Integrand size = 20, antiderivative size = 54 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=\frac {22}{343 (1-2 x)}+\frac {1}{98 (2+3 x)^2}-\frac {31}{343 (2+3 x)}-\frac {128 \log (1-2 x)}{2401}+\frac {128 \log (2+3 x)}{2401} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=\frac {22}{343 (1-2 x)}-\frac {31}{343 (3 x+2)}+\frac {1}{98 (3 x+2)^2}-\frac {128 \log (1-2 x)}{2401}+\frac {128 \log (3 x+2)}{2401} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {44}{343 (-1+2 x)^2}-\frac {256}{2401 (-1+2 x)}-\frac {3}{49 (2+3 x)^3}+\frac {93}{343 (2+3 x)^2}+\frac {384}{2401 (2+3 x)}\right ) \, dx \\ & = \frac {22}{343 (1-2 x)}+\frac {1}{98 (2+3 x)^2}-\frac {31}{343 (2+3 x)}-\frac {128 \log (1-2 x)}{2401}+\frac {128 \log (2+3 x)}{2401} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=\frac {-\frac {7 \left (59+576 x+768 x^2\right )}{(-1+2 x) (2+3 x)^2}-256 \log (1-2 x)+256 \log (4+6 x)}{4802} \]
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Time = 2.61 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {-\frac {384}{343} x^{2}-\frac {288}{343} x -\frac {59}{686}}{\left (-1+2 x \right ) \left (2+3 x \right )^{2}}-\frac {128 \ln \left (-1+2 x \right )}{2401}+\frac {128 \ln \left (2+3 x \right )}{2401}\) | \(44\) |
default | \(-\frac {22}{343 \left (-1+2 x \right )}-\frac {128 \ln \left (-1+2 x \right )}{2401}+\frac {1}{98 \left (2+3 x \right )^{2}}-\frac {31}{343 \left (2+3 x \right )}+\frac {128 \ln \left (2+3 x \right )}{2401}\) | \(45\) |
norman | \(\frac {-\frac {1464}{343} x^{2}-\frac {1296}{343} x^{3}+\frac {517}{686}}{\left (-1+2 x \right ) \left (2+3 x \right )^{2}}-\frac {128 \ln \left (-1+2 x \right )}{2401}+\frac {128 \ln \left (2+3 x \right )}{2401}\) | \(45\) |
parallelrisch | \(\frac {9216 \ln \left (\frac {2}{3}+x \right ) x^{3}-9216 \ln \left (x -\frac {1}{2}\right ) x^{3}+7238+7680 \ln \left (\frac {2}{3}+x \right ) x^{2}-7680 \ln \left (x -\frac {1}{2}\right ) x^{2}-36288 x^{3}-2048 \ln \left (\frac {2}{3}+x \right ) x +2048 \ln \left (x -\frac {1}{2}\right ) x -40992 x^{2}-2048 \ln \left (\frac {2}{3}+x \right )+2048 \ln \left (x -\frac {1}{2}\right )}{9604 \left (-1+2 x \right ) \left (2+3 x \right )^{2}}\) | \(91\) |
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none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.39 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=-\frac {5376 \, x^{2} - 256 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (3 \, x + 2\right ) + 256 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (2 \, x - 1\right ) + 4032 \, x + 413}{4802 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=\frac {- 768 x^{2} - 576 x - 59}{12348 x^{3} + 10290 x^{2} - 2744 x - 2744} - \frac {128 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {128 \log {\left (x + \frac {2}{3} \right )}}{2401} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=-\frac {768 \, x^{2} + 576 \, x + 59}{686 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} + \frac {128}{2401} \, \log \left (3 \, x + 2\right ) - \frac {128}{2401} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=-\frac {22}{343 \, {\left (2 \, x - 1\right )}} + \frac {6 \, {\left (\frac {203}{2 \, x - 1} + 90\right )}}{2401 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{2}} + \frac {128}{2401} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]
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Time = 1.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^3} \, dx=\frac {256\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}+\frac {\frac {64\,x^2}{1029}+\frac {16\,x}{343}+\frac {59}{12348}}{-x^3-\frac {5\,x^2}{6}+\frac {2\,x}{9}+\frac {2}{9}} \]
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