Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {16}{290521 (1-2 x)^2}+\frac {6464}{22370117 (1-2 x)}-\frac {27}{343 (2+3 x)^3}-\frac {3078}{2401 (2+3 x)^2}-\frac {333639}{16807 (2+3 x)}-\frac {15625}{1331 (3+5 x)}-\frac {761760 \log (1-2 x)}{1722499009}+\frac {15820110 \log (2+3 x)}{117649}-\frac {1968750 \log (3+5 x)}{14641} \]
16/290521/(1-2*x)^2+6464/22370117/(1-2*x)-27/343/(2+3*x)^3-3078/2401/(2+3* x)^2-333639/16807/(2+3*x)-15625/1331/(3+5*x)-761760/1722499009*ln(1-2*x)+1 5820110/117649*ln(2+3*x)-1968750/14641*ln(3+5*x)
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {2 \left (\frac {77}{2} \left (\frac {1232}{(1-2 x)^2}+\frac {6464}{1-2 x}-\frac {1760913}{(2+3 x)^3}-\frac {28677726}{(2+3 x)^2}-\frac {444073509}{2+3 x}-\frac {262609375}{3+5 x}\right )-380880 \log (1-2 x)+115811115255 \log (4+6 x)-115810734375 \log (6+10 x)\right )}{1722499009} \]
(2*((77*(1232/(1 - 2*x)^2 + 6464/(1 - 2*x) - 1760913/(2 + 3*x)^3 - 2867772 6/(2 + 3*x)^2 - 444073509/(2 + 3*x) - 262609375/(3 + 5*x)))/2 - 380880*Log [1 - 2*x] + 115811115255*Log[4 + 6*x] - 115810734375*Log[6 + 10*x]))/17224 99009
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.
| \(\displaystyle \int \frac {1}{(1-2 x)^3 (3 x+2)^4 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {47460330}{117649 (3 x+2)}-\frac {9843750}{14641 (5 x+3)}+\frac {1000917}{16807 (3 x+2)^2}+\frac {78125}{1331 (5 x+3)^2}+\frac {18468}{2401 (3 x+2)^3}+\frac {243}{343 (3 x+2)^4}-\frac {1523520}{1722499009 (2 x-1)}+\frac {12928}{22370117 (2 x-1)^2}-\frac {64}{290521 (2 x-1)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6464}{22370117 (1-2 x)}-\frac {333639}{16807 (3 x+2)}-\frac {15625}{1331 (5 x+3)}+\frac {16}{290521 (1-2 x)^2}-\frac {3078}{2401 (3 x+2)^2}-\frac {27}{343 (3 x+2)^3}-\frac {761760 \log (1-2 x)}{1722499009}+\frac {15820110 \log (3 x+2)}{117649}-\frac {1968750 \log (5 x+3)}{14641}\) |
16/(290521*(1 - 2*x)^2) + 6464/(22370117*(1 - 2*x)) - 27/(343*(2 + 3*x)^3) - 3078/(2401*(2 + 3*x)^2) - 333639/(16807*(2 + 3*x)) - 15625/(1331*(3 + 5 *x)) - (761760*Log[1 - 2*x])/1722499009 + (15820110*Log[2 + 3*x])/117649 - (1968750*Log[3 + 5*x])/14641
3.17.96.3.1 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 = 2.82 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75
| method | result | size |
| norman | \(\frac {-\frac {108296789400}{22370117} x^{5}-\frac {14955760620}{3195731} x^{4}-\frac {4446481815}{22370117} x +\frac {46403447130}{22370117} x^{3}+\frac {55829767905}{22370117} x^{2}-\frac {7606921499}{22370117}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {761760 \ln \left (-1+2 x \right )}{1722499009}+\frac {15820110 \ln \left (2+3 x \right )}{117649}-\frac {1968750 \ln \left (3+5 x \right )}{14641}\) | \(73\) |
| risch | \(\frac {-\frac {108296789400}{22370117} x^{5}-\frac {14955760620}{3195731} x^{4}-\frac {4446481815}{22370117} x +\frac {46403447130}{22370117} x^{3}+\frac {55829767905}{22370117} x^{2}-\frac {7606921499}{22370117}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {761760 \ln \left (-1+2 x \right )}{1722499009}+\frac {15820110 \ln \left (2+3 x \right )}{117649}-\frac {1968750 \ln \left (3+5 x \right )}{14641}\) | \(74\) |
