Optimal. Leaf size=20 \[ \left (\frac {4}{5}+e^x+x+x^2\right )^4-\frac {\log (x)}{5} \]
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Rubi [B] time = 0.56, antiderivative size = 199, normalized size of antiderivative = 9.95, number of steps used = 61, number of rules used = 5, integrand size = 143, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {12, 14, 2194, 2196, 2176} \begin {gather*} x^8+4 x^7+4 e^x x^6+\frac {46 x^6}{5}+12 e^x x^5+\frac {68 x^5}{5}+\frac {108 e^x x^4}{5}+6 e^{2 x} x^4+\frac {361 x^4}{25}+\frac {116 e^x x^3}{5}+12 e^{2 x} x^3+\frac {272 x^3}{25}+\frac {432 e^x x^2}{25}+\frac {78}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {736 x^2}{125}+\frac {192 e^x x}{25}+\frac {48}{5} e^{2 x} x+4 e^{3 x} x+\frac {256 x}{125}+\frac {256 e^x}{125}+\frac {96 e^{2 x}}{25}+\frac {16 e^{3 x}}{5}+e^{4 x}-\frac {\log (x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{125} \int \frac {-25+256 x+500 e^{4 x} x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8+e^{3 x} \left (1700 x+2500 x^2+1500 x^3\right )+e^{2 x} \left (2160 x+6300 x^2+8400 x^3+6000 x^4+1500 x^5\right )+e^x \left (1216 x+5280 x^2+10860 x^3+13700 x^4+10200 x^5+4500 x^6+500 x^7\right )}{x} \, dx\\ &=\frac {1}{125} \int \left (500 e^{4 x}+60 e^{2 x} \left (4+5 x+5 x^2\right ) \left (9+15 x+5 x^2\right )+4 e^x \left (4+5 x+5 x^2\right )^2 \left (19+35 x+5 x^2\right )+100 e^{3 x} \left (17+25 x+15 x^2\right )+\frac {-25+256 x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8}{x}\right ) \, dx\\ &=\frac {1}{125} \int \frac {-25+256 x+1472 x^2+4080 x^3+7220 x^4+8500 x^5+6900 x^6+3500 x^7+1000 x^8}{x} \, dx+\frac {4}{125} \int e^x \left (4+5 x+5 x^2\right )^2 \left (19+35 x+5 x^2\right ) \, dx+\frac {12}{25} \int e^{2 x} \left (4+5 x+5 x^2\right ) \left (9+15 x+5 x^2\right ) \, dx+\frac {4}{5} \int e^{3 x} \left (17+25 x+15 x^2\right ) \, dx+4 \int e^{4 x} \, dx\\ &=e^{4 x}+\frac {1}{125} \int \left (256-\frac {25}{x}+1472 x+4080 x^2+7220 x^3+8500 x^4+6900 x^5+3500 x^6+1000 x^7\right ) \, dx+\frac {4}{125} \int \left (304 e^x+1320 e^x x+2715 e^x x^2+3425 e^x x^3+2550 e^x x^4+1125 e^x x^5+125 e^x x^6\right ) \, dx+\frac {12}{25} \int \left (36 e^{2 x}+105 e^{2 x} x+140 e^{2 x} x^2+100 e^{2 x} x^3+25 e^{2 x} x^4\right ) \, dx+\frac {4}{5} \int \left (17 e^{3 x}+25 e^{3 x} x+15 e^{3 x} x^2\right ) \, dx\\ &=e^{4 x}+\frac {256 x}{125}+\frac {736 x^2}{125}+\frac {272 x^3}{25}+\frac {361 x^4}{25}+\frac {68 x^5}{5}+\frac {46 x^6}{5}+4 x^7+x^8-\frac {\log (x)}{5}+4 \int e^x x^6 \, dx+\frac {1216 \int e^x \, dx}{125}+12 \int e^{3 x} x^2 \, dx+12 \int e^{2 x} x^4 \, dx+\frac {68}{5} \int e^{3 x} \, dx+\frac {432}{25} \int e^{2 x} \, dx+20 \int e^{3 x} x \, dx+36 \int e^x x^5 \, dx+\frac {1056}{25} \int e^x x \, dx+48 \int e^{2 x} x^3 \, dx+\frac {252}{5} \int e^{2 x} x \, dx+\frac {336}{5} \int e^{2 x} x^2 \, dx+\frac {408}{5} \int e^x x^4 \, dx+\frac {2172}{25} \int e^x x^2 \, dx+\frac {548}{5} \int e^x x^3 \, dx\\ &=\frac {1216 e^x}{125}+\frac {216 e^{2 x}}{25}+\frac {68 e^{3 x}}{15}+e^{4 x}+\frac {256 x}{125}+\frac {1056 e^x x}{25}+\frac {126}{5} e^{2 x} x+\frac {20}{3} e^{3 x} x+\frac {736 x^2}{125}+\frac {2172 e^x x^2}{25}+\frac {168}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}+\frac {548 e^x x^3}{5}+24 e^{2 x} x^3+\frac {361 x^4}{25}+\frac {408 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+36 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}-\frac {20}{3} \int e^{3 x} \, dx-8 \int e^{3 x} x \, dx-24 \int e^{2 x} x^3 \, dx-24 \int e^x x^5 \, dx-\frac {126}{5} \int e^{2 x} \, dx-\frac {1056 \int e^x \, dx}{25}-\frac {336}{5} \int e^{2 x} x \, dx-72 \int e^{2 x} x^2 \, dx-\frac {4344}{25} \int e^x x \, dx-180 \int e^x x^4 \, dx-\frac {1632}{5} \int e^x x^3 \, dx-\frac {1644}{5} \int e^x x^2 \, dx\\ &=-\frac {4064 e^x}{125}-\frac {99 e^{2 x}}{25}+\frac {104 e^{3 x}}{45}+e^{4 x}+\frac {256 x}{125}-\frac {3288 e^x x}{25}-\frac {42}{5} e^{2 x} x+4 e^{3 x} x+\frac {736 x^2}{125}-\frac {6048 e^x x^2}{25}-\frac {12}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}-\frac {1084 e^x x^3}{5}+12 e^{2 x} x^3+\frac {361 x^4}{25}-\frac {492 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+12 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}+\frac {8}{3} \int e^{3 x} \, dx+\frac {168}{5} \int e^{2 x} \, dx+36 \int e^{2 x} x^2 \, dx+72 \int e^{2 x} x \, dx+120 \int e^x x^4 \, dx+\frac {4344 \int e^x \, dx}{25}+\frac {3288}{5} \int e^x x \, dx+720 \int e^x x^3 \, dx+\frac {4896}{5} \int e^x x^2 \, dx\\ &=\frac {17656 e^x}{125}+\frac {321 e^{2 x}}{25}+\frac {16 e^{3 x}}{5}+e^{4 x}+\frac {256 x}{125}+\frac {13152 e^x x}{25}+\frac {138}{5} e^{2 x} x+4 e^{3 x} x+\frac {736 x^2}{125}+\frac {18432 e^x x^2}{25}+\frac {78}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}+\frac {2516 e^x x^3}{5}+12 e^{2 x} x^3+\frac {361 x^4}{25}+\frac {108 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+12 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}-36 \int e^{2 x} \, dx-36 \int e^{2 x} x \, dx-480 \int e^x x^3 \, dx-\frac {3288 \int e^x \, dx}{5}-\frac {9792}{5} \int e^x x \, dx-2160 \int e^x x^2 \, dx\\ &=-\frac {64544 e^x}{125}-\frac {129 e^{2 x}}{25}+\frac {16 e^{3 x}}{5}+e^{4 x}+\frac {256 x}{125}-\frac {35808 e^x x}{25}+\frac {48}{5} e^{2 x} x+4 e^{3 x} x+\frac {736 x^2}{125}-\frac {35568 e^x x^2}{25}+\frac {78}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}+\frac {116 e^x x^3}{5}+12 e^{2 x} x^3+\frac {361 x^4}{25}+\frac {108 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+12 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}+18 \int e^{2 x} \, dx+1440 \int e^x x^2 \, dx+\frac {9792 \int e^x \, dx}{5}+4320 \int e^x x \, dx\\ &=\frac {180256 e^x}{125}+\frac {96 e^{2 x}}{25}+\frac {16 e^{3 x}}{5}+e^{4 x}+\frac {256 x}{125}+\frac {72192 e^x x}{25}+\frac {48}{5} e^{2 x} x+4 e^{3 x} x+\frac {736 x^2}{125}+\frac {432 e^x x^2}{25}+\frac {78}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}+\frac {116 e^x x^3}{5}+12 e^{2 x} x^3+\frac {361 x^4}{25}+\frac {108 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+12 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}-2880 \int e^x x \, dx-4320 \int e^x \, dx\\ &=-\frac {359744 e^x}{125}+\frac {96 e^{2 x}}{25}+\frac {16 e^{3 x}}{5}+e^{4 x}+\frac {256 x}{125}+\frac {192 e^x x}{25}+\frac {48}{5} e^{2 x} x+4 e^{3 x} x+\frac {736 x^2}{125}+\frac {432 e^x x^2}{25}+\frac {78}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}+\frac {116 e^x x^3}{5}+12 e^{2 x} x^3+\frac {361 x^4}{25}+\frac {108 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+12 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}+2880 \int e^x \, dx\\ &=\frac {256 e^x}{125}+\frac {96 e^{2 x}}{25}+\frac {16 e^{3 x}}{5}+e^{4 x}+\frac {256 x}{125}+\frac {192 e^x x}{25}+\frac {48}{5} e^{2 x} x+4 e^{3 x} x+\frac {736 x^2}{125}+\frac {432 e^x x^2}{25}+\frac {78}{5} e^{2 x} x^2+4 e^{3 x} x^2+\frac {272 x^3}{25}+\frac {116 e^x x^3}{5}+12 e^{2 x} x^3+\frac {361 x^4}{25}+\frac {108 e^x x^4}{5}+6 e^{2 x} x^4+\frac {68 x^5}{5}+12 e^x x^5+\frac {46 x^6}{5}+4 e^x x^6+4 x^7+x^8-\frac {\log (x)}{5}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.10, size = 106, normalized size = 5.30 \begin {gather*} \frac {1}{125} \left (125 e^{4 x}+100 e^{3 x} \left (4+5 x+5 x^2\right )+30 e^{2 x} \left (4+5 x+5 x^2\right )^2+4 e^x \left (4+5 x+5 x^2\right )^3+x \left (256+736 x+1360 x^2+1805 x^3+1700 x^4+1150 x^5+500 x^6+125 x^7\right )-25 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 121, normalized size = 6.05 \begin {gather*} x^{8} + 4 \, x^{7} + \frac {46}{5} \, x^{6} + \frac {68}{5} \, x^{5} + \frac {361}{25} \, x^{4} + \frac {272}{25} \, x^{3} + \frac {736}{125} \, x^{2} + \frac {4}{5} \, {\left (5 \, x^{2} + 5 \, x + 4\right )} e^{\left (3 \, x\right )} + \frac {6}{25} \, {\left (25 \, x^{4} + 50 \, x^{3} + 65 \, x^{2} + 40 \, x + 16\right )} e^{\left (2 \, x\right )} + \frac {4}{125} \, {\left (125 \, x^{6} + 375 \, x^{5} + 675 \, x^{4} + 725 \, x^{3} + 540 \, x^{2} + 240 \, x + 64\right )} e^{x} + \frac {256}{125} \, x + e^{\left (4 \, x\right )} - \frac {1}{5} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 151, normalized size = 7.55 \begin {gather*} x^{8} + 4 \, x^{7} + 4 \, x^{6} e^{x} + \frac {46}{5} \, x^{6} + 12 \, x^{5} e^{x} + \frac {68}{5} \, x^{5} + 6 \, x^{4} e^{\left (2 \, x\right )} + \frac {108}{5} \, x^{4} e^{x} + \frac {361}{25} \, x^{4} + 12 \, x^{3} e^{\left (2 \, x\right )} + \frac {116}{5} \, x^{3} e^{x} + \frac {272}{25} \, x^{3} + 4 \, x^{2} e^{\left (3 \, x\right )} + \frac {78}{5} \, x^{2} e^{\left (2 \, x\right )} + \frac {432}{25} \, x^{2} e^{x} + \frac {736}{125} \, x^{2} + 4 \, x e^{\left (3 \, x\right )} + \frac {48}{5} \, x e^{\left (2 \, x\right )} + \frac {192}{25} \, x e^{x} + \frac {256}{125} \, x + e^{\left (4 \, x\right )} + \frac {16}{5} \, e^{\left (3 \, x\right )} + \frac {96}{25} \, e^{\left (2 \, x\right )} + \frac {256}{125} \, e^{x} - \frac {1}{5} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 122, normalized size = 6.10
method | result | size |
risch | \(\frac {736 x^{2}}{125}+\frac {256 x}{125}-\frac {\ln \relax (x )}{5}+\frac {272 x^{3}}{25}+\frac {361 x^{4}}{25}+\frac {68 x^{5}}{5}+\frac {46 x^{6}}{5}+4 x^{7}+x^{8}+{\mathrm e}^{4 x}+\frac {\left (500 x^{2}+500 x +400\right ) {\mathrm e}^{3 x}}{125}+\frac {\left (750 x^{4}+1500 x^{3}+1950 x^{2}+1200 x +480\right ) {\mathrm e}^{2 x}}{125}+\frac {\left (500 x^{6}+1500 x^{5}+2700 x^{4}+2900 x^{3}+2160 x^{2}+960 x +256\right ) {\mathrm e}^{x}}{125}\) | \(122\) |
