Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=\frac {1}{1323 (2+3 x)^7}-\frac {103}{7938 (2+3 x)^6}+\frac {3469}{46305 (2+3 x)^5}-\frac {1331}{9604 (2+3 x)^4}-\frac {2662}{50421 (2+3 x)^3}-\frac {2662}{117649 (2+3 x)^2}-\frac {10648}{823543 (2+3 x)}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (2+3 x)}{5764801} \]
1/1323/(2+3*x)^7-103/7938/(2+3*x)^6+3469/46305/(2+3*x)^5-1331/9604/(2+3*x) ^4-2662/50421/(2+3*x)^3-2662/117649/(2+3*x)^2-10648/823543/(2+3*x)-21296/5 764801*ln(1-2*x)+21296/5764801*ln(2+3*x)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=\frac {4 \left (-\frac {7 \left (4309941128+29451465714 x+83293304778 x^2+127327486275 x^3+113990726520 x^4+57635760600 x^5+12575075040 x^6\right )}{16 (2+3 x)^7}-2156220 \log (1-2 x)+2156220 \log (4+6 x)\right )}{2334744405} \]
(4*((-7*(4309941128 + 29451465714*x + 83293304778*x^2 + 127327486275*x^3 + 113990726520*x^4 + 57635760600*x^5 + 12575075040*x^6))/(16*(2 + 3*x)^7) - 2156220*Log[1 - 2*x] + 2156220*Log[4 + 6*x]))/2334744405
Time = 0.21 (sec) , antiderivative size = 98, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x) (3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {63888}{5764801 (3 x+2)}+\frac {31944}{823543 (3 x+2)^2}+\frac {15972}{117649 (3 x+2)^3}+\frac {7986}{16807 (3 x+2)^4}+\frac {3993}{2401 (3 x+2)^5}-\frac {3469}{3087 (3 x+2)^6}+\frac {103}{441 (3 x+2)^7}-\frac {1}{63 (3 x+2)^8}-\frac {42592}{5764801 (2 x-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10648}{823543 (3 x+2)}-\frac {2662}{117649 (3 x+2)^2}-\frac {2662}{50421 (3 x+2)^3}-\frac {1331}{9604 (3 x+2)^4}+\frac {3469}{46305 (3 x+2)^5}-\frac {103}{7938 (3 x+2)^6}+\frac {1}{1323 (3 x+2)^7}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (3 x+2)}{5764801}\) |
1/(1323*(2 + 3*x)^7) - 103/(7938*(2 + 3*x)^6) + 3469/(46305*(2 + 3*x)^5) - 1331/(9604*(2 + 3*x)^4) - 2662/(50421*(2 + 3*x)^3) - 2662/(117649*(2 + 3* x)^2) - 10648/(823543*(2 + 3*x)) - (21296*Log[1 - 2*x])/5764801 + (21296*L og[2 + 3*x])/5764801
3.15.83.3.1 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 = 0.83 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.57
| method | result | size |
| norman | \(\frac {-\frac {4908577619}{222356610} x -\frac {1542468607}{24706290} x^{2}-\frac {314388855}{3294172} x^{3}-\frac {70364646}{823543} x^{4}-\frac {35577630}{823543} x^{5}-\frac {7762392}{823543} x^{6}-\frac {1077485282}{333534915}}{\left (2+3 x \right )^{7}}-\frac {21296 \ln \left (-1+2 x \right )}{5764801}+\frac {21296 \ln \left (2+3 x \right )}{5764801}\) | \(56\) |
| risch | \(\frac {-\frac {4908577619}{222356610} x -\frac {1542468607}{24706290} x^{2}-\frac {314388855}{3294172} x^{3}-\frac {70364646}{823543} x^{4}-\frac {35577630}{823543} x^{5}-\frac {7762392}{823543} x^{6}-\frac {1077485282}{333534915}}{\left (2+3 x \right )^{7}}-\frac {21296 \ln \left (-1+2 x \right )}{5764801}+\frac {21296 \ln \left (2+3 x \right )}{5764801}\) | \(57\) |
| default | \(-\frac {21296 \ln \left (-1+2 x \right )}{5764801}+\frac {1}{1323 \left (2+3 x \right )^{7}}-\frac {103}{7938 \left (2+3 x \right )^{6}}+\frac {3469}{46305 \left (2+3 x \right )^{5}}-\frac {1331}{9604 \left (2+3 x \right )^{4}}-\frac {2662}{50421 \left (2+3 x \right )^{3}}-\frac {2662}{117649 \left (2+3 x \right )^{2}}-\frac {10648}{823543 \left (2+3 x \right )}+\frac {21296 \ln \left (2+3 x \right )}{5764801}\) | \(81\) |
| parallelrisch | \(\frac {131105674560 x +618231398400 \ln \left (\frac {2}{3}+x \right ) x^{3}+247292559360 \ln \left (\frac {2}{3}+x \right ) x^{2}+54953902080 \ln \left (\frac {2}{3}+x \right ) x +5223888765144 x^{5}+2746699507692 x^{6}+610934154894 x^{7}+3167395752640 x^{3}+5389912615920 x^{4}+998470986240 x^{2}-927347097600 \ln \left (x -\frac {1}{2}\right ) x^{4}+927347097600 \ln \left (\frac {2}{3}+x \right ) x^{4}+5233704960 \ln \left (\frac {2}{3}+x \right )-618231398400 \ln \left (x -\frac {1}{2}\right ) x^{3}+89422755840 \ln \left (\frac {2}{3}+x \right ) x^{7}-247292559360 \ln \left (x -\frac {1}{2}\right ) x^{2}-54953902080 \ln \left (x -\frac {1}{2}\right ) x +834612387840 \ln \left (\frac {2}{3}+x \right ) x^{5}+417306193920 \ln \left (\frac {2}{3}+x \right ) x^{6}-5233704960 \ln \left (x -\frac {1}{2}\right )-89422755840 \ln \left (x -\frac {1}{2}\right ) x^{7}-417306193920 \ln \left (x -\frac {1}{2}\right ) x^{6}-834612387840 \ln \left (x -\frac {1}{2}\right ) x^{5}}{11068417920 \left (2+3 x \right )^{7}}\) | \(178\) |
