Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=\frac {64}{2033647 (1-2 x)}-\frac {27}{196 (2+3 x)^4}-\frac {666}{343 (2+3 x)^3}-\frac {107109}{4802 (2+3 x)^2}-\frac {5050944}{16807 (2+3 x)}-\frac {15625}{121 (3+5 x)}-\frac {15040 \log (1-2 x)}{156590819}+\frac {222359715 \log (2+3 x)}{117649}-\frac {2515625 \log (3+5 x)}{1331} \]
64/2033647/(1-2*x)-27/196/(2+3*x)^4-666/343/(2+3*x)^3-107109/4802/(2+3*x)^ 2-5050944/16807/(2+3*x)-15625/121/(3+5*x)-15040/156590819*ln(1-2*x)+222359 715/117649*ln(2+3*x)-2515625/1331*ln(3+5*x)
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=\frac {-\frac {77 \left (-77754195847-317609203475 x-132753874800 x^2+1064845635750 x^3+1771154199360 x^4+830228340600 x^5\right )}{(2+3 x)^4 \left (-3+x+10 x^2\right )}-60160 \log (3-6 x)+1183843122660 \log (2+3 x)-1183843062500 \log (-3 (3+5 x))}{626363276} \]
((-77*(-77754195847 - 317609203475*x - 132753874800*x^2 + 1064845635750*x^ 3 + 1771154199360*x^4 + 830228340600*x^5))/((2 + 3*x)^4*(-3 + x + 10*x^2)) - 60160*Log[3 - 6*x] + 1183843122660*Log[2 + 3*x] - 1183843062500*Log[-3* (3 + 5*x)])/626363276
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.
| \(\displaystyle \int \frac {1}{(1-2 x)^2 (3 x+2)^5 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {667079145}{117649 (3 x+2)}-\frac {12578125}{1331 (5 x+3)}+\frac {15152832}{16807 (3 x+2)^2}+\frac {78125}{121 (5 x+3)^2}+\frac {321327}{2401 (3 x+2)^3}+\frac {5994}{343 (3 x+2)^4}+\frac {81}{49 (3 x+2)^5}-\frac {30080}{156590819 (2 x-1)}+\frac {128}{2033647 (2 x-1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {64}{2033647 (1-2 x)}-\frac {5050944}{16807 (3 x+2)}-\frac {15625}{121 (5 x+3)}-\frac {107109}{4802 (3 x+2)^2}-\frac {666}{343 (3 x+2)^3}-\frac {27}{196 (3 x+2)^4}-\frac {15040 \log (1-2 x)}{156590819}+\frac {222359715 \log (3 x+2)}{117649}-\frac {2515625 \log (5 x+3)}{1331}\) |
64/(2033647*(1 - 2*x)) - 27/(196*(2 + 3*x)^4) - 666/(343*(2 + 3*x)^3) - 10 7109/(4802*(2 + 3*x)^2) - 5050944/(16807*(2 + 3*x)) - 15625/(121*(3 + 5*x) ) - (15040*Log[1 - 2*x])/156590819 + (222359715*Log[2 + 3*x])/117649 - (25 15625*Log[3 + 5*x])/1331
3.17.16.3.1 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 = 73, normalized size of antiderivative = 0.75
| method | result | size |
| norman | \(\frac {-\frac {532422817875}{4067294} x^{3}-\frac {207557085150}{2033647} x^{5}-\frac {63255507120}{290521} x^{4}+\frac {33188468700}{2033647} x^{2}+\frac {317609203475}{8134588} x +\frac {77754195847}{8134588}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4} \left (3+5 x \right )}-\frac {15040 \ln \left (-1+2 x \right )}{156590819}+\frac {222359715 \ln \left (2+3 x \right )}{117649}-\frac {2515625 \ln \left (3+5 x \right )}{1331}\) | \(73\) |
| risch | \(\frac {-\frac {532422817875}{4067294} x^{3}-\frac {207557085150}{2033647} x^{5}-\frac {63255507120}{290521} x^{4}+\frac {33188468700}{2033647} x^{2}+\frac {317609203475}{8134588} x +\frac {77754195847}{8134588}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4} \left (3+5 x \right )}-\frac {15040 \ln \left (-1+2 x \right )}{156590819}+\frac {222359715 \ln \left (2+3 x \right )}{117649}-\frac {2515625 \ln \left (3+5 x \right )}{1331}\) | \(74\) |
