Integrand size = 109, antiderivative size = 36 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{e^{x \left (5+\frac {x}{-x+\frac {5 (5-x)}{-\frac {3}{x}+x}}\right )} x^2} \]
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\[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=\int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \left (784+1638 x-731 x^2-27 x^3+21 x^4+2 x^5\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx \\ & = 2 \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \left (784+1638 x-731 x^2-27 x^3+21 x^4+2 x^5\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx \\ & = 2 \int \left (25 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )+\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x+2 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x^2-\frac {5 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \left (3920-1400 x-225 x^2+78 x^3\right )}{784-280 x-31 x^2+10 x^3+x^4}\right ) \, dx \\ & = 2 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \, dx+4 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x^2 \, dx-10 \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \left (3920-1400 x-225 x^2+78 x^3\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx+50 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \, dx \\ & = 2 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \, dx+4 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x^2 \, dx-10 \int \left (\frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) (-13300+3859 x)}{\left (-28+5 x+x^2\right )^2}+\frac {3 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) (-205+26 x)}{-28+5 x+x^2}\right ) \, dx+50 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \, dx \\ & = 2 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \, dx+4 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x^2 \, dx-10 \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) (-13300+3859 x)}{\left (-28+5 x+x^2\right )^2} \, dx-30 \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) (-205+26 x)}{-28+5 x+x^2} \, dx+50 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \, dx \\ & = 2 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \, dx+4 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x^2 \, dx-10 \int \left (-\frac {13300 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )}{\left (-28+5 x+x^2\right )^2}+\frac {3859 \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x}{\left (-28+5 x+x^2\right )^2}\right ) \, dx-30 \int \left (\frac {\left (26-\frac {540}{\sqrt {137}}\right ) \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )}{5-\sqrt {137}+2 x}+\frac {\left (26+\frac {540}{\sqrt {137}}\right ) \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )}{5+\sqrt {137}+2 x}\right ) \, dx+50 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \, dx \\ & = 2 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x \, dx+4 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x^2 \, dx+50 \int \exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) \, dx-38590 \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right ) x}{\left (-28+5 x+x^2\right )^2} \, dx+133000 \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )}{\left (-28+5 x+x^2\right )^2} \, dx-\frac {1}{137} \left (60 \left (1781-270 \sqrt {137}\right )\right ) \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )}{5-\sqrt {137}+2 x} \, dx-\frac {1}{137} \left (60 \left (1781+270 \sqrt {137}\right )\right ) \int \frac {\exp \left (e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}\right )}{5+\sqrt {137}+2 x} \, dx \\ & = 2 \int e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x \, dx+4 \int e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2 \, dx+50 \int e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \, dx-38590 \int \left (\frac {2 \left (-5+\sqrt {137}\right ) e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \left (-5+\sqrt {137}-2 x\right )^2}-\frac {10 e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \sqrt {137} \left (-5+\sqrt {137}-2 x\right )}+\frac {2 \left (-5-\sqrt {137}\right ) e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \left (5+\sqrt {137}+2 x\right )^2}-\frac {10 e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \sqrt {137} \left (5+\sqrt {137}+2 x\right )}\right ) \, dx+133000 \int \left (\frac {4 e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \left (-5+\sqrt {137}-2 x\right )^2}+\frac {4 e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \sqrt {137} \left (-5+\sqrt {137}-2 x\right )}+\frac {4 e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \left (5+\sqrt {137}+2 x\right )^2}+\frac {4 e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{137 \sqrt {137} \left (5+\sqrt {137}+2 x\right )}\right ) \, dx-\frac {1}{137} \left (60 \left (1781-270 \sqrt {137}\right )\right ) \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{5-\sqrt {137}+2 x} \, dx-\frac {1}{137} \left (60 \left (1781+270 \sqrt {137}\right )\right ) \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{5+\sqrt {137}+2 x} \, dx \\ & = 2 \int e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x \, dx+4 \int e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2 \, dx+50 \int e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \, dx+\frac {532000}{137} \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{\left (-5+\sqrt {137}-2 x\right )^2} \, dx+\frac {532000}{137} \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{\left (5+\sqrt {137}+2 x\right )^2} \, dx+\frac {385900 \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{-5+\sqrt {137}-2 x} \, dx}{137 \sqrt {137}}+\frac {385900 \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{5+\sqrt {137}+2 x} \, dx}{137 \sqrt {137}}+\frac {532000 \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{-5+\sqrt {137}-2 x} \, dx}{137 \sqrt {137}}+\frac {532000 \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{5+\sqrt {137}+2 x} \, dx}{137 \sqrt {137}}-\frac {1}{137} \left (60 \left (1781-270 \sqrt {137}\right )\right ) \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{5-\sqrt {137}+2 x} \, dx+\frac {1}{137} \left (77180 \left (5-\sqrt {137}\right )\right ) \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{\left (-5+\sqrt {137}-2 x\right )^2} \, dx+\frac {1}{137} \left (77180 \left (5+\sqrt {137}\right )\right ) \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{\left (5+\sqrt {137}+2 x\right )^2} \, dx-\frac {1}{137} \left (60 \left (1781+270 \sqrt {137}\right )\right ) \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}}}{5+\sqrt {137}+2 x} \, dx \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{e^{5+4 x-\frac {10 (-14+5 x)}{-28+5 x+x^2}} x^2} \]
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Time = 16.86 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
method | result | size |
risch | \({\mathrm e}^{x^{2} {\mathrm e}^{\frac {x \left (4 x^{2}+25 x -137\right )}{x^{2}+5 x -28}}}\) | \(29\) |
norman | \(\frac {x^{2} {\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}+5 x \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}-28 \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}}{x^{2}+5 x -28}\) | \(115\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (31) = 62\).
Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.61 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} + {\left (x^{4} + 5 \, x^{3} - 28 \, x^{2}\right )} e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} - 137 \, x}{x^{2} + 5 \, x - 28} - \frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} \]
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Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{x^{2} e^{\frac {4 x^{3} + 25 x^{2} - 137 x}{x^{2} + 5 x - 28}}} \]
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Time = 0.64 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{\left (x^{2} e^{\left (4 \, x - \frac {50 \, x}{x^{2} + 5 \, x - 28} + \frac {140}{x^{2} + 5 \, x - 28} + 5\right )}\right )} \]
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\[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=\int { \frac {2 \, {\left (2 \, x^{6} + 21 \, x^{5} - 27 \, x^{4} - 731 \, x^{3} + 1638 \, x^{2} + 784 \, x\right )} e^{\left (x^{2} e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} + \frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )}}{x^{4} + 10 \, x^{3} - 31 \, x^{2} - 280 \, x + 784} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx={\mathrm {e}}^{x^2\,{\mathrm {e}}^{\frac {4\,x^3+25\,x^2-137\,x}{x^2+5\,x-28}}} \]
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