Integrand size = 22, antiderivative size = 70 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {117649}{42592 (1-2 x)^2}-\frac {1563051}{234256 (1-2 x)}-\frac {729 x}{1000}-\frac {1}{1663750 (3+5 x)^2}-\frac {204}{9150625 (3+5 x)}-\frac {6950895 \log (1-2 x)}{2576816}+\frac {17547 \log (3+5 x)}{100656875} \] Output:
117649/42592/(1-2*x)^2-1563051/(234256-468512*x)-729/1000*x-1/1663750/(3+5 *x)^2-204/(27451875+45753125*x)-6950895/2576816*ln(1-2*x)+17547/100656875* ln(3+5*x)
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {-\frac {55 \left (2317121263+49588250 x-19656314001 x^2-21487765512 x^3+3700073520 x^4+4269315600 x^5\right )}{\left (-3+x+10 x^2\right )^2}-8688618750 \log (3-6 x)+561504 \log (-3 (3+5 x))}{3221020000} \] Input:
Integrate[(2 + 3*x)^6/((1 - 2*x)^3*(3 + 5*x)^3),x]
Output:
((-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 = 0.23 (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 |
| \(\displaystyle -\frac {729 x}{1000}-\frac {1563051}{234256 (1-2 x)}-\frac {204}{9150625 (5 x+3)}+\frac {117649}{42592 (1-2 x)^2}-\frac {1}{1663750 (5 x+3)^2}-\frac {6950895 \log (1-2 x)}{2576816}+\frac {17547 \log (5 x+3)}{100656875}\) |
Input:
Int[(2 + 3*x)^6/((1 - 2*x)^3*(3 + 5*x)^3),x]
Output:
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
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 = 0.23 (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 | \(-\frac {1303292812500 \ln \left (x -\frac {1}{2}\right ) x^{4}-84225600 \ln \left (x +\frac {3}{5}\right ) x^{4}+352218537000 x^{5}+197299050135+260658562500 \ln \left (x -\frac {1}{2}\right ) x^{3}-16845120 \ln \left (x +\frac {3}{5}\right ) x^{3}+373439909150 x^{4}-768942759375 \ln \left (x -\frac {1}{2}\right ) x^{2}+49693104 \ln \left (x +\frac {3}{5}\right ) x^{2}-1759103885990 x^{3}-78197568750 \ln \left (x -\frac {1}{2}\right ) x +5053536 \ln \left (x +\frac {3}{5}\right ) x -1661874372895 x^{2}+117296353125 \ln \left (x -\frac {1}{2}\right )-7580304 \ln \left (x +\frac {3}{5}\right )}{4831530000 \left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(119\) |
Input:
int((2+3*x)^6/(1-2*x)^3/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-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 = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.50 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {234812358000 \, x^{5} + 46962471600 \, x^{4} - 1213135417560 \, x^{3} - 988737742575 \, x^{2} - 561504 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 8688618750 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 12119848070 \, x + 113352927985}{3221020000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^3,x, algorithm="fricas")
Output:
-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 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=- \frac {729 x}{1000} - \frac {- 19538111388 x^{3} - 17720890929 x^{2} - 163877530 x + 2060962327}{5856400000 x^{4} + 1171280000 x^{3} - 3455276000 x^{2} - 351384000 x + 527076000} - \frac {6950895 \log {\left (x - \frac {1}{2} \right )}}{2576816} + \frac {17547 \log {\left (x + \frac {3}{5} \right )}}{100656875} \] Input:
integrate((2+3*x)**6/(1-2*x)**3/(3+5*x)**3,x)
Output:
-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 = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {729}{1000} \, x + \frac {19538111388 \, x^{3} + 17720890929 \, x^{2} + 163877530 \, x - 2060962327}{58564000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {17547}{100656875} \, \log \left (5 \, x + 3\right ) - \frac {6950895}{2576816} \, \log \left (2 \, x - 1\right ) \] Input:
integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^3,x, algorithm="maxima")
Output:
-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 = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=-\frac {729}{1000} \, x + \frac {19538111388 \, x^{3} + 17720890929 \, x^{2} + 163877530 \, x - 2060962327}{58564000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {17547}{100656875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {6950895}{2576816} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \] Input:
integrate((2+3*x)^6/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")
Output:
-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 = 1.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {17547\,\ln \left (x+\frac {3}{5}\right )}{100656875}-\frac {6950895\,\ln \left (x-\frac {1}{2}\right )}{2576816}-\frac {729\,x}{1000}+\frac {\frac {4884527847\,x^3}{1464100000}+\frac {17720890929\,x^2}{5856400000}+\frac {16387753\,x}{585640000}-\frac {2060962327}{5856400000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}} \] Input:
int(-(3*x + 2)^6/((2*x - 1)^3*(5*x + 3)^3),x)
Output:
(17547*log(x + 3/5))/100656875 - (6950895*log(x - 1/2))/2576816 - (729*x)/ 1000 + ((16387753*x)/585640000 + (17720890929*x^2)/5856400000 + (488452784 7*x^3)/1464100000 - 2060962327/5856400000)/(x^3/5 - (59*x^2)/100 - (3*x)/5 0 + x^4 + 9/100)
Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.04 \[ \int \frac {(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx=\frac {56150400 \,\mathrm {log}\left (5 x +3\right ) x^{4}+11230080 \,\mathrm {log}\left (5 x +3\right ) x^{3}-33128736 \,\mathrm {log}\left (5 x +3\right ) x^{2}-3369024 \,\mathrm {log}\left (5 x +3\right ) x +5053536 \,\mathrm {log}\left (5 x +3\right )-868861875000 \,\mathrm {log}\left (2 x -1\right ) x^{4}-173772375000 \,\mathrm {log}\left (2 x -1\right ) x^{3}+512628506250 \,\mathrm {log}\left (2 x -1\right ) x^{2}+52131712500 \,\mathrm {log}\left (2 x -1\right ) x -78197568750 \,\mathrm {log}\left (2 x -1\right )-234812358000 x^{5}-6112639559400 x^{4}+4567487224377 x^{2}+351820777198 x -659263865887}{322102000000 x^{4}+64420400000 x^{3}-190040180000 x^{2}-19326120000 x +28989180000} \] Input:
int((2+3*x)^6/(1-2*x)^3/(3+5*x)^3,x)
Output:
(56150400*log(5*x + 3)*x**4 + 11230080*log(5*x + 3)*x**3 - 33128736*log(5* x + 3)*x**2 - 3369024*log(5*x + 3)*x + 5053536*log(5*x + 3) - 868861875000 *log(2*x - 1)*x**4 - 173772375000*log(2*x - 1)*x**3 + 512628506250*log(2*x - 1)*x**2 + 52131712500*log(2*x - 1)*x - 78197568750*log(2*x - 1) - 23481 2358000*x**5 - 6112639559400*x**4 + 4567487224377*x**2 + 351820777198*x - 659263865887)/(3221020000*(100*x**4 + 20*x**3 - 59*x**2 - 6*x + 9))