3.1254 \(\int x^3 (d+e x^2)^2 (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=502 \[ -\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac {a b e^2 x}{4 c^7}+\frac {d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac {2 a b d e x}{3 c^5}-\frac {b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {a b d^2 x}{2 c^3}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}+\frac {b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}+\frac {71 b^2 e^2 x^2}{840 c^6}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}-\frac {8 b^2 d e x^2}{45 c^4}-\frac {3 b^2 e^2 x^4}{140 c^4}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac {b^2 d^2 x^2}{12 c^2}+\frac {b^2 d e x^4}{30 c^2}+\frac {b^2 e^2 x^6}{168 c^2}-\frac {22 b^2 e^2 \log \left (c^2 x^2+1\right )}{105 c^8}+\frac {23 b^2 d e \log \left (c^2 x^2+1\right )}{45 c^6}-\frac {b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^4} \]

[Out]

1/2*a*b*d^2*x/c^3+1/2*b^2*d^2*x*arctan(c*x)/c^3-1/6*b*d^2*x^3*(a+b*arctan(c*x))/c-1/4*d^2*(a+b*arctan(c*x))^2/
c^4+1/4*d^2*x^4*(a+b*arctan(c*x))^2-1/3*b^2*d^2*ln(c^2*x^2+1)/c^4-22/105*b^2*e^2*ln(c^2*x^2+1)/c^8+23/45*b^2*d
*e*ln(c^2*x^2+1)/c^6-1/8*e^2*(a+b*arctan(c*x))^2/c^8+1/8*e^2*x^8*(a+b*arctan(c*x))^2+1/4*b^2*e^2*x*arctan(c*x)
/c^7-1/12*b*e^2*x^3*(a+b*arctan(c*x))/c^5+1/20*b*e^2*x^5*(a+b*arctan(c*x))/c^3-1/28*b*e^2*x^7*(a+b*arctan(c*x)
)/c+1/4*a*b*e^2*x/c^7-8/45*b^2*d*e*x^2/c^4+1/30*b^2*d*e*x^4/c^2+71/840*b^2*e^2*x^2/c^6-3/140*b^2*e^2*x^4/c^4+1
/168*b^2*e^2*x^6/c^2+1/3*d*e*(a+b*arctan(c*x))^2/c^6+1/3*d*e*x^6*(a+b*arctan(c*x))^2+1/12*b^2*d^2*x^2/c^2-2/3*
a*b*d*e*x/c^5-2/3*b^2*d*e*x*arctan(c*x)/c^5+2/9*b*d*e*x^3*(a+b*arctan(c*x))/c^3-2/15*b*d*e*x^5*(a+b*arctan(c*x
))/c

________________________________________________________________________________________

Rubi [A]  time = 1.14, antiderivative size = 502, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4980, 4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac {a b d^2 x}{2 c^3}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {2 a b d e x}{3 c^5}+\frac {d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac {b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}+\frac {a b e^2 x}{4 c^7}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}+\frac {b^2 d^2 x^2}{12 c^2}-\frac {b^2 d^2 \log \left (c^2 x^2+1\right )}{3 c^4}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac {b^2 d e x^4}{30 c^2}-\frac {8 b^2 d e x^2}{45 c^4}+\frac {23 b^2 d e \log \left (c^2 x^2+1\right )}{45 c^6}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac {b^2 e^2 x^6}{168 c^2}-\frac {3 b^2 e^2 x^4}{140 c^4}+\frac {71 b^2 e^2 x^2}{840 c^6}-\frac {22 b^2 e^2 \log \left (c^2 x^2+1\right )}{105 c^8}+\frac {b^2 e^2 x \tan ^{-1}(c x)}{4 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*d^2*x)/(2*c^3) - (2*a*b*d*e*x)/(3*c^5) + (a*b*e^2*x)/(4*c^7) + (b^2*d^2*x^2)/(12*c^2) - (8*b^2*d*e*x^2)/(
45*c^4) + (71*b^2*e^2*x^2)/(840*c^6) + (b^2*d*e*x^4)/(30*c^2) - (3*b^2*e^2*x^4)/(140*c^4) + (b^2*e^2*x^6)/(168
*c^2) + (b^2*d^2*x*ArcTan[c*x])/(2*c^3) - (2*b^2*d*e*x*ArcTan[c*x])/(3*c^5) + (b^2*e^2*x*ArcTan[c*x])/(4*c^7)
- (b*d^2*x^3*(a + b*ArcTan[c*x]))/(6*c) + (2*b*d*e*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e^2*x^3*(a + b*ArcTan
[c*x]))/(12*c^5) - (2*b*d*e*x^5*(a + b*ArcTan[c*x]))/(15*c) + (b*e^2*x^5*(a + b*ArcTan[c*x]))/(20*c^3) - (b*e^
2*x^7*(a + b*ArcTan[c*x]))/(28*c) - (d^2*(a + b*ArcTan[c*x])^2)/(4*c^4) + (d*e*(a + b*ArcTan[c*x])^2)/(3*c^6)
