Integrand size = 22, antiderivative size = 72 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=-\frac {3579885909 x}{5000000}-\frac {118543581 x^2}{200000}-\frac {24660207 x^3}{50000}-\frac {6194313 x^4}{20000}-\frac {303993 x^5}{2500}-\frac {2187 x^6}{100}-\frac {1}{4296875 (3+5 x)}-\frac {5764801 \log (1-2 x)}{15488}+\frac {266 \log (3+5 x)}{47265625} \]
-3579885909/5000000*x-118543581/200000*x^2-24660207/50000*x^3-6194313/2000 0*x^4-303993/2500*x^5-2187/100*x^6-1/4296875/(3+5*x)-5764801/15488*ln(1-2* x)+266/47265625*ln(3+5*x)
Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=\frac {22 \left (-86057647830-196893724995 x-162997423875 x^2-135631138500 x^3-85171803750 x^4-33439230000 x^5-6014250000 x^6-\frac {64}{3+5 x}\right )-2251875390625 \log (3-6 x)+34048 \log (-3 (3+5 x))}{6050000000} \]
(22*(-86057647830 - 196893724995*x - 162997423875*x^2 - 135631138500*x^3 - 85171803750*x^4 - 33439230000*x^5 - 6014250000*x^6 - 64/(3 + 5*x)) - 2251 875390625*Log[3 - 6*x] + 34048*Log[-3*(3 + 5*x)])/6050000000
Time = 0.19 (sec) , antiderivative size = 72, 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.
\(\displaystyle \int \frac {(3 x+2)^8}{(1-2 x) (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {6561 x^5}{50}-\frac {303993 x^4}{500}-\frac {6194313 x^3}{5000}-\frac {73980621 x^2}{50000}-\frac {118543581 x}{100000}-\frac {5764801}{7744 (2 x-1)}+\frac {266}{9453125 (5 x+3)}+\frac {1}{859375 (5 x+3)^2}-\frac {3579885909}{5000000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2187 x^6}{100}-\frac {303993 x^5}{2500}-\frac {6194313 x^4}{20000}-\frac {24660207 x^3}{50000}-\frac {118543581 x^2}{200000}-\frac {3579885909 x}{5000000}-\frac {1}{4296875 (5 x+3)}-\frac {5764801 \log (1-2 x)}{15488}+\frac {266 \log (5 x+3)}{47265625}\) |
(-3579885909*x)/5000000 - (118543581*x^2)/200000 - (24660207*x^3)/50000 - (6194313*x^4)/20000 - (303993*x^5)/2500 - (2187*x^6)/100 - 1/(4296875*(3 + 5*x)) - (5764801*Log[1 - 2*x])/15488 + (266*Log[3 + 5*x])/47265625
3.16.1.3.1 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 = 0.84 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {2187 x^{6}}{100}-\frac {303993 x^{5}}{2500}-\frac {6194313 x^{4}}{20000}-\frac {24660207 x^{3}}{50000}-\frac {118543581 x^{2}}{200000}-\frac {3579885909 x}{5000000}-\frac {1}{21484375 \left (x +\frac {3}{5}\right )}-\frac {5764801 \ln \left (-1+2 x \right )}{15488}+\frac {266 \ln \left (3+5 x \right )}{47265625}\) | \(53\) |
default | \(-\frac {2187 x^{6}}{100}-\frac {303993 x^{5}}{2500}-\frac {6194313 x^{4}}{20000}-\frac {24660207 x^{3}}{50000}-\frac {118543581 x^{2}}{200000}-\frac {3579885909 x}{5000000}-\frac {1}{4296875 \left (3+5 x \right )}+\frac {266 \ln \left (3+5 x \right )}{47265625}-\frac {5764801 \ln \left (-1+2 x \right )}{15488}\) | \(55\) |
norman | \(\frac {-\frac {354408704927}{165000000} x -\frac {669754953}{125000} x^{2}-\frac {888640389}{200000} x^{3}-\frac {67903353}{20000} x^{4}-\frac {38267397}{20000} x^{5}-\frac {168399}{250} x^{6}-\frac {2187}{20} x^{7}}{3+5 x}-\frac {5764801 \ln \left (-1+2 x \right )}{15488}+\frac {266 \ln \left (3+5 x \right )}{47265625}\) | \(60\) |
parallelrisch | \(\frac {-1984702500000 x^{7}-12225767400000 x^{6}-34727662777500 x^{5}-61622292847500 x^{4}-80644115301750 x^{3}+510720 \ln \left (x +\frac {3}{5}\right ) x -33778130859375 \ln \left (x -\frac {1}{2}\right ) x -97248419175600 x^{2}+306432 \ln \left (x +\frac {3}{5}\right )-20266878515625 \ln \left (x -\frac {1}{2}\right )-38984957541970 x}{54450000000+90750000000 x}\) | \(70\) |
