Integrand size = 22, antiderivative size = 65 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=-\frac {1}{252 (2+3 x)^4}+\frac {68}{1323 (2+3 x)^3}-\frac {121}{686 (2+3 x)^2}-\frac {242}{2401 (2+3 x)}-\frac {484 \log (1-2 x)}{16807}+\frac {484 \log (2+3 x)}{16807} \]
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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} \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=-\frac {242}{2401 (3 x+2)}-\frac {121}{686 (3 x+2)^2}+\frac {68}{1323 (3 x+2)^3}-\frac {1}{252 (3 x+2)^4}-\frac {484 \log (1-2 x)}{16807}+\frac {484 \log (3 x+2)}{16807} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {968}{16807 (-1+2 x)}+\frac {1}{21 (2+3 x)^5}-\frac {68}{147 (2+3 x)^4}+\frac {363}{343 (2+3 x)^3}+\frac {726}{2401 (2+3 x)^2}+\frac {1452}{16807 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{252 (2+3 x)^4}+\frac {68}{1323 (2+3 x)^3}-\frac {121}{686 (2+3 x)^2}-\frac {242}{2401 (2+3 x)}-\frac {484 \log (1-2 x)}{16807}+\frac {484 \log (2+3 x)}{16807} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=\frac {2 \left (-\frac {7 \left (366413+1449768 x+1822986 x^2+705672 x^3\right )}{8 (2+3 x)^4}-6534 \log (1-2 x)+6534 \log (4+6 x)\right )}{453789} \]
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Time = 2.51 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.63
method | result | size |
norman | \(\frac {-\frac {120814}{21609} x -\frac {33759}{4802} x^{2}-\frac {6534}{2401} x^{3}-\frac {366413}{259308}}{\left (2+3 x \right )^{4}}-\frac {484 \ln \left (-1+2 x \right )}{16807}+\frac {484 \ln \left (2+3 x \right )}{16807}\) | \(41\) |
risch | \(\frac {-\frac {120814}{21609} x -\frac {33759}{4802} x^{2}-\frac {6534}{2401} x^{3}-\frac {366413}{259308}}{\left (2+3 x \right )^{4}}-\frac {484 \ln \left (-1+2 x \right )}{16807}+\frac {484 \ln \left (2+3 x \right )}{16807}\) | \(42\) |
default | \(-\frac {484 \ln \left (-1+2 x \right )}{16807}-\frac {1}{252 \left (2+3 x \right )^{4}}+\frac {68}{1323 \left (2+3 x \right )^{3}}-\frac {121}{686 \left (2+3 x \right )^{2}}-\frac {242}{2401 \left (2+3 x \right )}+\frac {484 \ln \left (2+3 x \right )}{16807}\) | \(54\) |
parallelrisch | \(\frac {2509056 \ln \left (\frac {2}{3}+x \right ) x^{4}-2509056 \ln \left (x -\frac {1}{2}\right ) x^{4}+6690816 \ln \left (\frac {2}{3}+x \right ) x^{3}-6690816 \ln \left (x -\frac {1}{2}\right ) x^{3}+7694673 x^{4}+6690816 \ln \left (\frac {2}{3}+x \right ) x^{2}-6690816 \ln \left (x -\frac {1}{2}\right ) x^{2}+17591896 x^{3}+2973696 \ln \left (\frac {2}{3}+x \right ) x -2973696 \ln \left (x -\frac {1}{2}\right ) x +12957112 x^{2}+495616 \ln \left (\frac {2}{3}+x \right )-495616 \ln \left (x -\frac {1}{2}\right )+3105760 x}{1075648 \left (2+3 x \right )^{4}}\) | \(109\) |
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.46 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=-\frac {4939704 \, x^{3} + 12760902 \, x^{2} - 52272 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 52272 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 10148376 \, x + 2564891}{1815156 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=- \frac {705672 x^{3} + 1822986 x^{2} + 1449768 x + 366413}{21003948 x^{4} + 56010528 x^{3} + 56010528 x^{2} + 24893568 x + 4148928} - \frac {484 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {484 \log {\left (x + \frac {2}{3} \right )}}{16807} \]
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Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=-\frac {705672 \, x^{3} + 1822986 \, x^{2} + 1449768 \, x + 366413}{259308 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {484}{16807} \, \log \left (3 \, x + 2\right ) - \frac {484}{16807} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=-\frac {242}{2401 \, {\left (3 \, x + 2\right )}} - \frac {121}{686 \, {\left (3 \, x + 2\right )}^{2}} + \frac {68}{1323 \, {\left (3 \, x + 2\right )}^{3}} - \frac {1}{252 \, {\left (3 \, x + 2\right )}^{4}} - \frac {484}{16807} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
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Time = 1.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx=\frac {968\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}-\frac {\frac {242\,x^3}{7203}+\frac {3751\,x^2}{43218}+\frac {120814\,x}{1750329}+\frac {366413}{21003948}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \]
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