Optimal. Leaf size=30 \[ 1+\frac {5 x \left (2 x-\frac {5 x}{1-x+4 \log (x)}\right )}{\log (x+\log (5))} \]
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Rubi [F] time = 5.20, 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 {15 x^2-5 x^3-10 x^4+\left (20 x^2+80 x^3\right ) \log (x)-160 x^2 \log ^2(x)+\left (70 x^2-15 x^3+20 x^4+\left (70 x-15 x^2+20 x^3\right ) \log (5)+\left (-40 x^2-160 x^3+\left (-40 x-160 x^2\right ) \log (5)\right ) \log (x)+\left (320 x^2+320 x \log (5)\right ) \log ^2(x)\right ) \log (x+\log (5))}{\left (x-2 x^2+x^3+\left (1-2 x+x^2\right ) \log (5)+\left (8 x-8 x^2+(8-8 x) \log (5)\right ) \log (x)+(16 x+16 \log (5)) \log ^2(x)\right ) \log ^2(x+\log (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 {5 x \left (-x \left (-3+x+2 x^2\right )+\left (14-3 x+4 x^2\right ) (x+\log (5)) \log (x+\log (5))-4 (1+4 x) \log (x) (-x+2 (x+\log (5)) \log (x+\log (5)))+32 \log ^2(x) (-x+2 (x+\log (5)) \log (x+\log (5)))\right )}{(x+\log (5)) (1-x+4 \log (x))^2 \log ^2(x+\log (5))} \, dx\\ &=5 \int \frac {x \left (-x \left (-3+x+2 x^2\right )+\left (14-3 x+4 x^2\right ) (x+\log (5)) \log (x+\log (5))-4 (1+4 x) \log (x) (-x+2 (x+\log (5)) \log (x+\log (5)))+32 \log ^2(x) (-x+2 (x+\log (5)) \log (x+\log (5)))\right )}{(x+\log (5)) (1-x+4 \log (x))^2 \log ^2(x+\log (5))} \, dx\\ &=5 \int \left (-\frac {x^2 (3+2 x-8 \log (x))}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))}+\frac {x \left (14-3 x+4 x^2-8 \log (x)-32 x \log (x)+64 \log ^2(x)\right )}{(-1+x-4 \log (x))^2 \log (x+\log (5))}\right ) \, dx\\ &=-\left (5 \int \frac {x^2 (3+2 x-8 \log (x))}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\right )+5 \int \frac {x \left (14-3 x+4 x^2-8 \log (x)-32 x \log (x)+64 \log ^2(x)\right )}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx\\ &=-\left (5 \int \left (\frac {x (3+2 x-8 \log (x))}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}-\frac {\log (5) (3+2 x-8 \log (x))}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}+\frac {\log ^2(5) (3+2 x-8 \log (x))}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))}\right ) \, dx\right )+5 \int \left (\frac {14 x}{(-1+x-4 \log (x))^2 \log (x+\log (5))}-\frac {3 x^2}{(-1+x-4 \log (x))^2 \log (x+\log (5))}+\frac {4 x^3}{(-1+x-4 \log (x))^2 \log (x+\log (5))}-\frac {8 x \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))}-\frac {32 x^2 \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))}+\frac {64 x \log ^2(x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))}\right ) \, dx\\ &=-\left (5 \int \frac {x (3+2 x-8 \log (x))}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\right )-15 \int \frac {x^2}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+20 \int \frac {x^3}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-40 \int \frac {x \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+70 \int \frac {x}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-160 \int \frac {x^2 \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+320 \int \frac {x \log ^2(x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+(5 \log (5)) \int \frac {3+2 x-8 \log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-\left (5 \log ^2(5)\right ) \int \frac {3+2 x-8 \log (x)}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\\ &=-\left (5 \int \left (\frac {3 x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}+\frac {2 x^2}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}-\frac {8 x \log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}\right ) \, dx\right )-15 \int \frac {x^2}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+20 \int \frac {x^3}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-40 \int \frac {x \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+70 \int \frac {x}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-160 \int \frac {x^2 \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+320 \int \frac {x \log ^2(x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+(5 \log (5)) \int \left (\frac {3}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}+\frac {2 x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}-\frac {8 \log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}\right ) \, dx-\left (5 \log ^2(5)\right ) \int \left (\frac {3}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))}+\frac {2 x}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))}-\frac {8 \log (x)}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))}\right ) \, dx\\ &=-\left (10 \int \frac {x^2}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\right )-15 \int \frac {x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-15 \int \frac {x^2}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+20 \int \frac {x^3}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+40 \int \frac {x \log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-40 \int \frac {x \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+70 \int \frac {x}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-160 \int \frac {x^2 \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+320 \int \frac {x \log ^2(x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+(10 \log (5)) \int \frac {x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+(15 \log (5)) \int \frac {1}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-(40 \log (5)) \int \frac {\log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-\left (10 \log ^2(5)\right ) \int \frac {x}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-\left (15 \log ^2(5)\right ) \int \frac {1}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+\left (40 \log ^2(5)\right ) \int \frac {\log (x)}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\\ &=-\left (10 \int \frac {x^2}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\right )-15 \int \frac {x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-15 \int \frac {x^2}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+20 \int \frac {x^3}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+40 \int \frac {x \log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-40 \int \frac {x \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+70 \int \frac {x}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-160 \int \frac {x^2 \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+320 \int \frac {x \log ^2(x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+(10 \log (5)) \int \frac {x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+(15 \log (5)) \int \frac {1}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-(40 \log (5)) \int \frac {\log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-\left (10 \log ^2(5)\right ) \int \left (\frac {1}{(-1+x-4 \log (x)) \log ^2(x+\log (5))}-\frac {\log (5)}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))}\right ) \, dx-\left (15 \log ^2(5)\right ) \int \frac {1}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+\left (40 \log ^2(5)\right ) \int \frac {\log (x)}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\\ &=-\left (10 \int \frac {x^2}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\right )-15 \int \frac {x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-15 \int \frac {x^2}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+20 \int \frac {x^3}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+40 \int \frac {x \log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-40 \int \frac {x \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+70 \int \frac {x}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx-160 \int \frac {x^2 \log (x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+320 \int \frac {x \log ^2(x)}{(-1+x-4 \log (x))^2 \log (x+\log (5))} \, dx+(10 \log (5)) \int \frac {x}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+(15 \log (5)) \int \frac {1}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-(40 \log (5)) \int \frac {\log (x)}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-\left (10 \log ^2(5)\right ) \int \frac {1}{(-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx-\left (15 \log ^2(5)\right ) \int \frac {1}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+\left (40 \log ^2(5)\right ) \int \frac {\log (x)}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx+\left (10 \log ^3(5)\right ) \int \frac {1}{(x+\log (5)) (-1+x-4 \log (x)) \log ^2(x+\log (5))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 30, normalized size = 1.00 \begin {gather*} \frac {5 x^2 (3+2 x-8 \log (x))}{(-1+x-4 \log (x)) \log (x+\log (5))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 36, normalized size = 1.20 \begin {gather*} \frac {5 \, {\left (2 \, x^{3} - 8 \, x^{2} \log \relax (x) + 3 \, x^{2}\right )}}{{\left (x - 4 \, \log \relax (x) - 1\right )} \log \left (x + \log \relax (5)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 46, normalized size = 1.53 \begin {gather*} \frac {5 \, {\left (2 \, x^{3} - 8 \, x^{2} \log \relax (x) + 3 \, x^{2}\right )}}{x \log \left (x + \log \relax (5)\right ) - 4 \, \log \left (x + \log \relax (5)\right ) \log \relax (x) - \log \left (x + \log \relax (5)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 31, normalized size = 1.03
method | result | size |
risch | \(\frac {5 \left (2 x -8 \ln \relax (x )+3\right ) x^{2}}{\left (x -4 \ln \relax (x )-1\right ) \ln \left (\ln \relax (5)+x \right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 36, normalized size = 1.20 \begin {gather*} \frac {5 \, {\left (2 \, x^{3} - 8 \, x^{2} \log \relax (x) + 3 \, x^{2}\right )}}{{\left (x - 4 \, \log \relax (x) - 1\right )} \log \left (x + \log \relax (5)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int -\frac {\ln \relax (x)\,\left (80\,x^3+20\,x^2\right )+\ln \left (x+\ln \relax (5)\right )\,\left (\ln \relax (5)\,\left (20\,x^3-15\,x^2+70\,x\right )-\ln \relax (x)\,\left (\ln \relax (5)\,\left (160\,x^2+40\,x\right )+40\,x^2+160\,x^3\right )+{\ln \relax (x)}^2\,\left (320\,x^2+320\,\ln \relax (5)\,x\right )+70\,x^2-15\,x^3+20\,x^4\right )-160\,x^2\,{\ln \relax (x)}^2+15\,x^2-5\,x^3-10\,x^4}{{\ln \left (x+\ln \relax (5)\right )}^2\,\left (x+{\ln \relax (x)}^2\,\left (16\,x+16\,\ln \relax (5)\right )-\ln \relax (x)\,\left (\ln \relax (5)\,\left (8\,x-8\right )-8\,x+8\,x^2\right )-2\,x^2+x^3+\ln \relax (5)\,\left (x^2-2\,x+1\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 32, normalized size = 1.07 \begin {gather*} \frac {10 x^{3} - 40 x^{2} \log {\relax (x )} + 15 x^{2}}{\left (x - 4 \log {\relax (x )} - 1\right ) \log {\left (x + \log {\relax (5 )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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