Integrand size = 22, antiderivative size = 65 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {823543}{1408 (1-2 x)}+\frac {370109547 x}{200000}+\frac {18237069 x^2}{20000}+\frac {853659 x^3}{2000}+\frac {13851 x^4}{100}+\frac {2187 x^5}{100}+\frac {5764801 \log (1-2 x)}{3872}+\frac {\log (3+5 x)}{1890625} \]
823543/1408/(1-2*x)+370109547/200000*x+18237069/20000*x^2+853659/2000*x^3+ 13851/100*x^4+2187/100*x^5+5764801/3872*ln(1-2*x)+1/1890625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {\frac {11 \left (-158719988357-14798867886 x+306816622200 x^2+153656514000 x^3+78666390000 x^4+28066500000 x^5+4811400000 x^6\right )}{-1+2 x}+1801500312500 \log (5-10 x)+640 \log (3+5 x)}{1210000000} \]
((11*(-158719988357 - 14798867886*x + 306816622200*x^2 + 153656514000*x^3 + 78666390000*x^4 + 28066500000*x^5 + 4811400000*x^6))/(-1 + 2*x) + 180150 0312500*Log[5 - 10*x] + 640*Log[3 + 5*x])/1210000000
Time = 0.19 (sec) , antiderivative size = 65, 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.
\(\displaystyle \int \frac {(3 x+2)^7}{(1-2 x)^2 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {2187 x^4}{20}+\frac {13851 x^3}{25}+\frac {2560977 x^2}{2000}+\frac {18237069 x}{10000}+\frac {5764801}{1936 (2 x-1)}+\frac {1}{378125 (5 x+3)}+\frac {823543}{704 (2 x-1)^2}+\frac {370109547}{200000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2187 x^5}{100}+\frac {13851 x^4}{100}+\frac {853659 x^3}{2000}+\frac {18237069 x^2}{20000}+\frac {370109547 x}{200000}+\frac {823543}{1408 (1-2 x)}+\frac {5764801 \log (1-2 x)}{3872}+\frac {\log (5 x+3)}{1890625}\) |
823543/(1408*(1 - 2*x)) + (370109547*x)/200000 + (18237069*x^2)/20000 + (8 53659*x^3)/2000 + (13851*x^4)/100 + (2187*x^5)/100 + (5764801*Log[1 - 2*x] )/3872 + Log[3 + 5*x]/1890625
3.16.89.3.1 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 = 0.87 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {2187 x^{5}}{100}+\frac {13851 x^{4}}{100}+\frac {853659 x^{3}}{2000}+\frac {18237069 x^{2}}{20000}+\frac {370109547 x}{200000}-\frac {823543}{2816 \left (x -\frac {1}{2}\right )}+\frac {5764801 \ln \left (-1+2 x \right )}{3872}+\frac {\ln \left (3+5 x \right )}{1890625}\) | \(48\) |
default | \(\frac {2187 x^{5}}{100}+\frac {13851 x^{4}}{100}+\frac {853659 x^{3}}{2000}+\frac {18237069 x^{2}}{20000}+\frac {370109547 x}{200000}+\frac {\ln \left (3+5 x \right )}{1890625}-\frac {823543}{1408 \left (-1+2 x \right )}+\frac {5764801 \ln \left (-1+2 x \right )}{3872}\) | \(50\) |
norman | \(\frac {-\frac {1661194223}{550000} x +\frac {139462101}{50000} x^{2}+\frac {6984387}{5000} x^{3}+\frac {715149}{1000} x^{4}+\frac {5103}{20} x^{5}+\frac {2187}{50} x^{6}}{-1+2 x}+\frac {5764801 \ln \left (-1+2 x \right )}{3872}+\frac {\ln \left (3+5 x \right )}{1890625}\) | \(55\) |
parallelrisch | \(\frac {2646270000 x^{6}+15436575000 x^{5}+43266514500 x^{4}+84511082700 x^{3}+64 \ln \left (x +\frac {3}{5}\right ) x +180150031250 \ln \left (x -\frac {1}{2}\right ) x +168749142210 x^{2}-32 \ln \left (x +\frac {3}{5}\right )-90075015625 \ln \left (x -\frac {1}{2}\right )-182731364530 x}{-60500000+121000000 x}\) | \(65\) |