- (e^2*(a + b*ArcTan[c*x])^2)/(8*c^8) + (d^2*x^4*(a + b*ArcTan[c*x])^2)/4 + (d*e*x^6*(a + b*ArcTan[c*x])^2)/3
+ (e^2*x^8*(a + b*ArcTan[c*x])^2)/8 - (b^2*d^2*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*d*e*Log[1 + c^2*x^2])/(45*c
^6) - (22*b^2*e^2*Log[1 + c^2*x^2])/(105*c^8)

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 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 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 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 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 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]

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 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])

Rubi steps

\begin {align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e^2 \int x^7 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} \left (b c d^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{3} (2 b c d e) \int \frac {x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{4} \left (b c e^2\right ) \int \frac {x^8 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (b d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac {\left (b d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c}-\frac {(2 b d e) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {(2 b d e) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}-\frac {\left (b e^2\right ) \int x^6 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c}+\frac {\left (b e^2\right ) \int \frac {x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{6} \left (b^2 d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3}+\frac {1}{15} \left (2 b^2 d e\right ) \int \frac {x^5}{1+c^2 x^2} \, dx+\frac {(2 b d e) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac {(2 b d e) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3}+\frac {1}{28} \left (b^2 e^2\right ) \int \frac {x^7}{1+c^2 x^2} \, dx+\frac {\left (b e^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c^3}-\frac {\left (b e^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{4 c^3}\\ &=\frac {a b d^2 x}{2 c^3}-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{12} \left (b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{2 c^3}+\frac {1}{15} \left (b^2 d e\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac {(2 b d e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^5}+\frac {(2 b d e) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac {\left (2 b^2 d e\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{9 c^2}+\frac {1}{56} \left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{1+c^2 x} \, dx,x,x^2\right )-\frac {\left (b e^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c^5}+\frac {\left (b e^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{4 c^5}-\frac {\left (b^2 e^2\right ) \int \frac {x^5}{1+c^2 x^2} \, dx}{20 c^2}\\ &=\frac {a b d^2 x}{2 c^3}-\frac {2 a b d e x}{3 c^5}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{12} \left (b^2 d^2\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^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2}+\frac {1}{15} \left (b^2 d 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 (2 b^2 d e\right ) \int \tan ^{-1}(c x) \, dx}{3 c^5}-\frac {\left (b^2 d e\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{9 c^2}+\frac {1}{56} \left (b^2 e^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^6}-\frac {x}{c^4}+\frac {x^2}{c^2}-\frac {1}{c^6 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {\left (b e^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{4 