Integrand size = 20, antiderivative size = 54 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=\frac {1}{63 (2+3 x)^3}-\frac {11}{98 (2+3 x)^2}-\frac {22}{343 (2+3 x)}-\frac {44 \log (1-2 x)}{2401}+\frac {44 \log (2+3 x)}{2401} \]
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Time = 0.01 (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} \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=-\frac {22}{343 (3 x+2)}-\frac {11}{98 (3 x+2)^2}+\frac {1}{63 (3 x+2)^3}-\frac {44 \log (1-2 x)}{2401}+\frac {44 \log (3 x+2)}{2401} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {88}{2401 (-1+2 x)}-\frac {1}{7 (2+3 x)^4}+\frac {33}{49 (2+3 x)^3}+\frac {66}{343 (2+3 x)^2}+\frac {132}{2401 (2+3 x)}\right ) \, dx \\ & = \frac {1}{63 (2+3 x)^3}-\frac {11}{98 (2+3 x)^2}-\frac {22}{343 (2+3 x)}-\frac {44 \log (1-2 x)}{2401}+\frac {44 \log (2+3 x)}{2401} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=\frac {-\frac {7 \left (2872+6831 x+3564 x^2\right )}{(2+3 x)^3}-792 \log (3-6 x)+792 \log (2+3 x)}{43218} \]
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Time = 2.50 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {-\frac {759}{686} x -\frac {198}{343} x^{2}-\frac {1436}{3087}}{\left (2+3 x \right )^{3}}-\frac {44 \ln \left (-1+2 x \right )}{2401}+\frac {44 \ln \left (2+3 x \right )}{2401}\) | \(36\) |
risch | \(\frac {-\frac {759}{686} x -\frac {198}{343} x^{2}-\frac {1436}{3087}}{\left (2+3 x \right )^{3}}-\frac {44 \ln \left (-1+2 x \right )}{2401}+\frac {44 \ln \left (2+3 x \right )}{2401}\) | \(37\) |
default | \(-\frac {44 \ln \left (-1+2 x \right )}{2401}+\frac {1}{63 \left (2+3 x \right )^{3}}-\frac {11}{98 \left (2+3 x \right )^{2}}-\frac {22}{343 \left (2+3 x \right )}+\frac {44 \ln \left (2+3 x \right )}{2401}\) | \(45\) |
parallelrisch | \(\frac {9504 \ln \left (\frac {2}{3}+x \right ) x^{3}-9504 \ln \left (x -\frac {1}{2}\right ) x^{3}+19008 \ln \left (\frac {2}{3}+x \right ) x^{2}-19008 \ln \left (x -\frac {1}{2}\right ) x^{2}+30156 x^{3}+12672 \ln \left (\frac {2}{3}+x \right ) x -12672 \ln \left (x -\frac {1}{2}\right ) x +49224 x^{2}+2816 \ln \left (\frac {2}{3}+x \right )-2816 \ln \left (x -\frac {1}{2}\right )+18956 x}{19208 \left (2+3 x \right )^{3}}\) | \(86\) |
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none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.39 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=-\frac {24948 \, x^{2} - 792 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 792 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 47817 \, x + 20104}{43218 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=- \frac {3564 x^{2} + 6831 x + 2872}{166698 x^{3} + 333396 x^{2} + 222264 x + 49392} - \frac {44 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {44 \log {\left (x + \frac {2}{3} \right )}}{2401} \]
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none
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=-\frac {3564 \, x^{2} + 6831 \, x + 2872}{6174 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {44}{2401} \, \log \left (3 \, x + 2\right ) - \frac {44}{2401} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=-\frac {3564 \, x^{2} + 6831 \, x + 2872}{6174 \, {\left (3 \, x + 2\right )}^{3}} + \frac {44}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {44}{2401} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx=\frac {88\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}-\frac {\frac {22\,x^2}{1029}+\frac {253\,x}{6174}+\frac {1436}{83349}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \]
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