3.132 \(\int \frac {x^5 (a+b x^2+c x^4)}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\)

Optimal. Leaf size=210 \[ -\frac {(d-e x)^{5/2} (d+e x)^{5/2} \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10}}+\frac {d^2 (d-e x)^{3/2} (d+e x)^{3/2} \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10}}-\frac {d^4 \sqrt {d-e x} \sqrt {d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10}}+\frac {(d-e x)^{7/2} (d+e x)^{7/2} \left (b e^2+4 c d^2\right )}{7 e^{10}}-\frac {c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}} \]

[Out]

1/3*d^2*(2*a*e^4+3*b*d^2*e^2+4*c*d^4)*(-e*x+d)^(3/2)*(e*x+d)^(3/2)/e^10-1/5*(a*e^4+3*b*d^2*e^2+6*c*d^4)*(-e*x+
d)^(5/2)*(e*x+d)^(5/2)/e^10+1/7*(b*e^2+4*c*d^2)*(-e*x+d)^(7/2)*(e*x+d)^(7/2)/e^10-1/9*c*(-e*x+d)^(9/2)*(e*x+d)
^(9/2)/e^10-d^4*(a*e^4+b*d^2*e^2+c*d^4)*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/e^10

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Rubi [A]  time = 0.31, antiderivative size = 278, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {520, 1251, 897, 1153} \[ -\frac {\left (d^2-e^2 x^2\right )^3 \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {d^2 \left (d^2-e^2 x^2\right )^2 \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^4 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^4 \left (b e^2+4 c d^2\right )}{7 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt {d-e x} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-((d^4*(c*d^4 + b*d^2*e^2 + a*e^4)*(d^2 - e^2*x^2))/(e^10*Sqrt[d - e*x]*Sqrt[d + e*x])) + (d^2*(4*c*d^4 + 3*b*
d^2*e^2 + 2*a*e^4)*(d^2 - e^2*x^2)^2)/(3*e^10*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((6*c*d^4 + 3*b*d^2*e^2 + a*e^4)*
(d^2 - e^2*x^2)^3)/(5*e^10*Sqrt[d - e*x]*Sqrt[d + e*x]) + ((4*c*d^2 + b*e^2)*(d^2 - e^2*x^2)^4)/(7*e^10*Sqrt[d
- e*x]*Sqrt[d + e*x]) - (c*(d^2 - e^2*x^2)^5)/(9*e^10*Sqrt[d - e*x]*Sqrt[d + e*x])

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {d^2-e^2 x^2} \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^2 \left (\frac {c d^4+b d^2 e^2+a e^4}{e^4}-\frac {\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac {c x^4}{e^4}\right ) \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \operatorname {Subst}\left (\int \left (\frac {c d^8+b d^6 e^2+a d^4 e^4}{e^8}-\frac {d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) x^2}{e^8}+\frac {\left (6 c d^4+3 b d^2 e^2+a e^4\right ) x^4}{e^8}-\frac {\left (4 c d^2+b e^2\right ) x^6}{e^8}+\frac {c x^8}{e^8}\right ) \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {d^4 \left (c d^4+b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )}{e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) \left (d^2-e^2 x^2\right )^2}{3 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (6 c d^4+3 b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )^3}{5 e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (4 c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )^4}{7 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 1.38, size = 265, normalized size = 1.26 \[ -\frac {\frac {630 d^{9/2} \sqrt {d+e x} \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt {\frac {e x}{d}+1}}-630 d^5 \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )+\sqrt {d-e x} \sqrt {d+e x} \left (21 a e^4 \left (8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right )+9 b \left (16 d^6 e^2+8 d^4 e^4 x^2+6 d^2 e^6 x^4+5 e^8 x^6\right )+c \left (128 d^8+64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+35 e^8 x^8\right )\right )}{315 e^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-1/315*(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*a*e^4*(8*d^4 + 4*d^2*e^2*x^2 + 3*e^4*x^4) + 9*b*(16*d^6*e^2 + 8*d^4*e^
4*x^2 + 6*d^2*e^6*x^4 + 5*e^8*x^6) + c*(128*d^8 + 64*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 40*d^2*e^6*x^6 + 35*e^8*x^
8)) + (630*d^(9/2)*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d + e*x]*ArcSin[Sqrt[d - e*x]/(Sqrt[2]*Sqrt[d])])/Sqrt[1 +
(e*x)/d] - 630*d^5*(c*d^4 + b*d^2*e^2 + a*e^4)*ArcTan[Sqrt[d - e*x]/Sqrt[d + e*x]])/e^10

