Optimal. Leaf size=22 \[ 25 (1+x) \left (-1+\frac {2}{1+x}+\log (5-x)\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(22)=44\).
time = 0.22, antiderivative size = 57, normalized size of antiderivative = 2.59, number of steps
used = 15, number of rules used = 9, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6874, 46, 78,
90, 2437, 2338, 2436, 2333, 2332} \begin {gather*} 25 x+\frac {100}{x+1}-25 (5-x) \log ^2(5-x)+150 \log ^2(5-x)+50 (5-x) \log (5-x)-200 \log (5-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 78
Rule 90
Rule 2332
Rule 2333
Rule 2338
Rule 2436
Rule 2437
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {425}{(-5+x) (1+x)^2}-\frac {275 x}{(-5+x) (1+x)^2}-\frac {125 x^2}{(-5+x) (1+x)^2}-\frac {25 x^3}{(-5+x) (1+x)^2}+\frac {300 \log (5-x)}{-5+x}+25 \log ^2(5-x)\right ) \, dx\\ &=-\left (25 \int \frac {x^3}{(-5+x) (1+x)^2} \, dx\right )+25 \int \log ^2(5-x) \, dx-125 \int \frac {x^2}{(-5+x) (1+x)^2} \, dx-275 \int \frac {x}{(-5+x) (1+x)^2} \, dx+300 \int \frac {\log (5-x)}{-5+x} \, dx+425 \int \frac {1}{(-5+x) (1+x)^2} \, dx\\ &=-\left (25 \int \left (1+\frac {125}{36 (-5+x)}+\frac {1}{6 (1+x)^2}-\frac {17}{36 (1+x)}\right ) \, dx\right )-25 \text {Subst}\left (\int \log ^2(x) \, dx,x,5-x\right )-125 \int \left (\frac {25}{36 (-5+x)}-\frac {1}{6 (1+x)^2}+\frac {11}{36 (1+x)}\right ) \, dx-275 \int \left (\frac {5}{36 (-5+x)}+\frac {1}{6 (1+x)^2}-\frac {5}{36 (1+x)}\right ) \, dx+300 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,5-x\right )+425 \int \left (\frac {1}{36 (-5+x)}-\frac {1}{6 (1+x)^2}-\frac {1}{36 (1+x)}\right ) \, dx\\ &=-25 x+\frac {100}{1+x}-200 \log (5-x)+150 \log ^2(5-x)-25 (5-x) \log ^2(5-x)+50 \text {Subst}(\int \log (x) \, dx,x,5-x)\\ &=25 x+\frac {100}{1+x}-200 \log (5-x)+50 (5-x) \log (5-x)+150 \log ^2(5-x)-25 (5-x) \log ^2(5-x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 1.82 \begin {gather*} \frac {25 \left (4+x+x^2-2 \left (-1+x^2\right ) \log (5-x)+(1+x)^2 \log ^2(5-x)\right )}{1+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs.
\(2(22)=44\).
time = 0.59, size = 61, normalized size = 2.77
method | result | size |
risch | \(\left (25 x +25\right ) \ln \left (5-x \right )^{2}-50 \ln \left (5-x \right ) x +\frac {50 \ln \left (x -5\right ) x +25 x^{2}+50 \ln \left (x -5\right )+25 x +100}{x +1}\) | \(51\) |
derivativedivides | \(-25 \ln \left (5-x \right )^{2} \left (5-x \right )+50 \left (5-x \right ) \ln \left (5-x \right )-125+25 x +150 \ln \left (5-x \right )^{2}-200 \ln \left (5-x \right )-\frac {100}{-x -1}\) | \(61\) |
default | \(-25 \ln \left (5-x \right )^{2} \left (5-x \right )+50 \left (5-x \right ) \ln \left (5-x \right )-125+25 x +150 \ln \left (5-x \right )^{2}-200 \ln \left (5-x \right )-\frac {100}{-x -1}\) | \(61\) |
norman | \(\frac {-75 x +50 \ln \left (5-x \right )+25 x^{2}+25 \ln \left (5-x \right )^{2}-50 x^{2} \ln \left (5-x \right )+50 \ln \left (5-x \right )^{2} x +25 \ln \left (5-x \right )^{2} x^{2}}{x +1}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs.
\(2 (22) = 44\).
time = 0.31, size = 67, normalized size = 3.05 \begin {gather*} \frac {25 \, {\left (36 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (-x + 5\right )^{2} + 36 \, x^{2} - {\left (72 \, x^{2} + 17 \, x - 55\right )} \log \left (-x + 5\right ) + 36 \, x + 42\right )}}{36 \, {\left (x + 1\right )}} + \frac {425}{6 \, {\left (x + 1\right )}} + \frac {425}{36} \, \log \left (x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 43, normalized size = 1.95 \begin {gather*} \frac {25 \, {\left ({\left (x^{2} + 2 \, x + 1\right )} \log \left (-x + 5\right )^{2} + x^{2} - 2 \, {\left (x^{2} - 1\right )} \log \left (-x + 5\right ) + x + 4\right )}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 34, normalized size = 1.55 \begin {gather*} - 50 x \log {\left (5 - x \right )} + 25 x + \left (25 x + 25\right ) \log {\left (5 - x \right )}^{2} + 50 \log {\left (x - 5 \right )} + \frac {100}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 44, normalized size = 2.00 \begin {gather*} 25 \, {\left (x + 1\right )} \log \left (-x + 5\right )^{2} - 50 \, {\left (x - 5\right )} \log \left (-x + 5\right ) + 25 \, x + \frac {100}{x + 1} - 200 \, \log \left (-x + 5\right ) - 125 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 46, normalized size = 2.09 \begin {gather*} 50\,\ln \left (x-5\right )+25\,{\ln \left (5-x\right )}^2+\frac {100}{x+1}+x\,\left (25\,{\ln \left (5-x\right )}^2-50\,\ln \left (5-x\right )+25\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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