Optimal. Leaf size=32 \[ e^{\frac {2}{-x+\frac {5 e^x}{3-(6-x) (3-\log (5))}}} \]
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Rubi [F] time = 69.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {-30+6 x+(12-2 x) \log (5)}{5 e^x+15 x-3 x^2+\left (-6 x+x^2\right ) \log (5)}\right ) \left (450-180 x+18 x^2+\left (-360+132 x-12 x^2\right ) \log (5)+\left (72-24 x+2 x^2\right ) \log ^2(5)+e^x (180-30 x+(-70+10 x) \log (5))\right )}{25 e^{2 x}+225 x^2-90 x^3+9 x^4+\left (-180 x^2+66 x^3-6 x^4\right ) \log (5)+\left (36 x^2-12 x^3+x^4\right ) \log ^2(5)+e^x \left (150 x-30 x^2+\left (-60 x+10 x^2\right ) \log (5)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2\ 5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) \left (5 e^x (18+x (-3+\log (5))-7 \log (5))+(15+x (-3+\log (5))-6 \log (5))^2\right )}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx\\ &=2 \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) \left (5 e^x (18+x (-3+\log (5))-7 \log (5))+(15+x (-3+\log (5))-6 \log (5))^2\right )}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx\\ &=2 \int \left (\frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) (18-x (3-\log (5))-7 \log (5))}{5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )}+\frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) \left (2 x^2 (18-7 \log (5)) (3-\log (5))-x^3 (3-\log (5))^2-9 x \left (40-31 \log (5)+6 \log ^2(5)\right )+9 (5-\log (25))^2\right )}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) (18-x (3-\log (5))-7 \log (5))}{5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )} \, dx+2 \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) \left (2 x^2 (18-7 \log (5)) (3-\log (5))-x^3 (3-\log (5))^2-9 x \left (40-31 \log (5)+6 \log ^2(5)\right )+9 (5-\log (25))^2\right )}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2} \, dx\\ &=2 \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) (18+x (-3+\log (5))-7 \log (5))}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))} \, dx+2 \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) \left (2 x^2 (18-7 \log (5)) (3-\log (5))-x^3 (3-\log (5))^2-9 x \left (40-31 \log (5)+6 \log ^2(5)\right )+9 (5-\log (25))^2\right )}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx\\ &=2 \int \left (\frac {18\ 5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) \left (1-\frac {7 \log (5)}{18}\right )}{5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )}+\frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x (-3+\log (5))}{5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )}\right ) \, dx+2 \int \left (\frac {2\ 5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x^2 (18-7 \log (5)) (3-\log (5))}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2}-\frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x^3 (3-\log (5))^2}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2}+\frac {9\ 5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) (5-\log (25))^2}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2}+\frac {9\ 5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x (5-\log (25)) (-8+\log (125))}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2}\right ) \, dx\\ &=(2 (18-7 \log (5))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right )}{5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )} \, dx+(4 (18-7 \log (5)) (3-\log (5))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x^2}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2} \, dx-\left (2 (3-\log (5))^2\right ) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x^3}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2} \, dx+(2 (-3+\log (5))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x}{5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )} \, dx+\left (18 (5-\log (25))^2\right ) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right )}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2} \, dx-(18 (5-\log (25)) (8-\log (125))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x}{\left (5 e^x+15 x \left (1-\frac {2 \log (5)}{5}\right )-3 x^2 \left (1-\frac {\log (5)}{3}\right )\right )^2} \, dx\\ &=(2 (18-7 \log (5))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right )}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))} \, dx+(4 (18-7 \log (5)) (3-\log (5))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x^2}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx-\left (2 (3-\log (5))^2\right ) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x^3}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx+(2 (-3+\log (5))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))} \, dx+\left (18 (5-\log (25))^2\right ) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right )}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx-(18 (5-\log (25)) (8-\log (125))) \int \frac {5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \exp \left (\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}\right ) x}{\left (5 e^x+x (15+x (-3+\log (5))-6 \log (5))\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.32, size = 59, normalized size = 1.84 \begin {gather*} 5^{-\frac {2 (-6+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} e^{\frac {6 (-5+x)}{5 e^x+x (15+x (-3+\log (5))-6 \log (5))}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 40, normalized size = 1.25 \begin {gather*} e^{\left (\frac {2 \, {\left ({\left (x - 6\right )} \log \relax (5) - 3 \, x + 15\right )}}{3 \, x^{2} - {\left (x^{2} - 6 \, x\right )} \log \relax (5) - 15 \, x - 5 \, e^{x}}\right )} \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*} \int \frac {2 \, {\left ({\left (x^{2} - 12 \, x + 36\right )} \log \relax (5)^{2} + 9 \, x^{2} + 5 \, {\left ({\left (x - 7\right )} \log \relax (5) - 3 \, x + 18\right )} e^{x} - 6 \, {\left (x^{2} - 11 \, x + 30\right )} \log \relax (5) - 90 \, x + 225\right )} e^{\left (\frac {2 \, {\left ({\left (x - 6\right )} \log \relax (5) - 3 \, x + 15\right )}}{3 \, x^{2} - {\left (x^{2} - 6 \, x\right )} \log \relax (5) - 15 \, x - 5 \, e^{x}}\right )}}{9 \, x^{4} - 90 \, x^{3} + {\left (x^{4} - 12 \, x^{3} + 36 \, x^{2}\right )} \log \relax (5)^{2} + 225 \, x^{2} - 10 \, {\left (3 \, x^{2} - {\left (x^{2} - 6 \, x\right )} \log \relax (5) - 15 \, x\right )} e^{x} - 6 \, {\left (x^{4} - 11 \, x^{3} + 30 \, x^{2}\right )} \log \relax (5) + 25 \, e^{\left (2 \, x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 43, normalized size = 1.34
method | result | size |
risch | \({\mathrm e}^{-\frac {2 \left (x \ln \relax (5)-6 \ln \relax (5)-3 x +15\right )}{x^{2} \ln \relax (5)-6 x \ln \relax (5)-3 x^{2}+5 \,{\mathrm e}^{x}+15 x}}\) | \(43\) |
norman | \(\frac {\left (-6 \ln \relax (5)+15\right ) x \,{\mathrm e}^{\frac {\left (-2 x +12\right ) \ln \relax (5)+6 x -30}{5 \,{\mathrm e}^{x}+\left (x^{2}-6 x \right ) \ln \relax (5)-3 x^{2}+15 x}}+\left (\ln \relax (5)-3\right ) x^{2} {\mathrm e}^{\frac {\left (-2 x +12\right ) \ln \relax (5)+6 x -30}{5 \,{\mathrm e}^{x}+\left (x^{2}-6 x \right ) \ln \relax (5)-3 x^{2}+15 x}}+5 \,{\mathrm e}^{x} {\mathrm e}^{\frac {\left (-2 x +12\right ) \ln \relax (5)+6 x -30}{5 \,{\mathrm e}^{x}+\left (x^{2}-6 x \right ) \ln \relax (5)-3 x^{2}+15 x}}}{x^{2} \ln \relax (5)-6 x \ln \relax (5)-3 x^{2}+5 \,{\mathrm e}^{x}+15 x}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.84, size = 112, normalized size = 3.50 \begin {gather*} e^{\left (-\frac {2 \, x \log \relax (5)}{x^{2} {\left (\log \relax (5) - 3\right )} - 3 \, x {\left (2 \, \log \relax (5) - 5\right )} + 5 \, e^{x}} + \frac {6 \, x}{x^{2} {\left (\log \relax (5) - 3\right )} - 3 \, x {\left (2 \, \log \relax (5) - 5\right )} + 5 \, e^{x}} + \frac {12 \, \log \relax (5)}{x^{2} {\left (\log \relax (5) - 3\right )} - 3 \, x {\left (2 \, \log \relax (5) - 5\right )} + 5 \, e^{x}} - \frac {30}{x^{2} {\left (\log \relax (5) - 3\right )} - 3 \, x {\left (2 \, \log \relax (5) - 5\right )} + 5 \, e^{x}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 92, normalized size = 2.88 \begin {gather*} {\left (\frac {1}{25}\right )}^{\frac {x-6}{15\,x+5\,{\mathrm {e}}^x-6\,x\,\ln \relax (5)+x^2\,\ln \relax (5)-3\,x^2}}\,{\mathrm {e}}^{\frac {6\,x}{15\,x+5\,{\mathrm {e}}^x-6\,x\,\ln \relax (5)+x^2\,\ln \relax (5)-3\,x^2}}\,{\mathrm {e}}^{-\frac {30}{15\,x+5\,{\mathrm {e}}^x-6\,x\,\ln \relax (5)+x^2\,\ln \relax (5)-3\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 37, normalized size = 1.16 \begin {gather*} e^{\frac {6 x + \left (12 - 2 x\right ) \log {\relax (5 )} - 30}{- 3 x^{2} + 15 x + \left (x^{2} - 6 x\right ) \log {\relax (5 )} + 5 e^{x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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