15*c) + b**2*d*x**2/(12*c**2) + b**2*e*x**4/(60*c**2) + b**2*d*x*atan(c*x)/(2*c**3)
+ b**2*e*x**3*atan(c*x)/(9*c**3) - b**2*d*log(x**2 + c**(-2))/(3*c**4) - b**2*d*atan(c*x)**2/(4*c**4) - 4*b**
2*e*x**2/(45*c**4) - b**2*e*x*atan(c*x)/(3*c**5) + 23*b**2*e*log(x**2 + c**(-2))/(90*c**6) + b**2*e*atan(c*x)*
*2/(6*c**6), Ne(c, 0)), (a**2*(d*x**4/4 + e*x**6/6), True))

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Giac [A]  time = 1.42377, size = 490, normalized size = 1.81 \begin{align*} \frac{30 \, b^{2} c^{6} x^{6} \arctan \left (c x\right )^{2} e + 60 \, a b c^{6} x^{6} \arctan \left (c x\right ) e + 45 \, b^{2} c^{6} d x^{4} \arctan \left (c x\right )^{2} + 30 \, a^{2} c^{6} x^{6} e + 90 \, a b c^{6} d x^{4} \arctan \left (c x\right ) - 12 \, b^{2} c^{5} x^{5} \arctan \left (c x\right ) e + 45 \, a^{2} c^{6} d x^{4} - 12 \, a b c^{5} x^{5} e - 30 \, b^{2} c^{5} d x^{3} \arctan \left (c x\right ) - 30 \, a b c^{5} d x^{3} + 3 \, b^{2} c^{4} x^{4} e + 20 \, b^{2} c^{3} x^{3} \arctan \left (c x\right ) e + 15 \, b^{2} c^{4} d x^{2} + 20 \, a b c^{3} x^{3} e + 90 \, b^{2} c^{3} d x \arctan \left (c x\right ) + 90 \, a b c^{3} d x - 45 \, b^{2} c^{2} d \arctan \left (c x\right )^{2} - 16 \, b^{2} c^{2} x^{2} e - 90 \, a b c^{2} d \arctan \left (c x\right ) - 60 \, b^{2} c x \arctan \left (c x\right ) e - 60 \, b^{2} c^{2} d \log \left (c^{2} x^{2} + 1\right ) - 60 \, \pi a b e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 60 \, a b c x e + 30 \, b^{2} \arctan \left (c x\right )^{2} e + 60 \, a b \arctan \left (c x\right ) e + 46 \, b^{2} e \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(30*b^2*c^6*x^6*arctan(c*x)^2*e + 60*a*b*c^6*x^6*arctan(c*x)*e + 45*b^2*c^6*d*x^4*arctan(c*x)^2 + 30*a^2
*c^6*x^6*e + 90*a*b*c^6*d*x^4*arctan(c*x) - 12*b^2*c^5*x^5*arctan(c*x)*e + 45*a^2*c^6*d*x^4 - 12*a*b*c^5*x^5*e
- 30*b^2*c^5*d*x^3*arctan(c*x) - 30*a*b*c^5*d*x^3 + 3*b^2*c^4*x^4*e + 20*b^2*c^3*x^3*arctan(c*x)*e + 15*b^2*c
^4*d*x^2 + 20*a*b*c^3*x^3*e + 90*b^2*c^3*d*x*arctan(c*x) + 90*a*b*c^3*d*x - 45*b^2*c^2*d*arctan(c*x)^2 - 16*b^
2*c^2*x^2*e - 90*a*b*c^2*d*arctan(c*x) - 60*b^2*c*x*arctan(c*x)*e - 60*b^2*c^2*d*log(c^2*x^2 + 1) - 60*pi*a*b*
e*sgn(c)*sgn(x) - 60*a*b*c*x*e + 30*b^2*arctan(c*x)^2*e + 60*a*b*arctan(c*x)*e + 46*b^2*e*log(c^2*x^2 + 1))/c^
6