Integrand size = 22, antiderivative size = 87 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=\frac {1}{1134 (2+3 x)^6}-\frac {103}{6615 (2+3 x)^5}+\frac {3469}{37044 (2+3 x)^4}-\frac {1331}{7203 (2+3 x)^3}-\frac {1331}{16807 (2+3 x)^2}-\frac {5324}{117649 (2+3 x)}-\frac {10648 \log (1-2 x)}{823543}+\frac {10648 \log (2+3 x)}{823543} \]
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Time = 0.02 (sec) , antiderivative size = 87, 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} \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=-\frac {5324}{117649 (3 x+2)}-\frac {1331}{16807 (3 x+2)^2}-\frac {1331}{7203 (3 x+2)^3}+\frac {3469}{37044 (3 x+2)^4}-\frac {103}{6615 (3 x+2)^5}+\frac {1}{1134 (3 x+2)^6}-\frac {10648 \log (1-2 x)}{823543}+\frac {10648 \log (3 x+2)}{823543} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {21296}{823543 (-1+2 x)}-\frac {1}{63 (2+3 x)^7}+\frac {103}{441 (2+3 x)^6}-\frac {3469}{3087 (2+3 x)^5}+\frac {3993}{2401 (2+3 x)^4}+\frac {7986}{16807 (2+3 x)^3}+\frac {15972}{117649 (2+3 x)^2}+\frac {31944}{823543 (2+3 x)}\right ) \, dx \\ & = \frac {1}{1134 (2+3 x)^6}-\frac {103}{6615 (2+3 x)^5}+\frac {3469}{37044 (2+3 x)^4}-\frac {1331}{7203 (2+3 x)^3}-\frac {1331}{16807 (2+3 x)^2}-\frac {5324}{117649 (2+3 x)}-\frac {10648 \log (1-2 x)}{823543}+\frac {10648 \log (2+3 x)}{823543} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=\frac {4 \left (-\frac {7 \left (733614062+4581535248 x+11211272235 x^2+13525968060 x^3+8208729540 x^4+2095845840 x^5\right )}{16 (2+3 x)^6}-1078110 \log (1-2 x)+1078110 \log (4+6 x)\right )}{333534915} \]
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Time = 3.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59
method | result | size |
norman | \(\frac {-\frac {127264868}{5294205} x -\frac {83046461}{1411788} x^{2}-\frac {8349363}{117649} x^{3}-\frac {5067117}{117649} x^{4}-\frac {1293732}{117649} x^{5}-\frac {366807031}{95295690}}{\left (2+3 x \right )^{6}}-\frac {10648 \ln \left (-1+2 x \right )}{823543}+\frac {10648 \ln \left (2+3 x \right )}{823543}\) | \(51\) |
risch | \(\frac {-\frac {127264868}{5294205} x -\frac {83046461}{1411788} x^{2}-\frac {8349363}{117649} x^{3}-\frac {5067117}{117649} x^{4}-\frac {1293732}{117649} x^{5}-\frac {366807031}{95295690}}{\left (2+3 x \right )^{6}}-\frac {10648 \ln \left (-1+2 x \right )}{823543}+\frac {10648 \ln \left (2+3 x \right )}{823543}\) | \(52\) |
default | \(-\frac {10648 \ln \left (-1+2 x \right )}{823543}+\frac {1}{1134 \left (2+3 x \right )^{6}}-\frac {103}{6615 \left (2+3 x \right )^{5}}+\frac {3469}{37044 \left (2+3 x \right )^{4}}-\frac {1331}{7203 \left (2+3 x \right )^{3}}-\frac {1331}{16807 \left (2+3 x \right )^{2}}-\frac {5324}{117649 \left (2+3 x \right )}+\frac {10648 \ln \left (2+3 x \right )}{823543}\) | \(72\) |
parallelrisch | \(\frac {16766742720 x +88318771200 \ln \left (\frac {2}{3}+x \right ) x^{3}+44159385600 \ln \left (\frac {2}{3}+x \right ) x^{2}+11775836160 \ln \left (\frac {2}{3}+x \right ) x +259918357356 x^{5}+69326528859 x^{6}+298608436000 x^{3}+394074806580 x^{4}+112399901040 x^{2}-99358617600 \ln \left (x -\frac {1}{2}\right ) x^{4}+99358617600 \ln \left (\frac {2}{3}+x \right ) x^{4}+1308426240 \ln \left (\frac {2}{3}+x \right )-88318771200 \ln \left (x -\frac {1}{2}\right ) x^{3}-44159385600 \ln \left (x -\frac {1}{2}\right ) x^{2}-11775836160 \ln \left (x -\frac {1}{2}\right ) x +59615170560 \ln \left (\frac {2}{3}+x \right ) x^{5}+14903792640 \ln \left (\frac {2}{3}+x \right ) x^{6}-1308426240 \ln \left (x -\frac {1}{2}\right )-14903792640 \ln \left (x -\frac {1}{2}\right ) x^{6}-59615170560 \ln \left (x -\frac {1}{2}\right ) x^{5}}{1581202560 \left (2+3 x \right )^{6}}\) | \(155\) |
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Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=-\frac {14670920880 \, x^{5} + 57461106780 \, x^{4} + 94681776420 \, x^{3} + 78478905645 \, x^{2} - 17249760 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 17249760 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 32070746736 \, x + 5135298434}{1334139660 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=- \frac {2095845840 x^{5} + 8208729540 x^{4} + 13525968060 x^{3} + 11211272235 x^{2} + 4581535248 x + 733614062}{138941116020 x^{6} + 555764464080 x^{5} + 926274106800 x^{4} + 823354761600 x^{3} + 411677380800 x^{2} + 109780634880 x + 12197848320} - \frac {10648 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {10648 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
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Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=-\frac {2095845840 \, x^{5} + 8208729540 \, x^{4} + 13525968060 \, x^{3} + 11211272235 \, x^{2} + 4581535248 \, x + 733614062}{190591380 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {10648}{823543} \, \log \left (3 \, x + 2\right ) - \frac {10648}{823543} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=-\frac {2095845840 \, x^{5} + 8208729540 \, x^{4} + 13525968060 \, x^{3} + 11211272235 \, x^{2} + 4581535248 \, x + 733614062}{190591380 \, {\left (3 \, x + 2\right )}^{6}} + \frac {10648}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {10648}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 1.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^7} \, dx=\frac {21296\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {5324\,x^5}{352947}+\frac {62557\,x^4}{1058841}+\frac {927707\,x^3}{9529569}+\frac {83046461\,x^2}{1029193452}+\frac {127264868\,x}{3859475445}+\frac {366807031}{69470558010}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \]
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