Integrand size = 22, antiderivative size = 76 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=-\frac {62934003 x}{500000}-\frac {9268263 x^2}{100000}-\frac {1535517 x^3}{25000}-\frac {264627 x^4}{10000}-\frac {6561 x^5}{1250}-\frac {1}{8593750 (3+5 x)^2}-\frac {266}{47265625 (3+5 x)}-\frac {5764801 \log (1-2 x)}{85184}+\frac {31024 \log (3+5 x)}{519921875} \]
-62934003/500000*x-9268263/100000*x^2-1535517/25000*x^3-264627/10000*x^4-6 561/1250*x^5-1/8593750/(3+5*x)^2-266/47265625/(3+5*x)-5764801/85184*ln(1-2 *x)+31024/519921875*ln(3+5*x)
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=\frac {22 \left (-85278446550-190375359075 x-140182477875 x^2-92898778500 x^3-40024833750 x^4-7938810000 x^5-\frac {176}{(3+5 x)^2}-\frac {8512}{3+5 x}\right )-2251875390625 \log (3-6 x)+1985536 \log (-3 (3+5 x))}{33275000000} \]
(22*(-85278446550 - 190375359075*x - 140182477875*x^2 - 92898778500*x^3 - 40024833750*x^4 - 7938810000*x^5 - 176/(3 + 5*x)^2 - 8512/(3 + 5*x)) - 225 1875390625*Log[3 - 6*x] + 1985536*Log[-3*(3 + 5*x)])/33275000000
Time = 0.20 (sec) , antiderivative size = 76, 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.
\(\displaystyle \int \frac {(3 x+2)^8}{(1-2 x) (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {6561 x^4}{250}-\frac {264627 x^3}{2500}-\frac {4606551 x^2}{25000}-\frac {9268263 x}{50000}-\frac {5764801}{42592 (2 x-1)}+\frac {31024}{103984375 (5 x+3)}+\frac {266}{9453125 (5 x+3)^2}+\frac {1}{859375 (5 x+3)^3}-\frac {62934003}{500000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {6561 x^5}{1250}-\frac {264627 x^4}{10000}-\frac {1535517 x^3}{25000}-\frac {9268263 x^2}{100000}-\frac {62934003 x}{500000}-\frac {266}{47265625 (5 x+3)}-\frac {1}{8593750 (5 x+3)^2}-\frac {5764801 \log (1-2 x)}{85184}+\frac {31024 \log (5 x+3)}{519921875}\) |
(-62934003*x)/500000 - (9268263*x^2)/100000 - (1535517*x^3)/25000 - (26462 7*x^4)/10000 - (6561*x^5)/1250 - 1/(8593750*(3 + 5*x)^2) - 266/(47265625*( 3 + 5*x)) - (5764801*Log[1 - 2*x])/85184 + (31024*Log[3 + 5*x])/519921875
3.16.16.3.1 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 = 2.59 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {6561 x^{5}}{1250}-\frac {264627 x^{4}}{10000}-\frac {1535517 x^{3}}{25000}-\frac {9268263 x^{2}}{100000}-\frac {62934003 x}{500000}+\frac {-\frac {266 x}{9453125}-\frac {1607}{94531250}}{\left (3+5 x \right )^{2}}-\frac {5764801 \ln \left (-1+2 x \right )}{85184}+\frac {31024 \ln \left (3+5 x \right )}{519921875}\) | \(55\) |
default | \(-\frac {6561 x^{5}}{1250}-\frac {264627 x^{4}}{10000}-\frac {1535517 x^{3}}{25000}-\frac {9268263 x^{2}}{100000}-\frac {62934003 x}{500000}-\frac {1}{8593750 \left (3+5 x \right )^{2}}-\frac {266}{47265625 \left (3+5 x \right )}+\frac {31024 \ln \left (3+5 x \right )}{519921875}-\frac {5764801 \ln \left (-1+2 x \right )}{85184}\) | \(59\) |
norman | \(\frac {-\frac {1028026913117}{907500000} x -\frac {2510245080613}{544500000} x^{2}-\frac {647996517}{100000} x^{3}-\frac {87957009}{20000} x^{4}-\frac {5941593}{2500} x^{5}-\frac {1638063}{2000} x^{6}-\frac {6561}{50} x^{7}}{\left (3+5 x \right )^{2}}-\frac {5764801 \ln \left (-1+2 x \right )}{85184}+\frac {31024 \ln \left (3+5 x \right )}{519921875}\) | \(60\) |
parallelrisch | \(\frac {-39297109500000 x^{7}-245279458462500 x^{6}-711743425470000 x^{5}-1317046263513750 x^{4}+446745600 \ln \left (x +\frac {3}{5}\right ) x^{2}-506671962890625 \ln \left (x -\frac {1}{2}\right ) x^{2}-1940587569285750 x^{3}+536094720 \ln \left (x +\frac {3}{5}\right ) x -608006355468750 \ln \left (x -\frac {1}{2}\right ) x -1380634794337150 x^{2}+160828416 \ln \left (x +\frac {3}{5}\right )-182401906640625 \ln \left (x -\frac {1}{2}\right )-339248881328610 x}{299475000000 \left (3+5 x \right )^{2}}\) | \(88\) |
