Integrand size = 22, antiderivative size = 53 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=-\frac {5}{22 (3+5 x)^2}+\frac {155}{121 (3+5 x)}-\frac {8 \log (1-2 x)}{9317}-\frac {27}{7} \log (2+3 x)+\frac {5135 \log (3+5 x)}{1331} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {84} \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=\frac {155}{121 (5 x+3)}-\frac {5}{22 (5 x+3)^2}-\frac {8 \log (1-2 x)}{9317}-\frac {27}{7} \log (3 x+2)+\frac {5135 \log (5 x+3)}{1331} \]
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Rule 84
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {16}{9317 (-1+2 x)}-\frac {81}{7 (2+3 x)}+\frac {25}{11 (3+5 x)^3}-\frac {775}{121 (3+5 x)^2}+\frac {25675}{1331 (3+5 x)}\right ) \, dx \\ & = -\frac {5}{22 (3+5 x)^2}+\frac {155}{121 (3+5 x)}-\frac {8 \log (1-2 x)}{9317}-\frac {27}{7} \log (2+3 x)+\frac {5135 \log (3+5 x)}{1331} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=\frac {\frac {1925 (35+62 x)}{(3+5 x)^2}-16 \log (1-2 x)-71874 \log (4+6 x)+71890 \log (6+10 x)}{18634} \]
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Time = 2.60 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {\frac {775 x}{121}+\frac {875}{242}}{\left (3+5 x \right )^{2}}-\frac {8 \ln \left (-1+2 x \right )}{9317}-\frac {27 \ln \left (2+3 x \right )}{7}+\frac {5135 \ln \left (3+5 x \right )}{1331}\) | \(40\) |
norman | \(\frac {-\frac {2050}{363} x -\frac {21875}{2178} x^{2}}{\left (3+5 x \right )^{2}}-\frac {8 \ln \left (-1+2 x \right )}{9317}-\frac {27 \ln \left (2+3 x \right )}{7}+\frac {5135 \ln \left (3+5 x \right )}{1331}\) | \(43\) |
default | \(-\frac {5}{22 \left (3+5 x \right )^{2}}+\frac {155}{121 \left (3+5 x \right )}+\frac {5135 \ln \left (3+5 x \right )}{1331}-\frac {8 \ln \left (-1+2 x \right )}{9317}-\frac {27 \ln \left (2+3 x \right )}{7}\) | \(44\) |
parallelrisch | \(-\frac {16171650 \ln \left (\frac {2}{3}+x \right ) x^{2}-16175250 \ln \left (x +\frac {3}{5}\right ) x^{2}+3600 \ln \left (x -\frac {1}{2}\right ) x^{2}+19405980 \ln \left (\frac {2}{3}+x \right ) x -19410300 \ln \left (x +\frac {3}{5}\right ) x +4320 \ln \left (x -\frac {1}{2}\right ) x +1684375 x^{2}+5821794 \ln \left (\frac {2}{3}+x \right )-5823090 \ln \left (x +\frac {3}{5}\right )+1296 \ln \left (x -\frac {1}{2}\right )+947100 x}{167706 \left (3+5 x \right )^{2}}\) | \(85\) |
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Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=\frac {71890 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 71874 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (3 \, x + 2\right ) - 16 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 119350 \, x + 67375}{18634 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=- \frac {- 1550 x - 875}{6050 x^{2} + 7260 x + 2178} - \frac {8 \log {\left (x - \frac {1}{2} \right )}}{9317} + \frac {5135 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{7} \]
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Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=\frac {25 \, {\left (62 \, x + 35\right )}}{242 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {5135}{1331} \, \log \left (5 \, x + 3\right ) - \frac {27}{7} \, \log \left (3 \, x + 2\right ) - \frac {8}{9317} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=\frac {25 \, {\left (62 \, x + 35\right )}}{242 \, {\left (5 \, x + 3\right )}^{2}} + \frac {5135}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {27}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {8}{9317} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx=\frac {5135\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {27\,\ln \left (x+\frac {2}{3}\right )}{7}-\frac {8\,\ln \left (x-\frac {1}{2}\right )}{9317}+\frac {\frac {31\,x}{121}+\frac {35}{242}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \]
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