| default | \(-\frac {15625}{121 \left (3+5 x \right )}-\frac {2515625 \ln \left (3+5 x \right )}{1331}-\frac {64}{2033647 \left (-1+2 x \right )}-\frac {15040 \ln \left (-1+2 x \right )}{156590819}-\frac {27}{196 \left (2+3 x \right )^{4}}-\frac {666}{343 \left (2+3 x \right )^{3}}-\frac {107109}{4802 \left (2+3 x \right )^{2}}-\frac {5050944}{16807 \left (2+3 x \right )}+\frac {222359715 \ln \left (2+3 x \right )}{117649}\) | \(80\) |
| parallelrisch | \(\frac {-454600261775968 x +22275191064000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+30003320100695040 \ln \left (\frac {2}{3}+x \right ) x^{3}+15456255024000000 \ln \left (x +\frac {3}{5}\right ) x -22275192195970560 \ln \left (\frac {2}{3}+x \right ) x^{2}-15456255809448960 \ln \left (\frac {2}{3}+x \right ) x +10348506825913179 x^{5}+4849529194977390 x^{6}-774494883376368 x^{3}+6224240959272567 x^{4}-1856274326185048 x^{2}-6159421440 \ln \left (x -\frac {1}{2}\right ) x^{4}+121206594270421440 \ln \left (\frac {2}{3}+x \right ) x^{4}-2727574554608640 \ln \left (\frac {2}{3}+x \right )-1524695040 \ln \left (x -\frac {1}{2}\right ) x^{3}+1131970560 \ln \left (x -\frac {1}{2}\right ) x^{2}+785448960 \ln \left (x -\frac {1}{2}\right ) x +2727574416000000 \ln \left (x +\frac {3}{5}\right )+127343637018290880 \ln \left (\frac {2}{3}+x \right ) x^{5}-30003318576000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-127343630547000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-121206588111000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+46027820609020800 \ln \left (\frac {2}{3}+x \right ) x^{6}-46027818270000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+138608640 \ln \left (x -\frac {1}{2}\right )-2339020800 \ln \left (x -\frac {1}{2}\right ) x^{6}-6471290880 \ln \left (x -\frac {1}{2}\right ) x^{5}}{30065437248 \left (-1+2 x \right ) \left (2+3 x \right )^{4} \left (3+5 x \right )}\) | \(227\) |
(-532422817875/4067294*x^3-207557085150/2033647*x^5-63255507120/290521*x^4 +33188468700/2033647*x^2+317609203475/8134588*x+77754195847/8134588)/(-1+2 *x)/(2+3*x)^4/(3+5*x)-15040/156590819*ln(-1+2*x)+222359715/117649*ln(2+3*x )-2515625/1331*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (79) = 158\).
Time = 0.22 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {63927582226200 \, x^{5} + 136378873350720 \, x^{4} + 81993113952750 \, x^{3} - 10222048359600 \, x^{2} + 1183843062500 \, {\left (810 \, x^{6} + 2241 \, x^{5} + 2133 \, x^{4} + 528 \, x^{3} - 392 \, x^{2} - 272 \, x - 48\right )} \log \left (5 \, x + 3\right ) - 1183843122660 \, {\left (810 \, x^{6} + 2241 \, x^{5} + 2133 \, x^{4} + 528 \, x^{3} - 392 \, x^{2} - 272 \, x - 48\right )} \log \left (3 \, x + 2\right ) + 60160 \, {\left (810 \, x^{6} + 2241 \, x^{5} + 2133 \, x^{4} + 528 \, x^{3} - 392 \, x^{2} - 272 \, x - 48\right )} \log \left (2 \, x - 1\right ) - 24455908667575 \, x - 5987073080219}{626363276 \, {\left (810 \, x^{6} + 2241 \, x^{5} + 2133 \, x^{4} + 528 \, x^{3} - 392 \, x^{2} - 272 \, x - 48\right )}} \]
-1/626363276*(63927582226200*x^5 + 136378873350720*x^4 + 81993113952750*x^ 3 - 10222048359600*x^2 + 1183843062500*(810*x^6 + 2241*x^5 + 2133*x^4 + 52 8*x^3 - 392*x^2 - 272*x - 48)*log(5*x + 3) - 1183843122660*(810*x^6 + 2241 *x^5 + 2133*x^4 + 528*x^3 - 392*x^2 - 272*x - 48)*log(3*x + 2) + 60160*(81 0*x^6 + 2241*x^5 + 2133*x^4 + 528*x^3 - 392*x^2 - 272*x - 48)*log(2*x - 1) - 24455908667575*x - 5987073080219)/(810*x^6 + 2241*x^5 + 2133*x^4 + 528* x^3 - 392*x^2 - 272*x - 48)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=\frac {- 830228340600 x^{5} - 1771154199360 x^{4} - 1064845635750 x^{3} + 132753874800 x^{2} + 317609203475 x + 77754195847}{6589016280 x^{6} + 18229611708 x^{5} + 17351076204 x^{4} + 4295062464 x^{3} - 3188758496 x^{2} - 2212607936 x - 390460224} - \frac {15040 \log {\left (x - \frac {1}{2} \right )}}{156590819} - \frac {2515625 \log {\left (x + \frac {3}{5} \right )}}{1331} + \frac {222359715 \log {\left (x + \frac {2}{3} \right )}}{117649} \]
