Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {1936}{823543 (1-2 x)}-\frac {1}{882 (2+3 x)^6}+\frac {22}{1715 (2+3 x)^5}-\frac {319}{9604 (2+3 x)^4}-\frac {1364}{50421 (2+3 x)^3}-\frac {2090}{117649 (2+3 x)^2}-\frac {11264}{823543 (2+3 x)}-\frac {4048 \log (1-2 x)}{823543}+\frac {4048 \log (2+3 x)}{823543} \]
1936/823543/(1-2*x)-1/882/(2+3*x)^6+22/1715/(2+3*x)^5-319/9604/(2+3*x)^4-1 364/50421/(2+3*x)^3-2090/117649/(2+3*x)^2-11264/823543/(2+3*x)-4048/823543 *ln(1-2*x)+4048/823543*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {4 \left (-\frac {7 \left (-18979078-60874336 x+48220029 x^2+459657990 x^3+795948120 x^4+604953360 x^5+177059520 x^6\right )}{16 (-1+2 x) (2+3 x)^6}-45540 \log (1-2 x)+45540 \log (4+6 x)\right )}{37059435} \]
(4*((-7*(-18979078 - 60874336*x + 48220029*x^2 + 459657990*x^3 + 795948120 *x^4 + 604953360*x^5 + 177059520*x^6))/(16*(-1 + 2*x)*(2 + 3*x)^6) - 45540 *Log[1 - 2*x] + 45540*Log[4 + 6*x]))/37059435
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x)^2 (3 x+2)^7} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {12144}{823543 (3 x+2)}+\frac {33792}{823543 (3 x+2)^2}+\frac {12540}{117649 (3 x+2)^3}+\frac {4092}{16807 (3 x+2)^4}+\frac {957}{2401 (3 x+2)^5}-\frac {66}{343 (3 x+2)^6}+\frac {1}{49 (3 x+2)^7}-\frac {8096}{823543 (2 x-1)}+\frac {3872}{823543 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1936}{823543 (1-2 x)}-\frac {11264}{823543 (3 x+2)}-\frac {2090}{117649 (3 x+2)^2}-\frac {1364}{50421 (3 x+2)^3}-\frac {319}{9604 (3 x+2)^4}+\frac {22}{1715 (3 x+2)^5}-\frac {1}{882 (3 x+2)^6}-\frac {4048 \log (1-2 x)}{823543}+\frac {4048 \log (3 x+2)}{823543}\) |
1936/(823543*(1 - 2*x)) - 1/(882*(2 + 3*x)^6) + 22/(1715*(2 + 3*x)^5) - 31 9/(9604*(2 + 3*x)^4) - 1364/(50421*(2 + 3*x)^3) - 2090/(117649*(2 + 3*x)^2 ) - 11264/(823543*(2 + 3*x)) - (4048*Log[1 - 2*x])/823543 + (4048*Log[2 + 3*x])/823543
3.16.69.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 2.64 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.64
method | result | size |
norman | \(\frac {-\frac {5357781}{2352980} x^{2}-\frac {4421934}{117649} x^{4}-\frac {3360852}{117649} x^{5}-\frac {983664}{117649} x^{6}+\frac {15218584}{5294205} x -\frac {5107311}{235298} x^{3}+\frac {9489539}{10588410}}{\left (-1+2 x \right ) \left (2+3 x \right )^{6}}-\frac {4048 \ln \left (-1+2 x \right )}{823543}+\frac {4048 \ln \left (2+3 x \right )}{823543}\) | \(63\) |
risch | \(\frac {-\frac {5357781}{2352980} x^{2}-\frac {4421934}{117649} x^{4}-\frac {3360852}{117649} x^{5}-\frac {983664}{117649} x^{6}+\frac {15218584}{5294205} x -\frac {5107311}{235298} x^{3}+\frac {9489539}{10588410}}{\left (-1+2 x \right ) \left (2+3 x \right )^{6}}-\frac {4048 \ln \left (-1+2 x \right )}{823543}+\frac {4048 \ln \left (2+3 x \right )}{823543}\) | \(64\) |
default | \(-\frac {1936}{823543 \left (-1+2 x \right )}-\frac {4048 \ln \left (-1+2 x \right )}{823543}-\frac {1}{882 \left (2+3 x \right )^{6}}+\frac {22}{1715 \left (2+3 x \right )^{5}}-\frac {319}{9604 \left (2+3 x \right )^{4}}-\frac {1364}{50421 \left (2+3 x \right )^{3}}-\frac {2090}{117649 \left (2+3 x \right )^{2}}-\frac {11264}{823543 \left (2+3 x \right )}+\frac {4048 \ln \left (2+3 x \right )}{823543}\) | \(81\) |
parallelrisch | \(\frac {-1791482560 x -2611445760 \ln \left (\frac {2}{3}+x \right ) x^{2}-1160642560 \ln \left (\frac {2}{3}+x \right ) x +35162023428 x^{5}+33257165571 x^{6}+10761137226 x^{7}-11440376640 x^{3}+8088980340 x^{4}-8639941520 x^{2}-9792921600 \ln \left (x -\frac {1}{2}\right ) x^{4}+9792921600 \ln \left (\frac {2}{3}+x \right ) x^{4}-165806080 \ln \left (\frac {2}{3}+x \right )+3777269760 \ln \left (\frac {2}{3}+x \right ) x^{7}+2611445760 \ln \left (x -\frac {1}{2}\right ) x^{2}+1160642560 \ln \left (x -\frac {1}{2}\right ) x +17627258880 \ln \left (\frac {2}{3}+x \right ) x^{5}+13220444160 \ln \left (\frac {2}{3}+x \right ) x^{6}+165806080 \ln \left (x -\frac {1}{2}\right )-3777269760 \ln \left (x -\frac {1}{2}\right ) x^{7}-13220444160 \ln \left (x -\frac {1}{2}\right ) x^{6}-17627258880 \ln \left (x -\frac {1}{2}\right ) x^{5}}{527067520 \left (-1+2 x \right ) \left (2+3 x \right )^{6}}\) | \(167\) |
