Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=\frac {1}{1323 (2+3 x)^7}-\frac {103}{7938 (2+3 x)^6}+\frac {3469}{46305 (2+3 x)^5}-\frac {1331}{9604 (2+3 x)^4}-\frac {2662}{50421 (2+3 x)^3}-\frac {2662}{117649 (2+3 x)^2}-\frac {10648}{823543 (2+3 x)}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (2+3 x)}{5764801} \]
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Time = 0.03 (sec) , antiderivative size = 98, 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} \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {10648}{823543 (3 x+2)}-\frac {2662}{117649 (3 x+2)^2}-\frac {2662}{50421 (3 x+2)^3}-\frac {1331}{9604 (3 x+2)^4}+\frac {3469}{46305 (3 x+2)^5}-\frac {103}{7938 (3 x+2)^6}+\frac {1}{1323 (3 x+2)^7}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (3 x+2)}{5764801} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {42592}{5764801 (-1+2 x)}-\frac {1}{63 (2+3 x)^8}+\frac {103}{441 (2+3 x)^7}-\frac {3469}{3087 (2+3 x)^6}+\frac {3993}{2401 (2+3 x)^5}+\frac {7986}{16807 (2+3 x)^4}+\frac {15972}{117649 (2+3 x)^3}+\frac {31944}{823543 (2+3 x)^2}+\frac {63888}{5764801 (2+3 x)}\right ) \, dx \\ & = \frac {1}{1323 (2+3 x)^7}-\frac {103}{7938 (2+3 x)^6}+\frac {3469}{46305 (2+3 x)^5}-\frac {1331}{9604 (2+3 x)^4}-\frac {2662}{50421 (2+3 x)^3}-\frac {2662}{117649 (2+3 x)^2}-\frac {10648}{823543 (2+3 x)}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (2+3 x)}{5764801} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=\frac {4 \left (-\frac {7 \left (4309941128+29451465714 x+83293304778 x^2+127327486275 x^3+113990726520 x^4+57635760600 x^5+12575075040 x^6\right )}{16 (2+3 x)^7}-2156220 \log (1-2 x)+2156220 \log (4+6 x)\right )}{2334744405} \]
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Time = 0.83 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.57
method | result | size |
norman | \(\frac {-\frac {4908577619}{222356610} x -\frac {1542468607}{24706290} x^{2}-\frac {314388855}{3294172} x^{3}-\frac {70364646}{823543} x^{4}-\frac {35577630}{823543} x^{5}-\frac {7762392}{823543} x^{6}-\frac {1077485282}{333534915}}{\left (2+3 x \right )^{7}}-\frac {21296 \ln \left (-1+2 x \right )}{5764801}+\frac {21296 \ln \left (2+3 x \right )}{5764801}\) | \(56\) |
risch | \(\frac {-\frac {4908577619}{222356610} x -\frac {1542468607}{24706290} x^{2}-\frac {314388855}{3294172} x^{3}-\frac {70364646}{823543} x^{4}-\frac {35577630}{823543} x^{5}-\frac {7762392}{823543} x^{6}-\frac {1077485282}{333534915}}{\left (2+3 x \right )^{7}}-\frac {21296 \ln \left (-1+2 x \right )}{5764801}+\frac {21296 \ln \left (2+3 x \right )}{5764801}\) | \(57\) |
default | \(-\frac {21296 \ln \left (-1+2 x \right )}{5764801}+\frac {1}{1323 \left (2+3 x \right )^{7}}-\frac {103}{7938 \left (2+3 x \right )^{6}}+\frac {3469}{46305 \left (2+3 x \right )^{5}}-\frac {1331}{9604 \left (2+3 x \right )^{4}}-\frac {2662}{50421 \left (2+3 x \right )^{3}}-\frac {2662}{117649 \left (2+3 x \right )^{2}}-\frac {10648}{823543 \left (2+3 x \right )}+\frac {21296 \ln \left (2+3 x \right )}{5764801}\) | \(81\) |
parallelrisch | \(\frac {131105674560 x +618231398400 \ln \left (\frac {2}{3}+x \right ) x^{3}+247292559360 \ln \left (\frac {2}{3}+x \right ) x^{2}+54953902080 \ln \left (\frac {2}{3}+x \right ) x +5223888765144 x^{5}+2746699507692 x^{6}+610934154894 x^{7}+3167395752640 x^{3}+5389912615920 x^{4}+998470986240 x^{2}-927347097600 \ln \left (x -\frac {1}{2}\right ) x^{4}+927347097600 \ln \left (\frac {2}{3}+x \right ) x^{4}+5233704960 \ln \left (\frac {2}{3}+x \right )-618231398400 \ln \left (x -\frac {1}{2}\right ) x^{3}+89422755840 \ln \left (\frac {2}{3}+x \right ) x^{7}-247292559360 \ln \left (x -\frac {1}{2}\right ) x^{2}-54953902080 \ln \left (x -\frac {1}{2}\right ) x +834612387840 \ln \left (\frac {2}{3}+x \right ) x^{5}+417306193920 \ln \left (\frac {2}{3}+x \right ) x^{6}-5233704960 \ln \left (x -\frac {1}{2}\right )-89422755840 \ln \left (x -\frac {1}{2}\right ) x^{7}-417306193920 \ln \left (x -\frac {1}{2}\right ) x^{6}-834612387840 \ln \left (x -\frac {1}{2}\right ) x^{5}}{11068417920 \left (2+3 x \right )^{7}}\) | \(178\) |
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Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {88025525280 \, x^{6} + 403450324200 \, x^{5} + 797935085640 \, x^{4} + 891292403925 \, x^{3} + 583053133446 \, x^{2} - 34499520 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 34499520 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 206160259998 \, x + 30169587896}{9338977620 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=- \frac {12575075040 x^{6} + 57635760600 x^{5} + 113990726520 x^{4} + 127327486275 x^{3} + 83293304778 x^{2} + 29451465714 x + 4309941128}{2917763436420 x^{7} + 13616229369960 x^{6} + 27232458739920 x^{5} + 30258287488800 x^{4} + 20172191659200 x^{3} + 8068876663680 x^{2} + 1793083703040 x + 170769876480} - \frac {21296 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {21296 \log {\left (x + \frac {2}{3} \right )}}{5764801} \]
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Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {12575075040 \, x^{6} + 57635760600 \, x^{5} + 113990726520 \, x^{4} + 127327486275 \, x^{3} + 83293304778 \, x^{2} + 29451465714 \, x + 4309941128}{1334139660 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {21296}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {21296}{5764801} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {12575075040 \, x^{6} + 57635760600 \, x^{5} + 113990726520 \, x^{4} + 127327486275 \, x^{3} + 83293304778 \, x^{2} + 29451465714 \, x + 4309941128}{1334139660 \, {\left (3 \, x + 2\right )}^{7}} + \frac {21296}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {21296}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=\frac {42592\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {10648\,x^6}{2470629}+\frac {146410\,x^5}{7411887}+\frac {2606098\,x^4}{66706983}+\frac {34932095\,x^3}{800483796}+\frac {1542468607\,x^2}{54032656230}+\frac {4908577619\,x}{486293906070}+\frac {1077485282}{729440859105}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \]
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