Optimal. Leaf size=28 \[ \frac {1}{9} x^2 \left (32 \left (3-e^x x\right )^5+e^{-x} \log (4)\right ) \]
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Rubi [B] time = 2.55, antiderivative size = 72, normalized size of antiderivative = 2.57, number of steps used = 77, number of rules used = 7, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6741, 12, 6688, 6742, 2196, 2176, 2194} \begin {gather*} -\frac {32}{9} e^{5 x} x^7+\frac {160}{3} e^{4 x} x^6-320 e^{3 x} x^5+960 e^{2 x} x^4-1440 e^x x^3+864 x^2+\frac {1}{9} e^{-x} x^2 \log (4) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2194
Rule 2196
Rule 6688
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-\left (6-2 e^x x\right )^5 \left (-6 e^x x+e^{2 x} \left (7 x^2+5 x^3\right )\right )-\left (-6 x+3 x^2\right ) \log (4)-e^x \left (2 x^2-x^3\right ) \log (4)\right )}{9 \left (3-e^x x\right )} \, dx\\ &=\frac {1}{9} \int \frac {e^{-x} \left (-\left (6-2 e^x x\right )^5 \left (-6 e^x x+e^{2 x} \left (7 x^2+5 x^3\right )\right )-\left (-6 x+3 x^2\right ) \log (4)-e^x \left (2 x^2-x^3\right ) \log (4)\right )}{3-e^x x} \, dx\\ &=\frac {1}{9} \int \frac {e^{-x} x \left (32 e^x \left (-3+e^x x\right )^5 \left (-6+e^x x (7+5 x)\right )-3 (-2+x) \log (4)+e^x (-2+x) x \log (4)\right )}{3-e^x x} \, dx\\ &=\frac {1}{9} \int \left (15552 x+17280 e^{2 x} x^3 (2+x)-12960 e^x x^2 (3+x)+960 e^{4 x} x^5 (3+2 x)-2880 e^{3 x} x^4 (5+3 x)-32 e^{5 x} x^6 (7+5 x)-e^{-x} (-2+x) x \log (4)\right ) \, dx\\ &=864 x^2-\frac {32}{9} \int e^{5 x} x^6 (7+5 x) \, dx+\frac {320}{3} \int e^{4 x} x^5 (3+2 x) \, dx-320 \int e^{3 x} x^4 (5+3 x) \, dx-1440 \int e^x x^2 (3+x) \, dx+1920 \int e^{2 x} x^3 (2+x) \, dx-\frac {1}{9} \log (4) \int e^{-x} (-2+x) x \, dx\\ &=864 x^2-\frac {32}{9} \int \left (7 e^{5 x} x^6+5 e^{5 x} x^7\right ) \, dx+\frac {320}{3} \int \left (3 e^{4 x} x^5+2 e^{4 x} x^6\right ) \, dx-320 \int \left (5 e^{3 x} x^4+3 e^{3 x} x^5\right ) \, dx-1440 \int \left (3 e^x x^2+e^x x^3\right ) \, dx+1920 \int \left (2 e^{2 x} x^3+e^{2 x} x^4\right ) \, dx-\frac {1}{9} \log (4) \int \left (-2 e^{-x} x+e^{-x} x^2\right ) \, dx\\ &=864 x^2-\frac {160}{9} \int e^{5 x} x^7 \, dx-\frac {224}{9} \int e^{5 x} x^6 \, dx+\frac {640}{3} \int e^{4 x} x^6 \, dx+320 \int e^{4 x} x^5 \, dx-960 \int e^{3 x} x^5 \, dx-1440 \int e^x x^3 \, dx-1600 \int e^{3 x} x^4 \, dx+1920 \int e^{2 x} x^4 \, dx+3840 \int e^{2 x} x^3 \, dx-4320 \int e^x x^2 \, dx-\frac {1}{9} \log (4) \int e^{-x} x^2 \, dx+\frac {1}{9} (2 \log (4)) \int e^{-x} x \, dx\\ &=864 x^2-4320 e^x x^2-1440 e^x x^3+1920 e^{2 x} x^3+960 e^{2 x} x^4-\frac {1600}{3} e^{3 x} x^4-320 e^{3 