Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=\frac {8}{26411 (1-2 x)^2}+\frac {2672}{2033647 (1-2 x)}+\frac {9}{343 (2+3 x)^3}+\frac {1107}{4802 (2+3 x)^2}+\frac {39393}{16807 (2+3 x)}-\frac {267760 \log (1-2 x)}{156590819}-\frac {1380915 \log (2+3 x)}{117649}+\frac {15625 \log (3+5 x)}{1331} \]
8/26411/(1-2*x)^2+2672/2033647/(1-2*x)+9/343/(2+3*x)^3+1107/4802/(2+3*x)^2 +39393/16807/(2+3*x)-267760/156590819*ln(1-2*x)-1380915/117649*ln(2+3*x)+1 5625/1331*ln(3+5*x)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=\frac {5 \left (\frac {77 \left (40167012-43096225 x-222614730 x^2+125249220 x^3+342903240 x^4\right )}{5 (1-2 x)^2 (2+3 x)^3}-107104 \log (5-10 x)-735199146 \log (5 (2+3 x))+735306250 \log (3+5 x)\right )}{313181638} \]
(5*((77*(40167012 - 43096225*x - 222614730*x^2 + 125249220*x^3 + 342903240 *x^4))/(5*(1 - 2*x)^2*(2 + 3*x)^3) - 107104*Log[5 - 10*x] - 735199146*Log[ 5*(2 + 3*x)] + 735306250*Log[3 + 5*x]))/313181638
Time = 0.21 (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.
\(\displaystyle \int \frac {1}{(1-2 x)^3 (3 x+2)^4 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {4142745}{117649 (3 x+2)}+\frac {78125}{1331 (5 x+3)}-\frac {118179}{16807 (3 x+2)^2}-\frac {3321}{2401 (3 x+2)^3}-\frac {81}{343 (3 x+2)^4}-\frac {535520}{156590819 (2 x-1)}+\frac {5344}{2033647 (2 x-1)^2}-\frac {32}{26411 (2 x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2672}{2033647 (1-2 x)}+\frac {39393}{16807 (3 x+2)}+\frac {8}{26411 (1-2 x)^2}+\frac {1107}{4802 (3 x+2)^2}+\frac {9}{343 (3 x+2)^3}-\frac {267760 \log (1-2 x)}{156590819}-\frac {1380915 \log (3 x+2)}{117649}+\frac {15625 \log (5 x+3)}{1331}\) |
8/(26411*(1 - 2*x)^2) + 2672/(2033647*(1 - 2*x)) + 9/(343*(2 + 3*x)^3) + 1 107/(4802*(2 + 3*x)^2) + 39393/(16807*(2 + 3*x)) - (267760*Log[1 - 2*x])/1 56590819 - (1380915*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/1331
3.17.83.3.1 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 = 61, normalized size of antiderivative = 0.71
method | result | size |
norman | \(\frac {-\frac {111307365}{2033647} x^{2}-\frac {43096225}{4067294} x +\frac {62624610}{2033647} x^{3}+\frac {171451620}{2033647} x^{4}+\frac {20083506}{2033647}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}-\frac {267760 \ln \left (-1+2 x \right )}{156590819}-\frac {1380915 \ln \left (2+3 x \right )}{117649}+\frac {15625 \ln \left (3+5 x \right )}{1331}\) | \(61\) |
risch | \(\frac {-\frac {111307365}{2033647} x^{2}-\frac {43096225}{4067294} x +\frac {62624610}{2033647} x^{3}+\frac {171451620}{2033647} x^{4}+\frac {20083506}{2033647}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}-\frac {267760 \ln \left (-1+2 x \right )}{156590819}-\frac {1380915 \ln \left (2+3 x \right )}{117649}+\frac {15625 \ln \left (3+5 x \right )}{1331}\) | \(62\) |
default | \(\frac {15625 \ln \left (3+5 x \right )}{1331}+\frac {8}{26411 \left (-1+2 x \right )^{2}}-\frac {2672}{2033647 \left (-1+2 x \right )}-\frac {267760 \ln \left (-1+2 x \right )}{156590819}+\frac {9}{343 \left (2+3 x \right )^{3}}+\frac {1107}{4802 \left (2+3 x \right )^{2}}+\frac {39393}{16807 \left (2+3 x \right )}-\frac {1380915 \ln \left (2+3 x \right )}{117649}\) | \(71\) |
parallelrisch | \(-\frac {-38918714296+852955250000 \ln \left (x +\frac {3}{5}\right ) x^{2}-661679231400 \ln \left (\frac {2}{3}+x \right ) x^{3}-58824500000 \ln \left (x +\frac {3}{5}\right ) x -852831009360 \ln \left (\frac {2}{3}+x \right ) x^{2}+58815931680 \ln \left (\frac {2}{3}+x \right ) x -358388207100 x^{5}+110751659865 x^{3}-464002405020 x^{4}+261033077690 x^{2}+231344640 \ln \left (x -\frac {1}{2}\right ) x^{4}+1588030155360 \ln \left (\frac {2}{3}+x \right ) x^{4}+117631863360 \ln \left (\frac {2}{3}+x \right )-96393600 \ln \left (x -\frac {1}{2}\right ) x^{3}-124240640 \ln \left (x -\frac {1}{2}\right ) x^{2}+8568320 \ln \left (x -\frac {1}{2}\right ) x -117649000000 \ln \left (x +\frac {3}{5}\right )+1588030155360 \ln \left (\frac {2}{3}+x \right ) x^{5}+661775625000 \ln \left (x +\frac {3}{5}\right ) x^{3}-1588261500000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1588261500000 \ln \left (x +\frac {3}{5}\right ) x^{4}+17136640 \ln \left (x -\frac {1}{2}\right )+231344640 \ln \left (x -\frac {1}{2}\right ) x^{5}}{1252726552 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{3}}\) | \(186\) |