c^7}-\frac {\left (b e^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{4 c^7}+\frac {\left (b^2 e^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{12 c^4}-\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )}{40 c^2}\\ &=\frac {a b d^2 x}{2 c^3}-\frac {2 a b d e x}{3 c^5}+\frac {a b e^2 x}{4 c^7}+\frac {b^2 d^2 x^2}{12 c^2}-\frac {b^2 d e x^2}{15 c^4}+\frac {b^2 e^2 x^2}{56 c^6}+\frac {b^2 d e x^4}{30 c^2}-\frac {b^2 e^2 x^4}{112 c^4}+\frac {b^2 e^2 x^6}{168 c^2}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {b^2 d e \log \left (1+c^2 x^2\right )}{15 c^6}-\frac {b^2 e^2 \log \left (1+c^2 x^2\right )}{56 c^8}+\frac {\left (2 b^2 d e\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 c^4}-\frac {\left (b^2 d 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 )}{9 c^2}+\frac {\left (b^2 e^2\right ) \int \tan ^{-1}(c x) \, dx}{4 c^7}+\frac {\left (b^2 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{24 c^4}-\frac {\left (b^2 e^2\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 )}{40 c^2}\\ &=\frac {a b d^2 x}{2 c^3}-\frac {2 a b d e x}{3 c^5}+\frac {a b e^2 x}{4 c^7}+\frac {b^2 d^2 x^2}{12 c^2}-\frac {8 b^2 d e x^2}{45 c^4}+\frac {3 b^2 e^2 x^2}{70 c^6}+\frac {b^2 d e x^4}{30 c^2}-\frac {3 b^2 e^2 x^4}{140 c^4}+\frac {b^2 e^2 x^6}{168 c^2}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac {b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {23 b^2 d e \log \left (1+c^2 x^2\right )}{45 c^6}-\frac {3 b^2 e^2 \log \left (1+c^2 x^2\right )}{70 c^8}-\frac {\left (b^2 e^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{4 c^6}+\frac {\left (b^2 e^2\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 )}{24 c^4}\\ &=\frac {a b d^2 x}{2 c^3}-\frac {2 a b d e x}{3 c^5}+\frac {a b e^2 x}{4 c^7}+\frac {b^2 d^2 x^2}{12 c^2}-\frac {8 b^2 d e x^2}{45 c^4}+\frac {71 b^2 e^2 x^2}{840 c^6}+\frac {b^2 d e x^4}{30 c^2}-\frac {3 b^2 e^2 x^4}{140 c^4}+\frac {b^2 e^2 x^6}{168 c^2}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}-\frac {2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac {b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {2 b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{12 c^5}-\frac {2 b d e x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{20 c^3}-\frac {b e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )}{28 c}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {d e \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^6}-\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b^2 d^2 \log \left (1+c^2 x^2\right )}{3 c^4}+\frac {23 b^2 d e \log \left (1+c^2 x^2\right )}{45 c^6}-\frac {22 b^2 e^2 \log \left (1+c^2 x^2\right )}{105 c^8}\\ \end {align*}

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Mathematica [A]  time = 0.50, size = 414, normalized size = 0.82 \[ \frac {c x \left (105 a^2 c^7 x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-2 a b \left (3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )-7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )+105 c^2 e \left (8 d+e x^2\right )-315 e^2\right )+b^2 c x \left (3 c^4 \left (70 d^2+28 d e x^2+5 e^2 x^4\right )-2 c^2 e \left (224 d+27 e x^2\right )+213 e^2\right )\right )+2 b \tan ^{-1}(c x) \left (105 a \left (c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-6 c^4 d^2+8 c^2 d e-3 e^2\right )+b c x \left (-3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )+7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )-105 c^2 e \left (8 d+e x^2\right )+315 e^2\right )\right )-8 b^2 \left (105 c^4 d^2-161 c^2 d e+66 e^2\right ) \log \left (c^2 x^2+1\right )+105 b^2 \tan ^{-1}(c x)^2 \left (c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )-6 c^4 d^2+8 c^2 d e-3 e^2\right )}{2520 c^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(105*a^2*c^7*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b^2*c*x*(213*e^2 - 2*c^2*e*(224*d + 27*e*x^2) + 3*c^4*
