3.14.13 \(\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx\)
Optimal. Leaf size=94 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c+6 \text {$\#$1}^4 c^2-4 \text {$\#$1}^2 c^3+b+c^4\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2 c+c^2}\& \right ]}{a} \]
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Rubi [F] time = 0.66, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(b + a^2*x^2),x]
[Out]
-1/2*Defer[Int][Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(Sqrt[-b] - a*x), x]/Sqrt[-b] - Defer[Int][Sqrt[c + Sq
rt[a*x + Sqrt[b + a^2*x^2]]]/(Sqrt[-b] + a*x), x]/(2*Sqrt[-b])
Rubi steps
\begin {align*} \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx &=\int \left (\frac {\sqrt {-b} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{2 b \left (\sqrt {-b}-a x\right )}+\frac {\sqrt {-b} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{2 b \left (\sqrt {-b}+a x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {-b}-a x} \, dx}{2 \sqrt {-b}}-\frac {\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {-b}+a x} \, dx}{2 \sqrt {-b}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 94, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c+6 \text {$\#$1}^4 c^2-4 \text {$\#$1}^2 c^3+b+c^4\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {a^2 x^2+b}+a x}+c}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2 c+c^2}\&\right ]}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(b + a^2*x^2),x]
[Out]
RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] -
#1]*#1)/(c^2 - 2*c*#1^2 + #1^4) & ]/a
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IntegrateAlgebraic [A] time = 0.00, size = 94, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(b + a^2*x^2),x]
[Out]
RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] -
#1]*#1)/(c^2 - 2*c*#1^2 + #1^4) & ]/a
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fricas [B] time = 0.70, size = 3517, normalized size = 37.41
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b),x, algorithm="fricas")
[Out]
1/4*(4*(sqrt(2)*a^8*b^4*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^4*b^2*c^2)*sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8
*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*((a^4*b^2*sqrt(-1/(a^8*b^3))
+ c^2)/(a^4*b^2))^(3/4)*(-1/(a^8*b^3))^(1/4)*arctan(1/4*(sqrt(2)*sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8
*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*((sqrt(2)*a^11*b^5*sqrt(-1/(
a^8*b^3)) - sqrt(2)*a^7*b^3*c^2)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*(-1/(a^8*b^3))^(1/4) - (sq
rt(2)*a^9*b^4*c*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^5*b^2*c^3)*(-1/(a^8*b^3))^(1/4))*sqrt((2*c^5 + sqrt(-4*a^6*b^2*
c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*
(sqrt(2)*(a^5*b^2*c^4 + a^5*b^3)*sqrt(-1/(a^8*b^3)) - (sqrt(2)*a^7*b^3*c^3*sqrt(-1/(a^8*b^3)) + sqrt(2)*a^3*b^
2*c)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))*((a^4*b^2*sqr
t(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(1/4) + 2*b*c + 2*(c^4 + b)*sqrt(a*x + sqrt(a^2*x^2 + b)) - 2*(a^2*b*c^4 + a
^2*b^2)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))/(c^4 + b))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^
4*b^2))^(3/4) - 2*sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*
a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*((sqrt(2)*a^11*b^5*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^7*b^3*c^2)*sqrt((a^4*b^2*s
qrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*(-1/(a^8*b^3))^(1/4) - (sqrt(2)*a^9*b^4*c*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^5
*b^2*c^3)*(-1/(a^8*b^3))^(1/4))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a
^4*b^2))^(3/4) + 4*(a^8*b^3*c^4 + a^8*b^4)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*(-1/(a^8*b^3))^(
3/4) - 4*(a^6*b^2*c^5 + a^6*b^3*c)*(-1/(a^8*b^3))^(3/4))/(c^4 + b)) + 4*(sqrt(2)*a^8*b^4*sqrt(-1/(a^8*b^3)) -
sqrt(2)*a^4*b^2*c^2)*sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) -
4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(3/4)*(-1/(a^8*b^3))^(1/4)
*arctan(1/4*(sqrt(2)*sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) -
4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*((sqrt(2)*a^11*b^5*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^7*b^3*c^2)*sqrt((a^4*b^
2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*(-1/(a^8*b^3))^(1/4) - (sqrt(2)*a^9*b^4*c*sqrt(-1/(a^8*b^3)) - sqrt(2)*
a^5*b^2*c^3)*(-1/(a^8*b^3))^(1/4))*sqrt((2*c^5 - sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^
4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*(sqrt(2)*(a^5*b^2*c^4 + a^5*b^3)*sqrt(-1/(a^8
*b^3)) - (sqrt(2)*a^7*b^3*c^3*sqrt(-1/(a^8*b^3)) + sqrt(2)*a^3*b^2*c)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/
(a^4*b^2)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(1/4) + 2*b
*c + 2*(c^4 + b)*sqrt(a*x + sqrt(a^2*x^2 + b)) - 2*(a^2*b*c^4 + a^2*b^2)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^
2)/(a^4*b^2)))/(c^4 + b))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(3/4) - 2*sqrt(-4*a^6*b^2*c*sqrt((a^4
*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*((sqrt(2)*a
