Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=-\frac {9}{35 (2+3 x)^5}-\frac {162}{49 (2+3 x)^4}-\frac {11457}{343 (2+3 x)^3}-\frac {806121}{2401 (2+3 x)^2}-\frac {70752609}{16807 (2+3 x)}-\frac {15625}{11 (3+5 x)}-\frac {128 \log (1-2 x)}{14235529}+\frac {2977686468 \log (2+3 x)}{117649}-\frac {3062500}{121} \log (3+5 x) \]
-9/35/(2+3*x)^5-162/49/(2+3*x)^4-11457/343/(2+3*x)^3-806121/2401/(2+3*x)^2 -70752609/16807/(2+3*x)-15625/11/(3+5*x)-128/14235529*ln(1-2*x)+2977686468 /117649*ln(2+3*x)-3062500/121*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=-\frac {9}{35 (2+3 x)^5}-\frac {162}{49 (2+3 x)^4}-\frac {11457}{343 (2+3 x)^3}-\frac {806121}{2401 (2+3 x)^2}-\frac {70752609}{16807 (2+3 x)}-\frac {15625}{33+55 x}-\frac {128 \log (1-2 x)}{14235529}+\frac {2977686468 \log (4+6 x)}{117649}-\frac {3062500}{121} \log (6+10 x) \]
-9/(35*(2 + 3*x)^5) - 162/(49*(2 + 3*x)^4) - 11457/(343*(2 + 3*x)^3) - 806 121/(2401*(2 + 3*x)^2) - 70752609/(16807*(2 + 3*x)) - 15625/(33 + 55*x) - (128*Log[1 - 2*x])/14235529 + (2977686468*Log[4 + 6*x])/117649 - (3062500* Log[6 + 10*x])/121
Time = 0.22 (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.
| \(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^6 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {8933059404}{117649 (3 x+2)}-\frac {15312500}{121 (5 x+3)}+\frac {212257827}{16807 (3 x+2)^2}+\frac {78125}{11 (5 x+3)^2}+\frac {4836726}{2401 (3 x+2)^3}+\frac {103113}{343 (3 x+2)^4}+\frac {1944}{49 (3 x+2)^5}+\frac {27}{7 (3 x+2)^6}-\frac {256}{14235529 (2 x-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {70752609}{16807 (3 x+2)}-\frac {15625}{11 (5 x+3)}-\frac {806121}{2401 (3 x+2)^2}-\frac {11457}{343 (3 x+2)^3}-\frac {162}{49 (3 x+2)^4}-\frac {9}{35 (3 x+2)^5}-\frac {128 \log (1-2 x)}{14235529}+\frac {2977686468 \log (3 x+2)}{117649}-\frac {3062500}{121} \log (5 x+3)\) |
-9/(35*(2 + 3*x)^5) - 162/(49*(2 + 3*x)^4) - 11457/(343*(2 + 3*x)^3) - 806 121/(2401*(2 + 3*x)^2) - 70752609/(16807*(2 + 3*x)) - 15625/(11*(3 + 5*x)) - (128*Log[1 - 2*x])/14235529 + (2977686468*Log[2 + 3*x])/117649 - (30625 00*Log[3 + 5*x])/121
3.16.15.3.1 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 = 2.60 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
| method | result | size |
| norman | \(\frac {-\frac {1650548638674}{184877} x^{3}-\frac {1088751934377}{184877} x^{2}-\frac {379016951220}{184877} x^{5}-\frac {358977035229}{184877} x -\frac {178679420046}{26411} x^{4}-\frac {236642515057}{924385}}{\left (2+3 x \right )^{5} \left (3+5 x \right )}-\frac {128 \ln \left (-1+2 x \right )}{14235529}+\frac {2977686468 \ln \left (2+3 x \right )}{117649}-\frac {3062500 \ln \left (3+5 x \right )}{121}\) | \(66\) |
| risch | \(\frac {-\frac {1650548638674}{184877} x^{3}-\frac {1088751934377}{184877} x^{2}-\frac {379016951220}{184877} x^{5}-\frac {358977035229}{184877} x -\frac {178679420046}{26411} x^{4}-\frac {236642515057}{924385}}{\left (2+3 x \right )^{5} \left (3+5 x \right )}-\frac {128 \ln \left (-1+2 x \right )}{14235529}+\frac {2977686468 \ln \left (2+3 x \right )}{117649}-\frac {3062500 \ln \left (3+5 x \right )}{121}\) | \(67\) |