(70*d^2 + 28*d*e*x^2 + 5*e^2*x^4)) - 2*a*b*(-315*e^2 + 105*c^2*e*(8*d + e*x^2) - 7*c^4*(90*d^2 + 40*d*e*x^2 +
9*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6))) + 2*b*(b*c*x*(315*e^2 - 105*c^2*e*(8*d + e*x^2) +
7*c^4*(90*d^2 + 40*d*e*x^2 + 9*e^2*x^4) - 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)) + 105*a*(-6*c^4*d^2 +
8*c^2*d*e - 3*e^2 + c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8)))*ArcTan[c*x] + 105*b^2*(-6*c^4*d^2 + 8*c^2*d*e -
3*e^2 + c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*ArcTan[c*x]^2 - 8*b^2*(105*c^4*d^2 - 161*c^2*d*e + 66*e^2)*Lo
g[1 + c^2*x^2])/(2520*c^8)

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fricas [A]  time = 0.55, size = 530, normalized size = 1.06 \[ \frac {315 \, a^{2} c^{8} e^{2} x^{8} - 90 \, a b c^{7} e^{2} x^{7} + 15 \, {\left (56 \, a^{2} c^{8} d e + b^{2} c^{6} e^{2}\right )} x^{6} - 42 \, {\left (8 \, a b c^{7} d e - 3 \, a b c^{5} e^{2}\right )} x^{5} + 6 \, {\left (105 \, a^{2} c^{8} d^{2} + 14 \, b^{2} c^{6} d e - 9 \, b^{2} c^{4} e^{2}\right )} x^{4} - 70 \, {\left (6 \, a b c^{7} d^{2} - 8 \, a b c^{5} d e + 3 \, a b c^{3} e^{2}\right )} x^{3} + {\left (210 \, b^{2} c^{6} d^{2} - 448 \, b^{2} c^{4} d e + 213 \, b^{2} c^{2} e^{2}\right )} x^{2} + 105 \, {\left (3 \, b^{2} c^{8} e^{2} x^{8} + 8 \, b^{2} c^{8} d e x^{6} + 6 \, b^{2} c^{8} d^{2} x^{4} - 6 \, b^{2} c^{4} d^{2} + 8 \, b^{2} c^{2} d e - 3 \, b^{2} e^{2}\right )} \arctan \left (c x\right )^{2} + 210 \, {\left (6 \, a b c^{5} d^{2} - 8 \, a b c^{3} d e + 3 \, a b c e^{2}\right )} x + 2 \, {\left (315 \, a b c^{8} e^{2} x^{8} + 840 \, a b c^{8} d e x^{6} - 45 \, b^{2} c^{7} e^{2} x^{7} + 630 \, a b c^{8} d^{2} x^{4} - 630 \, a b c^{4} d^{2} + 840 \, a b c^{2} d e - 21 \, {\left (8 \, b^{2} c^{7} d e - 3 \, b^{2} c^{5} e^{2}\right )} x^{5} - 315 \, a b e^{2} - 35 \, {\left (6 \, b^{2} c^{7} d^{2} - 8 \, b^{2} c^{5} d e + 3 \, b^{2} c^{3} e^{2}\right )} x^{3} + 105 \, {\left (6 \, b^{2} c^{5} d^{2} - 8 \, b^{2} c^{3} d e + 3 \, b^{2} c e^{2}\right )} x\right )} \arctan \left (c x\right ) - 8 \, {\left (105 \, b^{2} c^{4} d^{2} - 161 \, b^{2} c^{2} d e + 66 \, b^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{2520 \, c^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(315*a^2*c^8*e^2*x^8 - 90*a*b*c^7*e^2*x^7 + 15*(56*a^2*c^8*d*e + b^2*c^6*e^2)*x^6 - 42*(8*a*b*c^7*d*e -
3*a*b*c^5*e^2)*x^5 + 6*(105*a^2*c^8*d^2 + 14*b^2*c^6*d*e - 9*b^2*c^4*e^2)*x^4 - 70*(6*a*b*c^7*d^2 - 8*a*b*c^5
*d*e + 3*a*b*c^3*e^2)*x^3 + (210*b^2*c^6*d^2 - 448*b^2*c^4*d*e + 213*b^2*c^2*e^2)*x^2 + 105*(3*b^2*c^8*e^2*x^8
+ 8*b^2*c^8*d*e*x^6 + 6*b^2*c^8*d^2*x^4 - 6*b^2*c^4*d^2 + 8*b^2*c^2*d*e - 3*b^2*e^2)*arctan(c*x)^2 + 210*(6*a
*b*c^5*d^2 - 8*a*b*c^3*d*e + 3*a*b*c*e^2)*x + 2*(315*a*b*c^8*e^2*x^8 + 840*a*b*c^8*d*e*x^6 - 45*b^2*c^7*e^2*x^
7 + 630*a*b*c^8*d^2*x^4 - 630*a*b*c^4*d^2 + 840*a*b*c^2*d*e - 21*(8*b^2*c^7*d*e - 3*b^2*c^5*e^2)*x^5 - 315*a*b
*e^2 - 35*(6*b^2*c^7*d^2 - 8*b^2*c^5*d*e + 3*b^2*c^3*e^2)*x^3 + 105*(6*b^2*c^5*d^2 - 8*b^2*c^3*d*e + 3*b^2*c*e
