Integrand size = 22, antiderivative size = 65 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=\frac {1}{756 (2+3 x)^4}-\frac {103}{3969 (2+3 x)^3}+\frac {3469}{18522 (2+3 x)^2}-\frac {1331}{2401 (2+3 x)}-\frac {2662 \log (1-2 x)}{16807}+\frac {2662 \log (2+3 x)}{16807} \]
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Time = 0.02 (sec) , antiderivative size = 65, 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 = {90} \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=-\frac {1331}{2401 (3 x+2)}+\frac {3469}{18522 (3 x+2)^2}-\frac {103}{3969 (3 x+2)^3}+\frac {1}{756 (3 x+2)^4}-\frac {2662 \log (1-2 x)}{16807}+\frac {2662 \log (3 x+2)}{16807} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {5324}{16807 (-1+2 x)}-\frac {1}{63 (2+3 x)^5}+\frac {103}{441 (2+3 x)^4}-\frac {3469}{3087 (2+3 x)^3}+\frac {3993}{2401 (2+3 x)^2}+\frac {7986}{16807 (2+3 x)}\right ) \, dx \\ & = \frac {1}{756 (2+3 x)^4}-\frac {103}{3969 (2+3 x)^3}+\frac {3469}{18522 (2+3 x)^2}-\frac {1331}{2401 (2+3 x)}-\frac {2662 \log (1-2 x)}{16807}+\frac {2662 \log (2+3 x)}{16807} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=\frac {2 \left (-\frac {7 \left (2906507+13836972 x+21975894 x^2+11643588 x^3\right )}{8 (2+3 x)^4}-107811 \log (1-2 x)+107811 \log (4+6 x)\right )}{1361367} \]
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Time = 2.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.63
method | result | size |
norman | \(\frac {-\frac {1153081}{64827} x -\frac {406961}{14406} x^{2}-\frac {35937}{2401} x^{3}-\frac {2906507}{777924}}{\left (2+3 x \right )^{4}}-\frac {2662 \ln \left (-1+2 x \right )}{16807}+\frac {2662 \ln \left (2+3 x \right )}{16807}\) | \(41\) |
risch | \(\frac {-\frac {1153081}{64827} x -\frac {406961}{14406} x^{2}-\frac {35937}{2401} x^{3}-\frac {2906507}{777924}}{\left (2+3 x \right )^{4}}-\frac {2662 \ln \left (-1+2 x \right )}{16807}+\frac {2662 \ln \left (2+3 x \right )}{16807}\) | \(42\) |
default | \(-\frac {2662 \ln \left (-1+2 x \right )}{16807}+\frac {1}{756 \left (2+3 x \right )^{4}}-\frac {103}{3969 \left (2+3 x \right )^{3}}+\frac {3469}{18522 \left (2+3 x \right )^{2}}-\frac {1331}{2401 \left (2+3 x \right )}+\frac {2662 \ln \left (2+3 x \right )}{16807}\) | \(54\) |
parallelrisch | \(\frac {41399424 \ln \left (\frac {2}{3}+x \right ) x^{4}-41399424 \ln \left (x -\frac {1}{2}\right ) x^{4}+110398464 \ln \left (\frac {2}{3}+x \right ) x^{3}-110398464 \ln \left (x -\frac {1}{2}\right ) x^{3}+61036647 x^{4}+110398464 \ln \left (\frac {2}{3}+x \right ) x^{2}-110398464 \ln \left (x -\frac {1}{2}\right ) x^{2}+114465064 x^{3}+49065984 \ln \left (\frac {2}{3}+x \right ) x -49065984 \ln \left (x -\frac {1}{2}\right ) x +71605128 x^{2}+8177664 \ln \left (\frac {2}{3}+x \right )-8177664 \ln \left (x -\frac {1}{2}\right )+14941920 x}{3226944 \left (2+3 x \right )^{4}}\) | \(109\) |
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.46 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=-\frac {81505116 \, x^{3} + 153831258 \, x^{2} - 862488 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 862488 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 96858804 \, x + 20345549}{5445468 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=- \frac {11643588 x^{3} + 21975894 x^{2} + 13836972 x + 2906507}{63011844 x^{4} + 168031584 x^{3} + 168031584 x^{2} + 74680704 x + 12446784} - \frac {2662 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {2662 \log {\left (x + \frac {2}{3} \right )}}{16807} \]
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Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=-\frac {11643588 \, x^{3} + 21975894 \, x^{2} + 13836972 \, x + 2906507}{777924 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2662}{16807} \, \log \left (3 \, x + 2\right ) - \frac {2662}{16807} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=-\frac {1331}{2401 \, {\left (3 \, x + 2\right )}} + \frac {3469}{18522 \, {\left (3 \, x + 2\right )}^{2}} - \frac {103}{3969 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{756 \, {\left (3 \, x + 2\right )}^{4}} - \frac {2662}{16807} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx=\frac {5324\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}-\frac {\frac {1331\,x^3}{7203}+\frac {406961\,x^2}{1166886}+\frac {1153081\,x}{5250987}+\frac {2906507}{63011844}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \]
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