Optimal. Leaf size=333 \[ -\frac {3 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}-\frac {9 \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}-\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}+\frac {9 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {3 x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d^3}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {9 x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{2 a^3 d^2}-\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d}+\frac {3 x^2}{2 a^3 d^2}-\frac {3 x^3}{2 a^3 d}+\frac {x^4}{4 a^3}-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2} \]
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Rubi [A] time = 1.08, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 10, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191, 2279, 2391} \[ -\frac {3 x^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {9 x \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {6 x \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {3 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}-\frac {9 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}-\frac {6 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac {9 x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{2 a^3 d^2}-\frac {3 x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d^3}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}-\frac {x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d}+\frac {3 x^2}{2 a^3 d^2}-\frac {3 x^3}{2 a^3 d}+\frac {x^4}{4 a^3}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b e^{c+d x}\right )^3} \, dx &=\frac {\int \frac {x^3}{\left (a+b e^{c+d x}\right )^2} \, dx}{a}-\frac {b \int \frac {e^{c+d x} x^3}{\left (a+b e^{c+d x}\right )^3} \, dx}{a}\\ &=\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {\int \frac {x^3}{a+b e^{c+d x}} \, dx}{a^2}-\frac {b \int \frac {e^{c+d x} x^3}{\left (a+b e^{c+d x}\right )^2} \, dx}{a^2}-\frac {3 \int \frac {x^2}{\left (a+b e^{c+d x}\right )^2} \, dx}{2 a d}\\ &=\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^3}-\frac {b \int \frac {e^{c+d x} x^3}{a+b e^{c+d x}} \, dx}{a^3}-\frac {3 \int \frac {x^2}{a+b e^{c+d x}} \, dx}{2 a^2 d}-\frac {3 \int \frac {x^2}{a+b e^{c+d x}} \, dx}{a^2 d}+\frac {(3 b) \int \frac {e^{c+d x} x^2}{\left (a+b e^{c+d x}\right )^2} \, dx}{2 a^2 d}\\ &=-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^3}{2 a^3 d}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^3}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {3 \int \frac {x}{a+b e^{c+d x}} \, dx}{a^2 d^2}+\frac {3 \int x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d}+\frac {(3 b) \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}} \, dx}{2 a^3 d}+\frac {(3 b) \int \frac {e^{c+d x} x^2}{a+b e^{c+d x}} \, dx}{a^3 d}\\ &=\frac {3 x^2}{2 a^3 d^2}-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^3}{2 a^3 d}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^3}+\frac {9 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{2 a^3 d^2}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {3 \int x \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^2}-\frac {6 \int x \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^2}+\frac {6 \int x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^2}-\frac {(3 b) \int \frac {e^{c+d x} x}{a+b e^{c+d x}} \, dx}{a^3 d^2}\\ &=\frac {3 x^2}{2 a^3 d^2}-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^3}{2 a^3 d}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^3}-\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {9 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{2 a^3 d^2}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {9 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {3 \int \log \left (1+\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^3}-\frac {3 \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^3}-\frac {6 \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^3}-\frac {6 \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right ) \, dx}{a^3 d^3}\\ &=\frac {3 x^2}{2 a^3 d^2}-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^3}{2 a^3 d}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^3}-\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {9 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{2 a^3 d^2}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}+\frac {9 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}-\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}-\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}\\ &=\frac {3 x^2}{2 a^3 d^2}-\frac {3 x^2}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x^3}{2 a^3 d}+\frac {x^3}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x^3}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^4}{4 a^3}-\frac {3 x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}+\frac {9 x^2 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{2 a^3 d^2}-\frac {x^3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}-\frac {3 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}+\frac {9 x \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {3 x^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac {9 \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}+\frac {6 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^3}-\frac {6 \text {Li}_4\left (-\frac {b e^{c+d x}}{a}\right )}{a^3 d^4}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 241, normalized size = 0.72 \[ \frac {\frac {2 a^2 x^3}{d \left (a+b e^{c+d x}\right )^2}+\frac {12 (2 d x-3) \text {Li}_3\left (-\frac {b e^{c+d x}}{a}\right )}{d^4}-\frac {24 \text {Li}_4\left (-\frac {b e^{c+d x}}{a}\right )}{d^4}-\frac {12 x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{d^3}-\frac {6 a