-2187/100*x^6-303993/2500*x^5-6194313/20000*x^4-24660207/50000*x^3-1185435 81/200000*x^2-3579885909/5000000*x-1/21484375/(x+3/5)-5764801/15488*ln(-1+ 2*x)+266/47265625*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=-\frac {661567500000 \, x^{7} + 4075255800000 \, x^{6} + 11575887592500 \, x^{5} + 20540764282500 \, x^{4} + 26881371767250 \, x^{3} + 32416139725200 \, x^{2} - 34048 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 2251875390625 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 12994985849670 \, x + 1408}{6050000000 \, {\left (5 \, x + 3\right )}} \]
-1/6050000000*(661567500000*x^7 + 4075255800000*x^6 + 11575887592500*x^5 + 20540764282500*x^4 + 26881371767250*x^3 + 32416139725200*x^2 - 34048*(5*x + 3)*log(5*x + 3) + 2251875390625*(5*x + 3)*log(2*x - 1) + 12994985849670 *x + 1408)/(5*x + 3)
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=- \frac {2187 x^{6}}{100} - \frac {303993 x^{5}}{2500} - \frac {6194313 x^{4}}{20000} - \frac {24660207 x^{3}}{50000} - \frac {118543581 x^{2}}{200000} - \frac {3579885909 x}{5000000} - \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{15488} + \frac {266 \log {\left (x + \frac {3}{5} \right )}}{47265625} - \frac {1}{21484375 x + 12890625} \]
-2187*x**6/100 - 303993*x**5/2500 - 6194313*x**4/20000 - 24660207*x**3/500 00 - 118543581*x**2/200000 - 3579885909*x/5000000 - 5764801*log(x - 1/2)/1 5488 + 266*log(x + 3/5)/47265625 - 1/(21484375*x + 12890625)
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=-\frac {2187}{100} \, x^{6} - \frac {303993}{2500} \, x^{5} - \frac {6194313}{20000} \, x^{4} - \frac {24660207}{50000} \, x^{3} - \frac {118543581}{200000} \, x^{2} - \frac {3579885909}{5000000} \, x - \frac {1}{4296875 \, {\left (5 \, x + 3\right )}} + \frac {266}{47265625} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{15488} \, \log \left (2 \, x - 1\right ) \]
-2187/100*x^6 - 303993/2500*x^5 - 6194313/20000*x^4 - 24660207/50000*x^3 - 118543581/200000*x^2 - 3579885909/5000000*x - 1/4296875/(5*x + 3) + 266/4 7265625*log(5*x + 3) - 5764801/15488*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.38 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=-\frac {27}{125000000} \, {\left (5 \, x + 3\right )}^{6} {\left (\frac {63504}{5 \, x + 3} + \frac {466830}{{\left (5 \, x + 3\right )}^{2}} + \frac {3450300}{{\left (5 \, x + 3\right )}^{3}} + \frac {28481775}{{\left (5 \, x + 3\right )}^{4}} + \frac {313308485}{{\left (5 \, x + 3\right )}^{5}} + 6480\right )} - \frac {1}{4296875 \, {\left (5 \, x + 3\right )}} + \frac {18610540137}{50000000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {5764801}{15488} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]
-27/125000000*(5*x + 3)^6*(63504/(5*x + 3) + 466830/(5*x + 3)^2 + 3450300/ (5*x + 3)^3 + 28481775/(5*x + 3)^4 + 313308485/(5*x + 3)^5 + 6480) - 1/429 6875/(5*x + 3) + 18610540137/50000000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 5764801/15488*log(abs(-11/(5*x + 3) + 2))
Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^2} \, dx=\frac {266\,\ln \left (x+\frac {3}{5}\right )}{47265625}-\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{15488}-\frac {3579885909\,x}{5000000}-\frac {1}{21484375\,\left (x+\frac {3}{5}\right )}-\frac {118543581\,x^2}{200000}-\frac {24660207\,x^3}{50000}-\frac {6194313\,x^4}{20000}-\frac {303993\,x^5}{2500}-\frac {2187\,x^6}{100} \]