^11*b^5*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^7*b^3*c^2)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*(-1/(a^8*
b^3))^(1/4) - (sqrt(2)*a^9*b^4*c*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^5*b^2*c^3)*(-1/(a^8*b^3))^(1/4))*sqrt(c + sqrt
(a*x + sqrt(a^2*x^2 + b)))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(3/4) - 4*(a^8*b^3*c^4 + a^8*b^4)*sq
rt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*(-1/(a^8*b^3))^(3/4) + 4*(a^6*b^2*c^5 + a^6*b^3*c)*(-1/(a^8*b
^3))^(3/4))/(c^4 + b)) + sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3
)) - 4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*(sqrt(2)*(c^4 + b) + (sqrt(2)*a^6*b^3*c*sqrt(-1/(a^8*b^3)) - sqrt(2)*
a^2*b*c^3)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^
(1/4)*log(2*(2*c^5 + sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) -
4*a^4*b*c^2*sqrt(-1/(a^8*b^3)) + 4)*(sqrt(2)*(a^5*b^2*c^4 + a^5*b^3)*sqrt(-1/(a^8*b^3)) - (sqrt(2)*a^7*b^3*c^
3*sqrt(-1/(a^8*b^3)) + sqrt(2)*a^3*b^2*c)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))*sqrt(c + sqrt(a*
x + sqrt(a^2*x^2 + b)))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(1/4) + 2*b*c + 2*(c^4 + b)*sqrt(a*x +
sqrt(a^2*x^2 + b)) - 2*(a^2*b*c^4 + a^2*b^2)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))/(c^4 + b)) -
sqrt(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/
(a^8*b^3)) + 4)*(sqrt(2)*(c^4 + b) + (sqrt(2)*a^6*b^3*c*sqrt(-1/(a^8*b^3)) - sqrt(2)*a^2*b*c^3)*sqrt((a^4*b^2*
sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))*((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(1/4)*log(2*(2*c^5 - sqrt
(-4*a^6*b^2*c*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))*sqrt(-1/(a^8*b^3)) - 4*a^4*b*c^2*sqrt(-1/(a^8
*b^3)) + 4)*(sqrt(2)*(a^5*b^2*c^4 + a^5*b^3)*sqrt(-1/(a^8*b^3)) - (sqrt(2)*a^7*b^3*c^3*sqrt(-1/(a^8*b^3)) + sq
rt(2)*a^3*b^2*c)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))*(
(a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2))^(1/4) + 2*b*c + 2*(c^4 + b)*sqrt(a*x + sqrt(a^2*x^2 + b)) - 2*(a
^2*b*c^4 + a^2*b^2)*sqrt((a^4*b^2*sqrt(-1/(a^8*b^3)) + c^2)/(a^4*b^2)))/(c^4 + b)) - 4*(c^4 + b)*sqrt(-(a^2*b*
(-1/(a^8*b^3))^(1/4) + c)/(a^2*b))*log(2*a^5*b^2*sqrt(-(a^2*b*(-1/(a^8*b^3))^(1/4) + c)/(a^2*b))*sqrt(-1/(a^8*
b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) + 4*(c^4 + b)*sqrt(-(a^2*b*(-1/(a^8*b^3))^(1/4) + c)/(a^2*b
))*log(-2*a^5*b^2*sqrt(-(a^2*b*(-1/(a^8*b^3))^(1/4) + c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + s
qrt(a^2*x^2 + b)))) - 4*(c^4 + b)*sqrt((a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*log(2*a^5*b^2*sqrt((a^2*b*(-1
/(a^8*b^3))^(1/4) - c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) + 4*(c^4 + b)*
sqrt((a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*log(-2*a^5*b^2*sqrt((a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*s
qrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))))/(c^4 + b)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(4*a^2*x^2-4*a*x+4*b+1)]schur row 3 1.47052e-08Warning, choosing root of [1,0,%%%{-2,[2,0,2]%%%}+%%%{2,[1,0,1
]%%%}+%%%{-2,[0,1,0]%%%},%%%{4,[2,0,2]%%%}+%%%{4,[0,1,0]%%%},%%%{1,[4,0,4]%%%}+%%%{2,[3,0,3]%%%}+%%%{2,[2,1,2]
%%%}+%%%{2,[1,1,1]%%%}+%%%{1,[0,2,0]%%%}+%%%{-1,[0,1,0]%%%}] at parameters values [21,-68,89]Warning, choosing
root of [1,0,%%%{-2,[2,0,2]%%%}+%%%{2,[1,0,1]%%%}+%%%{-2,[0,1,0]%%%},%%%{4,[2,0,2]%%%}+%%%{4,[0,1,0]%%%},%%%{
1,[4,0,4]%%%}+%%%{2,[3,0,3]%%%}+%%%{2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{1,[0,2,0]%%%}+%%%{-1,[0,1,0]%%%}] at p
arameters values [71,-84,-83]Warning, integration of abs or sign assumes constant sign by intervals (correct i
f the argument is real):Check [abs(t_nostep)]Warning, need to choose a branch for the root of a polynomial wit
h parameters. This might be wrong.The choice was done assuming 0=[0]Warning, need to choose a branch for the r
oot of a polynomial with parameters. This might be wrong.The choice was done assuming 0=[0]Warning, need to ch
oose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming 0
=[0]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The cho
ice was done assuming 0=[0]Evaluation time: 23.11Conj Error: Bad Argument Type
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{a^{2} x^{2}+b}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b),x)
[Out]
int((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{a^{2} x^{2} + b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b),x, algorithm="maxima")
[Out]
integrate(sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))/(a^2*x^2 + b), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}}}{a^2\,x^2+b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2)/(b + a^2*x^2),x)
[Out]
int((c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2)/(b + a^2*x^2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{a^{2} x^{2} + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+(a*x+(a**2*x**2+b)**(1/2))**(1/2))**(1/2)/(a**2*x**2+b),x)
[Out]
Integral(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 + b)))/(a**2*x**2 + b), x)
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