Integrand size = 22, antiderivative size = 68 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {40089855591 x}{10000000}-\frac {7136193339 x^2}{2000000}-\frac {345533877 x^3}{100000}-\frac {111146499 x^4}{40000}-\frac {8018271 x^5}{5000}-\frac {114453 x^6}{200}-\frac {6561 x^7}{70}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (3+5 x)}{4296875} \]
-40089855591/10000000*x-7136193339/2000000*x^2-345533877/100000*x^3-111146 499/40000*x^4-8018271/5000*x^5-114453/200*x^6-6561/70*x^7-5764801/2816*ln( 1-2*x)+1/4296875*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {3 \left (40324556806+93542996379 x+83255588955 x^2+80624571300 x^3+64835457750 x^4+37418598000 x^5+13352850000 x^6+2187000000 x^7\right )}{70000000}-\frac {5764801 \log (3-6 x)}{2816}+\frac {\log (-3 (3+5 x))}{4296875} \]
(-3*(40324556806 + 93542996379*x + 83255588955*x^2 + 80624571300*x^3 + 648 35457750*x^4 + 37418598000*x^5 + 13352850000*x^6 + 2187000000*x^7))/700000 00 - (5764801*Log[3 - 6*x])/2816 + Log[-3*(3 + 5*x)]/4296875
Time = 0.20 (sec) , antiderivative size = 68, 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.
\(\displaystyle \int \frac {(3 x+2)^8}{(1-2 x) (5 x+3)} \, dx\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \int \left (-\frac {6561 x^6}{10}-\frac {343359 x^5}{100}-\frac {8018271 x^4}{1000}-\frac {111146499 x^3}{10000}-\frac {1036601631 x^2}{100000}-\frac {7136193339 x}{1000000}-\frac {5764801}{1408 (2 x-1)}+\frac {1}{859375 (5 x+3)}-\frac {40089855591}{10000000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {6561 x^7}{70}-\frac {114453 x^6}{200}-\frac {8018271 x^5}{5000}-\frac {111146499 x^4}{40000}-\frac {345533877 x^3}{100000}-\frac {7136193339 x^2}{2000000}-\frac {40089855591 x}{10000000}-\frac {5764801 \log (1-2 x)}{2816}+\frac {\log (5 x+3)}{4296875}\) |
(-40089855591*x)/10000000 - (7136193339*x^2)/2000000 - (345533877*x^3)/100 000 - (111146499*x^4)/40000 - (8018271*x^5)/5000 - (114453*x^6)/200 - (656 1*x^7)/70 - (5764801*Log[1 - 2*x])/2816 + Log[3 + 5*x]/4296875
3.15.85.3.1 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 = 0.83 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (x +\frac {3}{5}\right )}{4296875}-\frac {5764801 \ln \left (x -\frac {1}{2}\right )}{2816}\) | \(47\) |
default | \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\) | \(51\) |
norman | \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\) | \(51\) |
risch | \(-\frac {6561 x^{7}}{70}-\frac {114453 x^{6}}{200}-\frac {8018271 x^{5}}{5000}-\frac {111146499 x^{4}}{40000}-\frac {345533877 x^{3}}{100000}-\frac {7136193339 x^{2}}{2000000}-\frac {40089855591 x}{10000000}+\frac {\ln \left (3+5 x \right )}{4296875}-\frac {5764801 \ln \left (-1+2 x \right )}{2816}\) | \(51\) |
-6561/70*x^7-114453/200*x^6-8018271/5000*x^5-111146499/40000*x^4-345533877 /100000*x^3-7136193339/2000000*x^2-40089855591/10000000*x+1/4296875*ln(x+3 /5)-5764801/2816*ln(x-1/2)
Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{2816} \, \log \left (2 \, x - 1\right ) \]
-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 3 45533877/100000*x^3 - 7136193339/2000000*x^2 - 40089855591/10000000*x + 1/ 4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=- \frac {6561 x^{7}}{70} - \frac {114453 x^{6}}{200} - \frac {8018271 x^{5}}{5000} - \frac {111146499 x^{4}}{40000} - \frac {345533877 x^{3}}{100000} - \frac {7136193339 x^{2}}{2000000} - \frac {40089855591 x}{10000000} - \frac {5764801 \log {\left (x - \frac {1}{2} \right )}}{2816} + \frac {\log {\left (x + \frac {3}{5} \right )}}{4296875} \]
-6561*x**7/70 - 114453*x**6/200 - 8018271*x**5/5000 - 111146499*x**4/40000 - 345533877*x**3/100000 - 7136193339*x**2/2000000 - 40089855591*x/1000000 0 - 5764801*log(x - 1/2)/2816 + log(x + 3/5)/4296875
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac {5764801}{2816} \, \log \left (2 \, x - 1\right ) \]
-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 3 45533877/100000*x^3 - 7136193339/2000000*x^2 - 40089855591/10000000*x + 1/ 4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=-\frac {6561}{70} \, x^{7} - \frac {114453}{200} \, x^{6} - \frac {8018271}{5000} \, x^{5} - \frac {111146499}{40000} \, x^{4} - \frac {345533877}{100000} \, x^{3} - \frac {7136193339}{2000000} \, x^{2} - \frac {40089855591}{10000000} \, x + \frac {1}{4296875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5764801}{2816} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 3 45533877/100000*x^3 - 7136193339/2000000*x^2 - 40089855591/10000000*x + 1/ 4296875*log(abs(5*x + 3)) - 5764801/2816*log(abs(2*x - 1))
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{4296875}-\frac {5764801\,\ln \left (x-\frac {1}{2}\right )}{2816}-\frac {40089855591\,x}{10000000}-\frac {7136193339\,x^2}{2000000}-\frac {345533877\,x^3}{100000}-\frac {111146499\,x^4}{40000}-\frac {8018271\,x^5}{5000}-\frac {114453\,x^6}{200}-\frac {6561\,x^7}{70} \]