x} x^5+80 e^{4 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {224}{45} e^{5 x} x^6-\frac {32}{9} e^{5 x} x^7-\frac {2}{9} e^{-x} x \log (4)+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {224}{9} \int e^{5 x} x^6 \, dx+\frac {448}{15} \int e^{5 x} x^5 \, dx-320 \int e^{4 x} x^5 \, dx-400 \int e^{4 x} x^4 \, dx+1600 \int e^{3 x} x^4 \, dx+\frac {6400}{3} \int e^{3 x} x^3 \, dx-3840 \int e^{2 x} x^3 \, dx+4320 \int e^x x^2 \, dx-5760 \int e^{2 x} x^2 \, dx+8640 \int e^x x \, dx+\frac {1}{9} (2 \log (4)) \int e^{-x} \, dx-\frac {1}{9} (2 \log (4)) \int e^{-x} x \, dx\\ &=8640 e^x x+864 x^2-2880 e^{2 x} x^2-1440 e^x x^3+\frac {6400}{9} e^{3 x} x^3+960 e^{2 x} x^4-100 e^{4 x} x^4-320 e^{3 x} x^5+\frac {448}{75} e^{5 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7-\frac {2}{9} e^{-x} \log (4)+\frac {1}{9} e^{-x} x^2 \log (4)-\frac {448}{15} \int e^{5 x} x^4 \, dx-\frac {448}{15} \int e^{5 x} x^5 \, dx+400 \int e^{4 x} x^3 \, dx+400 \int e^{4 x} x^4 \, dx-\frac {6400}{3} \int e^{3 x} x^2 \, dx-\frac {6400}{3} \int e^{3 x} x^3 \, dx+5760 \int e^{2 x} x \, dx+5760 \int e^{2 x} x^2 \, dx-8640 \int e^x \, dx-8640 \int e^x x \, dx-\frac {1}{9} (2 \log (4)) \int e^{-x} \, dx\\ &=-8640 e^x+2880 e^{2 x} x+864 x^2-\frac {6400}{9} e^{3 x} x^2-1440 e^x x^3+100 e^{4 x} x^3+960 e^{2 x} x^4-\frac {448}{75} e^{5 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {1792}{75} \int e^{5 x} x^3 \, dx+\frac {448}{15} \int e^{5 x} x^4 \, dx-300 \int e^{4 x} x^2 \, dx-400 \int e^{4 x} x^3 \, dx+\frac {12800}{9} \int e^{3 x} x \, dx+\frac {6400}{3} \int e^{3 x} x^2 \, dx-2880 \int e^{2 x} \, dx-5760 \int e^{2 x} x \, dx+8640 \int e^x \, dx\\ &=-1440 e^{2 x}+\frac {12800}{27} e^{3 x} x+864 x^2-75 e^{4 x} x^2-1440 e^x x^3+\frac {1792}{375} e^{5 x} x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)-\frac {1792}{125} \int e^{5 x} x^2 \, dx-\frac {1792}{75} \int e^{5 x} x^3 \, dx+150 \int e^{4 x} x \, dx+300 \int e^{4 x} x^2 \, dx-\frac {12800}{27} \int e^{3 x} \, dx-\frac {12800}{9} \int e^{3 x} x \, dx+2880 \int e^{2 x} \, dx\\ &=-\frac {12800 e^{3 x}}{81}+\frac {75}{2} e^{4 x} x+864 x^2-\frac {1792}{625} e^{5 x} x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {3584}{625} \int e^{5 x} x \, dx+\frac {1792}{125} \int e^{5 x} x^2 \, dx-\frac {75}{2} \int e^{4 x} \, dx-150 \int e^{4 x} x \, dx+\frac {12800}{27} \int e^{3 x} \, dx\\ &=-\frac {75 e^{4 x}}{8}+\frac {3584 e^{5 x} x}{3125}+864 x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)-\frac {3584 \int e^{5 x} \, dx}{3125}-\frac {3584}{625} \int e^{5 x} x \, dx+\frac {75}{2} \int e^{4 x} \, dx\\ &=-\frac {3584 e^{5 x}}{15625}+864 