Optimal. Leaf size=26 \[ -x+\frac {8 \left (4 x+\log \left (2 x+x^2\right )\right )}{x^3 (3+x)} \]
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Rubi [B] time = 1.12, antiderivative size = 143, normalized size of antiderivative = 5.50, number of steps used = 44, number of rules used = 16, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6741, 6742, 44, 88, 72, 77, 2513, 2357, 2304, 2314, 31, 2418, 2395, 36, 29, 74} \begin {gather*} \frac {8 \log (x)}{3 x^3}+\frac {8 \log (x+2)}{3 x^3}-\frac {8 (\log (x)+\log (x+2)-\log (x (x+2)))}{(x+3) x^3}+\frac {32}{3 x^2}-\frac {8 \log (x)}{9 x^2}-\frac {8 \log (x+2)}{9 x^2}-x+\frac {32}{9 (x+3)}-\frac {32}{9 x}+\frac {8 x \log (x)}{81 (x+3)}-\frac {8 \log (x)}{81}-\frac {8 \log (x+2)}{27 (x+3)}+\frac {8 \log (x)}{27 x}+\frac {8 \log (x+2)}{27 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 72
Rule 74
Rule 77
Rule 88
Rule 2304
Rule 2314
Rule 2357
Rule 2395
Rule 2418
Rule 2513
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48-320 x-368 x^2-96 x^3-18 x^4-21 x^5-8 x^6-x^7+\left (-144-136 x-32 x^2\right ) \log \left (2 x+x^2\right )}{x^4 (2+x) (3+x)^2} \, dx\\ &=\int \left (-\frac {18}{(2+x) (3+x)^2}+\frac {48}{x^4 (2+x) (3+x)^2}-\frac {320}{x^3 (2+x) (3+x)^2}-\frac {368}{x^2 (2+x) (3+x)^2}-\frac {96}{x (2+x) (3+x)^2}-\frac {21 x}{(2+x) (3+x)^2}-\frac {8 x^2}{(2+x) (3+x)^2}-\frac {x^3}{(2+x) (3+x)^2}-\frac {8 (9+4 x) \log (x (2+x))}{x^4 (3+x)^2}\right ) \, dx\\ &=-\left (8 \int \frac {x^2}{(2+x) (3+x)^2} \, dx\right )-8 \int \frac {(9+4 x) \log (x (2+x))}{x^4 (3+x)^2} \, dx-18 \int \frac {1}{(2+x) (3+x)^2} \, dx-21 \int \frac {x}{(2+x) (3+x)^2} \, dx+48 \int \frac {1}{x^4 (2+x) (3+x)^2} \, dx-96 \int \frac {1}{x (2+x) (3+x)^2} \, dx-320 \int \frac {1}{x^3 (2+x) (3+x)^2} \, dx-368 \int \frac {1}{x^2 (2+x) (3+x)^2} \, dx-\int \frac {x^3}{(2+x) (3+x)^2} \, dx\\ &=-\left (8 \int \left (\frac {4}{2+x}-\frac {9}{(3+x)^2}-\frac {3}{3+x}\right ) \, dx\right )-8 \int \frac {(9+4 x) \log (x)}{x^4 (3+x)^2} \, dx-8 \int \frac {(9+4 x) \log (2+x)}{x^4 (3+x)^2} \, dx-18 \int \left (\frac {1}{-3-x}+\frac {1}{2+x}-\frac {1}{(3+x)^2}\right ) \, dx-21 \int \left (-\frac {2}{2+x}+\frac {3}{(3+x)^2}+\frac {2}{3+x}\right ) \, dx+48 \int \left (\frac {1}{18 x^4}-\frac {7}{108 x^3}+\frac {11}{216 x^2}-\frac {131}{3888 x}+\frac {1}{16 (2+x)}-\frac {1}{81 (3+x)^2}-\frac {7}{243 (3+x)}\right ) \, dx-96 \int \left (\frac {1}{18 x}-\frac {1}{2 (2+x)}+\frac {1}{3 (3+x)^2}+\frac {4}{9 (3+x)}\right ) \, dx-320 \int \left (\frac {1}{18 x^3}-\frac {7}{108 x^2}+\frac {11}{216 x}-\frac {1}{8 (2+x)}+\frac {1}{27 (3+x)^2}+\frac {2}{27 (3+x)}\right ) \, dx-368 \int \left (\frac {1}{18 x^2}-\frac {7}{108 x}+\frac {1}{4 (2+x)}-\frac {1}{9 (3+x)^2}-\frac {5}{27 (3+x)}\right ) \, dx+(8 (\log (x)+\log (2+x)-\log (x (2+x)))) \int \frac {9+4 x}{x^4 (3+x)^2} \, dx-\int \left (1-\frac {8}{2+x}+\frac {27}{(3+x)^2}\right ) \, dx\\ &=-\frac {8}{9 x^3}+\frac {94}{9 x^2}-\frac {74}{27 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}-\log (2+x)-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {32}{81} \log (3+x)-8 \int \left (\frac {\log (x)}{x^4}-\frac {2 \log (x)}{9 x^3}+\frac {\log (x)}{27 x^2}-\frac {\log (x)}{27 (3+x)^2}\right ) \, dx-8 \int \left (\frac {\log (2+x)}{x^4}-\frac {2 \log (2+x)}{9 x^3}+\frac {\log (2+x)}{27 x^2}-\frac {\log (2+x)}{27 (3+x)^2}\right ) \, dx\\ &=-\frac {8}{9 x^3}+\frac {94}{9 x^2}-\frac {74}{27 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}-\log (2+x)-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {32}{81} \log (3+x)-\frac {8}{27} \int \frac {\log (x)}{x^2} \, dx+\frac {8}{27} \int \frac {\log (x)}{(3+x)^2} \, dx-\frac {8}{27} \int \frac {\log (2+x)}{x^2} \, dx+\frac {8}{27} \int \frac {\log (2+x)}{(3+x)^2} \, dx+\frac {16}{9} \int \frac {\log (x)}{x^3} \, dx+\frac {16}{9} \int \frac {\log (2+x)}{x^3} \, dx-8 \int \frac {\log (x)}{x^4} \, dx-8 \int \frac {\log (2+x)}{x^4} \, dx\\ &=\frac {10}{x^2}-\frac {22}{9 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}+\frac {8 \log (x)}{3 x^3}-\frac {8 \log (x)}{9 x^2}+\frac {8 \log (x)}{27 x}+\frac {8 x \log (x)}{81 (3+x)}-\log (2+x)+\frac {8 \log (2+x)}{3 x^3}-\frac {8 \log (2+x)}{9 x^2}+\frac {8 \log (2+x)}{27 x}-\frac {8 \log (2+x)}{27 (3+x)}-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {32}{81} \log (3+x)-\frac {8}{81} \int \frac {1}{3+x} \, dx-\frac {8}{27} \int \frac {1}{x (2+x)} \, dx+\frac {8}{27} \int \frac {1}{(2+x) (3+x)} \, dx+\frac {8}{9} \int \frac {1}{x^2 (2+x)} \, dx-\frac {8}{3} \int \frac {1}{x^3 (2+x)} \, dx\\ &=\frac {10}{x^2}-\frac {22}{9 x}-x+\frac {32}{9 (3+x)}+\frac {49 \log (x)}{81}+\frac {8 \log (x)}{3 x^3}-\frac {8 \log (x)}{9 x^2}+\frac {8 \log (x)}{27 x}+\frac {8 x \log (x)}{81 (3+x)}-\log (2+x)+\frac {8 \log (2+x)}{3 x^3}-\frac {8 \log (2+x)}{9 x^2}+\frac {8 \log (2+x)}{27 x}-\frac {8 \log (2+x)}{27 (3+x)}-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}+\frac {8}{27} \log (3+x)-\frac {4}{27} \int \frac {1}{x} \, dx+\frac {4}{27} \int \frac {1}{2+x} \, dx+\frac {8}{27} \int \frac {1}{2+x} \, dx-\frac {8}{27} \int \frac {1}{3+x} \, dx+\frac {8}{9} \int \left (\frac {1}{2 x^2}-\frac {1}{4 x}+\frac {1}{4 (2+x)}\right ) \, dx-\frac {8}{3} \int \left (\frac {1}{2 x^3}-\frac {1}{4 x^2}+\frac {1}{8 x}-\frac {1}{8 (2+x)}\right ) \, dx\\ &=\frac {32}{3 x^2}-\frac {32}{9 x}-x+\frac {32}{9 (3+x)}-\frac {8 \log (x)}{81}+\frac {8 \log (x)}{3 x^3}-\frac {8 \log (x)}{9 x^2}+\frac {8 \log (x)}{27 x}+\frac {8 x \log (x)}{81 (3+x)}+\frac {8 \log (2+x)}{3 x^3}-\frac {8 \log (2+x)}{9 x^2}+\frac {8 \log (2+x)}{27 x}-\frac {8 \log (2+x)}{27 (3+x)}-\frac {8 (\log (x)+\log (2+x)-\log (x (2+x)))}{x^3 (3+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 30, normalized size = 1.15 \begin {gather*} -\frac {-32 x+3 x^4+x^5-8 \log (x (2+x))}{x^3 (3+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 35, normalized size = 1.35 \begin {gather*} -\frac {x^{5} + 3 \, x^{4} - 32 \, x - 8 \, \log \left (x^{2} + 2 \, x\right )}{x^{4} + 3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 48, normalized size = 1.85 \begin {gather*} -\frac {8}{27} \, {\left (\frac {1}{x + 3} - \frac {x^{2} - 3 \, x + 9}{x^{3}}\right )} \log \left (x^{2} + 2 \, x\right ) - x + \frac {32}{9 \, {\left (x + 3\right )}} - \frac {32 \, {\left (x - 3\right )}}{9 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 40, normalized size = 1.54
method | result | size |
risch | \(\frac {8 \ln \left (x^{2}+2 x \right )}{\left (3+x \right ) x^{3}}-\frac {x^{4}+3 x^{3}-32}{x^{2} \left (3+x \right )}\) | \(40\) |
default | \(-x -\frac {8}{9 x^{3}}+\frac {94}{9 x^{2}}-\frac {74}{27 x}-\ln \relax (x )-\ln \left (2+x \right )+\frac {32}{9 \left (3+x \right )}+\frac {\frac {8}{3}+x^{4} \ln \left (x^{2}+2 x \right )-\frac {22 x^{3}}{27}+\frac {14 x}{9}-\frac {20 x^{2}}{9}+3 x^{3} \ln \left (x^{2}+2 x \right )+8 \ln \left (x^{2}+2 x \right )}{\left (3+x \right ) x^{3}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 143, normalized size = 5.50 \begin {gather*} -x - \frac {66 \, x^{3} + 180 \, x^{2} - 81 \, {\left (x^{4} + 3 \, x^{3} + 8\right )} \log \left (x + 2\right ) + {\left (49 \, x^{4} + 147 \, x^{3} - 648\right )} \log \relax (x) - 126 \, x - 216}{81 \, {\left (x^{4} + 3 \, x^{3}\right )}} - \frac {2 \, {\left (25 \, x^{3} + 78 \, x^{2} - 51 \, x + 36\right )}}{27 \, {\left (x^{4} + 3 \, x^{3}\right )}} - \frac {80 \, {\left (x^{2} + 6 \, x - 3\right )}}{9 \, {\left (x^{3} + 3 \, x^{2}\right )}} - \frac {184 \, {\left (x - 3\right )}}{9 \, {\left (x^{2} + 3 \, x\right )}} + \frac {32}{x + 3} - \log \left (x + 2\right ) + \frac {49}{81} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.07, size = 33, normalized size = 1.27 \begin {gather*} \frac {32\,x+8\,\ln \left (x^2+2\,x\right )-3\,x^4-x^5}{x^3\,\left (x+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 29, normalized size = 1.12 \begin {gather*} - x + \frac {8 \log {\left (x^{2} + 2 x \right )}}{x^{4} + 3 x^{3}} + \frac {32}{x^{3} + 3 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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