| default | \(-\frac {15625}{1331 \left (3+5 x \right )}-\frac {1968750 \ln \left (3+5 x \right )}{14641}+\frac {16}{290521 \left (-1+2 x \right )^{2}}-\frac {6464}{22370117 \left (-1+2 x \right )}-\frac {761760 \ln \left (-1+2 x \right )}{1722499009}-\frac {27}{343 \left (2+3 x \right )^{3}}-\frac {3078}{2401 \left (2+3 x \right )^{2}}-\frac {333639}{16807 \left (2+3 x \right )}+\frac {15820110 \ln \left (2+3 x \right )}{117649}\) | \(80\) |
| parallelrisch | \(\frac {22241015287876 x +856072948500000 \ln \left (x +\frac {3}{5}\right ) x^{2}-2362546751202000 \ln \left (\frac {2}{3}+x \right ) x^{3}-289063593000000 \ln \left (x +\frac {3}{5}\right ) x -856075763964960 \ln \left (\frac {2}{3}+x \right ) x^{2}+289064543676480 \ln \left (\frac {2}{3}+x \right ) x +305940806674272 x^{5}+316295795928420 x^{6}-163182935758535 x^{3}-135480156793443 x^{4}+12970535953298 x^{2}-1809941760 \ln \left (x -\frac {1}{2}\right ) x^{4}+550334419691760 \ln \left (\frac {2}{3}+x \right ) x^{4}+133414404773760 \ln \left (\frac {2}{3}+x \right )+7769952000 \ln \left (x -\frac {1}{2}\right ) x^{3}+2815464960 \ln \left (x -\frac {1}{2}\right ) x^{2}-950676480 \ln \left (x -\frac {1}{2}\right ) x -133413966000000 \ln \left (x +\frac {3}{5}\right )+4802918571855360 \ln \left (\frac {2}{3}+x \right ) x^{5}+2362538981250000 \ln \left (x +\frac {3}{5}\right ) x^{3}-4802902776000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-550332609750000 \ln \left (x +\frac {3}{5}\right ) x^{4}+3001824107409600 \ln \left (\frac {2}{3}+x \right ) x^{6}-3001814235000000 \ln \left (x +\frac {3}{5}\right ) x^{6}-438773760 \ln \left (x -\frac {1}{2}\right )-9872409600 \ln \left (x -\frac {1}{2}\right ) x^{6}-15795855360 \ln \left (x -\frac {1}{2}\right ) x^{5}}{41339976216 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{3} \left (3+5 x \right )}\) | \(227\) |
(-108296789400/22370117*x^5-14955760620/3195731*x^4-4446481815/22370117*x+ 46403447130/22370117*x^3+55829767905/22370117*x^2-7606921499/22370117)/(-1 +2*x)^2/(2+3*x)^3/(3+5*x)-761760/1722499009*ln(-1+2*x)+15820110/117649*ln( 2+3*x)-1968750/14641*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 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {8338852783800 \, x^{5} + 8061154974180 \, x^{4} - 3573065429010 \, x^{3} - 4298892128685 \, x^{2} + 231621468750 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} \log \left (5 \, x + 3\right ) - 231622230510 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} \log \left (3 \, x + 2\right ) + 761760 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} \log \left (2 \, x - 1\right ) + 342379099755 \, x + 585732955423}{1722499009 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} \]
-1/1722499009*(8338852783800*x^5 + 8061154974180*x^4 - 3573065429010*x^3 - 4298892128685*x^2 + 231621468750*(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)*log(5*x + 3) - 231622230510*(540*x^6 + 864*x^5 + 99*x ^4 - 425*x^3 - 154*x^2 + 52*x + 24)*log(3*x + 2) + 761760*(540*x^6 + 864*x ^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)*log(2*x - 1) + 342379099755*x + 585732955423)/(540*x^6 + 864*x^5 + 99*x^4 - 425*x^3 - 154*x^2 + 52*x + 24)
