Integrand size = 20, antiderivative size = 76 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {88}{16807 (1-2 x)}+\frac {1}{196 (2+3 x)^4}-\frac {31}{1029 (2+3 x)^3}-\frac {64}{2401 (2+3 x)^2}-\frac {388}{16807 (2+3 x)}-\frac {1040 \log (1-2 x)}{117649}+\frac {1040 \log (2+3 x)}{117649} \]
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Time = 0.02 (sec) , antiderivative size = 76, 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} \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {88}{16807 (1-2 x)}-\frac {388}{16807 (3 x+2)}-\frac {64}{2401 (3 x+2)^2}-\frac {31}{1029 (3 x+2)^3}+\frac {1}{196 (3 x+2)^4}-\frac {1040 \log (1-2 x)}{117649}+\frac {1040 \log (3 x+2)}{117649} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {176}{16807 (-1+2 x)^2}-\frac {2080}{117649 (-1+2 x)}-\frac {3}{49 (2+3 x)^5}+\frac {93}{343 (2+3 x)^4}+\frac {384}{2401 (2+3 x)^3}+\frac {1164}{16807 (2+3 x)^2}+\frac {3120}{117649 (2+3 x)}\right ) \, dx \\ & = \frac {88}{16807 (1-2 x)}+\frac {1}{196 (2+3 x)^4}-\frac {31}{1029 (2+3 x)^3}-\frac {64}{2401 (2+3 x)^2}-\frac {388}{16807 (2+3 x)}-\frac {1040 \log (1-2 x)}{117649}+\frac {1040 \log (2+3 x)}{117649} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {4 \left (-\frac {7 \left (-52979-9230 x+429000 x^2+702000 x^3+336960 x^4\right )}{16 (-1+2 x) (2+3 x)^4}-780 \log (1-2 x)+780 \log (4+6 x)\right )}{352947} \]
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Time = 2.59 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
norman | \(\frac {-\frac {58500}{16807} x^{3}-\frac {35750}{16807} x^{2}-\frac {28080}{16807} x^{4}+\frac {4615}{100842} x +\frac {52979}{201684}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4}}-\frac {1040 \ln \left (-1+2 x \right )}{117649}+\frac {1040 \ln \left (2+3 x \right )}{117649}\) | \(53\) |
risch | \(\frac {-\frac {58500}{16807} x^{3}-\frac {35750}{16807} x^{2}-\frac {28080}{16807} x^{4}+\frac {4615}{100842} x +\frac {52979}{201684}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4}}-\frac {1040 \ln \left (-1+2 x \right )}{117649}+\frac {1040 \ln \left (2+3 x \right )}{117649}\) | \(54\) |
default | \(-\frac {88}{16807 \left (-1+2 x \right )}-\frac {1040 \ln \left (-1+2 x \right )}{117649}+\frac {1}{196 \left (2+3 x \right )^{4}}-\frac {31}{1029 \left (2+3 x \right )^{3}}-\frac {64}{2401 \left (2+3 x \right )^{2}}-\frac {388}{16807 \left (2+3 x \right )}+\frac {1040 \ln \left (2+3 x \right )}{117649}\) | \(63\) |
parallelrisch | \(\frac {10782720 \ln \left (\frac {2}{3}+x \right ) x^{5}-10782720 \ln \left (x -\frac {1}{2}\right ) x^{5}+23362560 \ln \left (\frac {2}{3}+x \right ) x^{4}-23362560 \ln \left (x -\frac {1}{2}\right ) x^{4}+20026062 x^{5}+14376960 \ln \left (\frac {2}{3}+x \right ) x^{3}-14376960 \ln \left (x -\frac {1}{2}\right ) x^{3}+30809961 x^{4}-1597440 \ln \left (\frac {2}{3}+x \right ) x^{2}+1597440 \ln \left (x -\frac {1}{2}\right ) x^{2}+493416 x^{3}-4259840 \ln \left (\frac {2}{3}+x \right ) x +4259840 \ln \left (x -\frac {1}{2}\right ) x -18982824 x^{2}-1064960 \ln \left (\frac {2}{3}+x \right )+1064960 \ln \left (x -\frac {1}{2}\right )-7566944 x}{7529536 \left (-1+2 x \right ) \left (2+3 x \right )^{4}}\) | \(139\) |
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Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.51 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=-\frac {2358720 \, x^{4} + 4914000 \, x^{3} + 3003000 \, x^{2} - 12480 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 12480 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) - 64610 \, x - 370853}{1411788 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {- 336960 x^{4} - 702000 x^{3} - 429000 x^{2} + 9230 x + 52979}{32672808 x^{5} + 70791084 x^{4} + 43563744 x^{3} - 4840416 x^{2} - 12907776 x - 3226944} - \frac {1040 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {1040 \log {\left (x + \frac {2}{3} \right )}}{117649} \]
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Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=-\frac {336960 \, x^{4} + 702000 \, x^{3} + 429000 \, x^{2} - 9230 \, x - 52979}{201684 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {1040}{117649} \, \log \left (3 \, x + 2\right ) - \frac {1040}{117649} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=-\frac {388}{16807 \, {\left (3 \, x + 2\right )}} + \frac {528}{117649 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} - \frac {64}{2401 \, {\left (3 \, x + 2\right )}^{2}} - \frac {31}{1029 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{196 \, {\left (3 \, x + 2\right )}^{4}} - \frac {1040}{117649} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {2080\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {\frac {520\,x^4}{50421}+\frac {3250\,x^3}{151263}+\frac {17875\,x^2}{1361367}-\frac {4615\,x}{16336404}-\frac {52979}{32672808}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \]
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