Integrand size = 22, antiderivative size = 80 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=\frac {4571416 x}{1953125}-\frac {915777 x^2}{390625}-\frac {81747 x^3}{15625}+\frac {74223 x^4}{12500}+\frac {134622 x^5}{15625}-\frac {3402 x^6}{625}-\frac {5832 x^7}{875}-\frac {1331}{19531250 (3+5 x)^2}-\frac {23232}{9765625 (3+5 x)}+\frac {166749 \log (3+5 x)}{9765625} \]
4571416/1953125*x-915777/390625*x^2-81747/15625*x^3+74223/12500*x^4+134622 /15625*x^5-3402/625*x^6-5832/875*x^7-1331/19531250/(3+5*x)^2-23232/9765625 /(3+5*x)+166749/9765625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=\frac {13353609877+73328526690 x+104273484075 x^2-80532567500 x^3-221653096875 x^4+170737481250 x^5+489359390625 x^6-10783125000 x^7-459421875000 x^8-227812500000 x^9+23344860 (3+5 x)^2 \log (6 (3+5 x))}{1367187500 (3+5 x)^2} \]
(13353609877 + 73328526690*x + 104273484075*x^2 - 80532567500*x^3 - 221653 096875*x^4 + 170737481250*x^5 + 489359390625*x^6 - 10783125000*x^7 - 45942 1875000*x^8 - 227812500000*x^9 + 23344860*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(1 367187500*(3 + 5*x)^2)
Time = 0.20 (sec) , antiderivative size = 80, 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.
| \(\displaystyle \int \frac {(1-2 x)^3 (3 x+2)^6}{(5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {5832 x^6}{125}-\frac {20412 x^5}{625}+\frac {134622 x^4}{3125}+\frac {74223 x^3}{3125}-\frac {245241 x^2}{15625}-\frac {1831554 x}{390625}+\frac {166749}{1953125 (5 x+3)}+\frac {23232}{1953125 (5 x+3)^2}+\frac {1331}{1953125 (5 x+3)^3}+\frac {4571416}{1953125}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5832 x^7}{875}-\frac {3402 x^6}{625}+\frac {134622 x^5}{15625}+\frac {74223 x^4}{12500}-\frac {81747 x^3}{15625}-\frac {915777 x^2}{390625}+\frac {4571416 x}{1953125}-\frac {23232}{9765625 (5 x+3)}-\frac {1331}{19531250 (5 x+3)^2}+\frac {166749 \log (5 x+3)}{9765625}\) |
(4571416*x)/1953125 - (915777*x^2)/390625 - (81747*x^3)/15625 + (74223*x^4 )/12500 + (134622*x^5)/15625 - (3402*x^6)/625 - (5832*x^7)/875 - 1331/(195 31250*(3 + 5*x)^2) - 23232/(9765625*(3 + 5*x)) + (166749*Log[3 + 5*x])/976 5625
3.15.19.3.1 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 = 0.81 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {5832 x^{7}}{875}-\frac {3402 x^{6}}{625}+\frac {134622 x^{5}}{15625}+\frac {74223 x^{4}}{12500}-\frac {81747 x^{3}}{15625}-\frac {915777 x^{2}}{390625}+\frac {4571416 x}{1953125}+\frac {-\frac {23232 x}{1953125}-\frac {140723}{19531250}}{\left (3+5 x \right )^{2}}+\frac {166749 \ln \left (3+5 x \right )}{9765625}\) | \(57\) |
| default | \(\frac {4571416 x}{1953125}-\frac {915777 x^{2}}{390625}-\frac {81747 x^{3}}{15625}+\frac {74223 x^{4}}{12500}+\frac {134622 x^{5}}{15625}-\frac {3402 x^{6}}{625}-\frac {5832 x^{7}}{875}-\frac {1331}{19531250 \left (3+5 x \right )^{2}}-\frac {23232}{9765625 \left (3+5 x \right )}+\frac {166749 \ln \left (3+5 x \right )}{9765625}\) | \(61\) |
| norman | \(\frac {\frac {123499259}{5859375} x +\frac {345497777}{7031250} x^{2}-\frac {4601861}{78125} x^{3}-\frac {10132713}{62500} x^{4}+\frac {3902571}{31250} x^{5}+\frac {4474143}{12500} x^{6}-\frac {34506}{4375} x^{7}-\frac {58806}{175} x^{8}-\frac {5832}{35} x^{9}}{\left (3+5 x \right )^{2}}+\frac {166749 \ln \left (3+5 x \right )}{9765625}\) | \(62\) |