(-111307365/2033647*x^2-43096225/4067294*x+62624610/2033647*x^3+171451620/ 2033647*x^4+20083506/2033647)/(-1+2*x)^2/(2+3*x)^3-267760/156590819*ln(-1+ 2*x)-1380915/117649*ln(2+3*x)+15625/1331*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 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=\frac {26403549480 \, x^{4} + 9644189940 \, x^{3} - 17141334210 \, x^{2} + 3676531250 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 3675995730 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 535520 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 3318409325 \, x + 3092859924}{313181638 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
1/313181638*(26403549480*x^4 + 9644189940*x^3 - 17141334210*x^2 + 36765312 50*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(5*x + 3) - 36759957 30*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(3*x + 2) - 535520*( 108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(2*x - 1) - 3318409325*x + 3092859924)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=- \frac {- 342903240 x^{4} - 125249220 x^{3} + 222614730 x^{2} + 43096225 x - 40167012}{439267752 x^{5} + 439267752 x^{4} - 183028230 x^{3} - 235903052 x^{2} + 16269176 x + 32538352} - \frac {267760 \log {\left (x - \frac {1}{2} \right )}}{156590819} + \frac {15625 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {1380915 \log {\left (x + \frac {2}{3} \right )}}{117649} \]
-(-342903240*x**4 - 125249220*x**3 + 222614730*x**2 + 43096225*x - 4016701 2)/(439267752*x**5 + 439267752*x**4 - 183028230*x**3 - 235903052*x**2 + 16 269176*x + 32538352) - 267760*log(x - 1/2)/156590819 + 15625*log(x + 3/5)/ 1331 - 1380915*log(x + 2/3)/117649
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=\frac {342903240 \, x^{4} + 125249220 \, x^{3} - 222614730 \, x^{2} - 43096225 \, x + 40167012}{4067294 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac {15625}{1331} \, \log \left (5 \, x + 3\right ) - \frac {1380915}{117649} \, \log \left (3 \, x + 2\right ) - \frac {267760}{156590819} \, \log \left (2 \, x - 1\right ) \]
1/4067294*(342903240*x^4 + 125249220*x^3 - 222614730*x^2 - 43096225*x + 40 167012)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8) + 15625/1331*log(5 *x + 3) - 1380915/117649*log(3*x + 2) - 267760/156590819*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=\frac {342903240 \, x^{4} + 125249220 \, x^{3} - 222614730 \, x^{2} - 43096225 \, x + 40167012}{4067294 \, {\left (3 \, x + 2\right )}^{3} {\left (2 \, x - 1\right )}^{2}} + \frac {15625}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {1380915}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {267760}{156590819} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
1/4067294*(342903240*x^4 + 125249220*x^3 - 222614730*x^2 - 43096225*x + 40 167012)/((3*x + 2)^3*(2*x - 1)^2) + 15625/1331*log(abs(5*x + 3)) - 1380915 /117649*log(abs(3*x + 2)) - 267760/156590819*log(abs(2*x - 1))
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx=\frac {15625\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {1380915\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {267760\,\ln \left (x-\frac {1}{2}\right )}{156590819}+\frac {\frac {1587515\,x^4}{2033647}+\frac {1159715\,x^3}{4067294}-\frac {4122495\,x^2}{8134588}-\frac {43096225\,x}{439267752}+\frac {3347251}{36605646}}{x^5+x^4-\frac {5\,x^3}{12}-\frac {29\,x^2}{54}+\frac {x}{27}+\frac {2}{27}} \]