(-830228340600*x**5 - 1771154199360*x**4 - 1064845635750*x**3 + 1327538748 00*x**2 + 317609203475*x + 77754195847)/(6589016280*x**6 + 18229611708*x** 5 + 17351076204*x**4 + 4295062464*x**3 - 3188758496*x**2 - 2212607936*x - 390460224) - 15040*log(x - 1/2)/156590819 - 2515625*log(x + 3/5)/1331 + 22 2359715*log(x + 2/3)/117649
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {830228340600 \, x^{5} + 1771154199360 \, x^{4} + 1064845635750 \, x^{3} - 132753874800 \, x^{2} - 317609203475 \, x - 77754195847}{8134588 \, {\left (810 \, x^{6} + 2241 \, x^{5} + 2133 \, x^{4} + 528 \, x^{3} - 392 \, x^{2} - 272 \, x - 48\right )}} - \frac {2515625}{1331} \, \log \left (5 \, x + 3\right ) + \frac {222359715}{117649} \, \log \left (3 \, x + 2\right ) - \frac {15040}{156590819} \, \log \left (2 \, x - 1\right ) \]
-1/8134588*(830228340600*x^5 + 1771154199360*x^4 + 1064845635750*x^3 - 132 753874800*x^2 - 317609203475*x - 77754195847)/(810*x^6 + 2241*x^5 + 2133*x ^4 + 528*x^3 - 392*x^2 - 272*x - 48) - 2515625/1331*log(5*x + 3) + 2223597 15/117649*log(3*x + 2) - 15040/156590819*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {15625}{121 \, {\left (5 \, x + 3\right )}} + \frac {25 \, {\left (\frac {6062344264539}{5 \, x + 3} + \frac {7964082495612}{{\left (5 \, x + 3\right )}^{2}} + \frac {3205106234076}{{\left (5 \, x + 3\right )}^{3}} + \frac {435889532968}{{\left (5 \, x + 3\right )}^{4}} - 1385260555122\right )}}{89480468 \, {\left (\frac {11}{5 \, x + 3} - 2\right )} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{4}} + \frac {222359715}{117649} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {15040}{156590819} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]
-15625/121/(5*x + 3) + 25/89480468*(6062344264539/(5*x + 3) + 796408249561 2/(5*x + 3)^2 + 3205106234076/(5*x + 3)^3 + 435889532968/(5*x + 3)^4 - 138 5260555122)/((11/(5*x + 3) - 2)*(1/(5*x + 3) + 3)^4) + 222359715/117649*lo g(abs(-1/(5*x + 3) - 3)) - 15040/156590819*log(abs(-11/(5*x + 3) + 2))
Time = 1.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)^2} \, dx=\frac {222359715\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {15040\,\ln \left (x-\frac {1}{2}\right )}{156590819}-\frac {2515625\,\ln \left (x+\frac {3}{5}\right )}{1331}+\frac {-\frac {256243315\,x^5}{2033647}-\frac {234279656\,x^4}{871563}-\frac {3943872725\,x^3}{24403764}+\frac {1106282290\,x^2}{54908469}+\frac {63521840695\,x}{1317803256}+\frac {77754195847}{6589016280}}{x^6+\frac {83\,x^5}{30}+\frac {79\,x^4}{30}+\frac {88\,x^3}{135}-\frac {196\,x^2}{405}-\frac {136\,x}{405}-\frac {8}{135}} \]
(222359715*log(x + 2/3))/117649 - (15040*log(x - 1/2))/156590819 - (251562 5*log(x + 3/5))/1331 + ((63521840695*x)/1317803256 + (1106282290*x^2)/5490 8469 - (3943872725*x^3)/24403764 - (234279656*x^4)/871563 - (256243315*x^5 )/2033647 + 77754195847/6589016280)/((88*x^3)/135 - (196*x^2)/405 - (136*x )/405 + (79*x^4)/30 + (83*x^5)/30 + x^6 - 8/135)