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=- \frac {108296789400 x^{5} + 104690324340 x^{4} - 46403447130 x^{3} - 55829767905 x^{2} + 4446481815 x + 7606921499}{12079863180 x^{6} + 19327781088 x^{5} + 2214641583 x^{4} - 9507299725 x^{3} - 3444998018 x^{2} + 1163246084 x + 536882808} - \frac {761760 \log {\left (x - \frac {1}{2} \right )}}{1722499009} - \frac {1968750 \log {\left (x + \frac {3}{5} \right )}}{14641} + \frac {15820110 \log {\left (x + \frac {2}{3} \right )}}{117649} \]
-(108296789400*x**5 + 104690324340*x**4 - 46403447130*x**3 - 55829767905*x **2 + 4446481815*x + 7606921499)/(12079863180*x**6 + 19327781088*x**5 + 22 14641583*x**4 - 9507299725*x**3 - 3444998018*x**2 + 1163246084*x + 5368828 08) - 761760*log(x - 1/2)/1722499009 - 1968750*log(x + 3/5)/14641 + 158201 10*log(x + 2/3)/117649
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {108296789400 \, x^{5} + 104690324340 \, x^{4} - 46403447130 \, x^{3} - 55829767905 \, x^{2} + 4446481815 \, x + 7606921499}{22370117 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} - \frac {1968750}{14641} \, \log \left (5 \, x + 3\right ) + \frac {15820110}{117649} \, \log \left (3 \, x + 2\right ) - \frac {761760}{1722499009} \, \log \left (2 \, x - 1\right ) \]
-1/22370117*(108296789400*x^5 + 104690324340*x^4 - 46403447130*x^3 - 55829 767905*x^2 + 4446481815*x + 7606921499)/(540*x^6 + 864*x^5 + 99*x^4 - 425* x^3 - 154*x^2 + 52*x + 24) - 1968750/14641*log(5*x + 3) + 15820110/117649* log(3*x + 2) - 761760/1722499009*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {15625}{1331 \, {\left (5 \, x + 3\right )}} - \frac {25 \, {\left (\frac {1535578147116}{5 \, x + 3} - \frac {3297944687832}{{\left (5 \, x + 3\right )}^{2}} - \frac {3224232263641}{{\left (5 \, x + 3\right )}^{3}} - \frac {689127341628}{{\left (5 \, x + 3\right )}^{4}} - 150040675728\right )}}{246071287 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{3}} + \frac {15820110}{117649} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {761760}{1722499009} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]
-15625/1331/(5*x + 3) - 25/246071287*(1535578147116/(5*x + 3) - 3297944687 832/(5*x + 3)^2 - 3224232263641/(5*x + 3)^3 - 689127341628/(5*x + 3)^4 - 1 50040675728)/((11/(5*x + 3) - 2)^2*(1/(5*x + 3) + 3)^3) + 15820110/117649* log(abs(-1/(5*x + 3) - 3)) - 761760/1722499009*log(abs(-11/(5*x + 3) + 2))
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)^2} \, dx=\frac {15820110\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {761760\,\ln \left (x-\frac {1}{2}\right )}{1722499009}-\frac {1968750\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {\frac {200549610\,x^5}{22370117}+\frac {27695853\,x^4}{3195731}-\frac {171864619\,x^3}{44740234}-\frac {1240661509\,x^2}{268441404}+\frac {98810707\,x}{268441404}+\frac {7606921499}{12079863180}}{x^6+\frac {8\,x^5}{5}+\frac {11\,x^4}{60}-\frac {85\,x^3}{108}-\frac {77\,x^2}{270}+\frac {13\,x}{135}+\frac {2}{45}} \]
(15820110*log(x + 2/3))/117649 - (761760*log(x - 1/2))/1722499009 - (19687 50*log(x + 3/5))/14641 - ((98810707*x)/268441404 - (1240661509*x^2)/268441 404 - (171864619*x^3)/44740234 + (27695853*x^4)/3195731 + (200549610*x^5)/ 22370117 + 7606921499/12079863180)/((13*x)/135 - (77*x^2)/270 - (85*x^3)/1 08 + (11*x^4)/60 + (8*x^5)/5 + x^6 + 2/45)