Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {32}{290521 (1-2 x)}-\frac {9}{49 (2+3 x)^3}-\frac {999}{343 (2+3 x)^2}-\frac {107109}{2401 (2+3 x)}-\frac {3125}{121 (3+5 x)}-\frac {6464 \log (1-2 x)}{22370117}+\frac {5050944 \log (2+3 x)}{16807}-\frac {400000 \log (3+5 x)}{1331} \]
32/290521/(1-2*x)-9/49/(2+3*x)^3-999/343/(2+3*x)^2-107109/2401/(2+3*x)-312 5/121/(3+5*x)-6464/22370117*ln(1-2*x)+5050944/16807*ln(2+3*x)-400000/1331* ln(3+5*x)
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {-\frac {77 \left (-220783501-570653522 x+479067048 x^2+2305013328 x^3+1571590080 x^4\right )}{(2+3 x)^3 \left (-3+x+10 x^2\right )}-6464 \log (3-6 x)+6722806464 \log (2+3 x)-6722800000 \log (-3 (3+5 x))}{22370117} \]
((-77*(-220783501 - 570653522*x + 479067048*x^2 + 2305013328*x^3 + 1571590 080*x^4))/((2 + 3*x)^3*(-3 + x + 10*x^2)) - 6464*Log[3 - 6*x] + 6722806464 *Log[2 + 3*x] - 6722800000*Log[-3*(3 + 5*x)])/22370117
Time = 0.22 (sec) , antiderivative size = 86, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^2 (3 x+2)^4 (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {15152832}{16807 (3 x+2)}-\frac {2000000}{1331 (5 x+3)}+\frac {321327}{2401 (3 x+2)^2}+\frac {15625}{121 (5 x+3)^2}+\frac {5994}{343 (3 x+2)^3}+\frac {81}{49 (3 x+2)^4}-\frac {12928}{22370117 (2 x-1)}+\frac {64}{290521 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {32}{290521 (1-2 x)}-\frac {107109}{2401 (3 x+2)}-\frac {3125}{121 (5 x+3)}-\frac {999}{343 (3 x+2)^2}-\frac {9}{49 (3 x+2)^3}-\frac {6464 \log (1-2 x)}{22370117}+\frac {5050944 \log (3 x+2)}{16807}-\frac {400000 \log (5 x+3)}{1331}\) |
32/(290521*(1 - 2*x)) - 9/(49*(2 + 3*x)^3) - 999/(343*(2 + 3*x)^2) - 10710 9/(2401*(2 + 3*x)) - 3125/(121*(3 + 5*x)) - (6464*Log[1 - 2*x])/22370117 + (5050944*Log[2 + 3*x])/16807 - (400000*Log[3 + 5*x])/1331
3.17.15.3.1 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 = 2.72 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79
method | result | size |
norman | \(\frac {-\frac {2305013328}{290521} x^{3}-\frac {1571590080}{290521} x^{4}-\frac {479067048}{290521} x^{2}+\frac {570653522}{290521} x +\frac {220783501}{290521}}{\left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {6464 \ln \left (-1+2 x \right )}{22370117}+\frac {5050944 \ln \left (2+3 x \right )}{16807}-\frac {400000 \ln \left (3+5 x \right )}{1331}\) | \(68\) |
risch | \(\frac {-\frac {2305013328}{290521} x^{3}-\frac {1571590080}{290521} x^{4}-\frac {479067048}{290521} x^{2}+\frac {570653522}{290521} x +\frac {220783501}{290521}}{\left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {6464 \ln \left (-1+2 x \right )}{22370117}+\frac {5050944 \ln \left (2+3 x \right )}{16807}-\frac {400000 \ln \left (3+5 x \right )}{1331}\) | \(69\) |
default | \(-\frac {3125}{121 \left (3+5 x \right )}-\frac {400000 \ln \left (3+5 x \right )}{1331}-\frac {32}{290521 \left (-1+2 x \right )}-\frac {6464 \ln \left (-1+2 x \right )}{22370117}-\frac {9}{49 \left (2+3 x \right )^{3}}-\frac {999}{343 \left (2+3 x \right )^{2}}-\frac {107109}{2401 \left (2+3 x \right )}+\frac {5050944 \ln \left (2+3 x \right )}{16807}\) | \(71\) |
parallelrisch | \(\frac {-645465249044 x +7421971200000 \ln \left (x +\frac {3}{5}\right ) x^{2}+53728669260288 \ln \left (\frac {2}{3}+x \right ) x^{3}+16134720000000 \ln \left (x +\frac {3}{5}\right ) x -7421978336256 \ln \left (\frac {2}{3}+x \right ) x^{2}-16134735513600 \ln \left (\frac {2}{3}+x \right ) x +4590088985790 x^{5}+1401445118997 x^{3}+6734888402319 x^{4}-1667331065246 x^{2}-87962112 \ln \left (x -\frac {1}{2}\right ) x^{4}+91483950362112 \ln \left (\frac {2}{3}+x \right ) x^{4}-3872336523264 \ln \left (\frac {2}{3}+x \right )-51660288 \ln \left (x -\frac {1}{2}\right ) x^{3}+7136256 \ln \left (x -\frac {1}{2}\right ) x^{2}+15513600 \ln \left (x -\frac {1}{2}\right ) x +3872332800000 \ln \left (x +\frac {3}{5}\right )+43563785886720 \ln \left (\frac {2}{3}+x \right ) x^{5}-53728617600000 \ln \left (x +\frac {3}{5}\right ) x^{3}-43563744000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-91483862400000 \ln \left (x +\frac {3}{5}\right ) x^{4}+3723264 \ln \left (x -\frac {1}{2}\right )-41886720 \ln \left (x -\frac {1}{2}\right ) x^{5}}{536882808 \left (-1+2 x \right ) \left (2+3 x \right )^{3} \left (3+5 x \right )}\) | \(195\) |
(-2305013328/290521*x^3-1571590080/290521*x^4-479067048/290521*x^2+5706535 22/290521*x+220783501/290521)/(-1+2*x)/(2+3*x)^3/(3+5*x)-6464/22370117*ln( -1+2*x)+5050944/16807*ln(2+3*x)-400000/1331*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).
Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {121012436160 \, x^{4} + 177486026256 \, x^{3} + 36888162696 \, x^{2} + 6722800000 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (5 \, x + 3\right ) - 6722806464 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (3 \, x + 2\right ) + 6464 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (2 \, x - 1\right ) - 43940321194 \, x - 17000329577}{22370117 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \]
-1/22370117*(121012436160*x^4 + 177486026256*x^3 + 36888162696*x^2 + 67228 00000*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(5*x + 3) - 6 722806464*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(3*x + 2) + 6464*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(2*x - 1) - 43940321194*x - 17000329577)/(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100* x - 24)
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {- 1571590080 x^{4} - 2305013328 x^{3} - 479067048 x^{2} + 570653522 x + 220783501}{78440670 x^{5} + 164725407 x^{4} + 96743493 x^{3} - 13363966 x^{2} - 29052100 x - 6972504} - \frac {6464 \log {\left (x - \frac {1}{2} \right )}}{22370117} - \frac {400000 \log {\left (x + \frac {3}{5} \right )}}{1331} + \frac {5050944 \log {\left (x + \frac {2}{3} \right )}}{16807} \]
(-1571590080*x**4 - 2305013328*x**3 - 479067048*x**2 + 570653522*x + 22078 3501)/(78440670*x**5 + 164725407*x**4 + 96743493*x**3 - 13363966*x**2 - 29 052100*x - 6972504) - 6464*log(x - 1/2)/22370117 - 400000*log(x + 3/5)/133 1 + 5050944*log(x + 2/3)/16807
Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {1571590080 \, x^{4} + 2305013328 \, x^{3} + 479067048 \, x^{2} - 570653522 \, x - 220783501}{290521 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} - \frac {400000}{1331} \, \log \left (5 \, x + 3\right ) + \frac {5050944}{16807} \, \log \left (3 \, x + 2\right ) - \frac {6464}{22370117} \, \log \left (2 \, x - 1\right ) \]
-1/290521*(1571590080*x^4 + 2305013328*x^3 + 479067048*x^2 - 570653522*x - 220783501)/(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24) - 400000/1 331*log(5*x + 3) + 5050944/16807*log(3*x + 2) - 6464/22370117*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {3125}{121 \, {\left (5 \, x + 3\right )}} + \frac {5 \, {\left (\frac {52083388017}{5 \, x + 3} + \frac {44729490744}{{\left (5 \, x + 3\right )}^{2}} + \frac {9228837286}{{\left (5 \, x + 3\right )}^{3}} - 11003835798\right )}}{3195731 \, {\left (\frac {11}{5 \, x + 3} - 2\right )} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{3}} + \frac {5050944}{16807} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {6464}{22370117} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]
-3125/121/(5*x + 3) + 5/3195731*(52083388017/(5*x + 3) + 44729490744/(5*x + 3)^2 + 9228837286/(5*x + 3)^3 - 11003835798)/((11/(5*x + 3) - 2)*(1/(5*x + 3) + 3)^3) + 5050944/16807*log(abs(-1/(5*x + 3) - 3)) - 6464/22370117*l og(abs(-11/(5*x + 3) + 2))
Time = 1.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {5050944\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {6464\,\ln \left (x-\frac {1}{2}\right )}{22370117}-\frac {400000\,\ln \left (x+\frac {3}{5}\right )}{1331}+\frac {\frac {5820704\,x^4}{290521}+\frac {42685432\,x^3}{1452605}+\frac {8871612\,x^2}{1452605}-\frac {285326761\,x}{39220335}-\frac {220783501}{78440670}}{-x^5-\frac {21\,x^4}{10}-\frac {37\,x^3}{30}+\frac {23\,x^2}{135}+\frac {10\,x}{27}+\frac {4}{45}} \]