2187/100*x^5+13851/100*x^4+853659/2000*x^3+18237069/20000*x^2+370109547/20 0000*x-823543/2816/(x-1/2)+5764801/3872*ln(-1+2*x)+1/1890625*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {10585080000 \, x^{6} + 61746300000 \, x^{5} + 173066058000 \, x^{4} + 338044330800 \, x^{3} + 674996568840 \, x^{2} + 128 \, {\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 360300062500 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 447832551870 \, x - 141546453125}{242000000 \, {\left (2 \, x - 1\right )}} \]
1/242000000*(10585080000*x^6 + 61746300000*x^5 + 173066058000*x^4 + 338044 330800*x^3 + 674996568840*x^2 + 128*(2*x - 1)*log(5*x + 3) + 360300062500* (2*x - 1)*log(2*x - 1) - 447832551870*x - 141546453125)/(2*x - 1)
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {2187 x^{5}}{100} + \frac {13851 x^{4}}{100} + \frac {853659 x^{3}}{2000} + \frac {18237069 x^{2}}{20000} + \frac {370109547 x}{200000} + \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{3872} + \frac {\log {\left (x + \frac {3}{5} \right )}}{1890625} - \frac {823543}{2816 x - 1408} \]
2187*x**5/100 + 13851*x**4/100 + 853659*x**3/2000 + 18237069*x**2/20000 + 370109547*x/200000 + 5764801*log(x - 1/2)/3872 + log(x + 3/5)/1890625 - 82 3543/(2816*x - 1408)
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {2187}{100} \, x^{5} + \frac {13851}{100} \, x^{4} + \frac {853659}{2000} \, x^{3} + \frac {18237069}{20000} \, x^{2} + \frac {370109547}{200000} \, x - \frac {823543}{1408 \, {\left (2 \, x - 1\right )}} + \frac {1}{1890625} \, \log \left (5 \, x + 3\right ) + \frac {5764801}{3872} \, \log \left (2 \, x - 1\right ) \]
2187/100*x^5 + 13851/100*x^4 + 853659/2000*x^3 + 18237069/20000*x^2 + 3701 09547/200000*x - 823543/1408/(2*x - 1) + 1/1890625*log(5*x + 3) + 5764801/ 3872*log(2*x - 1)
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {27}{400000} \, {\left (2 \, x - 1\right )}^{5} {\left (\frac {178875}{2 \, x - 1} + \frac {1404675}{{\left (2 \, x - 1\right )}^{2}} + \frac {6619260}{{\left (2 \, x - 1\right )}^{3}} + \frac {23397131}{{\left (2 \, x - 1\right )}^{4}} + 10125\right )} - \frac {823543}{1408 \, {\left (2 \, x - 1\right )}} - \frac {744421617}{500000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {1}{1890625} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]
27/400000*(2*x - 1)^5*(178875/(2*x - 1) + 1404675/(2*x - 1)^2 + 6619260/(2 *x - 1)^3 + 23397131/(2*x - 1)^4 + 10125) - 823543/1408/(2*x - 1) - 744421 617/500000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 1/1890625*log(abs(-11/(2*x - 1) - 5))
Time = 1.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^7}{(1-2 x)^2 (3+5 x)} \, dx=\frac {370109547\,x}{200000}+\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{3872}+\frac {\ln \left (x+\frac {3}{5}\right )}{1890625}-\frac {823543}{2816\,\left (x-\frac {1}{2}\right )}+\frac {18237069\,x^2}{20000}+\frac {853659\,x^3}{2000}+\frac {13851\,x^4}{100}+\frac {2187\,x^5}{100} \]