Optimal. Leaf size=24 \[ e^{\frac {9}{2 (-10+x) \left (2-\frac {x^2}{\log (5)}\right )}} \]
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Rubi [F] time = 2.36, 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 {5^{\frac {9}{20 x^2-2 x^3+(-40+4 x) \log (5)}} \left (\left (-180 x+27 x^2\right ) \log (5)-18 \log ^2(5)\right )}{200 x^4-40 x^5+2 x^6+\left (-800 x^2+160 x^3-8 x^4\right ) \log (5)+\left (800-160 x+8 x^2\right ) \log ^2(5)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9\ 5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \left (-20 x+3 x^2-2 \log (5)\right ) \log (5)}{2 (10-x)^2 \left (x^2-2 \log (5)\right )^2} \, dx\\ &=\frac {1}{2} (9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \left (-20 x+3 x^2-2 \log (5)\right )}{(10-x)^2 \left (x^2-2 \log (5)\right )^2} \, dx\\ &=\frac {1}{2} (9 \log (5)) \int \left (-\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 (-10+x)^2 (-50+\log (5))}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \left (x^2-2 \log (5)\right ) (-50+\log (5))}+\frac {2\ 5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} (5 x+\log (5))}{\left (x^2-2 \log (5)\right )^2 (-50+\log (5))}\right ) \, dx\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x^2-2 \log (5)} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} (5 x+\log (5))}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \left (-\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \sqrt {\log (25)}}{4 \log (5) \left (-x+\sqrt {\log (25)}\right )}-\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \sqrt {\log (25)}}{4 \log (5) \left (x+\sqrt {\log (25)}\right )}\right ) \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \left (\frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \log (5)}{\left (x^2-2 \log (5)\right )^2}\right ) \, dx}{50-\log (5)}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \left (\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{4 \left (-x+\sqrt {\log (25)}\right )^2 \log (25)}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{4 \left (x+\sqrt {\log (25)}\right )^2 \log (25)}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \log (25) \left (-x^2+\log (25)\right )}\right ) \, dx}{50-\log (5)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (-x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x^2+\log (25)} \, dx}{2 (50-\log (5)) \log (25)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (-x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \left (\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \left (-x+\sqrt {\log (25)}\right ) \sqrt {\log (25)}}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \left (x+\sqrt {\log (25)}\right ) \sqrt {\log (25)}}\right ) \, dx}{2 (50-\log (5)) \log (25)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{4 (50-\log (5)) \log ^{\frac {3}{2}}(25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{4 (50-\log (5)) \log ^{\frac {3}{2}}(25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (-x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.40, size = 21, normalized size = 0.88 \begin {gather*} 5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{5^{\frac {9}{2 \, {\left (x^{3} - 10 \, x^{2} - 2 \, {\left (x - 10\right )} \log \relax (5)\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 {9 \, {\left ({\left (3 \, x^{2} - 20 \, x\right )} \log \relax (5) - 2 \, \log \relax (5)^{2}\right )}}{2 \, {\left (x^{6} - 20 \, x^{5} + 100 \, x^{4} + 4 \, {\left (x^{2} - 20 \, x + 100\right )} \log \relax (5)^{2} - 4 \, {\left (x^{4} - 20 \, x^{3} + 100 \, x^{2}\right )} \log \relax (5)\right )} 5^{\frac {9}{2 \, {\left (x^{3} - 10 \, x^{2} - 2 \, {\left (x - 10\right )} \log \relax (5)\right )}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 22, normalized size = 0.92
method | result | size |
risch | \(5^{\frac {9}{2 \left (x -10\right ) \left (-x^{2}+2 \ln \relax (5)\right )}}\) | \(22\) |
gosper | \({\mathrm e}^{\frac {9 \ln \relax (5)}{2 \left (-x^{3}+2 x \ln \relax (5)+10 x^{2}-20 \ln \relax (5)\right )}}\) | \(28\) |
norman | \(\frac {10 x^{2} {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}-x^{3} {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}-20 \ln \relax (5) {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}+2 x \ln \relax (5) {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}}{\left (x -10\right ) \left (-x^{2}+2 \ln \relax (5)\right )}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 71, normalized size = 2.96 \begin {gather*} e^{\left (-\frac {9 \, x \log \relax (5)}{4 \, {\left (x^{2} {\left (\log \relax (5) - 50\right )} - 2 \, \log \relax (5)^{2} + 100 \, \log \relax (5)\right )}} - \frac {45 \, \log \relax (5)}{2 \, {\left (x^{2} {\left (\log \relax (5) - 50\right )} - 2 \, \log \relax (5)^{2} + 100 \, \log \relax (5)\right )}} + \frac {9 \, \log \relax (5)}{4 \, {\left (x {\left (\log \relax (5) - 50\right )} - 10 \, \log \relax (5) + 500\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 28, normalized size = 1.17 \begin {gather*} \frac {1}{5^{\frac {9}{2\,x^3-20\,x^2-4\,\ln \relax (5)\,x+40\,\ln \relax (5)}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 24, normalized size = 1.00 \begin {gather*} e^{\frac {9 \log {\relax (5 )}}{- 2 x^{3} + 20 x^{2} + \left (4 x - 40\right ) \log {\relax (5 )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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