Optimal. Leaf size=17 \[ \frac {2 x^2}{10+\frac {4}{x+\log (400)}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(17)=34\).
time = 0.05, antiderivative size = 36, normalized size of antiderivative = 2.12, number of steps
used = 4, number of rules used = 3, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2011, 27, 1864}
\begin {gather*} \frac {x^2}{5}-\frac {2 x}{25}-\frac {2 (2+5 \log (400))^2}{125 (5 x+2+5 \log (400))} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1864
Rule 2011
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 x^2+10 x^3+\left (4 x+20 x^2\right ) \log (400)+10 x \log ^2(400)}{25 x^2+10 x (2+5 \log (400))+(2+5 \log (400))^2} \, dx\\ &=\int \frac {6 x^2+10 x^3+\left (4 x+20 x^2\right ) \log (400)+10 x \log ^2(400)}{(2+5 x+5 \log (400))^2} \, dx\\ &=\int \left (-\frac {2}{25}+\frac {2 x}{5}+\frac {2 (2+5 \log (400))^2}{25 (2+5 x+5 \log (400))^2}\right ) \, dx\\ &=-\frac {2 x}{25}+\frac {x^2}{5}-\frac {2 (2+5 \log (400))^2}{125 (2+5 x+5 \log (400))}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(17)=34\).
time = 0.03, size = 59, normalized size = 3.47 \begin {gather*} \frac {125 x^3+125 x^2 \log (400)-(2+5 \log (400))^2 (6+5 \log (400))-5 x \left (12+40 \log (400)+25 \log ^2(400)\right )}{125 (2+5 x+5 \log (400))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.61, size = 35, normalized size = 2.06
method | result | size |
gosper | \(\frac {x^{2} \left (x +2 \ln \left (20\right )\right )}{10 \ln \left (20\right )+5 x +2}\) | \(22\) |
norman | \(\frac {x^{3}+2 x^{2} \ln \left (20\right )}{10 \ln \left (20\right )+5 x +2}\) | \(24\) |
default | \(-\frac {2 x}{25}+\frac {x^{2}}{5}-\frac {2 \left (\frac {4 \ln \left (20\right )^{2}}{5}+\frac {8 \ln \left (20\right )}{25}+\frac {4}{125}\right )}{10 \ln \left (20\right )+5 x +2}\) | \(35\) |
risch | \(\frac {x^{2}}{5}-\frac {2 x}{25}-\frac {8 \ln \left (2\right )^{2}}{25 \left (\ln \left (2\right )+\frac {\ln \left (5\right )}{2}+\frac {x}{4}+\frac {1}{10}\right )}-\frac {8 \ln \left (2\right ) \ln \left (5\right )}{25 \left (\ln \left (2\right )+\frac {\ln \left (5\right )}{2}+\frac {x}{4}+\frac {1}{10}\right )}-\frac {2 \ln \left (5\right )^{2}}{25 \left (\ln \left (2\right )+\frac {\ln \left (5\right )}{2}+\frac {x}{4}+\frac {1}{10}\right )}-\frac {8 \ln \left (2\right )}{125 \left (\ln \left (2\right )+\frac {\ln \left (5\right )}{2}+\frac {x}{4}+\frac {1}{10}\right )}-\frac {4 \ln \left (5\right )}{125 \left (\ln \left (2\right )+\frac {\ln \left (5\right )}{2}+\frac {x}{4}+\frac {1}{10}\right )}-\frac {2}{625 \left (\ln \left (2\right )+\frac {\ln \left (5\right )}{2}+\frac {x}{4}+\frac {1}{10}\right )}\) | \(116\) |
meijerg | \(\frac {16 \left (5 \ln \left (20\right )+1\right )^{3} \left (-\frac {5 x \left (-\frac {25 x^{2}}{2 \left (5 \ln \left (20\right )+1\right )^{2}}+\frac {15 x}{5 \ln \left (20\right )+1}+12\right )}{8 \left (5 \ln \left (20\right )+1\right ) \left (1+\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right )}\right )}+3 \ln \left (1+\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right )}\right )\right )}{625 \left (2 \ln \left (20\right )+\frac {2}{5}\right )}+\left (\frac {8 \ln \left (20\right )^{2}}{5}+\frac {8 \ln \left (20\right )}{25}\right ) \left (-\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right ) \left (1+\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right )}\right )}+\ln \left (1+\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right )}\right )\right )+\frac {4 \left (5 \ln \left (20\right )+1\right )^{2} \left (\frac {8 \ln \left (20\right )}{5}+\frac {6}{25}\right ) \left (\frac {5 x \left (\frac {15 x}{2 \left (5 \ln \left (20\right )+1\right )}+6\right )}{6 \left (5 \ln \left (20\right )+1\right ) \left (1+\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right )}\right )}-2 \ln \left (1+\frac {5 x}{2 \left (5 \ln \left (20\right )+1\right )}\right )\right )}{25 \left (2 \ln \left (20\right )+\frac {2}{5}\right )}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{5} \, x^{2} - \frac {2}{25} \, x - \frac {8 \, {\left (25 \, \log \left (20\right )^{2} + 10 \, \log \left (20\right ) + 1\right )}}{125 \, {\left (5 \, x + 10 \, \log \left (20\right ) + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (18) = 36\).
time = 0.37, size = 43, normalized size = 2.53 \begin {gather*} \frac {125 \, x^{3} + 10 \, {\left (25 \, x^{2} - 10 \, x - 8\right )} \log \left (20\right ) - 200 \, \log \left (20\right )^{2} - 20 \, x - 8}{125 \, {\left (5 \, x + 10 \, \log \left (20\right ) + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (14) = 28\).
time = 0.13, size = 34, normalized size = 2.00 \begin {gather*} \frac {x^{2}}{5} - \frac {2 x}{25} + \frac {- 200 \log {\left (20 \right )}^{2} - 80 \log {\left (20 \right )} - 8}{625 x + 250 + 1250 \log {\left (20 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{5} \, x^{2} - \frac {2}{25} \, x - \frac {8 \, {\left (25 \, \log \left (20\right )^{2} + 10 \, \log \left (20\right ) + 1\right )}}{125 \, {\left (5 \, x + 10 \, \log \left (20\right ) + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 34, normalized size = 2.00 \begin {gather*} \frac {x^2}{5}-\frac {16\,\ln \left (20\right )+40\,{\ln \left (20\right )}^2+\frac {8}{5}}{125\,x+250\,\ln \left (20\right )+50}-\frac {2\,x}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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