Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=\frac {16}{456533 (1-2 x)^2}+\frac {6528}{35153041 (1-2 x)}+\frac {243}{686 (2+3 x)^2}+\frac {26973}{2401 (2+3 x)}-\frac {3125}{2662 (3+5 x)^2}+\frac {290625}{14641 (3+5 x)}-\frac {776928 \log (1-2 x)}{2706784157}-\frac {1944972 \log (2+3 x)}{16807}+\frac {18637500 \log (3+5 x)}{161051} \]
16/456533/(1-2*x)^2+6528/35153041/(1-2*x)+243/686/(2+3*x)^2+26973/2401/(2+ 3*x)-3125/2662/(3+5*x)^2+290625/14641/(3+5*x)-776928/2706784157*ln(1-2*x)- 1944972/16807*ln(2+3*x)+18637500/161051*ln(3+5*x)
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=\frac {2 \left (\frac {77}{4} \left (\frac {2464}{(1-2 x)^2}+\frac {13056}{1-2 x}+\frac {24904341}{(2+3 x)^2}+\frac {789823386}{2+3 x}-\frac {82534375}{(3+5 x)^2}+\frac {1395581250}{3+5 x}\right )-388464 \log (1-2 x)-156619842786 \log (4+6 x)+156620231250 \log (6+10 x)\right )}{2706784157} \]
(2*((77*(2464/(1 - 2*x)^2 + 13056/(1 - 2*x) + 24904341/(2 + 3*x)^2 + 78982 3386/(2 + 3*x) - 82534375/(3 + 5*x)^2 + 1395581250/(3 + 5*x)))/4 - 388464* Log[1 - 2*x] - 156619842786*Log[4 + 6*x] + 156620231250*Log[6 + 10*x]))/27 06784157
Time = 0.23 (sec) , antiderivative size = 97, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^3 (3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {5834916}{16807 (3 x+2)}+\frac {93187500}{161051 (5 x+3)}-\frac {80919}{2401 (3 x+2)^2}-\frac {1453125}{14641 (5 x+3)^2}-\frac {729}{343 (3 x+2)^3}+\frac {15625}{1331 (5 x+3)^3}-\frac {1553856}{2706784157 (2 x-1)}+\frac {13056}{35153041 (2 x-1)^2}-\frac {64}{456533 (2 x-1)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6528}{35153041 (1-2 x)}+\frac {26973}{2401 (3 x+2)}+\frac {290625}{14641 (5 x+3)}+\frac {16}{456533 (1-2 x)^2}+\frac {243}{686 (3 x+2)^2}-\frac {3125}{2662 (5 x+3)^2}-\frac {776928 \log (1-2 x)}{2706784157}-\frac {1944972 \log (3 x+2)}{16807}+\frac {18637500 \log (5 x+3)}{161051}\) |
16/(456533*(1 - 2*x)^2) + 6528/(35153041*(1 - 2*x)) + 243/(686*(2 + 3*x)^2 ) + 26973/(2401*(2 + 3*x)) - 3125/(2662*(3 + 5*x)^2) + 290625/(14641*(3 + 5*x)) - (776928*Log[1 - 2*x])/2706784157 - (1944972*Log[2 + 3*x])/16807 + (18637500*Log[3 + 5*x])/161051
3.18.9.3.1 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 = 0.90 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75
| method | result | size |
| norman | \(\frac {-\frac {115595667228}{35153041} x^{2}-\frac {109477164252}{35153041} x^{3}+\frac {11597655386}{35153041} x +\frac {219659767560}{35153041} x^{4}+\frac {244072882800}{35153041} x^{5}+\frac {30858356237}{70306082}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {776928 \ln \left (-1+2 x \right )}{2706784157}-\frac {1944972 \ln \left (2+3 x \right )}{16807}+\frac {18637500 \ln \left (3+5 x \right )}{161051}\) | \(73\) |
