Optimal. Leaf size=34 \[ -2-x-5 (1+x)+\frac {2 x}{-3+2 x}+\frac {-2+2 x}{\log \left (\frac {x}{5}\right )} \]
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Rubi [A] time = 0.44, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {1594, 27, 6688, 12, 683, 2330, 2297, 2298, 2302, 30} \begin {gather*} -6 x-\frac {3}{3-2 x}+\frac {2 x}{\log \left (\frac {x}{5}\right )}-\frac {2}{\log \left (\frac {x}{5}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 30
Rule 683
Rule 1594
Rule 2297
Rule 2298
Rule 2302
Rule 2330
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18-42 x+32 x^2-8 x^3+\left (18 x-24 x^2+8 x^3\right ) \log \left (\frac {x}{5}\right )+\left (-60 x+72 x^2-24 x^3\right ) \log ^2\left (\frac {x}{5}\right )}{x \left (9-12 x+4 x^2\right ) \log ^2\left (\frac {x}{5}\right )} \, dx\\ &=\int \frac {18-42 x+32 x^2-8 x^3+\left (18 x-24 x^2+8 x^3\right ) \log \left (\frac {x}{5}\right )+\left (-60 x+72 x^2-24 x^3\right ) \log ^2\left (\frac {x}{5}\right )}{x (-3+2 x)^2 \log ^2\left (\frac {x}{5}\right )} \, dx\\ &=\int 2 \left (-\frac {6 \left (5-6 x+2 x^2\right )}{(3-2 x)^2}+\frac {-1+\frac {1}{x}}{\log ^2\left (\frac {x}{5}\right )}+\frac {1}{\log \left (\frac {x}{5}\right )}\right ) \, dx\\ &=2 \int \left (-\frac {6 \left (5-6 x+2 x^2\right )}{(3-2 x)^2}+\frac {-1+\frac {1}{x}}{\log ^2\left (\frac {x}{5}\right )}+\frac {1}{\log \left (\frac {x}{5}\right )}\right ) \, dx\\ &=2 \int \frac {-1+\frac {1}{x}}{\log ^2\left (\frac {x}{5}\right )} \, dx+2 \int \frac {1}{\log \left (\frac {x}{5}\right )} \, dx-12 \int \frac {5-6 x+2 x^2}{(3-2 x)^2} \, dx\\ &=10 \text {li}\left (\frac {x}{5}\right )+2 \int \left (-\frac {1}{\log ^2\left (\frac {x}{5}\right )}+\frac {1}{x \log ^2\left (\frac {x}{5}\right )}\right ) \, dx-12 \int \left (\frac {1}{2}+\frac {1}{2 (-3+2 x)^2}\right ) \, dx\\ &=-\frac {3}{3-2 x}-6 x+10 \text {li}\left (\frac {x}{5}\right )-2 \int \frac {1}{\log ^2\left (\frac {x}{5}\right )} \, dx+2 \int \frac {1}{x \log ^2\left (\frac {x}{5}\right )} \, dx\\ &=-\frac {3}{3-2 x}-6 x+\frac {2 x}{\log \left (\frac {x}{5}\right )}+10 \text {li}\left (\frac {x}{5}\right )-2 \int \frac {1}{\log \left (\frac {x}{5}\right )} \, dx+2 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {x}{5}\right )\right )\\ &=-\frac {3}{3-2 x}-6 x-\frac {2}{\log \left (\frac {x}{5}\right )}+\frac {2 x}{\log \left (\frac {x}{5}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [C] time = 0.13, size = 60, normalized size = 1.76 \begin {gather*} 2 \left (-\frac {3}{2 (3-2 x)}-\frac {3}{2} (-3+2 x)-5 \text {Ei}\left (\log \left (\frac {x}{5}\right )\right )-\frac {1}{\log \left (\frac {x}{5}\right )}+\frac {x}{\log \left (\frac {x}{5}\right )}+5 \text {li}\left (\frac {x}{5}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 40, normalized size = 1.18 \begin {gather*} \frac {4 \, x^{2} - 3 \, {\left (4 \, x^{2} - 6 \, x - 1\right )} \log \left (\frac {1}{5} \, x\right ) - 10 \, x + 6}{{\left (2 \, x - 3\right )} \log \left (\frac {1}{5} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 24, normalized size = 0.71 \begin {gather*} -6 \, x + \frac {2 \, {\left (x - 1\right )}}{\log \left (\frac {1}{5} \, x\right )} + \frac {3}{2 \, x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 0.91
method | result | size |
derivativedivides | \(-6 x +\frac {3}{2 x -3}+\frac {2 x}{\ln \left (\frac {x}{5}\right )}-\frac {2}{\ln \left (\frac {x}{5}\right )}\) | \(31\) |
default | \(-6 x +\frac {3}{2 x -3}+\frac {2 x}{\ln \left (\frac {x}{5}\right )}-\frac {2}{\ln \left (\frac {x}{5}\right )}\) | \(31\) |
risch | \(-\frac {3 \left (4 x^{2}-6 x -1\right )}{2 x -3}+\frac {2 x -2}{\ln \left (\frac {x}{5}\right )}\) | \(32\) |
norman | \(\frac {6+30 \ln \left (\frac {x}{5}\right )-10 x +4 x^{2}-12 \ln \left (\frac {x}{5}\right ) x^{2}}{\left (2 x -3\right ) \ln \left (\frac {x}{5}\right )}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 63, normalized size = 1.85 \begin {gather*} -\frac {4 \, x^{2} {\left (3 \, \log \relax (5) + 1\right )} - 2 \, x {\left (9 \, \log \relax (5) + 5\right )} - 3 \, {\left (4 \, x^{2} - 6 \, x - 1\right )} \log \relax (x) - 3 \, \log \relax (5) + 6}{2 \, x \log \relax (5) - {\left (2 \, x - 3\right )} \log \relax (x) - 3 \, \log \relax (5)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.80, size = 42, normalized size = 1.24 \begin {gather*} \frac {20\,x-12\,x^2}{2\,x-3}+\frac {4\,x^2-10\,x+6}{\ln \left (\frac {x}{5}\right )\,\left (2\,x-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 0.56 \begin {gather*} - 6 x + \frac {2 x - 2}{\log {\left (\frac {x}{5} \right )}} + \frac {3}{2 x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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