Integrand size = 22, antiderivative size = 98 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {1}{441 (2+3 x)^7}+\frac {34}{1323 (2+3 x)^6}-\frac {121}{1715 (2+3 x)^5}-\frac {121}{4802 (2+3 x)^4}-\frac {484}{50421 (2+3 x)^3}-\frac {484}{117649 (2+3 x)^2}-\frac {1936}{823543 (2+3 x)}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (2+3 x)}{5764801} \]
-1/441/(2+3*x)^7+34/1323/(2+3*x)^6-121/1715/(2+3*x)^5-121/4802/(2+3*x)^4-4 84/50421/(2+3*x)^3-484/117649/(2+3*x)^2-1936/823543/(2+3*x)-3872/5764801*l n(1-2*x)+3872/5764801*ln(2+3*x)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=\frac {8 \left (-\frac {7 \left (193528666+1098354408 x+2692491516 x^2+3858408675 x^3+3454264440 x^4+1746538200 x^5+381062880 x^6\right )}{16 (2+3 x)^7}-65340 \log (1-2 x)+65340 \log (4+6 x)\right )}{778248135} \]
(8*((-7*(193528666 + 1098354408*x + 2692491516*x^2 + 3858408675*x^3 + 3454 264440*x^4 + 1746538200*x^5 + 381062880*x^6))/(16*(2 + 3*x)^7) - 65340*Log [1 - 2*x] + 65340*Log[4 + 6*x]))/778248135
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.
| \(\displaystyle \int \frac {(5 x+3)^2}{(1-2 x) (3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {11616}{5764801 (3 x+2)}+\frac {5808}{823543 (3 x+2)^2}+\frac {2904}{117649 (3 x+2)^3}+\frac {1452}{16807 (3 x+2)^4}+\frac {726}{2401 (3 x+2)^5}+\frac {363}{343 (3 x+2)^6}-\frac {68}{147 (3 x+2)^7}+\frac {1}{21 (3 x+2)^8}-\frac {7744}{5764801 (2 x-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1936}{823543 (3 x+2)}-\frac {484}{117649 (3 x+2)^2}-\frac {484}{50421 (3 x+2)^3}-\frac {121}{4802 (3 x+2)^4}-\frac {121}{1715 (3 x+2)^5}+\frac {34}{1323 (3 x+2)^6}-\frac {1}{441 (3 x+2)^7}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (3 x+2)}{5764801}\) |
-1/441*1/(2 + 3*x)^7 + 34/(1323*(2 + 3*x)^6) - 121/(1715*(2 + 3*x)^5) - 12 1/(4802*(2 + 3*x)^4) - 484/(50421*(2 + 3*x)^3) - 484/(117649*(2 + 3*x)^2) - 1936/(823543*(2 + 3*x)) - (3872*Log[1 - 2*x])/5764801 + (3872*Log[2 + 3* x])/5764801
3.15.67.3.1 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 = 2.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.57
| method | result | size |
| norman | \(\frac {-\frac {183059068}{37059435} x -\frac {49860954}{4117715} x^{2}-\frac {28580805}{1647086} x^{3}-\frac {12793572}{823543} x^{4}-\frac {6468660}{823543} x^{5}-\frac {1411344}{823543} x^{6}-\frac {96764333}{111178305}}{\left (2+3 x \right )^{7}}-\frac {3872 \ln \left (-1+2 x \right )}{5764801}+\frac {3872 \ln \left (2+3 x \right )}{5764801}\) | \(56\) |
| risch | \(\frac {-\frac {183059068}{37059435} x -\frac {49860954}{4117715} x^{2}-\frac {28580805}{1647086} x^{3}-\frac {12793572}{823543} x^{4}-\frac {6468660}{823543} x^{5}-\frac {1411344}{823543} x^{6}-\frac {96764333}{111178305}}{\left (2+3 x \right )^{7}}-\frac {3872 \ln \left (-1+2 x \right )}{5764801}+\frac {3872 \ln \left (2+3 x \right )}{5764801}\) | \(57\) |
| default | \(-\frac {3872 \ln \left (-1+2 x \right )}{5764801}-\frac {1}{441 \left (2+3 x \right )^{7}}+\frac {34}{1323 \left (2+3 x \right )^{6}}-\frac {121}{1715 \left (2+3 x \right )^{5}}-\frac {121}{4802 \left (2+3 x \right )^{4}}-\frac {484}{50421 \left (2+3 x \right )^{3}}-\frac {484}{117649 \left (2+3 x \right )^{2}}-\frac {1936}{823543 \left (2+3 x \right )}+\frac {3872 \ln \left (2+3 x \right )}{5764801}\) | \(81\) |