| risch | \(\frac {-\frac {115595667228}{35153041} x^{2}-\frac {109477164252}{35153041} x^{3}+\frac {11597655386}{35153041} x +\frac {219659767560}{35153041} x^{4}+\frac {244072882800}{35153041} x^{5}+\frac {30858356237}{70306082}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {776928 \ln \left (-1+2 x \right )}{2706784157}-\frac {1944972 \ln \left (2+3 x \right )}{16807}+\frac {18637500 \ln \left (3+5 x \right )}{161051}\) | \(74\) |
| default | \(-\frac {3125}{2662 \left (3+5 x \right )^{2}}+\frac {290625}{14641 \left (3+5 x \right )}+\frac {18637500 \ln \left (3+5 x \right )}{161051}+\frac {16}{456533 \left (-1+2 x \right )^{2}}-\frac {6528}{35153041 \left (-1+2 x \right )}-\frac {776928 \ln \left (-1+2 x \right )}{2706784157}+\frac {243}{686 \left (2+3 x \right )^{2}}+\frac {26973}{2401 \left (2+3 x \right )}-\frac {1944972 \ln \left (2+3 x \right )}{16807}\) | \(80\) |
| parallelrisch | \(-\frac {135294446680932 x +5119602119100000 \ln \left (x +\frac {3}{5}\right ) x^{2}-15381321520327488 \ln \left (\frac {2}{3}+x \right ) x^{3}-1894478317200000 \ln \left (x +\frac {3}{5}\right ) x -5119589420988768 \ln \left (\frac {2}{3}+x \right ) x^{2}+1894473618339456 \ln \left (\frac {2}{3}+x \right ) x +1925868871500420 x^{5}+2138484087224100 x^{6}-1013554320816730 x^{3}-958799567455499 x^{4}+101489170445509 x^{2}+6097330944 \ln \left (x -\frac {1}{2}\right ) x^{4}+2458305052369056 \ln \left (\frac {2}{3}+x \right ) x^{4}+811917265002624 \ln \left (\frac {2}{3}+x \right )-38150272512 \ln \left (x -\frac {1}{2}\right ) x^{3}-12698111232 \ln \left (x -\frac {1}{2}\right ) x^{2}+4698860544 \ln \left (x -\frac {1}{2}\right ) x -811919278800000 \ln \left (x +\frac {3}{5}\right )+31123495158433920 \ln \left (\frac {2}{3}+x \right ) x^{5}+15381359670600000 \ln \left (x +\frac {3}{5}\right ) x^{3}-31123572354000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-2458311149700000 \ln \left (x +\frac {3}{5}\right ) x^{4}+20297931625065600 \ln \left (\frac {2}{3}+x \right ) x^{6}-20297981970000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+2013797376 \ln \left (x -\frac {1}{2}\right )+50344934400 \ln \left (x -\frac {1}{2}\right ) x^{6}+77195566080 \ln \left (x -\frac {1}{2}\right ) x^{5}}{194888459304 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(227\) |
(-115595667228/35153041*x^2-109477164252/35153041*x^3+11597655386/35153041 *x+219659767560/35153041*x^4+244072882800/35153041*x^5+30858356237/7030608 2)/(-1+2*x)^2/(2+3*x)^2/(3+5*x)^2-776928/2706784157*ln(-1+2*x)-1944972/168 07*ln(2+3*x)+18637500/161051*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (79) = 158\).
Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=\frac {37587223951200 \, x^{5} + 33827604204240 \, x^{4} - 16859483294808 \, x^{3} - 17801732753112 \, x^{2} + 626480925000 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 626479371144 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (3 \, x + 2\right ) - 1553856 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (2 \, x - 1\right ) + 1786038929444 \, x + 2376093430249}{5413568314 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \]
1/5413568314*(37587223951200*x^5 + 33827604204240*x^4 - 16859483294808*x^3 - 17801732753112*x^2 + 626480925000*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x ^3 - 227*x^2 + 84*x + 36)*log(5*x + 3) - 626479371144*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(3*x + 2) - 1553856*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(2*x - 1) + 1786 038929444*x + 2376093430249)/(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227 *x^2 + 84*x + 36)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=- \frac {- 488145765600 x^{5} - 439319535120 x^{4} + 218954328504 x^{3} + 231191334456 x^{2} - 23195310772 x - 30858356237}{63275473800 x^{6} + 97022393160 x^{5} + 7663362938 x^{4} - 47948747924 x^{3} - 15959480614 x^{2} + 5905710888 x + 2531018952} - \frac {776928 \log {\left (x - \frac {1}{2} \right )}}{2706784157} + \frac {18637500 \log {\left (x + \frac {3}{5} \right )}}{161051} - \frac {1944972 \log {\left (x + \frac {2}{3} \right )}}{16807} \]
-(-488145765600*x**5 - 439319535120*x**4 + 218954328504*x**3 + 23119133445 6*x**2 - 23195310772*x - 30858356237)/(63275473800*x**6 + 97022393160*x**5 + 7663362938*x**4 - 47948747924*x**3 - 15959480614*x**2 + 5905710888*x + 2531018952) - 776928*log(x - 1/2)/2706784157 + 18637500*log(x + 3/5)/16105 1 - 1944972*log(x + 2/3)/16807
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=\frac {488145765600 \, x^{5} + 439319535120 \, x^{4} - 218954328504 \, x^{3} - 231191334456 \, x^{2} + 23195310772 \, x + 30858356237}{70306082 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} + \frac {18637500}{161051} \, \log \left (5 \, x + 3\right ) - \frac {1944972}{16807} \, \log \left (3 \, x + 2\right ) - \frac {776928}{2706784157} \, \log \left (2 \, x - 1\right ) \]
1/70306082*(488145765600*x^5 + 439319535120*x^4 - 218954328504*x^3 - 23119 1334456*x^2 + 23195310772*x + 30858356237)/(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36) + 18637500/161051*log(5*x + 3) - 1944972/1 6807*log(3*x + 2) - 776928/2706784157*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=\frac {488145765600 \, x^{5} + 439319535120 \, x^{4} - 218954328504 \, x^{3} - 231191334456 \, x^{2} + 23195310772 \, x + 30858356237}{70306082 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}^{2}} + \frac {18637500}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {1944972}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {776928}{2706784157} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
1/70306082*(488145765600*x^5 + 439319535120*x^4 - 218954328504*x^3 - 23119 1334456*x^2 + 23195310772*x + 30858356237)/(30*x^3 + 23*x^2 - 7*x - 6)^2 + 18637500/161051*log(abs(5*x + 3)) - 1944972/16807*log(abs(3*x + 2)) - 776 928/2706784157*log(abs(2*x - 1))
Time = 1.38 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^3} \, dx=\frac {18637500\,\ln \left (x+\frac {3}{5}\right )}{161051}-\frac {1944972\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {776928\,\ln \left (x-\frac {1}{2}\right )}{2706784157}+\frac {\frac {271192092\,x^5}{35153041}+\frac {1220332042\,x^4}{175765205}-\frac {9123097021\,x^3}{2636478075}-\frac {9632972269\,x^2}{2636478075}+\frac {5798827693\,x}{15818868450}+\frac {30858356237}{63275473800}}{x^6+\frac {23\,x^5}{15}+\frac {109\,x^4}{900}-\frac {341\,x^3}{450}-\frac {227\,x^2}{900}+\frac {7\,x}{75}+\frac {1}{25}} \]
(18637500*log(x + 3/5))/161051 - (1944972*log(x + 2/3))/16807 - (776928*lo g(x - 1/2))/2706784157 + ((5798827693*x)/15818868450 - (9632972269*x^2)/26 36478075 - (9123097021*x^3)/2636478075 + (1220332042*x^4)/175765205 + (271 192092*x^5)/35153041 + 30858356237/63275473800)/((7*x)/75 - (227*x^2)/900 - (341*x^3)/450 + (109*x^4)/900 + (23*x^5)/15 + x^6 + 1/25)