(-5357781/2352980*x^2-4421934/117649*x^4-3360852/117649*x^5-983664/117649* x^6+15218584/5294205*x-5107311/235298*x^3+9489539/10588410)/(-1+2*x)/(2+3* x)^6-4048/823543*ln(-1+2*x)+4048/823543*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.43 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=-\frac {1239416640 \, x^{6} + 4234673520 \, x^{5} + 5571636840 \, x^{4} + 3217605930 \, x^{3} + 337540203 \, x^{2} - 728640 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (3 \, x + 2\right ) + 728640 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (2 \, x - 1\right ) - 426120352 \, x - 132853546}{148237740 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} \]
-1/148237740*(1239416640*x^6 + 4234673520*x^5 + 5571636840*x^4 + 321760593 0*x^3 + 337540203*x^2 - 728640*(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(3*x + 2) + 728640*(1458*x^7 + 5103*x^6 + 6804 *x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(2*x - 1) - 426120352*x - 1328 53546)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {- 177059520 x^{6} - 604953360 x^{5} - 795948120 x^{4} - 459657990 x^{3} - 48220029 x^{2} + 60874336 x + 18979078}{30875803560 x^{7} + 108065312460 x^{6} + 144087083280 x^{5} + 80048379600 x^{4} - 21346234560 x^{2} - 9487215360 x - 1355316480} - \frac {4048 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {4048 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
(-177059520*x**6 - 604953360*x**5 - 795948120*x**4 - 459657990*x**3 - 4822 0029*x**2 + 60874336*x + 18979078)/(30875803560*x**7 + 108065312460*x**6 + 144087083280*x**5 + 80048379600*x**4 - 21346234560*x**2 - 9487215360*x - 1355316480) - 4048*log(x - 1/2)/823543 + 4048*log(x + 2/3)/823543
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=-\frac {177059520 \, x^{6} + 604953360 \, x^{5} + 795948120 \, x^{4} + 459657990 \, x^{3} + 48220029 \, x^{2} - 60874336 \, x - 18979078}{21176820 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} + \frac {4048}{823543} \, \log \left (3 \, x + 2\right ) - \frac {4048}{823543} \, \log \left (2 \, x - 1\right ) \]
-1/21176820*(177059520*x^6 + 604953360*x^5 + 795948120*x^4 + 459657990*x^3 + 48220029*x^2 - 60874336*x - 18979078)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64) + 4048/823543*log(3*x + 2) - 4048/82354 3*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=-\frac {1936}{823543 \, {\left (2 \, x - 1\right )}} + \frac {4 \, {\left (\frac {407084454}{2 \, x - 1} + \frac {2053765665}{{\left (2 \, x - 1\right )}^{2}} + \frac {5220014100}{{\left (2 \, x - 1\right )}^{3}} + \frac {6680782500}{{\left (2 \, x - 1\right )}^{4}} + \frac {3440056760}{{\left (2 \, x - 1\right )}^{5}} + 32498901\right )}}{28824005 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{6}} + \frac {4048}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]
-1936/823543/(2*x - 1) + 4/28824005*(407084454/(2*x - 1) + 2053765665/(2*x - 1)^2 + 5220014100/(2*x - 1)^3 + 6680782500/(2*x - 1)^4 + 3440056760/(2* x - 1)^5 + 32498901)/(7/(2*x - 1) + 3)^6 + 4048/823543*log(abs(-7/(2*x - 1 ) - 3))
Time = 1.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx=\frac {8096\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {2024\,x^6}{352947}+\frac {20746\,x^5}{1058841}+\frac {245663\,x^4}{9529569}+\frac {567479\,x^3}{38118276}+\frac {595309\,x^2}{381182760}-\frac {7609292\,x}{3859475445}-\frac {9489539}{15437901780}}{x^7+\frac {7\,x^6}{2}+\frac {14\,x^5}{3}+\frac {70\,x^4}{27}-\frac {56\,x^2}{81}-\frac {224\,x}{729}-\frac {32}{729}} \]