^2)*x)*arctan(c*x) - 8*(105*b^2*c^4*d^2 - 161*b^2*c^2*d*e + 66*b^2*e^2)*log(c^2*x^2 + 1))/c^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.06, size = 621, normalized size = 1.24 \[ \frac {2 a b \,x^{3} d e}{9 c^{3}}-\frac {2 a b e d \,x^{5}}{15 c}+\frac {2 a b \arctan \left (c x \right ) e d \,x^{6}}{3}+\frac {2 a b \arctan \left (c x \right ) d e}{3 c^{6}}-\frac {2 b^{2} \arctan \left (c x \right ) e d \,x^{5}}{15 c}+\frac {2 b^{2} \arctan \left (c x \right ) x^{3} d e}{9 c^{3}}-\frac {2 a b d e x}{3 c^{5}}-\frac {2 b^{2} d e x \arctan \left (c x \right )}{3 c^{5}}+\frac {a b \,d^{2} x}{2 c^{3}}+\frac {b^{2} d^{2} x \arctan \left (c x \right )}{2 c^{3}}+\frac {b^{2} d^{2} x^{2}}{12 c^{2}}-\frac {b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}-\frac {22 b^{2} e^{2} \ln \left (c^{2} x^{2}+1\right )}{105 c^{8}}-\frac {a b \,e^{2} x^{7}}{28 c}-\frac {b^{2} \arctan \left (c x \right ) d^{2} x^{3}}{6 c}-\frac {a b \arctan \left (c x \right ) e^{2}}{4 c^{8}}+\frac {a b \arctan \left (c x \right ) e^{2} x^{8}}{4}-\frac {a b \,d^{2} x^{3}}{6 c}+\frac {a b \,x^{5} e^{2}}{20 c^{3}}-\frac {a b \,x^{3} e^{2}}{12 c^{5}}+\frac {a b \,e^{2} x}{4 c^{7}}-\frac {8 b^{2} d e \,x^{2}}{45 c^{4}}+\frac {b^{2} d e \,x^{4}}{30 c^{2}}+\frac {b^{2} e^{2} x \arctan \left (c x \right )}{4 c^{7}}+\frac {71 b^{2} e^{2} x^{2}}{840 c^{6}}-\frac {3 b^{2} e^{2} x^{4}}{140 c^{4}}+\frac {b^{2} e^{2} x^{6}}{168 c^{2}}+\frac {b^{2} \arctan \left (c x \right ) x^{5} e^{2}}{20 c^{3}}-\frac {b^{2} \arctan \left (c x \right ) x^{3} e^{2}}{12 c^{5}}-\frac {a b \arctan \left (c x \right ) d^{2}}{2 c^{4}}+\frac {b^{2} \arctan \left (c x \right )^{2} e d \,x^{6}}{3}+\frac {a b \arctan \left (c x \right ) d^{2} x^{4}}{2}-\frac {b^{2} \arctan \left (c x \right ) e^{2} x^{7}}{28 c}+\frac {b^{2} \arctan \left (c x \right )^{2} d e}{3 c^{6}}+\frac {b^{2} \arctan \left (c x \right )^{2} e^{2} x^{8}}{8}-\frac {b^{2} \arctan \left (c x \right )^{2} d^{2}}{4 c^{4}}-\frac {b^{2} \arctan \left (c x \right )^{2} e^{2}}{8 c^{8}}+\frac {a^{2} e d \,x^{6}}{3}+\frac {b^{2} \arctan \left (c x \right )^{2} d^{2} x^{4}}{4}+\frac {23 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{45 c^{6}}+\frac {a^{2} e^{2} x^{8}}{8}+\frac {a^{2} x^{4} d^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9/c^3*a*b*x^3*d*e-2/15/c*a*b*e*d*x^5+2/3*a*b*arctan(c*x)*e*d*x^6+2/3/c^6*a*b*arctan(c*x)*d*e-2/15/c*b^2*arct
an(c*x)*e*d*x^5+2/9/c^3*b^2*arctan(c*x)*x^3*d*e-2/3*a*b*d*e*x/c^5-2/3*b^2*d*e*x*arctan(c*x)/c^5+1/2*a*b*d^2*x/
c^3+1/2*b^2*d^2*x*arctan(c*x)/c^3+1/12*b^2*d^2*x^2/c^2-1/3*b^2*d^2*ln(c^2*x^2+1)/c^4-22/105*b^2*e^2*ln(c^2*x^2
+1)/c^8-1/28/c*a*b*e^2*x^7-1/6/c*b^2*arctan(c*x)*d^2*x^3-1/4/c^8*a*b*arctan(c*x)*e^2+1/4*a*b*arctan(c*x)*e^2*x
^8-1/6/c*a*b*d^2*x^3+1/20/c^3*a*b*x^5*e^2-1/12/c^5*a*b*x^3*e^2+1/4*a*b*e^2*x/c^7-8/45*b^2*d*e*x^2/c^4+1/30*b^2
*d*e*x^4/c^2+1/4*b^2*e^2*x*arctan(c*x)/c^7+71/840*b^2*e^2*x^2/c^6-3/140*b^2*e^2*x^4/c^4+1/168*b^2*e^2*x^6/c^2+
1/20/c^3*b^2*arctan(c*x)*x^5*e^2-1/12/c^5*b^2*arctan(c*x)*x^3*e^2-1/2/c^4*a*b*arctan(c*x)*d^2+1/3*b^2*arctan(c
*x)^2*e*d*x^6+1/2*a*b*arctan(c*x)*d^2*x^4-1/28/c*b^2*arctan(c*x)*e^2*x^7+1/3/c^6*b^2*arctan(c*x)^2*d*e+1/8*b^2
*arctan(c*x)^2*e^2*x^8-1/4/c^4*b^2*arctan(c*x)^2*d^2-1/8/c^8*b^2*arctan(c*x)^2*e^2+1/3*a^2*e*d*x^6+1/4*b^2*arc
tan(c*x)^2*d^2*x^4+23/45*b^2*d*e*ln(c^2*x^2+1)/c^6+1/8*a^2*e^2*x^8+1/4*a^2*x^4*d^2