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fricas [A]  time = 1.02, size = 138, normalized size = 0.66 \[ -\frac {{\left (35 \, c e^{8} x^{8} + 128 \, c d^{8} + 144 \, b d^{6} e^{2} + 168 \, a d^{4} e^{4} + 5 \, {\left (8 \, c d^{2} e^{6} + 9 \, b e^{8}\right )} x^{6} + 3 \, {\left (16 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 21 \, a e^{8}\right )} x^{4} + 4 \, {\left (16 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 21 \, a d^{2} e^{6}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, e^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(35*c*e^8*x^8 + 128*c*d^8 + 144*b*d^6*e^2 + 168*a*d^4*e^4 + 5*(8*c*d^2*e^6 + 9*b*e^8)*x^6 + 3*(16*c*d^4
*e^4 + 18*b*d^2*e^6 + 21*a*e^8)*x^4 + 4*(16*c*d^6*e^2 + 18*b*d^4*e^4 + 21*a*d^2*e^6)*x^2)*sqrt(e*x + d)*sqrt(-
e*x + d)/e^10

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giac [A]  time = 0.89, size = 269, normalized size = 1.28 \[ -\frac {1}{315} \, {\left ({\left ({\left ({\left ({\left (5 \, {\left ({\left (7 \, {\left ({\left (x e + d\right )} c e^{\left (-9\right )} - 8 \, c d e^{\left (-9\right )}\right )} {\left (x e + d\right )} + 3 \, {\left (68 \, c d^{2} e^{81} + 3 \, b e^{83}\right )} e^{\left (-90\right )}\right )} {\left (x e + d\right )} - 2 \, {\left (220 \, c d^{3} e^{81} + 27 \, b d e^{83}\right )} e^{\left (-90\right )}\right )} {\left (x e + d\right )} + {\left (3098 \, c d^{4} e^{81} + 729 \, b d^{2} e^{83} + 63 \, a e^{85}\right )} e^{\left (-90\right )}\right )} {\left (x e + d\right )} - 36 \, {\left (82 \, c d^{5} e^{81} + 31 \, b d^{3} e^{83} + 7 \, a d e^{85}\right )} e^{\left (-90\right )}\right )} {\left (x e + d\right )} + 21 \, {\left (92 \, c d^{6} e^{81} + 51 \, b d^{4} e^{83} + 22 \, a d^{2} e^{85}\right )} e^{\left (-90\right )}\right )} {\left (x e + d\right )} - 210 \, {\left (4 \, c d^{7} e^{81} + 3 \, b d^{5} e^{83} + 2 \, a d^{3} e^{85}\right )} e^{\left (-90\right )}\right )} {\left (x e + d\right )} + 315 \, {\left (c d^{8} e^{81} + b d^{6} e^{83} + a d^{4} e^{85}\right )} e^{\left (-90\right )}\right )} \sqrt {x e + d} \sqrt {-x e + d} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-1/315*(((((5*((7*((x*e + d)*c*e^(-9) - 8*c*d*e^(-9))*(x*e + d) + 3*(68*c*d^2*e^81 + 3*b*e^83)*e^(-90))*(x*e +
d) - 2*(220*c*d^3*e^81 + 27*b*d*e^83)*e^(-90))*(x*e + d) + (3098*c*d^4*e^81 + 729*b*d^2*e^83 + 63*a*e^85)*e^(
-90))*(x*e + d) - 36*(82*c*d^5*e^81 + 31*b*d^3*e^83 + 7*a*d*e^85)*e^(-90))*(x*e + d) + 21*(92*c*d^6*e^81 + 51*
b*d^4*e^83 + 22*a*d^2*e^85)*e^(-90))*(x*e + d) - 210*(4*c*d^7*e^81 + 3*b*d^5*e^83 + 2*a*d^3*e^85)*e^(-90))*(x*
e + d) + 315*(c*d^8*e^81 + b*d^6*e^83 + a*d^4*e^85)*e^(-90))*sqrt(x*e + d)*sqrt(-x*e + d)*e^(-1)

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maple [A]  time = 0.01, size = 145, normalized size = 0.69 \[ -\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/315*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(35*c*e^8*x^8+45*b*e^8*x^6+40*c*d^2*e^6*x^6+63*a*e^8*x^4+54*b*d^2*e^6*x^4+
48*c*d^4*e^4*x^4+84*a*d^2*e^6*x^2+72*b*d^4*e^4*x^2+64*c*d^6*e^2*x^2+168*a*d^4*e^4+144*b*d^6*e^2+128*c*d^8)/e^1
0

