Integrand size = 22, antiderivative size = 77 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {823543}{85184 (1-2 x)^2}-\frac {7411887}{234256 (1-2 x)}-\frac {95499 x}{10000}-\frac {2187 x^2}{2000}-\frac {1}{8318750 (3+5 x)^2}-\frac {237}{45753125 (3+5 x)}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (3+5 x)}{503284375} \]
823543/85184/(1-2*x)^2-7411887/234256/(1-2*x)-95499/10000*x-2187/2000*x^2- 1/8318750/(3+5*x)^2-237/45753125/(3+5*x)-25059237/1288408*ln(1-2*x)+24279/ 503284375*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {11 \left (734029874011+410862940766 x-5105353973121 x^2-5957126547060 x^3+2092320420300 x^4+2860441452000 x^5+320198670000 x^6\right )}{\left (-3+x+10 x^2\right )^2}-626480925000 \log (3-6 x)+1553856 \log (-3 (3+5 x))}{32210200000} \]
((-11*(734029874011 + 410862940766*x - 5105353973121*x^2 - 5957126547060*x ^3 + 2092320420300*x^4 + 2860441452000*x^5 + 320198670000*x^6))/(-3 + x + 10*x^2)^2 - 626480925000*Log[3 - 6*x] + 1553856*Log[-3*(3 + 5*x)])/3221020 0000
Time = 0.21 (sec) , antiderivative size = 77, 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.
\(\displaystyle \int \frac {(3 x+2)^7}{(1-2 x)^3 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {2187 x}{1000}-\frac {25059237}{644204 (2 x-1)}+\frac {24279}{100656875 (5 x+3)}-\frac {7411887}{117128 (2 x-1)^2}+\frac {237}{9150625 (5 x+3)^2}-\frac {823543}{21296 (2 x-1)^3}+\frac {1}{831875 (5 x+3)^3}-\frac {95499}{10000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2187 x^2}{2000}-\frac {95499 x}{10000}-\frac {7411887}{234256 (1-2 x)}-\frac {237}{45753125 (5 x+3)}+\frac {823543}{85184 (1-2 x)^2}-\frac {1}{8318750 (5 x+3)^2}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (5 x+3)}{503284375}\) |
823543/(85184*(1 - 2*x)^2) - 7411887/(234256*(1 - 2*x)) - (95499*x)/10000 - (2187*x^2)/2000 - 1/(8318750*(3 + 5*x)^2) - 237/(45753125*(3 + 5*x)) - ( 25059237*Log[1 - 2*x])/1288408 + (24279*Log[3 + 5*x])/503284375
3.17.99.3.1 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 = 2.80 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {2187 x^{2}}{2000}-\frac {95499 x}{10000}+\frac {\frac {115810726791}{73205000} x^{3}+\frac {3950432948061}{2928200000} x^{2}-\frac {131252111833}{1464100000} x -\frac {579053717731}{2928200000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {25059237 \ln \left (-1+2 x \right )}{1288408}+\frac {24279 \ln \left (3+5 x \right )}{503284375}\) | \(57\) |
norman | \(\frac {-\frac {16762567687}{91506250} x +\frac {79694882889}{36602500} x^{3}+\frac {483860615643}{366025000} x^{2}-\frac {48843}{50} x^{5}-\frac {2187}{20} x^{6}-\frac {68215129523}{366025000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {25059237 \ln \left (-1+2 x \right )}{1288408}+\frac {24279 \ln \left (3+5 x \right )}{503284375}\) | \(58\) |
default | \(-\frac {2187 x^{2}}{2000}-\frac {95499 x}{10000}-\frac {1}{8318750 \left (3+5 x \right )^{2}}-\frac {237}{45753125 \left (3+5 x \right )}+\frac {24279 \ln \left (3+5 x \right )}{503284375}+\frac {823543}{85184 \left (-1+2 x \right )^{2}}+\frac {7411887}{234256 \left (-1+2 x \right )}-\frac {25059237 \ln \left (-1+2 x \right )}{1288408}\) | \(62\) |
parallelrisch | \(\frac {-1320819513750 x^{6}+58269600 \ln \left (x +\frac {3}{5}\right ) x^{4}-23493034687500 \ln \left (x -\frac {1}{2}\right ) x^{4}-11799320989500 x^{5}-5570087676285+11653920 \ln \left (x +\frac {3}{5}\right ) x^{3}-4698606937500 \ln \left (x -\frac {1}{2}\right ) x^{3}-36877648911400 x^{4}-34379064 \ln \left (x +\frac {3}{5}\right ) x^{2}+13860890465625 \ln \left (x -\frac {1}{2}\right ) x^{2}+18923781571090 x^{3}-3496176 \ln \left (x +\frac {3}{5}\right ) x +1409582081250 \ln \left (x -\frac {1}{2}\right ) x +37725213173945 x^{2}+5244264 \ln \left (x +\frac {3}{5}\right )-2114373121875 \ln \left (x -\frac {1}{2}\right )}{12078825000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(124\) |
