Integrand size = 22, antiderivative size = 84 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {5764801}{170368 (1-2 x)^2}-\frac {130943337}{937024 (1-2 x)}-\frac {242028 x}{3125}-\frac {330237 x^2}{20000}-\frac {2187 x^3}{1000}-\frac {1}{41593750 (3+5 x)^2}-\frac {54}{45753125 (3+5 x)}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (3+5 x)}{100656875} \]
5764801/170368/(1-2*x)^2-130943337/937024/(1-2*x)-242028/3125*x-330237/200 00*x^2-2187/1000*x^3-1/41593750/(3+5*x)^2-54/45753125/(3+5*x)-595421589/51 53632*ln(1-2*x)+1284/100656875*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {11 \left (39754322426279+31893783102814 x-254889143270829 x^2-314407515766380 x^3+148045752548100 x^4+232677700200000 x^5+49630793850000 x^6+6403973400000 x^7\right )}{\left (-3+x+10 x^2\right )^2}-37213849312500 \log (3-6 x)+4108800 \log (-3 (3+5 x))}{322102000000} \]
((-11*(39754322426279 + 31893783102814*x - 254889143270829*x^2 - 314407515 766380*x^3 + 148045752548100*x^4 + 232677700200000*x^5 + 49630793850000*x^ 6 + 6403973400000*x^7))/(-3 + x + 10*x^2)^2 - 37213849312500*Log[3 - 6*x] + 4108800*Log[-3*(3 + 5*x)])/322102000000
Time = 0.21 (sec) , antiderivative size = 84, 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.
\(\displaystyle \int \frac {(3 x+2)^8}{(1-2 x)^3 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {6561 x^2}{1000}-\frac {330237 x}{10000}-\frac {595421589}{2576816 (2 x-1)}+\frac {1284}{20131375 (5 x+3)}-\frac {130943337}{468512 (2 x-1)^2}+\frac {54}{9150625 (5 x+3)^2}-\frac {5764801}{42592 (2 x-1)^3}+\frac {1}{4159375 (5 x+3)^3}-\frac {242028}{3125}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2187 x^3}{1000}-\frac {330237 x^2}{20000}-\frac {242028 x}{3125}-\frac {130943337}{937024 (1-2 x)}-\frac {54}{45753125 (5 x+3)}+\frac {5764801}{170368 (1-2 x)^2}-\frac {1}{41593750 (5 x+3)^2}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (5 x+3)}{100656875}\) |
5764801/(170368*(1 - 2*x)^2) - 130943337/(937024*(1 - 2*x)) - (242028*x)/3 125 - (330237*x^2)/20000 - (2187*x^3)/1000 - 1/(41593750*(3 + 5*x)^2) - 54 /(45753125*(3 + 5*x)) - (595421589*Log[1 - 2*x])/5153632 + (1284*Log[3 + 5 *x])/100656875
3.17.98.3.1 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 = 0.90 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {2187 x^{3}}{1000}-\frac {330237 x^{2}}{20000}-\frac {242028 x}{3125}+\frac {\frac {2045989633713}{292820000} x^{3}+\frac {167989904414289}{29282000000} x^{2}-\frac {9689497987007}{14641000000} x -\frac {27910387088759}{29282000000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {595421589 \ln \left (-1+2 x \right )}{5153632}+\frac {1284 \ln \left (3+5 x \right )}{100656875}\) | \(62\) |
norman | \(\frac {-\frac {407765282557}{292820000} x +\frac {86004166569}{7320500} x^{3}+\frac {3350842985349}{585640000} x^{2}-\frac {79461}{10} x^{5}-\frac {67797}{40} x^{6}-\frac {2187}{10} x^{7}-\frac {528604093939}{585640000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {595421589 \ln \left (-1+2 x \right )}{5153632}+\frac {1284 \ln \left (3+5 x \right )}{100656875}\) | \(63\) |
default | \(-\frac {2187 x^{3}}{1000}-\frac {330237 x^{2}}{20000}-\frac {242028 x}{3125}-\frac {1}{41593750 \left (3+5 x \right )^{2}}-\frac {54}{45753125 \left (3+5 x \right )}+\frac {1284 \ln \left (3+5 x \right )}{100656875}+\frac {5764801}{170368 \left (-1+2 x \right )^{2}}+\frac {130943337}{937024 \left (-1+2 x \right )}-\frac {595421589 \ln \left (-1+2 x \right )}{5153632}\) | \(67\) |
parallelrisch | \(\frac {-2113311222000 x^{7}-16378161970500 x^{6}+12326400 \ln \left (x +\frac {3}{5}\right ) x^{4}-111641547937500 \ln \left (x -\frac {1}{2}\right ) x^{4}-76783641066000 x^{5}-28906349036565+2465280 \ln \left (x +\frac {3}{5}\right ) x^{3}-22328309587500 \ln \left (x -\frac {1}{2}\right ) x^{3}-224270905406350 x^{4}-7272576 \ln \left (x +\frac {3}{5}\right ) x^{2}+65868513283125 \ln \left (x -\frac {1}{2}\right ) x^{2}+68671318789810 x^{3}-739584 \ln \left (x +\frac {3}{5}\right ) x +6698492876250 \ln \left (x -\frac {1}{2}\right ) x +187608743448005 x^{2}+1109376 \ln \left (x +\frac {3}{5}\right )-10047739314375 \ln \left (x -\frac {1}{2}\right )}{9663060000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(129\) |
