Integrand size = 20, antiderivative size = 98 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=\frac {1}{147 (2+3 x)^7}-\frac {11}{294 (2+3 x)^6}-\frac {22}{1715 (2+3 x)^5}-\frac {11}{2401 (2+3 x)^4}-\frac {88}{50421 (2+3 x)^3}-\frac {88}{117649 (2+3 x)^2}-\frac {352}{823543 (2+3 x)}-\frac {704 \log (1-2 x)}{5764801}+\frac {704 \log (2+3 x)}{5764801} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=-\frac {352}{823543 (3 x+2)}-\frac {88}{117649 (3 x+2)^2}-\frac {88}{50421 (3 x+2)^3}-\frac {11}{2401 (3 x+2)^4}-\frac {22}{1715 (3 x+2)^5}-\frac {11}{294 (3 x+2)^6}+\frac {1}{147 (3 x+2)^7}-\frac {704 \log (1-2 x)}{5764801}+\frac {704 \log (3 x+2)}{5764801} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1408}{5764801 (-1+2 x)}-\frac {1}{7 (2+3 x)^8}+\frac {33}{49 (2+3 x)^7}+\frac {66}{343 (2+3 x)^6}+\frac {132}{2401 (2+3 x)^5}+\frac {264}{16807 (2+3 x)^4}+\frac {528}{117649 (2+3 x)^3}+\frac {1056}{823543 (2+3 x)^2}+\frac {2112}{5764801 (2+3 x)}\right ) \, dx \\ & = \frac {1}{147 (2+3 x)^7}-\frac {11}{294 (2+3 x)^6}-\frac {22}{1715 (2+3 x)^5}-\frac {11}{2401 (2+3 x)^4}-\frac {88}{50421 (2+3 x)^3}-\frac {88}{117649 (2+3 x)^2}-\frac {352}{823543 (2+3 x)}-\frac {704 \log (1-2 x)}{5764801}+\frac {704 \log (2+3 x)}{5764801} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.61 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=\frac {-\frac {7 \left (5811068+25308459 x+54393768 x^2+77947650 x^3+69783120 x^4+35283600 x^5+7698240 x^6\right )}{(2+3 x)^7}-21120 \log (3-6 x)+21120 \log (2+3 x)}{172944030} \]
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Time = 2.50 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.57
method | result | size |
norman | \(\frac {-\frac {9065628}{4117715} x^{2}-\frac {8436153}{8235430} x -\frac {2598255}{823543} x^{3}-\frac {2326104}{823543} x^{4}-\frac {1176120}{823543} x^{5}-\frac {256608}{823543} x^{6}-\frac {2905534}{12353145}}{\left (2+3 x \right )^{7}}-\frac {704 \ln \left (-1+2 x \right )}{5764801}+\frac {704 \ln \left (2+3 x \right )}{5764801}\) | \(56\) |
risch | \(\frac {-\frac {9065628}{4117715} x^{2}-\frac {8436153}{8235430} x -\frac {2598255}{823543} x^{3}-\frac {2326104}{823543} x^{4}-\frac {1176120}{823543} x^{5}-\frac {256608}{823543} x^{6}-\frac {2905534}{12353145}}{\left (2+3 x \right )^{7}}-\frac {704 \ln \left (-1+2 x \right )}{5764801}+\frac {704 \ln \left (2+3 x \right )}{5764801}\) | \(57\) |
default | \(-\frac {704 \ln \left (-1+2 x \right )}{5764801}+\frac {1}{147 \left (2+3 x \right )^{7}}-\frac {11}{294 \left (2+3 x \right )^{6}}-\frac {22}{1715 \left (2+3 x \right )^{5}}-\frac {11}{2401 \left (2+3 x \right )^{4}}-\frac {88}{50421 \left (2+3 x \right )^{3}}-\frac {88}{117649 \left (2+3 x \right )^{2}}-\frac {352}{823543 \left (2+3 x \right )}+\frac {704 \ln \left (2+3 x \right )}{5764801}\) | \(81\) |
parallelrisch | \(\frac {5332358080 x +6812467200 \ln \left (\frac {2}{3}+x \right ) x^{3}+2724986880 \ln \left (\frac {2}{3}+x \right ) x^{2}+605552640 \ln \left (\frac {2}{3}+x \right ) x +133115755752 x^{5}+68042782836 x^{6}+14826940002 x^{7}+90867057120 x^{3}+143339913360 x^{4}+32880093120 x^{2}-10218700800 \ln \left (x -\frac {1}{2}\right ) x^{4}+10218700800 \ln \left (\frac {2}{3}+x \right ) x^{4}+57671680 \ln \left (\frac {2}{3}+x \right )-6812467200 \ln \left (x -\frac {1}{2}\right ) x^{3}+985374720 \ln \left (\frac {2}{3}+x \right ) x^{7}-2724986880 \ln \left (x -\frac {1}{2}\right ) x^{2}-605552640 \ln \left (x -\frac {1}{2}\right ) x +9196830720 \ln \left (\frac {2}{3}+x \right ) x^{5}+4598415360 \ln \left (\frac {2}{3}+x \right ) x^{6}-57671680 \ln \left (x -\frac {1}{2}\right )-985374720 \ln \left (x -\frac {1}{2}\right ) x^{7}-4598415360 \ln \left (x -\frac {1}{2}\right ) x^{6}-9196830720 \ln \left (x -\frac {1}{2}\right ) x^{5}}{3689472640 \left (2+3 x \right )^{7}}\) | \(178\) |
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Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=-\frac {53887680 \, x^{6} + 246985200 \, x^{5} + 488481840 \, x^{4} + 545633550 \, x^{3} + 380756376 \, x^{2} - 21120 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 21120 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 177159213 \, x + 40677476}{172944030 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=- \frac {7698240 x^{6} + 35283600 x^{5} + 69783120 x^{4} + 77947650 x^{3} + 54393768 x^{2} + 25308459 x + 5811068}{54032656230 x^{7} + 252152395740 x^{6} + 504304791480 x^{5} + 560338657200 x^{4} + 373559104800 x^{3} + 149423641920 x^{2} + 33205253760 x + 3162405120} - \frac {704 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {704 \log {\left (x + \frac {2}{3} \right )}}{5764801} \]
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Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=-\frac {7698240 \, x^{6} + 35283600 \, x^{5} + 69783120 \, x^{4} + 77947650 \, x^{3} + 54393768 \, x^{2} + 25308459 \, x + 5811068}{24706290 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {704}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {704}{5764801} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=-\frac {7698240 \, x^{6} + 35283600 \, x^{5} + 69783120 \, x^{4} + 77947650 \, x^{3} + 54393768 \, x^{2} + 25308459 \, x + 5811068}{24706290 \, {\left (3 \, x + 2\right )}^{7}} + \frac {704}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {704}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^8} \, dx=\frac {1408\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {352\,x^6}{2470629}+\frac {4840\,x^5}{7411887}+\frac {86152\,x^4}{66706983}+\frac {288695\,x^3}{200120949}+\frac {335764\,x^2}{333534915}+\frac {2812051\,x}{6003628470}+\frac {2905534}{27016328115}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \]
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