Optimal. Leaf size=436 \[ \frac {3 b^2 e^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {3 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}-\frac {3 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}-\frac {3 b^3 e^3 n^3 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}+\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}+\frac {6 b^3 e^3 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}+\frac {6 b^3 e^3 n^3 \text {Li}_3\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3} \]
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Rubi [A]
time = 0.59, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used =
{2501, 2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31}
\begin {gather*} \frac {6 b^2 e^3 n^2 \text {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {3 b^3 e^3 n^3 \text {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}+\frac {6 b^3 e^3 n^3 \text {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right )}{d^3}+\frac {6 b^3 e^3 n^3 \text {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}+\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {6 b^2 e^3 n^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {3 b^2 e^2 n^2 \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {3 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}-\frac {3 b e^2 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+\frac {b^3 e^3 n^3 \log (x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2445
Rule 2458
Rule 2501
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx &=3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )\\ &=-\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^4} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3-(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3-(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3-\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}\\ &=\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\left (3 b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}\\ &=-\frac {3 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3-\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {\left (3 b e^3 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\left (3 b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}+\frac {\left (6 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {\left (3 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}\\ &=\frac {3 b^2 e^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {3 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {3 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}+\frac {\left (3 e^3\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {\left (3 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\left (3 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {\left (6 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\left (3 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\left (6 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac {3 b^2 e^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {3 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d^3}-\frac {3 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {3 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}+\frac {6 b^3 e^3 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {\left (3 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {\left (6 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac {3 b^2 e^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {3 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d^3}-\frac {3 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+\frac {3 b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {3 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}+\frac {9 b^3 e^3 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}+\frac {6 b^3 e^3 n^3 \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 675, normalized size = 1.55 \begin {gather*} \frac {-6 b d e^2 n \sqrt [3]{x} \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+3 b d^2 e n x^{2/3} \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+6 b d^3 n x \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+2 d^3 x \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3+6 b e^3 n \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \log \left (e+d \sqrt [3]{x}\right )+6 b^2 n^2 \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (\left (e^3+d^3 x\right ) \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )+e^2 \left (d \sqrt [3]{x}+3 e \log \left (-\frac {e}{d \sqrt [3]{x}}\right )\right )+e \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (-3 e^2-2 d e \sqrt [3]{x}+d^2 x^{2/3}-2 e^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )\right )-2 e^3 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )\right )-b^3 n^3 \left (-6 e^3 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )-6 d e^2 \sqrt [3]{x} \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+9 e^3 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )+6 d e^2 \sqrt [3]{x} \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )-3 d^2 e x^{2/3} \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )-2 e^3 \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )-2 d^3 x \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )+6 e^3 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-18 e^3 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+6 e^3 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+6 e^3 \left (-3+2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )-12 e^3 \text {Li}_3\left (1+\frac {e}{d \sqrt [3]{x}}\right )\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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