Integrand size = 20, antiderivative size = 87 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {44}{16807 (1-2 x)^2}+\frac {1040}{117649 (1-2 x)}+\frac {3}{1372 (2+3 x)^4}-\frac {29}{2401 (2+3 x)^3}-\frac {279}{16807 (2+3 x)^2}-\frac {2280}{117649 (2+3 x)}-\frac {7680 \log (1-2 x)}{823543}+\frac {7680 \log (2+3 x)}{823543} \]
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Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {1040}{117649 (1-2 x)}-\frac {2280}{117649 (3 x+2)}+\frac {44}{16807 (1-2 x)^2}-\frac {279}{16807 (3 x+2)^2}-\frac {29}{2401 (3 x+2)^3}+\frac {3}{1372 (3 x+2)^4}-\frac {7680 \log (1-2 x)}{823543}+\frac {7680 \log (3 x+2)}{823543} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {176}{16807 (-1+2 x)^3}+\frac {2080}{117649 (-1+2 x)^2}-\frac {15360}{823543 (-1+2 x)}-\frac {9}{343 (2+3 x)^5}+\frac {261}{2401 (2+3 x)^4}+\frac {1674}{16807 (2+3 x)^3}+\frac {6840}{117649 (2+3 x)^2}+\frac {23040}{823543 (2+3 x)}\right ) \, dx \\ & = \frac {44}{16807 (1-2 x)^2}+\frac {1040}{117649 (1-2 x)}+\frac {3}{1372 (2+3 x)^4}-\frac {29}{2401 (2+3 x)^3}-\frac {279}{16807 (2+3 x)^2}-\frac {2280}{117649 (2+3 x)}-\frac {7680 \log (1-2 x)}{823543}+\frac {7680 \log (2+3 x)}{823543} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {4 \left (-\frac {7 \left (28275-403584 x-1101440 x^2+384000 x^3+2626560 x^4+1658880 x^5\right )}{16 (1-2 x)^2 (2+3 x)^4}-1920 \log (1-2 x)+1920 \log (4+6 x)\right )}{823543} \]
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Time = 2.74 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {-\frac {656640}{117649} x^{4}-\frac {414720}{117649} x^{5}-\frac {96000}{117649} x^{3}+\frac {100896}{117649} x +\frac {275360}{117649} x^{2}-\frac {28275}{470596}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}-\frac {7680 \ln \left (-1+2 x \right )}{823543}+\frac {7680 \ln \left (2+3 x \right )}{823543}\) | \(58\) |
risch | \(\frac {-\frac {656640}{117649} x^{4}-\frac {414720}{117649} x^{5}-\frac {96000}{117649} x^{3}+\frac {100896}{117649} x +\frac {275360}{117649} x^{2}-\frac {28275}{470596}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}-\frac {7680 \ln \left (-1+2 x \right )}{823543}+\frac {7680 \ln \left (2+3 x \right )}{823543}\) | \(59\) |
default | \(\frac {44}{16807 \left (-1+2 x \right )^{2}}-\frac {1040}{117649 \left (-1+2 x \right )}-\frac {7680 \ln \left (-1+2 x \right )}{823543}+\frac {3}{1372 \left (2+3 x \right )^{4}}-\frac {29}{2401 \left (2+3 x \right )^{3}}-\frac {279}{16807 \left (2+3 x \right )^{2}}-\frac {2280}{117649 \left (2+3 x \right )}+\frac {7680 \ln \left (2+3 x \right )}{823543}\) | \(72\) |
parallelrisch | \(\frac {51535008 x -129761280 \ln \left (\frac {2}{3}+x \right ) x^{3}-51118080 \ln \left (\frac {2}{3}+x \right ) x^{2}+15728640 \ln \left (\frac {2}{3}+x \right ) x -78915060 x^{5}+64127700 x^{6}-95260200 x^{3}-278142795 x^{4}+102777080 x^{2}-39813120 \ln \left (x -\frac {1}{2}\right ) x^{4}+39813120 \ln \left (\frac {2}{3}+x \right ) x^{4}+7864320 \ln \left (\frac {2}{3}+x \right )+129761280 \ln \left (x -\frac {1}{2}\right ) x^{3}+51118080 \ln \left (x -\frac {1}{2}\right ) x^{2}-15728640 \ln \left (x -\frac {1}{2}\right ) x +265420800 \ln \left (\frac {2}{3}+x \right ) x^{5}+159252480 \ln \left (\frac {2}{3}+x \right ) x^{6}-7864320 \ln \left (x -\frac {1}{2}\right )-159252480 \ln \left (x -\frac {1}{2}\right ) x^{6}-265420800 \ln \left (x -\frac {1}{2}\right ) x^{5}}{52706752 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}\) | \(162\) |
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Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {11612160 \, x^{5} + 18385920 \, x^{4} + 2688000 \, x^{3} - 7710080 \, x^{2} - 30720 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 30720 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 2825088 \, x + 197925}{3294172 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=- \frac {1658880 x^{5} + 2626560 x^{4} + 384000 x^{3} - 1101440 x^{2} - 403584 x + 28275}{152473104 x^{6} + 254121840 x^{5} + 38118276 x^{4} - 124237344 x^{3} - 48941984 x^{2} + 15059072 x + 7529536} - \frac {7680 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {7680 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
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Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {1658880 \, x^{5} + 2626560 \, x^{4} + 384000 \, x^{3} - 1101440 \, x^{2} - 403584 \, x + 28275}{470596 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac {7680}{823543} \, \log \left (3 \, x + 2\right ) - \frac {7680}{823543} \, \log \left (2 \, x - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {2280}{117649 \, {\left (3 \, x + 2\right )}} + \frac {48 \, {\left (\frac {1141}{3 \, x + 2} - 293\right )}}{823543 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} - \frac {279}{16807 \, {\left (3 \, x + 2\right )}^{2}} - \frac {29}{2401 \, {\left (3 \, x + 2\right )}^{3}} + \frac {3}{1372 \, {\left (3 \, x + 2\right )}^{4}} - \frac {7680}{823543} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {15360\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {1280\,x^5}{117649}+\frac {6080\,x^4}{352947}+\frac {8000\,x^3}{3176523}-\frac {68840\,x^2}{9529569}-\frac {8408\,x}{3176523}+\frac {9425}{50824368}}{x^6+\frac {5\,x^5}{3}+\frac {x^4}{4}-\frac {22\,x^3}{27}-\frac {26\,x^2}{81}+\frac {8\,x}{81}+\frac {4}{81}} \]
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