| default | \(-\frac {15625}{11 \left (3+5 x \right )}-\frac {3062500 \ln \left (3+5 x \right )}{121}-\frac {128 \ln \left (-1+2 x \right )}{14235529}-\frac {9}{35 \left (2+3 x \right )^{5}}-\frac {162}{49 \left (2+3 x \right )^{4}}-\frac {11457}{343 \left (2+3 x \right )^{3}}-\frac {806121}{2401 \left (2+3 x \right )^{2}}-\frac {70752609}{16807 \left (2+3 x \right )}+\frac {2977686468 \ln \left (2+3 x \right )}{117649}\) | \(80\) |
| parallelrisch | \(\frac {2767105598198480 x -581091940800000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+1182937165620249600 \ln \left (\frac {2}{3}+x \right ) x^{3}-152190746400000000 \ln \left (x +\frac {3}{5}\right ) x +581091941006438400 \ln \left (\frac {2}{3}+x \right ) x^{2}+152190746454067200 \ln \left (\frac {2}{3}+x \right ) x +73071956101128831 x^{5}+22139090496157635 x^{6}+63630602144829720 x^{3}+96446199198714750 x^{4}+20983880000973120 x^{2}-481075200 \ln \left (x -\frac {1}{2}\right ) x^{4}+1354151755381075200 \ln \left (\frac {2}{3}+x \right ) x^{4}+16602626885898240 \ln \left (\frac {2}{3}+x \right )-420249600 \ln \left (x -\frac {1}{2}\right ) x^{3}-206438400 \ln \left (x -\frac {1}{2}\right ) x^{2}-54067200 \ln \left (x -\frac {1}{2}\right ) x -16602626880000000 \ln \left (x +\frac {3}{5}\right )+826499519663621760 \ln \left (\frac {2}{3}+x \right ) x^{5}-1182937165200000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-826499519370000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1354151754900000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+210126996524649600 \ln \left (\frac {2}{3}+x \right ) x^{6}-210126996450000000 \ln \left (x +\frac {3}{5}\right ) x^{6}-5898240 \ln \left (x -\frac {1}{2}\right )-74649600 \ln \left (x -\frac {1}{2}\right ) x^{6}-293621760 \ln \left (x -\frac {1}{2}\right ) x^{5}}{6833053920 \left (2+3 x \right )^{5} \left (3+5 x \right )}\) | \(220\) |
(-1650548638674/184877*x^3-1088751934377/184877*x^2-379016951220/184877*x^ 5-358977035229/184877*x-178679420046/26411*x^4-236642515057/924385)/(2+3*x )^5/(3+5*x)-128/14235529*ln(-1+2*x)+2977686468/117649*ln(2+3*x)-3062500/12 1*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 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=-\frac {145921526219700 \, x^{5} + 481541037023970 \, x^{4} + 635461225889490 \, x^{3} + 419169494735145 \, x^{2} + 1801500312500 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 1801500313140 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 640 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (2 \, x - 1\right ) + 138206158563165 \, x + 18221473659389}{71177645 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]
-1/71177645*(145921526219700*x^5 + 481541037023970*x^4 + 635461225889490*x ^3 + 419169494735145*x^2 + 1801500312500*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 1801500313140*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(3*x + 2) + 6 40*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log (2*x - 1) + 138206158563165*x + 18221473659389)/(1215*x^6 + 4779*x^5 + 783 0*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=- \frac {1895084756100 x^{5} + 6253779701610 x^{4} + 8252743193370 x^{3} + 5443759671885 x^{2} + 1794885176145 x + 236642515057}{1123127775 x^{6} + 4417635915 x^{5} + 7237934550 x^{4} + 6322793400 x^{3} + 3105933600 x^{2} + 813458800 x + 88740960} - \frac {128 \log {\left (x - \frac {1}{2} \right )}}{14235529} - \frac {3062500 \log {\left (x + \frac {3}{5} \right )}}{121} + \frac {2977686468 \log {\left (x + \frac {2}{3} \right )}}{117649} \]