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maxima [A]  time = 1.03, size = 295, normalized size = 1.40 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{8}}{9 \, e^{2}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{6}}{63 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{6}}{7 \, e^{2}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4} x^{4}}{105 \, e^{6}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2} x^{4}}{35 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a x^{4}}{5 \, e^{2}} - \frac {64 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{6} x^{2}}{315 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{4} x^{2}}{35 \, e^{6}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{2} x^{2}}{15 \, e^{4}} - \frac {128 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{8}}{315 \, e^{10}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{6}}{35 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{4}}{15 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-1/9*sqrt(-e^2*x^2 + d^2)*c*x^8/e^2 - 8/63*sqrt(-e^2*x^2 + d^2)*c*d^2*x^6/e^4 - 1/7*sqrt(-e^2*x^2 + d^2)*b*x^6
/e^2 - 16/105*sqrt(-e^2*x^2 + d^2)*c*d^4*x^4/e^6 - 6/35*sqrt(-e^2*x^2 + d^2)*b*d^2*x^4/e^4 - 1/5*sqrt(-e^2*x^2
+ d^2)*a*x^4/e^2 - 64/315*sqrt(-e^2*x^2 + d^2)*c*d^6*x^2/e^8 - 8/35*sqrt(-e^2*x^2 + d^2)*b*d^4*x^2/e^6 - 4/15
*sqrt(-e^2*x^2 + d^2)*a*d^2*x^2/e^4 - 128/315*sqrt(-e^2*x^2 + d^2)*c*d^8/e^10 - 16/35*sqrt(-e^2*x^2 + d^2)*b*d
^6/e^8 - 8/15*sqrt(-e^2*x^2 + d^2)*a*d^4/e^6

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mupad [B]  time = 1.65, size = 287, normalized size = 1.37 \[ -\frac {\sqrt {d-e\,x}\,\left (\frac {128\,c\,d^9+144\,b\,d^7\,e^2+168\,a\,d^5\,e^4}{315\,e^{10}}+\frac {x^7\,\left (40\,c\,d^2\,e^7+45\,b\,e^9\right )}{315\,e^{10}}+\frac {x^2\,\left (64\,c\,d^7\,e^2+72\,b\,d^5\,e^4+84\,a\,d^3\,e^6\right )}{315\,e^{10}}+\frac {x^3\,\left (64\,c\,d^6\,e^3+72\,b\,d^4\,e^5+84\,a\,d^2\,e^7\right )}{315\,e^{10}}+\frac {c\,x^9}{9\,e}+\frac {x^5\,\left (48\,c\,d^4\,e^5+54\,b\,d^2\,e^7+63\,a\,e^9\right )}{315\,e^{10}}+\frac {x\,\left (128\,c\,d^8\,e+144\,b\,d^6\,e^3+168\,a\,d^4\,e^5\right )}{315\,e^{10}}+\frac {x^6\,\left (40\,c\,d^3\,e^6+45\,b\,d\,e^8\right )}{315\,e^{10}}+\frac {x^4\,\left (48\,c\,d^5\,e^4+54\,b\,d^3\,e^6+63\,a\,d\,e^8\right )}{315\,e^{10}}+\frac {c\,d\,x^8}{9\,e^2}\right )}{\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a + b*x^2 + c*x^4))/((d + e*x)^(1/2)*(d - e*x)^(1/2)),x)

[Out]

-((d - e*x)^(1/2)*((128*c*d^9 + 168*a*d^5*e^4 + 144*b*d^7*e^2)/(315*e^10) + (x^7*(45*b*e^9 + 40*c*d^2*e^7))/(3
15*e^10) + (x^2*(84*a*d^3*e^6 + 72*b*d^5*e^4 + 64*c*d^7*e^2))/(315*e^10) + (x^3*(84*a*d^2*e^7 + 72*b*d^4*e^5 +
64*c*d^6*e^3))/(315*e^10) + (c*x^9)/(9*e) + (x^5*(63*a*e^9 + 54*b*d^2*e^7 + 48*c*d^4*e^5))/(315*e^10) + (x*(1
68*a*d^4*e^5 + 144*b*d^6*e^3 + 128*c*d^8*e))/(315*e^10) + (x^6*(40*c*d^3*e^6 + 45*b*d*e^8))/(315*e^10) + (x^4*
(54*b*d^3*e^6 + 48*c*d^5*e^4 + 63*a*d*e^8))/(315*e^10) + (c*d*x^8)/(9*e^2)))/(d + e*x)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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