Integrand size = 11, antiderivative size = 53 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=-\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {9}{128 (2+3 x)^2}+\frac {27}{128 (2+3 x)}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (2+3 x) \]
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Time = 0.02 (sec) , antiderivative size = 53, 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} \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=-\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {27}{128 (3 x+2)}+\frac {9}{128 (3 x+2)^2}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (3 x+2) \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{64 x^3}-\frac {9}{128 x^2}+\frac {27}{128 x}-\frac {27}{64 (2+3 x)^3}-\frac {81}{128 (2+3 x)^2}-\frac {81}{128 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {9}{128 (2+3 x)^2}+\frac {27}{128 (2+3 x)}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (2+3 x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=\frac {1}{128} \left (\frac {2 \left (-2+12 x+81 x^2+81 x^3\right )}{x^2 (2+3 x)^2}+27 \log (x)-27 \log (2+3 x)\right ) \]
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Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75
method | result | size |
norman | \(\frac {-\frac {1}{32}-\frac {81}{32} x^{3}-\frac {729}{256} x^{4}+\frac {3}{16} x}{x^{2} \left (2+3 x \right )^{2}}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2+3 x \right )}{128}\) | \(40\) |
risch | \(\frac {\frac {81}{64} x^{3}+\frac {81}{64} x^{2}+\frac {3}{16} x -\frac {1}{32}}{x^{2} \left (2+3 x \right )^{2}}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2+3 x \right )}{128}\) | \(41\) |
default | \(-\frac {1}{128 x^{2}}+\frac {9}{128 x}+\frac {9}{128 \left (2+3 x \right )^{2}}+\frac {27}{128 \left (2+3 x \right )}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2+3 x \right )}{128}\) | \(42\) |
meijerg | \(-\frac {1}{128 x^{2}}+\frac {9}{128 x}+\frac {63}{512}+\frac {27 \ln \left (x \right )}{128}+\frac {27 \ln \left (3\right )}{128}-\frac {27 \ln \left (2\right )}{128}-\frac {27 x \left (8+\frac {21 x}{2}\right )}{1024 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {27 \ln \left (1+\frac {3 x}{2}\right )}{128}\) | \(48\) |
parallelrisch | \(\frac {486 \ln \left (x \right ) x^{4}-486 \ln \left (\frac {2}{3}+x \right ) x^{4}-8+648 \ln \left (x \right ) x^{3}-648 \ln \left (\frac {2}{3}+x \right ) x^{3}-729 x^{4}+216 \ln \left (x \right ) x^{2}-216 \ln \left (\frac {2}{3}+x \right ) x^{2}-648 x^{3}+48 x}{256 x^{2} \left (2+3 x \right )^{2}}\) | \(76\) |
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Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=\frac {162 \, x^{3} + 162 \, x^{2} - 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) + 24 \, x - 4}{128 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=\frac {27 \log {\left (x \right )}}{128} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{128} + \frac {81 x^{3} + 81 x^{2} + 12 x - 2}{576 x^{4} + 768 x^{3} + 256 x^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=\frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} - \frac {27}{128} \, \log \left (3 \, x + 2\right ) + \frac {27}{128} \, \log \left (x\right ) \]
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Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=\frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (3 \, x^{2} + 2 \, x\right )}^{2}} - \frac {27}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {27}{128} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^3 (4+6 x)^3} \, dx=\frac {\frac {9\,x^3}{64}+\frac {9\,x^2}{64}+\frac {x}{48}-\frac {1}{288}}{x^4+\frac {4\,x^3}{3}+\frac {4\,x^2}{9}}-\frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{64} \]
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