-2187/2000*x^2-95499/10000*x+100*(115810726791/7320500000*x^3+395043294806 1/292820000000*x^2-131252111833/146410000000*x-579053717731/292820000000)/ (-1+2*x)^2/(3+5*x)^2-25059237/1288408*ln(-1+2*x)+24279/503284375*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.43 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {3522185370000 \, x^{6} + 31464855972000 \, x^{5} + 4073994411300 \, x^{4} - 69316698060060 \, x^{3} - 44983390879251 \, x^{2} - 1553856 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 626480925000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 5655984161146 \, x + 6369590895041}{32210200000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/32210200000*(3522185370000*x^6 + 31464855972000*x^5 + 4073994411300*x^4 - 69316698060060*x^3 - 44983390879251*x^2 - 1553856*(100*x^4 + 20*x^3 - 5 9*x^2 - 6*x + 9)*log(5*x + 3) + 626480925000*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 5655984161146*x + 6369590895041)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {2187 x^{2}}{2000} - \frac {95499 x}{10000} - \frac {- 4632429071640 x^{3} - 3950432948061 x^{2} + 262504223666 x + 579053717731}{292820000000 x^{4} + 58564000000 x^{3} - 172763800000 x^{2} - 17569200000 x + 26353800000} - \frac {25059237 \log {\left (x - \frac {1}{2} \right )}}{1288408} + \frac {24279 \log {\left (x + \frac {3}{5} \right )}}{503284375} \]
-2187*x**2/2000 - 95499*x/10000 - (-4632429071640*x**3 - 3950432948061*x** 2 + 262504223666*x + 579053717731)/(292820000000*x**4 + 58564000000*x**3 - 172763800000*x**2 - 17569200000*x + 26353800000) - 25059237*log(x - 1/2)/ 1288408 + 24279*log(x + 3/5)/503284375
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.83 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{2000} \, x^{2} - \frac {95499}{10000} \, x + \frac {4632429071640 \, x^{3} + 3950432948061 \, x^{2} - 262504223666 \, x - 579053717731}{2928200000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {24279}{503284375} \, \log \left (5 \, x + 3\right ) - \frac {25059237}{1288408} \, \log \left (2 \, x - 1\right ) \]
-2187/2000*x^2 - 95499/10000*x + 1/2928200000*(4632429071640*x^3 + 3950432 948061*x^2 - 262504223666*x - 579053717731)/(100*x^4 + 20*x^3 - 59*x^2 - 6 *x + 9) + 24279/503284375*log(5*x + 3) - 25059237/1288408*log(2*x - 1)
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {2187}{2000} \, x^{2} - \frac {95499}{10000} \, x + \frac {4632429071640 \, x^{3} + 3950432948061 \, x^{2} - 262504223666 \, x - 579053717731}{2928200000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {24279}{503284375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {25059237}{1288408} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-2187/2000*x^2 - 95499/10000*x + 1/2928200000*(4632429071640*x^3 + 3950432 948061*x^2 - 262504223666*x - 579053717731)/((5*x + 3)^2*(2*x - 1)^2) + 24 279/503284375*log(abs(5*x + 3)) - 25059237/1288408*log(abs(2*x - 1))
Time = 1.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {24279\,\ln \left (x+\frac {3}{5}\right )}{503284375}-\frac {25059237\,\ln \left (x-\frac {1}{2}\right )}{1288408}-\frac {95499\,x}{10000}-\frac {-\frac {115810726791\,x^3}{7320500000}-\frac {3950432948061\,x^2}{292820000000}+\frac {131252111833\,x}{146410000000}+\frac {579053717731}{292820000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {2187\,x^2}{2000} \]