3.1.66 \(\int (3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^3 \, dx\) [66]

Optimal. Leaf size=189 \[ -\frac {46077855 (1-4 x) \sqrt {3-x+2 x^2}}{33554432}-\frac {667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}-\frac {1059790665 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{67108864 \sqrt {2}} \]

[Out]

-667795/2097152*(1-4*x)*(2*x^2-x+3)^(3/2)-4625907/2293760*(2*x^2-x+3)^(5/2)-81685/114688*x*(2*x^2-x+3)^(5/2)+3
84739/43008*x^2*(2*x^2-x+3)^(5/2)+27785/1536*x^3*(2*x^2-x+3)^(5/2)+725/48*x^4*(2*x^2-x+3)^(5/2)+25/4*x^5*(2*x^
2-x+3)^(5/2)-1059790665/134217728*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-46077855/33554432*(1-4*x)*(2*x^2-x+3)
^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1675, 654, 626, 633, 221} \begin {gather*} \frac {384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac {81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac {4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac {667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac {46077855 (1-4 x) \sqrt {2 x^2-x+3}}{33554432}+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac {725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac {27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}-\frac {1059790665 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{67108864 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^3,x]

[Out]

(-46077855*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/33554432 - (667795*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/2097152 - (46259
07*(3 - x + 2*x^2)^(5/2))/2293760 - (81685*x*(3 - x + 2*x^2)^(5/2))/114688 + (384739*x^2*(3 - x + 2*x^2)^(5/2)
)/43008 + (27785*x^3*(3 - x + 2*x^2)^(5/2))/1536 + (725*x^4*(3 - x + 2*x^2)^(5/2))/48 + (25*x^5*(3 - x + 2*x^2
)^(5/2))/4 - (1059790665*ArcSinh[(1 - 4*x)/Sqrt[23]])/(67108864*Sqrt[2])

Rule 221

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

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 654

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

Rule 1675

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {1}{20} \int \left (3-x+2 x^2\right )^{3/2} \left (160+720 x+2280 x^2+4140 x^3+3825 x^4+\frac {10875 x^5}{2}\right ) \, dx\\ &=\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {1}{360} \int \left (3-x+2 x^2\right )^{3/2} \left (2880+12960 x+41040 x^2+9270 x^3+\frac {416775 x^4}{4}\right ) \, dx\\ &=\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \left (3-x+2 x^2\right )^{3/2} \left (46080+207360 x-\frac {1124415 x^2}{4}+\frac {5771085 x^3}{8}\right ) \, dx}{5760}\\ &=\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \left (645120-\frac {5701095 x}{4}-\frac {11027475 x^2}{16}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{80640}\\ &=-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \left (\frac {156945465}{16}-\frac {624497445 x}{32}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{967680}\\ &=-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {667795 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{131072}\\ &=-\frac {667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {46077855 \int \sqrt {3-x+2 x^2} \, dx}{4194304}\\ &=-\frac {46077855 (1-4 x) \sqrt {3-x+2 x^2}}{33554432}-\frac {667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {1059790665 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{67108864}\\ &=-\frac {46077855 (1-4 x) \sqrt {3-x+2 x^2}}{33554432}-\frac {667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac {\left (46077855 \sqrt {\frac {23}{2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{67108864}\\ &=-\frac {46077855 (1-4 x) \sqrt {3-x+2 x^2}}{33554432}-\frac {667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}-\frac {1059790665 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{67108864 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.78, size = 95, normalized size = 0.50 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (-72152399943+53985432012 x+199615064544 x^2+389257196928 x^3+487891884032 x^4+571298324480 x^5+430820229120 x^6+328328806400 x^7+124780544000 x^8+88080384000 x^9\right )-111278019825 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{14092861440} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^3,x]

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-72152399943 + 53985432012*x + 199615064544*x^2 + 389257196928*x^3 + 487891884032*x^4
+ 571298324480*x^5 + 430820229120*x^6 + 328328806400*x^7 + 124780544000*x^8 + 88080384000*x^9) - 111278019825*
Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/14092861440

