Integrand size = 11, antiderivative size = 60 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=-\frac {1}{192 x^3}+\frac {9}{256 x^2}-\frac {27}{128 x}-\frac {27}{256 (2+3 x)^2}-\frac {27}{64 (2+3 x)}-\frac {135 \log (x)}{256}+\frac {135}{256} \log (2+3 x) \]
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Time = 0.01 (sec) , antiderivative size = 60, 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} \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=-\frac {1}{192 x^3}+\frac {9}{256 x^2}-\frac {27}{128 x}-\frac {27}{64 (3 x+2)}-\frac {27}{256 (3 x+2)^2}-\frac {135 \log (x)}{256}+\frac {135}{256} \log (3 x+2) \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{64 x^4}-\frac {9}{128 x^3}+\frac {27}{128 x^2}-\frac {135}{256 x}+\frac {81}{128 (2+3 x)^3}+\frac {81}{64 (2+3 x)^2}+\frac {405}{256 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{192 x^3}+\frac {9}{256 x^2}-\frac {27}{128 x}-\frac {27}{256 (2+3 x)^2}-\frac {27}{64 (2+3 x)}-\frac {135 \log (x)}{256}+\frac {135}{256} \log (2+3 x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=\frac {1}{768} \left (-\frac {2 \left (8-30 x+180 x^2+1215 x^3+1215 x^4\right )}{x^3 (2+3 x)^2}-405 \log (x)+405 \log (2+3 x)\right ) \]
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Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.75
method | result | size |
norman | \(\frac {-\frac {1}{48}+\frac {405}{64} x^{4}+\frac {3645}{512} x^{5}+\frac {5}{64} x -\frac {15}{32} x^{2}}{x^{3} \left (2+3 x \right )^{2}}-\frac {135 \ln \left (x \right )}{256}+\frac {135 \ln \left (2+3 x \right )}{256}\) | \(45\) |
risch | \(\frac {-\frac {405}{128} x^{4}-\frac {405}{128} x^{3}-\frac {15}{32} x^{2}+\frac {5}{64} x -\frac {1}{48}}{x^{3} \left (2+3 x \right )^{2}}-\frac {135 \ln \left (x \right )}{256}+\frac {135 \ln \left (2+3 x \right )}{256}\) | \(46\) |
default | \(-\frac {1}{192 x^{3}}+\frac {9}{256 x^{2}}-\frac {27}{128 x}-\frac {27}{256 \left (2+3 x \right )^{2}}-\frac {27}{64 \left (2+3 x \right )}-\frac {135 \ln \left (x \right )}{256}+\frac {135 \ln \left (2+3 x \right )}{256}\) | \(47\) |
meijerg | \(-\frac {1}{192 x^{3}}+\frac {9}{256 x^{2}}-\frac {27}{128 x}-\frac {243}{1024}-\frac {135 \ln \left (x \right )}{256}-\frac {135 \ln \left (3\right )}{256}+\frac {135 \ln \left (2\right )}{256}+\frac {81 x \left (\frac {27 x}{2}+10\right )}{2048 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {135 \ln \left (1+\frac {3 x}{2}\right )}{256}\) | \(53\) |
parallelrisch | \(-\frac {7290 \ln \left (x \right ) x^{5}-7290 \ln \left (\frac {2}{3}+x \right ) x^{5}+32+9720 \ln \left (x \right ) x^{4}-9720 \ln \left (\frac {2}{3}+x \right ) x^{4}-10935 x^{5}+3240 \ln \left (x \right ) x^{3}-3240 \ln \left (\frac {2}{3}+x \right ) x^{3}-9720 x^{4}+720 x^{2}-120 x}{1536 x^{3} \left (2+3 x \right )^{2}}\) | \(81\) |
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Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=-\frac {2430 \, x^{4} + 2430 \, x^{3} + 360 \, x^{2} - 405 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (3 \, x + 2\right ) + 405 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (x\right ) - 60 \, x + 16}{768 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=- \frac {135 \log {\left (x \right )}}{256} + \frac {135 \log {\left (x + \frac {2}{3} \right )}}{256} + \frac {- 1215 x^{4} - 1215 x^{3} - 180 x^{2} + 30 x - 8}{3456 x^{5} + 4608 x^{4} + 1536 x^{3}} \]
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Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=-\frac {1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} + \frac {135}{256} \, \log \left (3 \, x + 2\right ) - \frac {135}{256} \, \log \left (x\right ) \]
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Time = 0.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=-\frac {1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \, {\left (3 \, x + 2\right )}^{2} x^{3}} + \frac {135}{256} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {135}{256} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 (4+6 x)^3} \, dx=\frac {135\,\mathrm {atanh}\left (3\,x+1\right )}{128}-\frac {\frac {45\,x^4}{128}+\frac {45\,x^3}{128}+\frac {5\,x^2}{96}-\frac {5\,x}{576}+\frac {1}{432}}{x^5+\frac {4\,x^4}{3}+\frac {4\,x^3}{9}} \]
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