Integrand size = 22, antiderivative size = 76 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=\frac {5764801}{5632 (1-2 x)^2}-\frac {188591347}{30976 (1-2 x)}-\frac {2941619571 x}{400000}-\frac {110180817 x^2}{40000}-\frac {124416 x^3}{125}-\frac {408969 x^4}{1600}-\frac {6561 x^5}{200}-\frac {2644396573 \log (1-2 x)}{340736}+\frac {\log (3+5 x)}{20796875} \]
5764801/5632/(1-2*x)^2-188591347/30976/(1-2*x)-2941619571/400000*x-1101808 17/40000*x^2-124416/125*x^3-408969/1600*x^4-6561/200*x^5-2644396573/340736 *ln(1-2*x)+1/20796875*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.29 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=\frac {5764801}{5632 (1-2 x)^2}+\frac {188591347}{30976 (-1+2 x)}-\frac {631722537 (2+3 x)}{400000}-\frac {5992353 (2+3 x)^2}{40000}-\frac {17019 (2+3 x)^3}{1000}-\frac {2889 (2+3 x)^4}{1600}-\frac {27}{200} (2+3 x)^5-\frac {2644396573 \log (3-6 x)}{340736}+\frac {\log (-3 (3+5 x))}{20796875} \]
5764801/(5632*(1 - 2*x)^2) + 188591347/(30976*(-1 + 2*x)) - (631722537*(2 + 3*x))/400000 - (5992353*(2 + 3*x)^2)/40000 - (17019*(2 + 3*x)^3)/1000 - (2889*(2 + 3*x)^4)/1600 - (27*(2 + 3*x)^5)/200 - (2644396573*Log[3 - 6*x]) /340736 + Log[-3*(3 + 5*x)]/20796875
Time = 0.20 (sec) , antiderivative size = 76, 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.
\(\displaystyle \int \frac {(3 x+2)^8}{(1-2 x)^3 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {6561 x^4}{40}-\frac {408969 x^3}{400}-\frac {373248 x^2}{125}-\frac {110180817 x}{20000}-\frac {2644396573}{170368 (2 x-1)}+\frac {1}{4159375 (5 x+3)}-\frac {188591347}{15488 (2 x-1)^2}-\frac {5764801}{1408 (2 x-1)^3}-\frac {2941619571}{400000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {6561 x^5}{200}-\frac {408969 x^4}{1600}-\frac {124416 x^3}{125}-\frac {110180817 x^2}{40000}-\frac {2941619571 x}{400000}-\frac {188591347}{30976 (1-2 x)}+\frac {5764801}{5632 (1-2 x)^2}-\frac {2644396573 \log (1-2 x)}{340736}+\frac {\log (5 x+3)}{20796875}\) |
5764801/(5632*(1 - 2*x)^2) - 188591347/(30976*(1 - 2*x)) - (2941619571*x)/ 400000 - (110180817*x^2)/40000 - (124416*x^3)/125 - (408969*x^4)/1600 - (6 561*x^5)/200 - (2644396573*Log[1 - 2*x])/340736 + Log[3 + 5*x]/20796875
3.17.71.3.1 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 = 55, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {6561 x^{5}}{200}-\frac {408969 x^{4}}{1600}-\frac {124416 x^{3}}{125}-\frac {110180817 x^{2}}{40000}-\frac {2941619571 x}{400000}+\frac {\frac {188591347 x}{15488}-\frac {313769883}{61952}}{\left (-1+2 x \right )^{2}}-\frac {2644396573 \ln \left (-1+2 x \right )}{340736}+\frac {\ln \left (3+5 x \right )}{20796875}\) | \(55\) |
default | \(-\frac {6561 x^{5}}{200}-\frac {408969 x^{4}}{1600}-\frac {124416 x^{3}}{125}-\frac {110180817 x^{2}}{40000}-\frac {2941619571 x}{400000}+\frac {\ln \left (3+5 x \right )}{20796875}+\frac {5764801}{5632 \left (-1+2 x \right )^{2}}+\frac {188591347}{30976 \left (-1+2 x \right )}-\frac {2644396573 \ln \left (-1+2 x \right )}{340736}\) | \(59\) |
norman | \(\frac {-\frac {747118893091}{48400000} x +\frac {2270955968169}{48400000} x^{2}-\frac {1939344201}{100000} x^{3}-\frac {291695013}{40000} x^{4}-\frac {5983389}{2000} x^{5}-\frac {356481}{400} x^{6}-\frac {6561}{50} x^{7}}{\left (-1+2 x \right )^{2}}-\frac {2644396573 \ln \left (-1+2 x \right )}{340736}+\frac {\ln \left (3+5 x \right )}{20796875}\) | \(60\) |
