Optimal. Leaf size=19 \[ 3 \left (\frac {5625 x^2}{(1+x)^2}-\log (x)\right )^2 \]
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Rubi [B] time = 0.43, antiderivative size = 58, normalized size of antiderivative = 3.05, number of steps used = 14, number of rules used = 10, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6688, 12, 6742, 1620, 2357, 2301, 2319, 44, 2314, 31} \begin {gather*} -\frac {379687500}{x+1}+\frac {569531250}{(x+1)^2}-\frac {379687500}{(x+1)^3}+\frac {94921875}{(x+1)^4}+3 \log ^2(x)-\frac {67500 x \log (x)}{x+1}-\frac {33750 \log (x)}{(x+1)^2}+33750 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 44
Rule 1620
Rule 2301
Rule 2314
Rule 2319
Rule 2357
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \left (1+3 x-11247 x^2+x^3\right ) \left (-5625 x^2+(1+x)^2 \log (x)\right )}{x (1+x)^5} \, dx\\ &=6 \int \frac {\left (1+3 x-11247 x^2+x^3\right ) \left (-5625 x^2+(1+x)^2 \log (x)\right )}{x (1+x)^5} \, dx\\ &=6 \int \left (-\frac {5625 x \left (1+3 x-11247 x^2+x^3\right )}{(1+x)^5}+\frac {\left (1+3 x-11247 x^2+x^3\right ) \log (x)}{x (1+x)^3}\right ) \, dx\\ &=6 \int \frac {\left (1+3 x-11247 x^2+x^3\right ) \log (x)}{x (1+x)^3} \, dx-33750 \int \frac {x \left (1+3 x-11247 x^2+x^3\right )}{(1+x)^5} \, dx\\ &=6 \int \left (\frac {\log (x)}{x}+\frac {11250 \log (x)}{(1+x)^3}-\frac {11250 \log (x)}{(1+x)^2}\right ) \, dx-33750 \int \left (\frac {11250}{(1+x)^5}-\frac {33750}{(1+x)^4}+\frac {33750}{(1+x)^3}-\frac {11251}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-33750 \log (1+x)+6 \int \frac {\log (x)}{x} \, dx+67500 \int \frac {\log (x)}{(1+x)^3} \, dx-67500 \int \frac {\log (x)}{(1+x)^2} \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)-33750 \log (1+x)+33750 \int \frac {1}{x (1+x)^2} \, dx+67500 \int \frac {1}{1+x} \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379721250}{1+x}-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)+33750 \log (1+x)+33750 \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx\\ &=\frac {94921875}{(1+x)^4}-\frac {379687500}{(1+x)^3}+\frac {569531250}{(1+x)^2}-\frac {379687500}{1+x}+33750 \log (x)-\frac {33750 \log (x)}{(1+x)^2}-\frac {67500 x \log (x)}{1+x}+3 \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.07, size = 41, normalized size = 2.16 \begin {gather*} 3 \left (-\frac {31640625 \left (1+4 x+6 x^2+4 x^3\right )}{(1+x)^4}-\frac {11250 x^2 \log (x)}{(1+x)^2}+\log ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 77, normalized size = 4.05 \begin {gather*} -\frac {3 \, {\left (126562500 \, x^{3} - {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )} \log \relax (x)^{2} + 189843750 \, x^{2} + 11250 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (x) + 126562500 \, x + 31640625\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \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 {6 \, {\left (5625 \, x^{5} - 63264375 \, x^{4} + 16875 \, x^{3} + 5625 \, x^{2} - {\left (x^{5} - 11245 \, x^{4} - 22490 \, x^{3} - 11240 \, x^{2} + 5 \, x + 1\right )} \log \relax (x)\right )}}{x^{6} + 5 \, x^{5} + 10 \, x^{4} + 10 \, x^{3} + 5 \, x^{2} + x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 59, normalized size = 3.11
method | result | size |
default | \(\frac {569531250}{\left (x +1\right )^{2}}+\frac {94921875}{\left (x +1\right )^{4}}-\frac {379687500}{x +1}-\frac {379687500}{\left (x +1\right )^{3}}+3 \ln \relax (x )^{2}+\frac {33750 \ln \relax (x ) x \left (2+x \right )}{\left (x +1\right )^{2}}-\frac {67500 \ln \relax (x ) x}{x +1}\) | \(59\) |
norman | \(\frac {5625 \ln \relax (x )+94921875 x^{4}-45000 x^{3} \ln \relax (x )-28125 x^{4} \ln \relax (x )+22500 x \ln \relax (x )+3 \ln \relax (x )^{2}+12 x \ln \relax (x )^{2}+18 x^{2} \ln \relax (x )^{2}+12 x^{3} \ln \relax (x )^{2}+3 x^{4} \ln \relax (x )^{2}}{\left (x +1\right )^{4}}-5625 \ln \relax (x )\) | \(81\) |
risch | \(3 \ln \relax (x )^{2}+\frac {33750 \left (2 x +1\right ) \ln \relax (x )}{x^{2}+2 x +1}-\frac {16875 \left (2 x^{4} \ln \relax (x )+8 x^{3} \ln \relax (x )+12 x^{2} \ln \relax (x )+22500 x^{3}+8 x \ln \relax (x )+33750 x^{2}+2 \ln \relax (x )+22500 x +5625\right )}{\left (x^{2}+2 x +1\right )^{2}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.78, size = 172, normalized size = 9.05 \begin {gather*} -\frac {5625 \, {\left (48 \, x^{3} + 108 \, x^{2} + 88 \, x + 25\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} - \frac {189793125 \, {\left (4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} + \frac {16875 \, {\left (6 \, x^{2} + 4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} - \frac {3 \, {\left (11250 \, x^{2} \log \relax (x) - {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} - 11250 \, x - 11250\right )}}{x^{2} + 2 \, x + 1} + \frac {5625 \, {\left (4 \, x + 1\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 28, normalized size = 1.47 \begin {gather*} \frac {3\,{\left (\ln \relax (x)+x^2\,\ln \relax (x)+2\,x\,\ln \relax (x)-5625\,x^2\right )}^2}{{\left (x+1\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 61, normalized size = 3.21 \begin {gather*} \frac {\left (67500 x + 33750\right ) \log {\relax (x )}}{x^{2} + 2 x + 1} - \frac {379687500 x^{3} + 569531250 x^{2} + 379687500 x + 94921875}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1} + 3 \log {\relax (x )}^{2} - 33750 \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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