default | \(\frac {256 x}{125}+{\mathrm e}^{4 x}+12 x^{5} {\mathrm e}^{x}+\frac {16 \,{\mathrm e}^{3 x}}{5}+\frac {96 \,{\mathrm e}^{2 x}}{25}+4 x^{7}+x^{8}-\frac {\ln \relax (x )}{5}+\frac {361 x^{4}}{25}+\frac {272 x^{3}}{25}+\frac {736 x^{2}}{125}+\frac {256 \,{\mathrm e}^{x}}{125}+\frac {46 x^{6}}{5}+\frac {68 x^{5}}{5}+4 x^{2} {\mathrm e}^{3 x}+12 \,{\mathrm e}^{2 x} x^{3}+\frac {78 \,{\mathrm e}^{2 x} x^{2}}{5}+\frac {48 x \,{\mathrm e}^{2 x}}{5}+4 x^{6} {\mathrm e}^{x}+4 x \,{\mathrm e}^{3 x}+\frac {108 \,{\mathrm e}^{x} x^{4}}{5}+\frac {192 \,{\mathrm e}^{x} x}{25}+\frac {432 \,{\mathrm e}^{x} x^{2}}{25}+\frac {116 \,{\mathrm e}^{x} x^{3}}{5}+6 \,{\mathrm e}^{2 x} x^{4}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 279, normalized size = 13.95 \begin {gather*} x^{8} + 4 \, x^{7} + \frac {46}{5} \, x^{6} + \frac {68}{5} \, x^{5} + \frac {361}{25} \, x^{4} + \frac {272}{25} \, x^{3} + \frac {736}{125} \, x^{2} + \frac {4}{9} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {20}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} + 3 \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} + 6 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + \frac {84}{5} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {63}{5} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} + 36 \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + \frac {408}{5} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + \frac {548}{5} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + \frac {2172}{25} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac {1056}{25} \, {\left (x - 1\right )} e^{x} + \frac {256}{125} \, x + e^{\left (4 \, x\right )} + \frac {68}{15} \, e^{\left (3 \, x\right )} + \frac {216}{25} \, e^{\left (2 \, x\right )} + \frac {1216}{125} \, e^{x} - \frac {1}{5} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 151, normalized size = 7.55 \begin {gather*} \frac {256\,x}{125}+\frac {96\,{\mathrm {e}}^{2\,x}}{25}+\frac {16\,{\mathrm {e}}^{3\,x}}{5}+{\mathrm {e}}^{4\,x}+\frac {256\,{\mathrm {e}}^x}{125}-\frac {\ln \relax (x)}{5}+\frac {48\,x\,{\mathrm {e}}^{2\,x}}{5}+4\,x\,{\mathrm {e}}^{3\,x}+\frac {432\,x^2\,{\mathrm {e}}^x}{25}+\frac {116\,x^3\,{\mathrm {e}}^x}{5}+\frac {108\,x^4\,{\mathrm {e}}^x}{5}+12\,x^5\,{\mathrm {e}}^x+4\,x^6\,{\mathrm {e}}^x+\frac {78\,x^2\,{\mathrm {e}}^{2\,x}}{5}+4\,x^2\,{\mathrm {e}}^{3\,x}+12\,x^3\,{\mathrm {e}}^{2\,x}+6\,x^4\,{\mathrm {e}}^{2\,x}+\frac {192\,x\,{\mathrm {e}}^x}{25}+\frac {736\,x^2}{125}+\frac {272\,x^3}{25}+\frac {361\,x^4}{25}+\frac {68\,x^5}{5}+\frac {46\,x^6}{5}+4\,x^7+x^8 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.27, size = 136, normalized size = 6.80 \begin {gather*} x^{8} + 4 x^{7} + \frac {46 x^{6}}{5} + \frac {68 x^{5}}{5} + \frac {361 x^{4}}{25} + \frac {272 x^{3}}{25} + \frac {736 x^{2}}{125} + \frac {256 x}{125} + \frac {\left (62500 x^{2} + 62500 x + 50000\right ) e^{3 x}}{15625} + \frac {\left (93750 x^{4} + 187500 x^{3} + 243750 x^{2} + 150000 x + 60000\right ) e^{2 x}}{15625} + \frac {\left (62500 x^{6} + 187500 x^{5} + 337500 x^{4} + 362500 x^{3} + 270000 x^{2} + 120000 x + 32000\right ) e^{x}}{15625} + e^{4 x} - \frac {\log {\relax (x )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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