parallelrisch | \(\frac {-698615280000 x^{7}-4744762110000 x^{6}-15927781518000 x^{5}-38824606230300 x^{4}+1024 \ln \left (x +\frac {3}{5}\right ) x^{2}-165274785812500 \ln \left (x -\frac {1}{2}\right ) x^{2}-103250685261240 x^{3}-1024 \ln \left (x +\frac {3}{5}\right ) x +165274785812500 \ln \left (x -\frac {1}{2}\right ) x +249805156498590 x^{2}+256 \ln \left (x +\frac {3}{5}\right )-41318696453125 \ln \left (x -\frac {1}{2}\right )-82183078240010 x}{5324000000 \left (-1+2 x \right )^{2}}\) | \(88\) |
-6561/200*x^5-408969/1600*x^4-124416/125*x^3-110180817/40000*x^2-294161957 1/400000*x+4*(188591347/61952*x-313769883/247808)/(-1+2*x)^2-2644396573/34 0736*ln(-1+2*x)+1/20796875*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {1397230560000 \, x^{7} + 9489524220000 \, x^{6} + 31855563036000 \, x^{5} + 77649212460600 \, x^{4} + 206501370522480 \, x^{3} - 283893518434680 \, x^{2} - 512 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 82637392906250 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 51350638082480 \, x + 53929198640625}{10648000000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/10648000000*(1397230560000*x^7 + 9489524220000*x^6 + 31855563036000*x^5 + 77649212460600*x^4 + 206501370522480*x^3 - 283893518434680*x^2 - 512*(4 *x^2 - 4*x + 1)*log(5*x + 3) + 82637392906250*(4*x^2 - 4*x + 1)*log(2*x - 1) - 51350638082480*x + 53929198640625)/(4*x^2 - 4*x + 1)
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=- \frac {6561 x^{5}}{200} - \frac {408969 x^{4}}{1600} - \frac {124416 x^{3}}{125} - \frac {110180817 x^{2}}{40000} - \frac {2941619571 x}{400000} - \frac {313769883 - 754365388 x}{247808 x^{2} - 247808 x + 61952} - \frac {2644396573 \log {\left (x - \frac {1}{2} \right )}}{340736} + \frac {\log {\left (x + \frac {3}{5} \right )}}{20796875} \]
-6561*x**5/200 - 408969*x**4/1600 - 124416*x**3/125 - 110180817*x**2/40000 - 2941619571*x/400000 - (313769883 - 754365388*x)/(247808*x**2 - 247808*x + 61952) - 2644396573*log(x - 1/2)/340736 + log(x + 3/5)/20796875
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {6561}{200} \, x^{5} - \frac {408969}{1600} \, x^{4} - \frac {124416}{125} \, x^{3} - \frac {110180817}{40000} \, x^{2} - \frac {2941619571}{400000} \, x + \frac {823543 \, {\left (916 \, x - 381\right )}}{61952 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{20796875} \, \log \left (5 \, x + 3\right ) - \frac {2644396573}{340736} \, \log \left (2 \, x - 1\right ) \]
-6561/200*x^5 - 408969/1600*x^4 - 124416/125*x^3 - 110180817/40000*x^2 - 2 941619571/400000*x + 823543/61952*(916*x - 381)/(4*x^2 - 4*x + 1) + 1/2079 6875*log(5*x + 3) - 2644396573/340736*log(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=-\frac {6561}{200} \, x^{5} - \frac {408969}{1600} \, x^{4} - \frac {124416}{125} \, x^{3} - \frac {110180817}{40000} \, x^{2} - \frac {2941619571}{400000} \, x + \frac {823543 \, {\left (916 \, x - 381\right )}}{61952 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{20796875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {2644396573}{340736} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-6561/200*x^5 - 408969/1600*x^4 - 124416/125*x^3 - 110180817/40000*x^2 - 2 941619571/400000*x + 823543/61952*(916*x - 381)/(2*x - 1)^2 + 1/20796875*l og(abs(5*x + 3)) - 2644396573/340736*log(abs(2*x - 1))
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)} \, dx=\frac {\ln \left (x+\frac {3}{5}\right )}{20796875}-\frac {2644396573\,\ln \left (x-\frac {1}{2}\right )}{340736}-\frac {2941619571\,x}{400000}+\frac {\frac {188591347\,x}{61952}-\frac {313769883}{247808}}{x^2-x+\frac {1}{4}}-\frac {110180817\,x^2}{40000}-\frac {124416\,x^3}{125}-\frac {408969\,x^4}{1600}-\frac {6561\,x^5}{200} \]