-6561/1250*x^5-264627/10000*x^4-1535517/25000*x^3-9268263/100000*x^2-62934 003/500000*x+25*(-266/236328125*x-1607/2363281250)/(3+5*x)^2-5764801/85184 *ln(-1+2*x)+31024/519921875*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=-\frac {4366345500000 \, x^{7} + 27253273162500 \, x^{6} + 79082602830000 \, x^{5} + 146338473723750 \, x^{4} + 215620841031750 \, x^{3} + 153403867608750 \, x^{2} - 1985536 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 2251875390625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 37694322033170 \, x + 565664}{33275000000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/33275000000*(4366345500000*x^7 + 27253273162500*x^6 + 79082602830000*x^ 5 + 146338473723750*x^4 + 215620841031750*x^3 + 153403867608750*x^2 - 1985 536*(25*x^2 + 30*x + 9)*log(5*x + 3) + 2251875390625*(25*x^2 + 30*x + 9)*l og(2*x - 1) + 37694322033170*x + 565664)/(25*x^2 + 30*x + 9)
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=- \frac {6561 x^{5}}{1250} - \frac {264627 x^{4}}{10000} - \frac {1535517 x^{3}}{25000} - \frac {9268263 x^{2}}{100000} - \frac {62934003 x}{500000} - \frac {2660 x + 1607}{2363281250 x^{2} + 2835937500 x + 850781250} - \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{85184} + \frac {31024 \log {\left (x + \frac {3}{5} \right )}}{519921875} \]
-6561*x**5/1250 - 264627*x**4/10000 - 1535517*x**3/25000 - 9268263*x**2/10 0000 - 62934003*x/500000 - (2660*x + 1607)/(2363281250*x**2 + 2835937500*x + 850781250) - 5764801*log(x - 1/2)/85184 + 31024*log(x + 3/5)/519921875
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=-\frac {6561}{1250} \, x^{5} - \frac {264627}{10000} \, x^{4} - \frac {1535517}{25000} \, x^{3} - \frac {9268263}{100000} \, x^{2} - \frac {62934003}{500000} \, x - \frac {2660 \, x + 1607}{94531250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {31024}{519921875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{85184} \, \log \left (2 \, x - 1\right ) \]
-6561/1250*x^5 - 264627/10000*x^4 - 1535517/25000*x^3 - 9268263/100000*x^2 - 62934003/500000*x - 1/94531250*(2660*x + 1607)/(25*x^2 + 30*x + 9) + 31 024/519921875*log(5*x + 3) - 5764801/85184*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=-\frac {6561}{1250} \, x^{5} - \frac {264627}{10000} \, x^{4} - \frac {1535517}{25000} \, x^{3} - \frac {9268263}{100000} \, x^{2} - \frac {62934003}{500000} \, x - \frac {2660 \, x + 1607}{94531250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {31024}{519921875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5764801}{85184} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-6561/1250*x^5 - 264627/10000*x^4 - 1535517/25000*x^3 - 9268263/100000*x^2 - 62934003/500000*x - 1/94531250*(2660*x + 1607)/(5*x + 3)^2 + 31024/5199 21875*log(abs(5*x + 3)) - 5764801/85184*log(abs(2*x - 1))
Time = 1.42 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)^3} \, dx=\frac {31024\,\ln \left (x+\frac {3}{5}\right )}{519921875}-\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{85184}-\frac {62934003\,x}{500000}-\frac {\frac {266\,x}{236328125}+\frac {1607}{2363281250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {9268263\,x^2}{100000}-\frac {1535517\,x^3}{25000}-\frac {264627\,x^4}{10000}-\frac {6561\,x^5}{1250} \]