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maxima [A]  time = 0.48, size = 516, normalized size = 1.03 \[ \frac {1}{8} \, b^{2} e^{2} x^{8} \arctan \left (c x\right )^{2} + \frac {1}{8} \, a^{2} e^{2} x^{8} + \frac {1}{3} \, b^{2} d e x^{6} \arctan \left (c x\right )^{2} + \frac {1}{3} \, a^{2} d e x^{6} + \frac {1}{4} \, b^{2} d^{2} x^{4} \arctan \left (c x\right )^{2} + \frac {1}{4} \, a^{2} d^{2} x^{4} + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b d^{2} - \frac {1}{12} \, {\left (2 \, c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d^{2} + \frac {2}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b d e - \frac {1}{90} \, {\left (4 \, c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac {3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} d e + \frac {1}{420} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} a b e^{2} - \frac {1}{840} \, {\left (2 \, c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )} \arctan \left (c x\right ) - \frac {5 \, c^{6} x^{6} - 18 \, c^{4} x^{4} + 71 \, c^{2} x^{2} + 105 \, \arctan \left (c x\right )^{2} - 176 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )} b^{2} e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*e^2*x^8*arctan(c*x)^2 + 1/8*a^2*e^2*x^8 + 1/3*b^2*d*e*x^6*arctan(c*x)^2 + 1/3*a^2*d*e*x^6 + 1/4*b^2*d^
2*x^4*arctan(c*x)^2 + 1/4*a^2*d^2*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^2 - 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^2 + 2/45*(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*d*e - 1/90*(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*d*e + 1/420*(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))*a*b*e^2 - 1/840*(2*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)*arctan(c*x) - (5*c^6*x^6 - 18*c^4*x^4 + 71*c^2*x^2 + 105
*arctan(c*x)^2 - 176*log(c^2*x^2 + 1))/c^8)*b^2*e^2

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mupad [B]  time = 6.90, size = 929, normalized size = 1.85 \[ \frac {a^2\,d^2\,x^4}{4}+\frac {a^2\,e^2\,x^8}{8}-\frac {b^2\,d^2\,\ln \left (c^2\,x^2+1\right )}{3\,c^4}-\frac {22\,b^2\,e^2\,\ln \left (c^2\,x^2+1\right )}{105\,c^8}+\frac {b^2\,d^2\,x^2}{12\,c^2}+\frac {b^2\,e^2\,x^6}{168\,c^2}-\frac {3\,b^2\,e^2\,x^4}{140\,c^4}+\frac {71\,b^2\,e^2\,x^2}{840\,c^6}-\frac {b^2\,d^2\,{\mathrm {atan}\left (c\,x\right )}^2}{4\,c^4}-\frac {b^2\,e^2\,{\mathrm {atan}\left (c\,x\right )}^2}{8\,c^8}+\frac {b^2\,d^2\,x^4\,{\mathrm {atan}\left (c\,x\right )}^2}{4}+\frac {b^2\,e^2\,x^8\,{\mathrm {atan}\left (c\,x\right )}^2}{8}+\frac {a^2\,d\,e\,x^6}{3}-\frac {b^2\,d^2\,x^3\,\mathrm {atan}\left (c\,x\right )}{6\,c}-\frac {b^2\,e^2\,x^7\,\mathrm {atan}\left (c\,x\right )}{28\,c}+\frac {b^2\,e^2\,x^5\,\mathrm {atan}\left (c\,x\right )}{20\,c^3}-\frac {b^2\,e^2\,x^3\,\mathrm {atan}\left (c\,x\right )}{12\,c^5}+\frac {a\,b\,d^2\,x}{2\,c^3}+\frac {a\,b\,e^2\,x}{4\,c^7}+\frac {a\,b\,d^2\,x^4\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {a\,b\,e^2\,x^8\,\mathrm {atan}\left (c\,x\right )}{4}+\frac {23\,b^2\,d\,e\,\ln \left (c^2\,x^2+1\right )}{45\,c^6}-\frac {a\,b\,d^2\,x^3}{6\,c}-\frac {a\,b\,e^2\,x^7}{28\,c}+\frac {a\,b\,e^2\,x^5}{20\,c^3}-\frac {a\,b\,e^2\,x^3}{12\,c^5}+\frac {b^2\,d\,e\,x^4}{30\,c^2}-\frac {8\,b^2\,d\,e\,x^2}{45\,c^4}+\frac {b^2\,d\,e\,{\mathrm {atan}\left (c\,x\right )}^2}{3\,c^6}+\frac {b^2\,d^2\,x\,\mathrm {atan}\left (c\,x\right )}{2\,c^3}+\frac {b^2\,e^2\,x\,\mathrm {atan}\left (c\,x\right )}{4\,c^7}+\frac {b^2\,d\,e\,x^6\,{\mathrm {atan}\left (c\,x\right )}^2}{3}-\frac {a\,b\,d^2\,\mathrm {atan}\left (\frac {3\,b\,c\,e^2\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}+\frac {6\,b\,c^5\,d^2\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}-\frac {8\,b\,c^3\,d\,e\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}\right )}{2\,c^4}-\frac {a\,b\,e^2\,\mathrm {atan}\left (\frac {3\,b\,c\,e^2\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}+\frac {6\,b\,c^5\,d^2\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}-\frac {8\,b\,c^3\,d\,e\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}\right )}{4\,c^8}-\frac {2\,b^2\,d\,e\,x^5\,\mathrm {atan}\left (c\,x\right )}{15\,c}+\frac {2\,b^2\,d\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{9\,c^3}-\frac {2\,a\,b\,d\,e\,x}{3\,c^5}+\frac {2\,a\,b\,d\,e\,x^6\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {2\,a\,b\,d\,e\,x^5}{15\,c}+\frac {2\,a\,b\,d\,e\,x^3}{9\,c^3}-\frac {2\,b^2\,d\,e\,x\,\mathrm {atan}\left (c\,x\right )}{3\,c^5}+\frac {2\,a\,b\,d\,e\,\mathrm {atan}\left (\frac {3\,b\,c\,e^2\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}+\frac {6\,b\,c^5\,d^2\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}-\frac {8\,b\,c^3\,d\,e\,x}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}\right )}{3\,c^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*d^2*x^4)/4 + (a^2*e^2*x^8)/8 - (b^2*d^2*log(c^2*x^2 + 1))/(3*c^4) - (22*b^2*e^2*log(c^2*x^2 + 1))/(105*c^
8) + (b^2*d^2*x^2)/(12*c^2) + (b^2*e^2*x^6)/(168*c^2) - (3*b^2*e^2*x^4)/(140*c^4) + (71*b^2*e^2*x^2)/(840*c^6)
- (b^2*d^2*atan(c*x)^2)/(4*c^4) - (b^2*e^2*atan(c*x)^2)/(8*c^8) + (b^2*d^2*x^4*atan(c*x)^2)/4 + (b^2*e^2*x^8*
atan(c*x)^2)/8 + (a^2*d*e*x^6)/3 - (b^2*d^2*x^3*atan(c*x))/(6*c) - (b^2*e^2*x^7*atan(c*x))/(28*c) + (b^2*e^2*x
^5*atan(c*x))/(20*c^3) - (b^2*e^2*x^3*atan(c*x))/(12*c^5) + (a*b*d^2*x)/(2*c^3) + (a*b*e^2*x)/(4*c^7) + (a*b*d
^2*x^4*atan(c*x))/2 + (a*b*e^2*x^8*atan(c*x))/4 + (23*b^2*d*e*log(c^2*x^2 + 1))/(45*c^6) - (a*b*d^2*x^3)/(6*c)
- (a*b*e^2*x^7)/(28*c) + (a*b*e^2*x^5)/(20*c^3) - (a*b*e^2*x^3)/(12*c^5) + (b^2*d*e*x^4)/(30*c^2) - (8*b^2*d*
e*x^2)/(45*c^4) + (b^2*d*e*atan(c*x)^2)/(3*c^6) + (b^2*d^2*x*atan(c*x))/(2*c^3) + (b^2*e^2*x*atan(c*x))/(4*c^7
) + (b^2*d*e*x^6*atan(c*x)^2)/3 - (a*b*d^2*atan((3*b*c*e^2*x)/(3*b*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e) + (6*b*c^5
*d^2*x)/(3*b*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e) - (8*b*c^3*d*e*x)/(3*b*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e)))/(2*c^4
) - (a*b*e^2*atan((3*b*c*e^2*x)/(3*b*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e) + (6*b*c^5*d^2*x)/(3*b*e^2 + 6*b*c^4*d^2
- 8*b*c^2*d*e) - (8*b*c^3*d*e*x)/(3*b*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e)))/(4*c^8) - (2*b^2*d*e*x^5*atan(c*x))/
(15*c) + (2*b^2*d*e*x^3*atan(c*x))/(9*c^3) - (2*a*b*d*e*x)/(3*c^5) + (2*a*b*d*e*x^6*atan(c*x))/3 - (2*a*b*d*e*
x^5)/(15*c) + (2*a*b*d*e*x^3)/(9*c^3) - (2*b^2*d*e*x*atan(c*x))/(3*c^5) + (2*a*b*d*e*atan((3*b*c*e^2*x)/(3*b*e
^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e) + (6*b*c^5*d^2*x)/(3*b*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e) - (8*b*c^3*d*e*x)/(3*b
*e^2 + 6*b*c^4*d^2 - 8*b*c^2*d*e)))/(3*c^6)