| parallelrisch | \(\frac {15492447040 x +37468569600 \ln \left (\frac {2}{3}+x \right ) x^{3}+14987427840 \ln \left (\frac {2}{3}+x \right ) x^{2}+3330539520 \ln \left (\frac {2}{3}+x \right ) x +483097253436 x^{5}+249715603998 x^{6}+54865376811 x^{7}+315295182160 x^{3}+511659075480 x^{4}+107051059360 x^{2}-56202854400 \ln \left (x -\frac {1}{2}\right ) x^{4}+56202854400 \ln \left (\frac {2}{3}+x \right ) x^{4}+317194240 \ln \left (\frac {2}{3}+x \right )-37468569600 \ln \left (x -\frac {1}{2}\right ) x^{3}+5419560960 \ln \left (\frac {2}{3}+x \right ) x^{7}-14987427840 \ln \left (x -\frac {1}{2}\right ) x^{2}-3330539520 \ln \left (x -\frac {1}{2}\right ) x +50582568960 \ln \left (\frac {2}{3}+x \right ) x^{5}+25291284480 \ln \left (\frac {2}{3}+x \right ) x^{6}-317194240 \ln \left (x -\frac {1}{2}\right )-5419560960 \ln \left (x -\frac {1}{2}\right ) x^{7}-25291284480 \ln \left (x -\frac {1}{2}\right ) x^{6}-50582568960 \ln \left (x -\frac {1}{2}\right ) x^{5}}{3689472640 \left (2+3 x \right )^{7}}\) | \(178\) |
(-183059068/37059435*x-49860954/4117715*x^2-28580805/1647086*x^3-12793572/ 823543*x^4-6468660/823543*x^5-1411344/823543*x^6-96764333/111178305)/(2+3* x)^7-3872/5764801*ln(-1+2*x)+3872/5764801*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {2667440160 \, x^{6} + 12225767400 \, x^{5} + 24179851080 \, x^{4} + 27008860725 \, x^{3} + 18847440612 \, x^{2} - 1045440 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 1045440 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 7688480856 \, x + 1354700662}{1556496270 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
-1/1556496270*(2667440160*x^6 + 12225767400*x^5 + 24179851080*x^4 + 270088 60725*x^3 + 18847440612*x^2 - 1045440*(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) + 1045440*(2 187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344* x + 128)*log(2*x - 1) + 7688480856*x + 1354700662)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=- \frac {381062880 x^{6} + 1746538200 x^{5} + 3454264440 x^{4} + 3858408675 x^{3} + 2692491516 x^{2} + 1098354408 x + 193528666}{486293906070 x^{7} + 2269371561660 x^{6} + 4538743123320 x^{5} + 5043047914800 x^{4} + 3362031943200 x^{3} + 1344812777280 x^{2} + 298847283840 x + 28461646080} - \frac {3872 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {3872 \log {\left (x + \frac {2}{3} \right )}}{5764801} \]
-(381062880*x**6 + 1746538200*x**5 + 3454264440*x**4 + 3858408675*x**3 + 2 692491516*x**2 + 1098354408*x + 193528666)/(486293906070*x**7 + 2269371561 660*x**6 + 4538743123320*x**5 + 5043047914800*x**4 + 3362031943200*x**3 + 1344812777280*x**2 + 298847283840*x + 28461646080) - 3872*log(x - 1/2)/576 4801 + 3872*log(x + 2/3)/5764801
Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {3872}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {3872}{5764801} \, \log \left (2 \, x - 1\right ) \]
-1/222356610*(381062880*x^6 + 1746538200*x^5 + 3454264440*x^4 + 3858408675 *x^3 + 2692491516*x^2 + 1098354408*x + 193528666)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 3872/576480 1*log(3*x + 2) - 3872/5764801*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=-\frac {381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \, {\left (3 \, x + 2\right )}^{7}} + \frac {3872}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3872}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-1/222356610*(381062880*x^6 + 1746538200*x^5 + 3454264440*x^4 + 3858408675 *x^3 + 2692491516*x^2 + 1098354408*x + 193528666)/(3*x + 2)^7 + 3872/57648 01*log(abs(3*x + 2)) - 3872/5764801*log(abs(2*x - 1))
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx=\frac {7744\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {1936\,x^6}{2470629}+\frac {26620\,x^5}{7411887}+\frac {473836\,x^4}{66706983}+\frac {3175645\,x^3}{400241898}+\frac {1846702\,x^2}{333534915}+\frac {183059068\,x}{81048984345}+\frac {96764333}{243146953035}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \]
(7744*atanh((12*x)/7 + 1/7))/5764801 - ((183059068*x)/81048984345 + (18467 02*x^2)/333534915 + (3175645*x^3)/400241898 + (473836*x^4)/66706983 + (266 20*x^5)/7411887 + (1936*x^6)/2470629 + 96764333/243146953035)/((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)