Integrand size = 11, antiderivative size = 45 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=-\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (2+3 x) \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^5 (4+6 x)} \, dx=-\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (3 x+2) \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4 x^5}-\frac {3}{8 x^4}+\frac {9}{16 x^3}-\frac {27}{32 x^2}+\frac {81}{64 x}-\frac {243}{64 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (2+3 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=-\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (2+3 x) \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73
method | result | size |
norman | \(\frac {-\frac {1}{16}+\frac {1}{8} x -\frac {9}{32} x^{2}+\frac {27}{32} x^{3}}{x^{4}}+\frac {81 \ln \left (x \right )}{64}-\frac {81 \ln \left (2+3 x \right )}{64}\) | \(33\) |
risch | \(\frac {-\frac {1}{16}+\frac {1}{8} x -\frac {9}{32} x^{2}+\frac {27}{32} x^{3}}{x^{4}}+\frac {81 \ln \left (x \right )}{64}-\frac {81 \ln \left (2+3 x \right )}{64}\) | \(33\) |
default | \(-\frac {1}{16 x^{4}}+\frac {1}{8 x^{3}}-\frac {9}{32 x^{2}}+\frac {27}{32 x}+\frac {81 \ln \left (x \right )}{64}-\frac {81 \ln \left (2+3 x \right )}{64}\) | \(34\) |
parallelrisch | \(\frac {81 \ln \left (x \right ) x^{4}-81 \ln \left (\frac {2}{3}+x \right ) x^{4}-4+54 x^{3}-18 x^{2}+8 x}{64 x^{4}}\) | \(37\) |
meijerg | \(-\frac {1}{16 x^{4}}+\frac {1}{8 x^{3}}-\frac {9}{32 x^{2}}+\frac {27}{32 x}+\frac {81 \ln \left (x \right )}{64}+\frac {81 \ln \left (3\right )}{64}-\frac {81 \ln \left (2\right )}{64}-\frac {81 \ln \left (1+\frac {3 x}{2}\right )}{64}\) | \(42\) |
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none
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=-\frac {81 \, x^{4} \log \left (3 \, x + 2\right ) - 81 \, x^{4} \log \left (x\right ) - 54 \, x^{3} + 18 \, x^{2} - 8 \, x + 4}{64 \, x^{4}} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=\frac {81 \log {\left (x \right )}}{64} - \frac {81 \log {\left (x + \frac {2}{3} \right )}}{64} + \frac {27 x^{3} - 9 x^{2} + 4 x - 2}{32 x^{4}} \]
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none
Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=\frac {27 \, x^{3} - 9 \, x^{2} + 4 \, x - 2}{32 \, x^{4}} - \frac {81}{64} \, \log \left (3 \, x + 2\right ) + \frac {81}{64} \, \log \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=\frac {27 \, x^{3} - 9 \, x^{2} + 4 \, x - 2}{32 \, x^{4}} - \frac {81}{64} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {81}{64} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^5 (4+6 x)} \, dx=\frac {\frac {27\,x^3}{32}-\frac {9\,x^2}{32}+\frac {x}{8}-\frac {1}{16}}{x^4}-\frac {81\,\mathrm {atanh}\left (3\,x+1\right )}{32} \]
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