-(1895084756100*x**5 + 6253779701610*x**4 + 8252743193370*x**3 + 544375967 1885*x**2 + 1794885176145*x + 236642515057)/(1123127775*x**6 + 4417635915* x**5 + 7237934550*x**4 + 6322793400*x**3 + 3105933600*x**2 + 813458800*x + 88740960) - 128*log(x - 1/2)/14235529 - 3062500*log(x + 3/5)/121 + 297768 6468*log(x + 2/3)/117649
Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=-\frac {1895084756100 \, x^{5} + 6253779701610 \, x^{4} + 8252743193370 \, x^{3} + 5443759671885 \, x^{2} + 1794885176145 \, x + 236642515057}{924385 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - \frac {3062500}{121} \, \log \left (5 \, x + 3\right ) + \frac {2977686468}{117649} \, \log \left (3 \, x + 2\right ) - \frac {128}{14235529} \, \log \left (2 \, x - 1\right ) \]
-1/924385*(1895084756100*x^5 + 6253779701610*x^4 + 8252743193370*x^3 + 544 3759671885*x^2 + 1794885176145*x + 236642515057)/(1215*x^6 + 4779*x^5 + 78 30*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 3062500/121*log(5*x + 3) + 29 77686468/117649*log(3*x + 2) - 128/14235529*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=-\frac {15625}{11 \, {\left (5 \, x + 3\right )}} + \frac {135 \, {\left (\frac {1627470333}{5 \, x + 3} + \frac {915260769}{{\left (5 \, x + 3\right )}^{2}} + \frac {234430752}{{\left (5 \, x + 3\right )}^{3}} + \frac {23397131}{{\left (5 \, x + 3\right )}^{4}} + 1103836896\right )}}{16807 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{5}} + \frac {2977686468}{117649} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {128}{14235529} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]
-15625/11/(5*x + 3) + 135/16807*(1627470333/(5*x + 3) + 915260769/(5*x + 3 )^2 + 234430752/(5*x + 3)^3 + 23397131/(5*x + 3)^4 + 1103836896)/(1/(5*x + 3) + 3)^5 + 2977686468/117649*log(abs(-1/(5*x + 3) - 3)) - 128/14235529*l og(abs(-11/(5*x + 3) + 2))
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx=\frac {2977686468\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {128\,\ln \left (x-\frac {1}{2}\right )}{14235529}-\frac {3062500\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {\frac {311948108\,x^5}{184877}+\frac {2205918766\,x^4}{396165}+\frac {61131431062\,x^3}{8319465}+\frac {120972437153\,x^2}{24958395}+\frac {119659011743\,x}{74875185}+\frac {236642515057}{1123127775}}{x^6+\frac {59\,x^5}{15}+\frac {58\,x^4}{9}+\frac {152\,x^3}{27}+\frac {224\,x^2}{81}+\frac {176\,x}{243}+\frac {32}{405}} \]
(2977686468*log(x + 2/3))/117649 - (128*log(x - 1/2))/14235529 - (3062500* log(x + 3/5))/121 - ((119659011743*x)/74875185 + (120972437153*x^2)/249583 95 + (61131431062*x^3)/8319465 + (2205918766*x^4)/396165 + (311948108*x^5) /184877 + 236642515057/1123127775)/((176*x)/243 + (224*x^2)/81 + (152*x^3) /27 + (58*x^4)/9 + (59*x^5)/15 + x^6 + 32/405)