Integrand size = 22, antiderivative size = 54 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=-\frac {1}{189 (2+3 x)^3}+\frac {34}{441 (2+3 x)^2}-\frac {121}{343 (2+3 x)}-\frac {242 \log (1-2 x)}{2401}+\frac {242 \log (2+3 x)}{2401} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=-\frac {121}{343 (3 x+2)}+\frac {34}{441 (3 x+2)^2}-\frac {1}{189 (3 x+2)^3}-\frac {242 \log (1-2 x)}{2401}+\frac {242 \log (3 x+2)}{2401} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {484}{2401 (-1+2 x)}+\frac {1}{21 (2+3 x)^4}-\frac {68}{147 (2+3 x)^3}+\frac {363}{343 (2+3 x)^2}+\frac {726}{2401 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{189 (2+3 x)^3}+\frac {34}{441 (2+3 x)^2}-\frac {121}{343 (2+3 x)}-\frac {242 \log (1-2 x)}{2401}+\frac {242 \log (2+3 x)}{2401} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=\frac {-\frac {7 \left (11689+37062 x+29403 x^2\right )}{(2+3 x)^3}-6534 \log (1-2 x)+6534 \log (4+6 x)}{64827} \]
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Time = 2.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {-\frac {4118}{1029} x -\frac {1089}{343} x^{2}-\frac {11689}{9261}}{\left (2+3 x \right )^{3}}-\frac {242 \ln \left (-1+2 x \right )}{2401}+\frac {242 \ln \left (2+3 x \right )}{2401}\) | \(36\) |
risch | \(\frac {-\frac {4118}{1029} x -\frac {1089}{343} x^{2}-\frac {11689}{9261}}{\left (2+3 x \right )^{3}}-\frac {242 \ln \left (-1+2 x \right )}{2401}+\frac {242 \ln \left (2+3 x \right )}{2401}\) | \(37\) |
default | \(-\frac {242 \ln \left (-1+2 x \right )}{2401}-\frac {1}{189 \left (2+3 x \right )^{3}}+\frac {34}{441 \left (2+3 x \right )^{2}}-\frac {121}{343 \left (2+3 x \right )}+\frac {242 \ln \left (2+3 x \right )}{2401}\) | \(45\) |
parallelrisch | \(\frac {52272 \ln \left (\frac {2}{3}+x \right ) x^{3}-52272 \ln \left (x -\frac {1}{2}\right ) x^{3}+104544 \ln \left (\frac {2}{3}+x \right ) x^{2}-104544 \ln \left (x -\frac {1}{2}\right ) x^{2}+81823 x^{3}+69696 \ln \left (\frac {2}{3}+x \right ) x -69696 \ln \left (x -\frac {1}{2}\right ) x +102662 x^{2}+15488 \ln \left (\frac {2}{3}+x \right )-15488 \ln \left (x -\frac {1}{2}\right )+32228 x}{19208 \left (2+3 x \right )^{3}}\) | \(86\) |
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Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.39 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=-\frac {205821 \, x^{2} - 6534 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 6534 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 259434 \, x + 81823}{64827 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=- \frac {29403 x^{2} + 37062 x + 11689}{250047 x^{3} + 500094 x^{2} + 333396 x + 74088} - \frac {242 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {242 \log {\left (x + \frac {2}{3} \right )}}{2401} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=-\frac {29403 \, x^{2} + 37062 \, x + 11689}{9261 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {242}{2401} \, \log \left (3 \, x + 2\right ) - \frac {242}{2401} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=-\frac {29403 \, x^{2} + 37062 \, x + 11689}{9261 \, {\left (3 \, x + 2\right )}^{3}} + \frac {242}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {242}{2401} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^4} \, dx=\frac {484\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}-\frac {\frac {121\,x^2}{1029}+\frac {4118\,x}{27783}+\frac {11689}{250047}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \]
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