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Maple [A]
time = 0.14, size = 151, normalized size = 0.80

method	result	size
risch	\(\frac {\left (88080384000 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x -72152399943\right ) \sqrt {2 x^{2}-x +3}}{3523215360}+\frac {1059790665 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{134217728}\)	\(75\)
trager	\(\left (25 x^{9}+\frac {425}{12} x^{8}+\frac {35785}{384} x^{7}+\frac {438253}{3584} x^{6}+\frac {13947713}{86016} x^{5}+\frac {34032637}{245760} x^{4}+\frac {1013690617}{9175040} x^{3}+\frac {297046227}{5242880} x^{2}+\frac {4498786001}{293601280} x -\frac {24050799981}{1174405120}\right ) \sqrt {2 x^{2}-x +3}+\frac {1059790665 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{134217728}\)	\(101\)
default	\(\frac {46077855 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{33554432}+\frac {1059790665 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{134217728}+\frac {667795 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{2097152}-\frac {81685 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{114688}+\frac {384739 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{43008}+\frac {27785 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{1536}+\frac {725 x^{4} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{48}+\frac {25 x^{5} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{4}-\frac {4625907 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{2293760}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)

[Out]

46077855/33554432*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/134217728*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+667795
/2097152*(4*x-1)*(2*x^2-x+3)^(3/2)-81685/114688*x*(2*x^2-x+3)^(5/2)+384739/43008*x^2*(2*x^2-x+3)^(5/2)+27785/1
536*x^3*(2*x^2-x+3)^(5/2)+725/48*x^4*(2*x^2-x+3)^(5/2)+25/4*x^5*(2*x^2-x+3)^(5/2)-4625907/2293760*(2*x^2-x+3)^
(5/2)

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Maxima [A]
time = 0.51, size = 172, normalized size = 0.91 \begin {gather*} \frac {25}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{5} + \frac {725}{48} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{4} + \frac {27785}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {384739}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {81685}{114688} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {4625907}{2293760} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {667795}{524288} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {667795}{2097152} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {46077855}{8388608} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {1059790665}{134217728} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {46077855}{33554432} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^3,x, algorithm="maxima")

[Out]

25/4*(2*x^2 - x + 3)^(5/2)*x^5 + 725/48*(2*x^2 - x + 3)^(5/2)*x^4 + 27785/1536*(2*x^2 - x + 3)^(5/2)*x^3 + 384
739/43008*(2*x^2 - x + 3)^(5/2)*x^2 - 81685/114688*(2*x^2 - x + 3)^(5/2)*x - 4625907/2293760*(2*x^2 - x + 3)^(
5/2) + 667795/524288*(2*x^2 - x + 3)^(3/2)*x - 667795/2097152*(2*x^2 - x + 3)^(3/2) + 46077855/8388608*sqrt(2*
x^2 - x + 3)*x + 1059790665/134217728*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 46077855/33554432*sqrt(2*x^2
- x + 3)

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Fricas [A]
time = 4.44, size = 98, normalized size = 0.52 \begin {gather*} \frac {1}{3523215360} \, {\left (88080384000 \, x^{9} + 124780544000 \, x^{8} + 328328806400 \, x^{7} + 430820229120 \, x^{6} + 571298324480 \, x^{5} + 487891884032 \, x^{4} + 389257196928 \, x^{3} + 199615064544 \, x^{2} + 53985432012 \, x - 72152399943\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1059790665}{268435456} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^3,x, algorithm="fricas")

[Out]

1/3523215360*(88080384000*x^9 + 124780544000*x^8 + 328328806400*x^7 + 430820229120*x^6 + 571298324480*x^5 + 48
7891884032*x^4 + 389257196928*x^3 + 199615064544*x^2 + 53985432012*x - 72152399943)*sqrt(2*x^2 - x + 3) + 1059
790665/268435456*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-x+3)**(3/2)*(5*x**2+3*x+2)**3,x)

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**3, x)

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Giac [A]
time = 4.64, size = 93, normalized size = 0.49 \begin {gather*} \frac {1}{3523215360} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (8 \, {\left (140 \, {\left (160 \, {\left (12 \, x + 17\right )} x + 7157\right )} x + 1314759\right )} x + 13947713\right )} x + 238228459\right )} x + 3041071851\right )} x + 6237970767\right )} x + 13496358003\right )} x - 72152399943\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {1059790665}{134217728} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^3,x, algorithm="giac")

[Out]

1/3523215360*(4*(8*(4*(16*(20*(8*(140*(160*(12*x + 17)*x + 7157)*x + 1314759)*x + 13947713)*x + 238228459)*x +
3041071851)*x + 6237970767)*x + 13496358003)*x - 72152399943)*sqrt(2*x^2 - x + 3) - 1059790665/134217728*sqrt
(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^3,x)

[Out]

int((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^3, x)

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