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sympy [A]  time = 7.49, size = 758, normalized size = 1.51 \[ \begin {cases} \frac {a^{2} d^{2} x^{4}}{4} + \frac {a^{2} d e x^{6}}{3} + \frac {a^{2} e^{2} x^{8}}{8} + \frac {a b d^{2} x^{4} \operatorname {atan}{\left (c x \right )}}{2} + \frac {2 a b d e x^{6} \operatorname {atan}{\left (c x \right )}}{3} + \frac {a b e^{2} x^{8} \operatorname {atan}{\left (c x \right )}}{4} - \frac {a b d^{2} x^{3}}{6 c} - \frac {2 a b d e x^{5}}{15 c} - \frac {a b e^{2} x^{7}}{28 c} + \frac {a b d^{2} x}{2 c^{3}} + \frac {2 a b d e x^{3}}{9 c^{3}} + \frac {a b e^{2} x^{5}}{20 c^{3}} - \frac {a b d^{2} \operatorname {atan}{\left (c x \right )}}{2 c^{4}} - \frac {2 a b d e x}{3 c^{5}} - \frac {a b e^{2} x^{3}}{12 c^{5}} + \frac {2 a b d e \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {a b e^{2} x}{4 c^{7}} - \frac {a b e^{2} \operatorname {atan}{\left (c x \right )}}{4 c^{8}} + \frac {b^{2} d^{2} x^{4} \operatorname {atan}^{2}{\left (c x \right )}}{4} + \frac {b^{2} d e x^{6} \operatorname {atan}^{2}{\left (c x \right )}}{3} + \frac {b^{2} e^{2} x^{8} \operatorname {atan}^{2}{\left (c x \right )}}{8} - \frac {b^{2} d^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{6 c} - \frac {2 b^{2} d e x^{5} \operatorname {atan}{\left (c x \right )}}{15 c} - \frac {b^{2} e^{2} x^{7} \operatorname {atan}{\left (c x \right )}}{28 c} + \frac {b^{2} d^{2} x^{2}}{12 c^{2}} + \frac {b^{2} d e x^{4}}{30 c^{2}} + \frac {b^{2} e^{2} x^{6}}{168 c^{2}} + \frac {b^{2} d^{2} x \operatorname {atan}{\left (c x \right )}}{2 c^{3}} + \frac {2 b^{2} d e x^{3} \operatorname {atan}{\left (c x \right )}}{9 c^{3}} + \frac {b^{2} e^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{20 c^{3}} - \frac {b^{2} d^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{4}} - \frac {b^{2} d^{2} \operatorname {atan}^{2}{\left (c x \right )}}{4 c^{4}} - \frac {8 b^{2} d e x^{2}}{45 c^{4}} - \frac {3 b^{2} e^{2} x^{4}}{140 c^{4}} - \frac {2 b^{2} d e x \operatorname {atan}{\left (c x \right )}}{3 c^{5}} - \frac {b^{2} e^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{12 c^{5}} + \frac {23 b^{2} d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{45 c^{6}} + \frac {b^{2} d e \operatorname {atan}^{2}{\left (c x \right )}}{3 c^{6}} + \frac {71 b^{2} e^{2} x^{2}}{840 c^{6}} + \frac {b^{2} e^{2} x \operatorname {atan}{\left (c x \right )}}{4 c^{7}} - \frac {22 b^{2} e^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{105 c^{8}} - \frac {b^{2} e^{2} \operatorname {atan}^{2}{\left (c x \right )}}{8 c^{8}} & \text {for}\: c \neq 0 \\a^{2} \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x**4/4 + a**2*d*e*x**6/3 + a**2*e**2*x**8/8 + a*b*d**2*x**4*atan(c*x)/2 + 2*a*b*d*e*x**6*
atan(c*x)/3 + a*b*e**2*x**8*atan(c*x)/4 - a*b*d**2*x**3/(6*c) - 2*a*b*d*e*x**5/(15*c) - a*b*e**2*x**7/(28*c) +
a*b*d**2*x/(2*c**3) + 2*a*b*d*e*x**3/(9*c**3) + a*b*e**2*x**5/(20*c**3) - a*b*d**2*atan(c*x)/(2*c**4) - 2*a*b
*d*e*x/(3*c**5) - a*b*e**2*x**3/(12*c**5) + 2*a*b*d*e*atan(c*x)/(3*c**6) + a*b*e**2*x/(4*c**7) - a*b*e**2*atan
(c*x)/(4*c**8) + b**2*d**2*x**4*atan(c*x)**2/4 + b**2*d*e*x**6*atan(c*x)**2/3 + b**2*e**2*x**8*atan(c*x)**2/8
- b**2*d**2*x**3*atan(c*x)/(6*c) - 2*b**2*d*e*x**5*atan(c*x)/(15*c) - b**2*e**2*x**7*atan(c*x)/(28*c) + b**2*d
**2*x**2/(12*c**2) + b**2*d*e*x**4/(30*c**2) + b**2*e**2*x**6/(168*c**2) + b**2*d**2*x*atan(c*x)/(2*c**3) + 2*
b**2*d*e*x**3*atan(c*x)/(9*c**3) + b**2*e**2*x**5*atan(c*x)/(20*c**3) - b**2*d**2*log(x**2 + c**(-2))/(3*c**4)
- b**2*d**2*atan(c*x)**2/(4*c**4) - 8*b**2*d*e*x**2/(45*c**4) - 3*b**2*e**2*x**4/(140*c**4) - 2*b**2*d*e*x*at
an(c*x)/(3*c**5) - b**2*e**2*x**3*atan(c*x)/(12*c**5) + 23*b**2*d*e*log(x**2 + c**(-2))/(45*c**6) + b**2*d*e*a
tan(c*x)**2/(3*c**6) + 71*b**2*e**2*x**2/(840*c**6) + b**2*e**2*x*atan(c*x)/(4*c**7) - 22*b**2*e**2*log(x**2 +
c**(-2))/(105*c**8) - b**2*e**2*atan(c*x)**2/(8*c**8), Ne(c, 0)), (a**2*(d**2*x**4/4 + d*e*x**6/3 + e**2*x**8
/8), True))

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