3.21.48 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [2048]

Optimal. Leaf size=171 \[ \frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 c^3 d^3 (d+e x)^{7/2}}+\frac {8 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}} \]

[Out]

16/693*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/(e*x+d)^(7/2)+8/99*(-a*e^2+c*d^2)*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/(e*x+d)^(5/2)+2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*
x+d)^(3/2)

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Rubi [A]
time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \begin {gather*} \frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 c^3 d^3 (d+e x)^{7/2}}+\frac {8 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*c^3*d^3*(d + e*x)^(7/2)) + (8*(c*d^2
- a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*c^2*d^2*(d + e*x)^(5/2)) + (2*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(3/2))

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (4 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{11 d}\\ &=\frac {8 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{99 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 c^3 d^3 (d+e x)^{7/2}}+\frac {8 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 c^2 d^2 (d+e x)^{5/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 98, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (8 a^2 e^4-4 a c d e^2 (11 d+7 e x)+c^2 d^2 \left (99 d^2+154 d e x+63 e^2 x^2\right )\right )}{693 c^3 d^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^4 - 4*a*c*d*e^2*(11*d + 7*e*x) + c^2*d^2*(99*d^2 + 1
54*d*e*x + 63*e^2*x^2)))/(693*c^3*d^3*Sqrt[d + e*x])

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Maple [A]
time = 0.72, size = 102, normalized size = 0.60

method	result	size
default	\(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (63 e^{2} x^{2} c^{2} d^{2}-28 a c d \,e^{3} x +154 c^{2} d^{3} e x +8 a^{2} e^{4}-44 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right )}{693 \sqrt {e x +d}\, c^{3} d^{3}}\)	\(102\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (63 e^{2} x^{2} c^{2} d^{2}-28 a c d \,e^{3} x +154 c^{2} d^{3} e x +8 a^{2} e^{4}-44 a c \,d^{2} e^{2}+99 c^{2} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/693*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^3*(63*c^2*d^2*e^2*x^2-28*a*c*d*e^3*x+154*c^2*d^3*e
*x+8*a^2*e^4-44*a*c*d^2*e^2+99*c^2*d^4)/c^3/d^3

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Maxima [A]
time = 0.30, size = 208, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (63 \, c^{5} d^{5} x^{5} e^{2} + 99 \, a^{3} c^{2} d^{4} e^{3} - 44 \, a^{4} c d^{2} e^{5} + 8 \, a^{5} e^{7} + 7 \, {\left (22 \, c^{5} d^{6} e + 23 \, a c^{4} d^{4} e^{3}\right )} x^{4} + {\left (99 \, c^{5} d^{7} + 418 \, a c^{4} d^{5} e^{2} + 113 \, a^{2} c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (99 \, a c^{4} d^{6} e + 110 \, a^{2} c^{3} d^{4} e^{3} + a^{3} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (297 \, a^{2} c^{3} d^{5} e^{2} + 22 \, a^{3} c^{2} d^{3} e^{4} - 4 \, a^{4} c d e^{6}\right )} x\right )} \sqrt {c d x + a e}}{693 \, c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*c^5*d^5*x^5*e^2 + 99*a^3*c^2*d^4*e^3 - 44*a^4*c*d^2*e^5 + 8*a^5*e^7 + 7*(22*c^5*d^6*e + 23*a*c^4*d^4
*e^3)*x^4 + (99*c^5*d^7 + 418*a*c^4*d^5*e^2 + 113*a^2*c^3*d^3*e^4)*x^3 + 3*(99*a*c^4*d^6*e + 110*a^2*c^3*d^4*e
^3 + a^3*c^2*d^2*e^5)*x^2 + (297*a^2*c^3*d^5*e^2 + 22*a^3*c^2*d^3*e^4 - 4*a^4*c*d*e^6)*x)*sqrt(c*d*x + a*e)/(c
^3*d^3)

