Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {1}{441 (2+3 x)^7}+\frac {34}{1323 (2+3 x)^6}-\frac {121}{1715 (2+3 x)^5}-\frac {121}{4802 (2+3 x)^4}-\frac {484}{50421 (2+3 x)^3}-\frac {484}{117649 (2+3 x)^2}-\frac {1936}{823543 (2+3 x)}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (2+3 x)}{5764801} \]
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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} \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {1936}{823543 (3 x+2)}-\frac {484}{117649 (3 x+2)^2}-\frac {484}{50421 (3 x+2)^3}-\frac {121}{4802 (3 x+2)^4}-\frac {121}{1715 (3 x+2)^5}+\frac {34}{1323 (3 x+2)^6}-\frac {1}{441 (3 x+2)^7}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (3 x+2)}{5764801} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {7744}{5764801 (-1+2 x)}+\frac {1}{21 (2+3 x)^8}-\frac {68}{147 (2+3 x)^7}+\frac {363}{343 (2+3 x)^6}+\frac {726}{2401 (2+3 x)^5}+\frac {1452}{16807 (2+3 x)^4}+\frac {2904}{117649 (2+3 x)^3}+\frac {5808}{823543 (2+3 x)^2}+\frac {11616}{5764801 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{441 (2+3 x)^7}+\frac {34}{1323 (2+3 x)^6}-\frac {121}{1715 (2+3 x)^5}-\frac {121}{4802 (2+3 x)^4}-\frac {484}{50421 (2+3 x)^3}-\frac {484}{117649 (2+3 x)^2}-\frac {1936}{823543 (2+3 x)}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (2+3 x)}{5764801} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=\frac {8 \left (-\frac {7 \left (193528666+1098354408 x+2692491516 x^2+3858408675 x^3+3454264440 x^4+1746538200 x^5+381062880 x^6\right )}{16 (2+3 x)^7}-65340 \log (1-2 x)+65340 \log (4+6 x)\right )}{778248135} \]
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Time = 2.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.57
method | result | size |
norman | \(\frac {-\frac {183059068}{37059435} x -\frac {49860954}{4117715} x^{2}-\frac {28580805}{1647086} x^{3}-\frac {12793572}{823543} x^{4}-\frac {6468660}{823543} x^{5}-\frac {1411344}{823543} x^{6}-\frac {96764333}{111178305}}{\left (2+3 x \right )^{7}}-\frac {3872 \ln \left (-1+2 x \right )}{5764801}+\frac {3872 \ln \left (2+3 x \right )}{5764801}\) | \(56\) |
risch | \(\frac {-\frac {183059068}{37059435} x -\frac {49860954}{4117715} x^{2}-\frac {28580805}{1647086} x^{3}-\frac {12793572}{823543} x^{4}-\frac {6468660}{823543} x^{5}-\frac {1411344}{823543} x^{6}-\frac {96764333}{111178305}}{\left (2+3 x \right )^{7}}-\frac {3872 \ln \left (-1+2 x \right )}{5764801}+\frac {3872 \ln \left (2+3 x \right )}{5764801}\) | \(57\) |
default | \(-\frac {3872 \ln \left (-1+2 x \right )}{5764801}-\frac {1}{441 \left (2+3 x \right )^{7}}+\frac {34}{1323 \left (2+3 x \right )^{6}}-\frac {121}{1715 \left (2+3 x \right )^{5}}-\frac {121}{4802 \left (2+3 x \right )^{4}}-\frac {484}{50421 \left (2+3 x \right )^{3}}-\frac {484}{117649 \left (2+3 x \right )^{2}}-\frac {1936}{823543 \left (2+3 x \right )}+\frac {3872 \ln \left (2+3 x \right )}{5764801}\) | \(81\) |
parallelrisch | \(\frac {15492447040 x +37468569600 \ln \left (\frac {2}{3}+x \right ) x^{3}+14987427840 \ln \left (\frac {2}{3}+x \right ) x^{2}+3330539520 \ln \left (\frac {2}{3}+x \right ) x +483097253436 x^{5}+249715603998 x^{6}+54865376811 x^{7}+315295182160 x^{3}+511659075480 x^{4}+107051059360 x^{2}-56202854400 \ln \left (x -\frac {1}{2}\right ) x^{4}+56202854400 \ln \left (\frac {2}{3}+x \right ) x^{4}+317194240 \ln \left (\frac {2}{3}+x \right )-37468569600 \ln \left (x -\frac {1}{2}\right ) x^{3}+5419560960 \ln \left (\frac {2}{3}+x \right ) x^{7}-14987427840 \ln \left (x -\frac {1}{2}\right ) x^{2}-3330539520 \ln \left (x -\frac {1}{2}\right ) x +50582568960 \ln \left (\frac {2}{3}+x \right ) x^{5}+25291284480 \ln \left (\frac {2}{3}+x \right ) x^{6}-317194240 \ln \left (x -\frac {1}{2}\right )-5419560960 \ln \left (x -\frac {1}{2}\right ) x^{7}-25291284480 \ln \left (x -\frac {1}{2}\right ) x^{6}-50582568960 \ln \left (x -\frac {1}{2}\right ) x^{5}}{3689472640 \left (2+3 x \right )^{7}}\) | \(178\) |
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Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {2667440160 \, x^{6} + 12225767400 \, x^{5} + 24179851080 \, x^{4} + 27008860725 \, x^{3} + 18847440612 \, x^{2} - 1045440 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 1045440 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 7688480856 \, x + 1354700662}{1556496270 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=- \frac {381062880 x^{6} + 1746538200 x^{5} + 3454264440 x^{4} + 3858408675 x^{3} + 2692491516 x^{2} + 1098354408 x + 193528666}{486293906070 x^{7} + 2269371561660 x^{6} + 4538743123320 x^{5} + 5043047914800 x^{4} + 3362031943200 x^{3} + 1344812777280 x^{2} + 298847283840 x + 28461646080} - \frac {3872 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {3872 \log {\left (x + \frac {2}{3} \right )}}{5764801} \]
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Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {3872}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {3872}{5764801} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \, {\left (3 \, x + 2\right )}^{7}} + \frac {3872}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3872}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=\frac {7744\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {1936\,x^6}{2470629}+\frac {26620\,x^5}{7411887}+\frac {473836\,x^4}{66706983}+\frac {3175645\,x^3}{400241898}+\frac {1846702\,x^2}{333534915}+\frac {183059068\,x}{81048984345}+\frac {96764333}{243146953035}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \]
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