(-4908577619/222356610*x-1542468607/24706290*x^2-314388855/3294172*x^3-703 64646/823543*x^4-35577630/823543*x^5-7762392/823543*x^6-1077485282/3335349 15)/(2+3*x)^7-21296/5764801*ln(-1+2*x)+21296/5764801*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {88025525280 \, x^{6} + 403450324200 \, x^{5} + 797935085640 \, x^{4} + 891292403925 \, x^{3} + 583053133446 \, x^{2} - 34499520 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 34499520 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 206160259998 \, x + 30169587896}{9338977620 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
-1/9338977620*(88025525280*x^6 + 403450324200*x^5 + 797935085640*x^4 + 891 292403925*x^3 + 583053133446*x^2 - 34499520*(2187*x^7 + 10206*x^6 + 20412* x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 3449 9520*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(2*x - 1) + 206160259998*x + 30169587896)/(2187*x^7 + 1 0206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=- \frac {12575075040 x^{6} + 57635760600 x^{5} + 113990726520 x^{4} + 127327486275 x^{3} + 83293304778 x^{2} + 29451465714 x + 4309941128}{2917763436420 x^{7} + 13616229369960 x^{6} + 27232458739920 x^{5} + 30258287488800 x^{4} + 20172191659200 x^{3} + 8068876663680 x^{2} + 1793083703040 x + 170769876480} - \frac {21296 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {21296 \log {\left (x + \frac {2}{3} \right )}}{5764801} \]
-(12575075040*x**6 + 57635760600*x**5 + 113990726520*x**4 + 127327486275*x **3 + 83293304778*x**2 + 29451465714*x + 4309941128)/(2917763436420*x**7 + 13616229369960*x**6 + 27232458739920*x**5 + 30258287488800*x**4 + 2017219 1659200*x**3 + 8068876663680*x**2 + 1793083703040*x + 170769876480) - 2129 6*log(x - 1/2)/5764801 + 21296*log(x + 2/3)/5764801
Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {12575075040 \, x^{6} + 57635760600 \, x^{5} + 113990726520 \, x^{4} + 127327486275 \, x^{3} + 83293304778 \, x^{2} + 29451465714 \, x + 4309941128}{1334139660 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {21296}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {21296}{5764801} \, \log \left (2 \, x - 1\right ) \]
-1/1334139660*(12575075040*x^6 + 57635760600*x^5 + 113990726520*x^4 + 1273 27486275*x^3 + 83293304778*x^2 + 29451465714*x + 4309941128)/(2187*x^7 + 1 0206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 21296/5764801*log(3*x + 2) - 21296/5764801*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=-\frac {12575075040 \, x^{6} + 57635760600 \, x^{5} + 113990726520 \, x^{4} + 127327486275 \, x^{3} + 83293304778 \, x^{2} + 29451465714 \, x + 4309941128}{1334139660 \, {\left (3 \, x + 2\right )}^{7}} + \frac {21296}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {21296}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-1/1334139660*(12575075040*x^6 + 57635760600*x^5 + 113990726520*x^4 + 1273 27486275*x^3 + 83293304778*x^2 + 29451465714*x + 4309941128)/(3*x + 2)^7 + 21296/5764801*log(abs(3*x + 2)) - 21296/5764801*log(abs(2*x - 1))
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx=\frac {42592\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {10648\,x^6}{2470629}+\frac {146410\,x^5}{7411887}+\frac {2606098\,x^4}{66706983}+\frac {34932095\,x^3}{800483796}+\frac {1542468607\,x^2}{54032656230}+\frac {4908577619\,x}{486293906070}+\frac {1077485282}{729440859105}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \]
(42592*atanh((12*x)/7 + 1/7))/5764801 - ((4908577619*x)/486293906070 + (15 42468607*x^2)/54032656230 + (34932095*x^3)/800483796 + (2606098*x^4)/66706 983 + (146410*x^5)/7411887 + (10648*x^6)/2470629 + 1077485282/729440859105 )/((448*x)/729 + (224*x^2)/81 + (560*x^3)/81 + (280*x^4)/27 + (28*x^5)/3 + (14*x^6)/3 + x^7 + 128/2187)