x^2}{d^2 \left (a+b e^{c+d x}\right )}+\frac {18 x^2 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{d^2}-\frac {12 \left (d^2 x^2-3 d x+1\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a}\right )}{d^4}+\frac {4 a x^3}{a d+b d e^{c+d x}}-\frac {4 x^3 \log \left (\frac {b e^{c+d x}}{a}+1\right )}{d}+\frac {6 x^2}{d^2}-\frac {6 x^3}{d}+x^4}{4 a^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 702, normalized size = 2.11 \[ \frac {a^{2} d^{4} x^{4} - a^{2} c^{4} - 6 \, a^{2} c^{3} - 6 \, a^{2} c^{2} - 12 \, {\left (a^{2} d^{2} x^{2} - 3 \, a^{2} d x + a^{2} + {\left (b^{2} d^{2} x^{2} - 3 \, b^{2} d x + b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b d^{2} x^{2} - 3 \, a b d x + a b\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right ) + {\left (b^{2} d^{4} x^{4} - 6 \, b^{2} d^{3} x^{3} - b^{2} c^{4} + 6 \, b^{2} d^{2} x^{2} - 6 \, b^{2} c^{3} - 6 \, b^{2} c^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b d^{4} x^{4} - 4 \, a b d^{3} x^{3} - a b c^{4} + 3 \, a b d^{2} x^{2} - 6 \, a b c^{3} - 6 \, a b c^{2}\right )} e^{\left (d x + c\right )} + 2 \, {\left (2 \, a^{2} c^{3} + 9 \, a^{2} c^{2} + 6 \, a^{2} c + {\left (2 \, b^{2} c^{3} + 9 \, b^{2} c^{2} + 6 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b c^{3} + 9 \, a b c^{2} + 6 \, a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \, {\left (2 \, a^{2} d^{3} x^{3} - 9 \, a^{2} d^{2} x^{2} + 2 \, a^{2} c^{3} + 9 \, a^{2} c^{2} + 6 \, a^{2} d x + 6 \, a^{2} c + {\left (2 \, b^{2} d^{3} x^{3} - 9 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{3} + 9 \, b^{2} c^{2} + 6 \, b^{2} d x + 6 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b d^{3} x^{3} - 9 \, a b d^{2} x^{2} + 2 \, a b c^{3} + 9 \, a b c^{2} + 6 \, a b d x + 6 \, a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right ) - 24 \, {\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} {\rm polylog}\left (4, -\frac {b e^{\left (d x + c\right )}}{a}\right ) + 12 \, {\left (2 \, a^{2} d x - 3 \, a^{2} + {\left (2 \, b^{2} d x - 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b d x - 3 \, a b\right )} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -\frac {b e^{\left (d x + c\right )}}{a}\right )}{4 \, {\left (a^{3} b^{2} d^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{4} e^{\left (d x + c\right )} + a^{5} d^{4}\right )}} \]
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 \[ \int \frac {x^{3}}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 548, normalized size = 1.65 \[ \frac {x^{4}}{4 a^{3}}-\frac {x^{3} \ln \left (\frac {b \,{\mathrm e}^{d x +c}}{a}+1\right )}{a^{3} d}-\frac {3 x^{3}}{2 a^{3} d}+\frac {\left (2 b d x \,{\mathrm e}^{d x +c}+3 a d x -3 b \,{\mathrm e}^{d x +c}-3 a \right ) x^{2}}{2 \left (b \,{\mathrm e}^{d x +c}+a \right )^{2} a^{2} d^{2}}+\frac {c^{3} x}{a^{3} d^{3}}-\frac {3 x^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{2}}+\frac {9 x^{2} \ln \left (\frac {b \,{\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{3} d^{2}}+\frac {3 c^{4}}{4 a^{3} d^{4}}-\frac {c^{3} \ln \left (\frac {b \,{\mathrm e}^{d x +c}}{a}+1\right )}{a^{3} d^{4}}+\frac {c^{3} \ln \left (b \,{\mathrm e}^{d x +c}+a \right )}{a^{3} d^{4}}-\frac {c^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{a^{3} d^{4}}+\frac {9 c^{2} x}{2 a^{3} d^{3}}+\frac {3 x^{2}}{2 a^{3} d^{2}}+\frac {3 c^{3}}{a^{3} d^{4}}-\frac {9 c^{2} \ln \left (\frac {b \,{\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{3} d^{4}}+\frac {9 c^{2} \ln \left (b \,{\mathrm e}^{d x +c}+a \right )}{2 a^{3} d^{4}}-\frac {9 c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{2 a^{3} d^{4}}+\frac {3 c x}{a^{3} d^{3}}+\frac {9 x \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{3}}+\frac {6 x \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{3}}-\frac {3 x \ln \left (\frac {b \,{\mathrm e}^{d x +c}}{a}+1\right )}{a^{3} d^{3}}+\frac {3 c^{2}}{2 a^{3} d^{4}}-\frac {3 c \ln \left (\frac {b \,{\mathrm e}^{d x +c}}{a}+1\right )}{a^{3} d^{4}}+\frac {3 c \ln \left (b \,{\mathrm e}^{d x +c}+a \right )}{a^{3} d^{4}}-\frac {3 c \ln \left ({\mathrm e}^{d x +c}\right )}{a^{3} d^{4}}-\frac {3 \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{4}}-\frac {9 \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{4}}-\frac {6 \polylog \left (4, -\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 303, normalized size = 0.91 \[ \frac {3 \, a d x^{3} - 3 \, a x^{2} + {\left (2 \, b d x^{3} e^{c} - 3 \, b x^{2} e^{c}\right )} e^{\left (d x\right )}}{2 \, {\left (a^{2} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac {d^{4} x^{4} - 6 \, d^{3} x^{3} + 6 \, d^{2} x^{2}}{4 \, a^{3} d^{4}} - \frac {d^{3} x^{3} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 6 \, d x {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a}) + 6 \, {\rm Li}_{4}(-\frac {b e^{\left (d x + c\right )}}{a})}{a^{3} d^{4}} + \frac {9 \, {\left (d^{2} x^{2} \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right ) - 2 \, {\rm Li}_{3}(-\frac {b e^{\left (d x + c\right )}}{a})\right )}}{2 \, a^{3} d^{4}} - \frac {3 \, {\left (d x \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right )\right )}}{a^{3} d^{4}} \]
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 \[ \int \frac {x^3}{{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^3} \,d x \]
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 \[ \frac {3 a d x^{3} - 3 a x^{2} + \left (2 b d x^{3} - 3 b x^{2}\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac {\int \frac {6 x}{a + b e^{c} e^{d x}}\, dx + \int \left (- \frac {9 d x^{2}}{a + b e^{c} e^{d x}}\right )\, dx + \int \frac {2 d^{2} x^{3}}{a + b e^{c} e^{d x}}\, dx}{2 a^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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