Integrand size = 20, antiderivative size = 54 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=\frac {11}{98 (1-2 x)^2}+\frac {31}{343 (1-2 x)}+\frac {3}{343 (2+3 x)}-\frac {87 \log (1-2 x)}{2401}+\frac {87 \log (2+3 x)}{2401} \]
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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} \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=\frac {31}{343 (1-2 x)}+\frac {3}{343 (3 x+2)}+\frac {11}{98 (1-2 x)^2}-\frac {87 \log (1-2 x)}{2401}+\frac {87 \log (3 x+2)}{2401} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {22}{49 (-1+2 x)^3}+\frac {62}{343 (-1+2 x)^2}-\frac {174}{2401 (-1+2 x)}-\frac {9}{343 (2+3 x)^2}+\frac {261}{2401 (2+3 x)}\right ) \, dx \\ & = \frac {11}{98 (1-2 x)^2}+\frac {31}{343 (1-2 x)}+\frac {3}{343 (2+3 x)}-\frac {87 \log (1-2 x)}{2401}+\frac {87 \log (2+3 x)}{2401} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=\frac {\frac {7 \left (284+145 x-348 x^2\right )}{(1-2 x)^2 (2+3 x)}-174 \log (1-2 x)+174 \log (4+6 x)}{4802} \]
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Time = 0.87 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {-\frac {174}{343} x^{2}+\frac {145}{686} x +\frac {142}{343}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )}-\frac {87 \ln \left (-1+2 x \right )}{2401}+\frac {87 \ln \left (2+3 x \right )}{2401}\) | \(44\) |
default | \(\frac {11}{98 \left (-1+2 x \right )^{2}}-\frac {31}{343 \left (-1+2 x \right )}-\frac {87 \ln \left (-1+2 x \right )}{2401}+\frac {3}{343 \left (2+3 x \right )}+\frac {87 \ln \left (2+3 x \right )}{2401}\) | \(45\) |
norman | \(\frac {\frac {110}{343} x^{2}-\frac {852}{343} x^{3}+\frac {855}{686} x}{\left (-1+2 x \right )^{2} \left (2+3 x \right )}-\frac {87 \ln \left (-1+2 x \right )}{2401}+\frac {87 \ln \left (2+3 x \right )}{2401}\) | \(47\) |
parallelrisch | \(\frac {2088 \ln \left (\frac {2}{3}+x \right ) x^{3}-2088 \ln \left (x -\frac {1}{2}\right ) x^{3}-696 \ln \left (\frac {2}{3}+x \right ) x^{2}+696 \ln \left (x -\frac {1}{2}\right ) x^{2}-11928 x^{3}-870 \ln \left (\frac {2}{3}+x \right ) x +870 \ln \left (x -\frac {1}{2}\right ) x +1540 x^{2}+348 \ln \left (\frac {2}{3}+x \right )-348 \ln \left (x -\frac {1}{2}\right )+5985 x}{4802 \left (-1+2 x \right )^{2} \left (2+3 x \right )}\) | \(93\) |
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Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.39 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=-\frac {2436 \, x^{2} - 174 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 174 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (2 \, x - 1\right ) - 1015 \, x - 1988}{4802 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=- \frac {348 x^{2} - 145 x - 284}{8232 x^{3} - 2744 x^{2} - 3430 x + 1372} - \frac {87 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {87 \log {\left (x + \frac {2}{3} \right )}}{2401} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=-\frac {348 \, x^{2} - 145 \, x - 284}{686 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} + \frac {87}{2401} \, \log \left (3 \, x + 2\right ) - \frac {87}{2401} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=\frac {3}{343 \, {\left (3 \, x + 2\right )}} + \frac {6 \, {\left (\frac {448}{3 \, x + 2} - 95\right )}}{2401 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} - \frac {87}{2401} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx=\frac {174\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}-\frac {-\frac {29\,x^2}{686}+\frac {145\,x}{8232}+\frac {71}{2058}}{-x^3+\frac {x^2}{3}+\frac {5\,x}{12}-\frac {1}{6}} \]
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