Integrand size = 22, antiderivative size = 61 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=-\frac {1127138733 x}{1000000}-\frac {187738857 x^2}{200000}-\frac {7889751 x^3}{10000}-\frac {2006937 x^4}{4000}-\frac {99873 x^5}{500}-\frac {729 x^6}{20}-\frac {823543 \log (1-2 x)}{1408}+\frac {\log (3+5 x)}{859375} \]
-1127138733/1000000*x-187738857/200000*x^2-7889751/10000*x^3-2006937/4000* x^4-99873/500*x^5-729/20*x^6-823543/1408*ln(1-2*x)+1/859375*ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=-\frac {823543 \log (3-6 x)}{1408}+\frac {-165 \left (163998254+375712911 x+312898095 x^2+262991700 x^3+167244750 x^4+66582000 x^5+12150000 x^6\right )+64 \log (-3 (3+5 x))}{55000000} \]
(-823543*Log[3 - 6*x])/1408 + (-165*(163998254 + 375712911*x + 312898095*x ^2 + 262991700*x^3 + 167244750*x^4 + 66582000*x^5 + 12150000*x^6) + 64*Log [-3*(3 + 5*x)])/55000000
Time = 0.18 (sec) , antiderivative size = 61, 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 = {93, 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.
\(\displaystyle \int \frac {(3 x+2)^7}{(1-2 x) (5 x+3)} \, dx\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \int \left (-\frac {2187 x^5}{10}-\frac {99873 x^4}{100}-\frac {2006937 x^3}{1000}-\frac {23669253 x^2}{10000}-\frac {187738857 x}{100000}-\frac {823543}{704 (2 x-1)}+\frac {1}{171875 (5 x+3)}-\frac {1127138733}{1000000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {729 x^6}{20}-\frac {99873 x^5}{500}-\frac {2006937 x^4}{4000}-\frac {7889751 x^3}{10000}-\frac {187738857 x^2}{200000}-\frac {1127138733 x}{1000000}-\frac {823543 \log (1-2 x)}{1408}+\frac {\log (5 x+3)}{859375}\) |
(-1127138733*x)/1000000 - (187738857*x^2)/200000 - (7889751*x^3)/10000 - ( 2006937*x^4)/4000 - (99873*x^5)/500 - (729*x^6)/20 - (823543*Log[1 - 2*x]) /1408 + Log[3 + 5*x]/859375
3.15.86.3.1 Defintions of rubi rules used
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Time = 2.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(-\frac {729 x^{6}}{20}-\frac {99873 x^{5}}{500}-\frac {2006937 x^{4}}{4000}-\frac {7889751 x^{3}}{10000}-\frac {187738857 x^{2}}{200000}-\frac {1127138733 x}{1000000}+\frac {\ln \left (x +\frac {3}{5}\right )}{859375}-\frac {823543 \ln \left (x -\frac {1}{2}\right )}{1408}\) | \(42\) |
default | \(-\frac {729 x^{6}}{20}-\frac {99873 x^{5}}{500}-\frac {2006937 x^{4}}{4000}-\frac {7889751 x^{3}}{10000}-\frac {187738857 x^{2}}{200000}-\frac {1127138733 x}{1000000}+\frac {\ln \left (3+5 x \right )}{859375}-\frac {823543 \ln \left (-1+2 x \right )}{1408}\) | \(46\) |
norman | \(-\frac {729 x^{6}}{20}-\frac {99873 x^{5}}{500}-\frac {2006937 x^{4}}{4000}-\frac {7889751 x^{3}}{10000}-\frac {187738857 x^{2}}{200000}-\frac {1127138733 x}{1000000}+\frac {\ln \left (3+5 x \right )}{859375}-\frac {823543 \ln \left (-1+2 x \right )}{1408}\) | \(46\) |
risch | \(-\frac {729 x^{6}}{20}-\frac {99873 x^{5}}{500}-\frac {2006937 x^{4}}{4000}-\frac {7889751 x^{3}}{10000}-\frac {187738857 x^{2}}{200000}-\frac {1127138733 x}{1000000}+\frac {\ln \left (3+5 x \right )}{859375}-\frac {823543 \ln \left (-1+2 x \right )}{1408}\) | \(46\) |
-729/20*x^6-99873/500*x^5-2006937/4000*x^4-7889751/10000*x^3-187738857/200 000*x^2-1127138733/1000000*x+1/859375*ln(x+3/5)-823543/1408*ln(x-1/2)
Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=-\frac {729}{20} \, x^{6} - \frac {99873}{500} \, x^{5} - \frac {2006937}{4000} \, x^{4} - \frac {7889751}{10000} \, x^{3} - \frac {187738857}{200000} \, x^{2} - \frac {1127138733}{1000000} \, x + \frac {1}{859375} \, \log \left (5 \, x + 3\right ) - \frac {823543}{1408} \, \log \left (2 \, x - 1\right ) \]
-729/20*x^6 - 99873/500*x^5 - 2006937/4000*x^4 - 7889751/10000*x^3 - 18773 8857/200000*x^2 - 1127138733/1000000*x + 1/859375*log(5*x + 3) - 823543/14 08*log(2*x - 1)
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=- \frac {729 x^{6}}{20} - \frac {99873 x^{5}}{500} - \frac {2006937 x^{4}}{4000} - \frac {7889751 x^{3}}{10000} - \frac {187738857 x^{2}}{200000} - \frac {1127138733 x}{1000000} - \frac {823543 \log {\left (x - \frac {1}{2} \right )}}{1408} + \frac {\log {\left (x + \frac {3}{5} \right )}}{859375} \]
-729*x**6/20 - 99873*x**5/500 - 2006937*x**4/4000 - 7889751*x**3/10000 - 1 87738857*x**2/200000 - 1127138733*x/1000000 - 823543*log(x - 1/2)/1408 + l og(x + 3/5)/859375
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=-\frac {729}{20} \, x^{6} - \frac {99873}{500} \, x^{5} - \frac {2006937}{4000} \, x^{4} - \frac {7889751}{10000} \, x^{3} - \frac {187738857}{200000} \, x^{2} - \frac {1127138733}{1000000} \, x + \frac {1}{859375} \, \log \left (5 \, x + 3\right ) - \frac {823543}{1408} \, \log \left (2 \, x - 1\right ) \]
-729/20*x^6 - 99873/500*x^5 - 2006937/4000*x^4 - 7889751/10000*x^3 - 18773 8857/200000*x^2 - 1127138733/1000000*x + 1/859375*log(5*x + 3) - 823543/14 08*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=-\frac {729}{20} \, x^{6} - \frac {99873}{500} \, x^{5} - \frac {2006937}{4000} \, x^{4} - \frac {7889751}{10000} \, x^{3} - \frac {187738857}{200000} \, x^{2} - \frac {1127138733}{1000000} \, x + \frac {1}{859375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {823543}{1408} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-729/20*x^6 - 99873/500*x^5 - 2006937/4000*x^4 - 7889751/10000*x^3 - 18773 8857/200000*x^2 - 1127138733/1000000*x + 1/859375*log(abs(5*x + 3)) - 8235 43/1408*log(abs(2*x - 1))
Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{859375}-\frac {823543\,\ln \left (x-\frac {1}{2}\right )}{1408}-\frac {1127138733\,x}{1000000}-\frac {187738857\,x^2}{200000}-\frac {7889751\,x^3}{10000}-\frac {2006937\,x^4}{4000}-\frac {99873\,x^5}{500}-\frac {729\,x^6}{20} \]