Integrand size = 22, antiderivative size = 70 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {117649}{42592 (1-2 x)^2}-\frac {1563051}{234256 (1-2 x)}-\frac {729 x}{1000}-\frac {1}{1663750 (3+5 x)^2}-\frac {204}{9150625 (3+5 x)}-\frac {6950895 \log (1-2 x)}{2576816}+\frac {17547 \log (3+5 x)}{100656875} \]
117649/42592/(1-2*x)^2-1563051/234256/(1-2*x)-729/1000*x-1/1663750/(3+5*x) ^2-204/9150625/(3+5*x)-6950895/2576816*ln(1-2*x)+17547/100656875*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {55 \left (2317121263+49588250 x-19656314001 x^2-21487765512 x^3+3700073520 x^4+4269315600 x^5\right )}{\left (-3+x+10 x^2\right )^2}-8688618750 \log (3-6 x)+561504 \log (-3 (3+5 x))}{3221020000} \]
((-55*(2317121263 + 49588250*x - 19656314001*x^2 - 21487765512*x^3 + 37000 73520*x^4 + 4269315600*x^5))/(-3 + x + 10*x^2)^2 - 8688618750*Log[3 - 6*x] + 561504*Log[-3*(3 + 5*x)])/3221020000
Time = 0.20 (sec) , antiderivative size = 70, 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.
\(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^3 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {17547}{20131375 (5 x+3)}+\frac {204}{1830125 (5 x+3)^2}+\frac {1}{166375 (5 x+3)^3}-\frac {6950895}{1288408 (2 x-1)}-\frac {1563051}{117128 (2 x-1)^2}-\frac {117649}{10648 (2 x-1)^3}-\frac {729}{1000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {729 x}{1000}-\frac {1563051}{234256 (1-2 x)}-\frac {204}{9150625 (5 x+3)}+\frac {117649}{42592 (1-2 x)^2}-\frac {1}{1663750 (5 x+3)^2}-\frac {6950895 \log (1-2 x)}{2576816}+\frac {17547 \log (5 x+3)}{100656875}\) |
117649/(42592*(1 - 2*x)^2) - 1563051/(234256*(1 - 2*x)) - (729*x)/1000 - 1 /(1663750*(3 + 5*x)^2) - 204/(9150625*(3 + 5*x)) - (6950895*Log[1 - 2*x])/ 2576816 + (17547*Log[3 + 5*x])/100656875
3.17.100.3.1 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 = 2.82 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.74
method | result | size |
risch | \(-\frac {729 x}{1000}+\frac {\frac {4884527847}{14641000} x^{3}+\frac {17720890929}{58564000} x^{2}+\frac {16387753}{5856400} x -\frac {2060962327}{58564000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {6950895 \ln \left (-1+2 x \right )}{2576816}+\frac {17547 \ln \left (3+5 x \right )}{100656875}\) | \(52\) |
norman | \(\frac {-\frac {678981653}{146410000} x +\frac {2778472527}{7320500} x^{3}+\frac {87366353121}{292820000} x^{2}-\frac {729}{10} x^{5}-\frac {9920573231}{292820000}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {6950895 \ln \left (-1+2 x \right )}{2576816}+\frac {17547 \ln \left (3+5 x \right )}{100656875}\) | \(53\) |
default | \(-\frac {729 x}{1000}-\frac {1}{1663750 \left (3+5 x \right )^{2}}-\frac {204}{9150625 \left (3+5 x \right )}+\frac {17547 \ln \left (3+5 x \right )}{100656875}+\frac {117649}{42592 \left (-1+2 x \right )^{2}}+\frac {1563051}{234256 \left (-1+2 x \right )}-\frac {6950895 \ln \left (-1+2 x \right )}{2576816}\) | \(57\) |
parallelrisch | \(\frac {84225600 \ln \left (x +\frac {3}{5}\right ) x^{4}-1303292812500 \ln \left (x -\frac {1}{2}\right ) x^{4}-352218537000 x^{5}-197299050135+16845120 \ln \left (x +\frac {3}{5}\right ) x^{3}-260658562500 \ln \left (x -\frac {1}{2}\right ) x^{3}-373439909150 x^{4}-49693104 \ln \left (x +\frac {3}{5}\right ) x^{2}+768942759375 \ln \left (x -\frac {1}{2}\right ) x^{2}+1759103885990 x^{3}-5053536 \ln \left (x +\frac {3}{5}\right ) x +78197568750 \ln \left (x -\frac {1}{2}\right ) x +1661874372895 x^{2}+7580304 \ln \left (x +\frac {3}{5}\right )-117296353125 \ln \left (x -\frac {1}{2}\right )}{4831530000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(119\) |
-729/1000*x+100*(4884527847/1464100000*x^3+17720890929/5856400000*x^2+1638 7753/585640000*x-2060962327/5856400000)/(-1+2*x)^2/(3+5*x)^2-6950895/25768 16*ln(-1+2*x)+17547/100656875*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.50 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {234812358000 \, x^{5} + 46962471600 \, x^{4} - 1213135417560 \, x^{3} - 988737742575 \, x^{2} - 561504 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 8688618750 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 12119848070 \, x + 113352927985}{3221020000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/3221020000*(234812358000*x^5 + 46962471600*x^4 - 1213135417560*x^3 - 98 8737742575*x^2 - 561504*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 8688618750*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 1211984 8070*x + 113352927985)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {729 x}{1000} - \frac {- 19538111388 x^{3} - 17720890929 x^{2} - 163877530 x + 2060962327}{5856400000 x^{4} + 1171280000 x^{3} - 3455276000 x^{2} - 351384000 x + 527076000} - \frac {6950895 \log {\left (x - \frac {1}{2} \right )}}{2576816} + \frac {17547 \log {\left (x + \frac {3}{5} \right )}}{100656875} \]
-729*x/1000 - (-19538111388*x**3 - 17720890929*x**2 - 163877530*x + 206096 2327)/(5856400000*x**4 + 1171280000*x**3 - 3455276000*x**2 - 351384000*x + 527076000) - 6950895*log(x - 1/2)/2576816 + 17547*log(x + 3/5)/100656875
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {729}{1000} \, x + \frac {19538111388 \, x^{3} + 17720890929 \, x^{2} + 163877530 \, x - 2060962327}{58564000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {17547}{100656875} \, \log \left (5 \, x + 3\right ) - \frac {6950895}{2576816} \, \log \left (2 \, x - 1\right ) \]
-729/1000*x + 1/58564000*(19538111388*x^3 + 17720890929*x^2 + 163877530*x - 2060962327)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 17547/100656875*log( 5*x + 3) - 6950895/2576816*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {729}{1000} \, x + \frac {19538111388 \, x^{3} + 17720890929 \, x^{2} + 163877530 \, x - 2060962327}{58564000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {17547}{100656875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {6950895}{2576816} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-729/1000*x + 1/58564000*(19538111388*x^3 + 17720890929*x^2 + 163877530*x - 2060962327)/((5*x + 3)^2*(2*x - 1)^2) + 17547/100656875*log(abs(5*x + 3) ) - 6950895/2576816*log(abs(2*x - 1))
Time = 1.38 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {17547\,\ln \left (x+\frac {3}{5}\right )}{100656875}-\frac {6950895\,\ln \left (x-\frac {1}{2}\right )}{2576816}-\frac {729\,x}{1000}+\frac {\frac {4884527847\,x^3}{1464100000}+\frac {17720890929\,x^2}{5856400000}+\frac {16387753\,x}{585640000}-\frac {2060962327}{5856400000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}} \]