x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)+\frac {3584 \int e^{5 x} \, dx}{3125}\\ &=864 x^2-1440 e^x x^3+960 e^{2 x} x^4-320 e^{3 x} x^5+\frac {160}{3} e^{4 x} x^6-\frac {32}{9} e^{5 x} x^7+\frac {1}{9} e^{-x} x^2 \log (4)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.48, size = 63, normalized size = 2.25 \begin {gather*} \frac {1}{9} x^2 \left (7776-12960 e^x x+8640 e^{2 x} x^2-2880 e^{3 x} x^3+480 e^{4 x} x^4-32 e^{5 x} x^5+e^{-x} \log (4)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 66, normalized size = 2.36 \begin {gather*} -\frac {2}{9} \, {\left (16 \, x^{7} e^{\left (6 \, x\right )} - 240 \, x^{6} e^{\left (5 \, x\right )} + 1440 \, x^{5} e^{\left (4 \, x\right )} - 4320 \, x^{4} e^{\left (3 \, x\right )} + 6480 \, x^{3} e^{\left (2 \, x\right )} - 3888 \, x^{2} e^{x} - x^{2} \log \relax (2)\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 66, normalized size = 2.36 \begin {gather*} -\frac {2}{9} \, {\left (16 \, x^{7} e^{\left (6 \, x\right )} - 240 \, x^{6} e^{\left (5 \, x\right )} + 1440 \, x^{5} e^{\left (4 \, x\right )} - 4320 \, x^{4} e^{\left (3 \, x\right )} + 6480 \, x^{3} e^{\left (2 \, x\right )} - 3888 \, x^{2} e^{x} - x^{2} \log \relax (2)\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 61, normalized size = 2.18
method | result | size |
risch | \(864 x^{2}-\frac {32 x^{7} {\mathrm e}^{5 x}}{9}+\frac {160 x^{6} {\mathrm e}^{4 x}}{3}-320 x^{5} {\mathrm e}^{3 x}+960 \,{\mathrm e}^{2 x} x^{4}-1440 \,{\mathrm e}^{x} x^{3}+\frac {2 \ln \relax (2) x^{2} {\mathrm e}^{-x}}{9}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 60, normalized size = 2.14 \begin {gather*} -\frac {32}{9} \, x^{7} e^{\left (5 \, x\right )} + \frac {160}{3} \, x^{6} e^{\left (4 \, x\right )} - 320 \, x^{5} e^{\left (3 \, x\right )} + 960 \, x^{4} e^{\left (2 \, x\right )} - 1440 \, x^{3} e^{x} + \frac {2}{9} \, x^{2} e^{\left (-x\right )} \log \relax (2) + 864 \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 55, normalized size = 1.96 \begin {gather*} \frac {2\,x^2\,\left ({\mathrm {e}}^{-x}\,\ln \relax (2)+4320\,x^2\,{\mathrm {e}}^{2\,x}-1440\,x^3\,{\mathrm {e}}^{3\,x}+240\,x^4\,{\mathrm {e}}^{4\,x}-16\,x^5\,{\mathrm {e}}^{5\,x}-6480\,x\,{\mathrm {e}}^x+3888\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 70, normalized size = 2.50 \begin {gather*} - \frac {32 x^{7} e^{5 x}}{9} + \frac {160 x^{6} e^{4 x}}{3} - 320 x^{5} e^{3 x} + 960 x^{4} e^{2 x} - 1440 x^{3} e^{x} + 864 x^{2} + \frac {2 x^{2} e^{- x} \log {\relax (2 )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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