-2187/1000*x^3-330237/20000*x^2-242028/3125*x+100*(2045989633713/292820000 00*x^3+167989904414289/2928200000000*x^2-9689497987007/1464100000000*x-279 10387088759/2928200000000)/(-1+2*x)^2/(3+5*x)^2-595421589/5153632*ln(-1+2* x)+1284/100656875*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.37 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {70443707400000 \, x^{7} + 545938732350000 \, x^{6} + 2559454702200000 \, x^{5} + 180911181221100 \, x^{4} - 3748001092791780 \, x^{3} - 1949701238862399 \, x^{2} - 4108800 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 37213849312500 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 437687139939434 \, x + 307014257976349}{322102000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/322102000000*(70443707400000*x^7 + 545938732350000*x^6 + 25594547022000 00*x^5 + 180911181221100*x^4 - 3748001092791780*x^3 - 1949701238862399*x^2 - 4108800*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 3721384931 2500*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 437687139939434* x + 307014257976349)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {2187 x^{3}}{1000} - \frac {330237 x^{2}}{20000} - \frac {242028 x}{3125} - \frac {- 204598963371300 x^{3} - 167989904414289 x^{2} + 19378995974014 x + 27910387088759}{2928200000000 x^{4} + 585640000000 x^{3} - 1727638000000 x^{2} - 175692000000 x + 263538000000} - \frac {595421589 \log {\left (x - \frac {1}{2} \right )}}{5153632} + \frac {1284 \log {\left (x + \frac {3}{5} \right )}}{100656875} \]
-2187*x**3/1000 - 330237*x**2/20000 - 242028*x/3125 - (-204598963371300*x* *3 - 167989904414289*x**2 + 19378995974014*x + 27910387088759)/(2928200000 000*x**4 + 585640000000*x**3 - 1727638000000*x**2 - 175692000000*x + 26353 8000000) - 595421589*log(x - 1/2)/5153632 + 1284*log(x + 3/5)/100656875
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{1000} \, x^{3} - \frac {330237}{20000} \, x^{2} - \frac {242028}{3125} \, x + \frac {204598963371300 \, x^{3} + 167989904414289 \, x^{2} - 19378995974014 \, x - 27910387088759}{29282000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {1284}{100656875} \, \log \left (5 \, x + 3\right ) - \frac {595421589}{5153632} \, \log \left (2 \, x - 1\right ) \]
-2187/1000*x^3 - 330237/20000*x^2 - 242028/3125*x + 1/29282000000*(2045989 63371300*x^3 + 167989904414289*x^2 - 19378995974014*x - 27910387088759)/(1 00*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 1284/100656875*log(5*x + 3) - 595421 589/5153632*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{1000} \, x^{3} - \frac {330237}{20000} \, x^{2} - \frac {242028}{3125} \, x + \frac {204598963371300 \, x^{3} + 167989904414289 \, x^{2} - 19378995974014 \, x - 27910387088759}{29282000000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {1284}{100656875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {595421589}{5153632} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-2187/1000*x^3 - 330237/20000*x^2 - 242028/3125*x + 1/29282000000*(2045989 63371300*x^3 + 167989904414289*x^2 - 19378995974014*x - 27910387088759)/(( 5*x + 3)^2*(2*x - 1)^2) + 1284/100656875*log(abs(5*x + 3)) - 595421589/515 3632*log(abs(2*x - 1))
Time = 1.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {1284\,\ln \left (x+\frac {3}{5}\right )}{100656875}-\frac {595421589\,\ln \left (x-\frac {1}{2}\right )}{5153632}-\frac {242028\,x}{3125}-\frac {-\frac {2045989633713\,x^3}{29282000000}-\frac {167989904414289\,x^2}{2928200000000}+\frac {9689497987007\,x}{1464100000000}+\frac {27910387088759}{2928200000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {330237\,x^2}{20000}-\frac {2187\,x^3}{1000} \]
(1284*log(x + 3/5))/100656875 - (595421589*log(x - 1/2))/5153632 - (242028 *x)/3125 - ((9689497987007*x)/1464100000000 - (167989904414289*x^2)/292820 0000000 - (2045989633713*x^3)/29282000000 + 27910387088759/2928200000000)/ (x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100) - (330237*x^2)/20000 - (21 87*x^3)/1000