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Fricas [A]
time = 2.88, size = 251, normalized size = 1.47 \begin {gather*} \frac {2 \, {\left (99 \, c^{5} d^{7} x^{3} - 4 \, a^{4} c d x e^{6} + 8 \, a^{5} e^{7} + {\left (3 \, a^{3} c^{2} d^{2} x^{2} - 44 \, a^{4} c d^{2}\right )} e^{5} + {\left (113 \, a^{2} c^{3} d^{3} x^{3} + 22 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (161 \, a c^{4} d^{4} x^{4} + 330 \, a^{2} c^{3} d^{4} x^{2} + 99 \, a^{3} c^{2} d^{4}\right )} e^{3} + {\left (63 \, c^{5} d^{5} x^{5} + 418 \, a c^{4} d^{5} x^{3} + 297 \, a^{2} c^{3} d^{5} x\right )} e^{2} + 11 \, {\left (14 \, c^{5} d^{6} x^{4} + 27 \, a c^{4} d^{6} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{693 \, {\left (c^{3} d^{3} x e + c^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(99*c^5*d^7*x^3 - 4*a^4*c*d*x*e^6 + 8*a^5*e^7 + (3*a^3*c^2*d^2*x^2 - 44*a^4*c*d^2)*e^5 + (113*a^2*c^3*d^
3*x^3 + 22*a^3*c^2*d^3*x)*e^4 + (161*a*c^4*d^4*x^4 + 330*a^2*c^3*d^4*x^2 + 99*a^3*c^2*d^4)*e^3 + (63*c^5*d^5*x
^5 + 418*a*c^4*d^5*x^3 + 297*a^2*c^3*d^5*x)*e^2 + 11*(14*c^5*d^6*x^4 + 27*a*c^4*d^6*x^2)*e)*sqrt(c*d^2*x + a*x
*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c^3*d^3*x*e + c^3*d^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(5/2)/sqrt(d + e*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1964 vs. \(2 (157) = 314\).
time = 2.09, size = 1964, normalized size = 11.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(33*c^2*d^4*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^
2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*c*d*e - c*d^2
*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*d*e - c*d^2*e
+ a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^(-1) - 462*a*c*d^3*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 -
3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^
2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2))*e^(-1) - 22*c^2*d^3*((35*sqrt(-c*d^2*e
+ a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt
(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e
- c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d
*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4)) + 1155*a^2
*d^2*(((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*e^(-1)/(c*d) + (sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e +
a*e^3)*a*e^2)/(c*d))*e + 132*a*c*d^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4
*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e +
d)*c*d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e + c^2*d^2*((315*sqrt(-c*d^2*e + a*e^3)*c^5*d^10 - 35*s
qrt(-c*d^2*e + a*e^3)*a*c^4*d^8*e^2 - 40*sqrt(-c*d^2*e + a*e^3)*a^2*c^3*d^6*e^4 - 48*sqrt(-c*d^2*e + a*e^3)*a^
3*c^2*d^4*e^6 - 64*sqrt(-c*d^2*e + a*e^3)*a^4*c*d^2*e^8 - 128*sqrt(-c*d^2*e + a*e^3)*a^5*e^10)*e^(-4)/(c^5*d^5
) + (1155*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*e^12 - 2772*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*
a^3*e^9 + 2970*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*e^6 - 1540*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9
/2)*a*e^3 + 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(11/2))*e^(-9)/(c^5*d^5))*e - 22*a*c*d*((35*sqrt(-c*d^2*e
+ a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(
-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)*e^(-3)/(c^4*d^4) + (105*((x*e + d)*c*d*e
- c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((x*e + d)*c*d*
e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))*e^(-7)/(c^4*d^4))*e^2 - 462*a
^2*d*((5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))*e^(-2)
/(c^2*d^2) + (3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)
*a^2*e^4)/(c^2*d^2))*e + 33*a^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 -
4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)*e^(-2)/(c^3*d^3) + (35*((x*e + d)*
c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((x*e + d)*c*
d*e - c*d^2*e + a*e^3)^(7/2))*e^(-5)/(c^3*d^3))*e^3)*e^(-1)

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Mupad [B]
time = 1.20, size = 241, normalized size = 1.41 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,a^5\,e^7-88\,a^4\,c\,d^2\,e^5+198\,a^3\,c^2\,d^4\,e^3}{693\,c^3\,d^3}+\frac {x^3\,\left (226\,a^2\,c^3\,d^3\,e^4+836\,a\,c^4\,d^5\,e^2+198\,c^5\,d^7\right )}{693\,c^3\,d^3}+\frac {2\,c^2\,d^2\,e^2\,x^5}{11}+\frac {2\,c\,d\,e\,x^4\,\left (22\,c\,d^2+23\,a\,e^2\right )}{99}+\frac {2\,a\,e\,x^2\,\left (a^2\,e^4+110\,a\,c\,d^2\,e^2+99\,c^2\,d^4\right )}{231\,c\,d}+\frac {2\,a^2\,e^2\,x\,\left (-4\,a^2\,e^4+22\,a\,c\,d^2\,e^2+297\,c^2\,d^4\right )}{693\,c^2\,d^2}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*a^5*e^7 - 88*a^4*c*d^2*e^5 + 198*a^3*c^2*d^4*e^3)/(693*c^3
*d^3) + (x^3*(198*c^5*d^7 + 836*a*c^4*d^5*e^2 + 226*a^2*c^3*d^3*e^4))/(693*c^3*d^3) + (2*c^2*d^2*e^2*x^5)/11 +
(2*c*d*e*x^4*(23*a*e^2 + 22*c*d^2))/99 + (2*a*e*x^2*(a^2*e^4 + 99*c^2*d^4 + 110*a*c*d^2*e^2))/(231*c*d) + (2*
a^2*e^2*x*(297*c^2*d^4 - 4*a^2*e^4 + 22*a*c*d^2*e^2))/(693*c^2*d^2)))/(d + e*x)^(1/2)

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