| parallelrisch | \(\frac {-410062500000 x^{9}-826959375000 x^{8}-19409625000 x^{7}+880846903125 x^{6}+307327466250 x^{5}-398975574375 x^{4}+1050518700 \ln \left (x +\frac {3}{5}\right ) x^{2}-144958621500 x^{3}+1260622440 \ln \left (x +\frac {3}{5}\right ) x +120924221950 x^{2}+378186732 \ln \left (x +\frac {3}{5}\right )+51869688780 x}{2460937500 \left (3+5 x \right )^{2}}\) | \(76\) |
| meijerg | \(\frac {32 x^{2}}{9 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {334611 x \left (\frac {125000}{729} x^{6}-\frac {43750}{243} x^{5}+\frac {17500}{81} x^{4}-\frac {8750}{27} x^{3}+\frac {7000}{9} x^{2}+2100 x +840\right )}{1562500 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {166749 \ln \left (1+\frac {5 x}{3}\right )}{9765625}+\frac {39366 x \left (-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{390625 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {2125764 x \left (\frac {3906250}{6561} x^{8}-\frac {390625}{729} x^{7}+\frac {125000}{243} x^{6}-\frac {43750}{81} x^{5}+\frac {17500}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{3} x^{2}+6300 x +2520\right )}{68359375 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {12474 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {26973 x \left (-\frac {21875}{243} x^{5}+\frac {8750}{81} x^{4}-\frac {4375}{27} x^{3}+\frac {3500}{9} x^{2}+1050 x +420\right )}{62500 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {88 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {112 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{25 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {32 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {378 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}\) | \(297\) |
-5832/875*x^7-3402/625*x^6+134622/15625*x^5+74223/12500*x^4-81747/15625*x^ 3-915777/390625*x^2+4571416/1953125*x+25*(-23232/48828125*x-140723/4882812 50)/(3+5*x)^2+166749/9765625*ln(3+5*x)
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=-\frac {45562500000 \, x^{9} + 91884375000 \, x^{8} + 2156625000 \, x^{7} - 97871878125 \, x^{6} - 34147496250 \, x^{5} + 44330619375 \, x^{4} + 16106513500 \, x^{3} - 13430552100 \, x^{2} - 4668972 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 5756731680 \, x + 1970122}{273437500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/273437500*(45562500000*x^9 + 91884375000*x^8 + 2156625000*x^7 - 9787187 8125*x^6 - 34147496250*x^5 + 44330619375*x^4 + 16106513500*x^3 - 134305521 00*x^2 - 4668972*(25*x^2 + 30*x + 9)*log(5*x + 3) - 5756731680*x + 1970122 )/(25*x^2 + 30*x + 9)
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=- \frac {5832 x^{7}}{875} - \frac {3402 x^{6}}{625} + \frac {134622 x^{5}}{15625} + \frac {74223 x^{4}}{12500} - \frac {81747 x^{3}}{15625} - \frac {915777 x^{2}}{390625} + \frac {4571416 x}{1953125} - \frac {232320 x + 140723}{488281250 x^{2} + 585937500 x + 175781250} + \frac {166749 \log {\left (5 x + 3 \right )}}{9765625} \]
-5832*x**7/875 - 3402*x**6/625 + 134622*x**5/15625 + 74223*x**4/12500 - 81 747*x**3/15625 - 915777*x**2/390625 + 4571416*x/1953125 - (232320*x + 1407 23)/(488281250*x**2 + 585937500*x + 175781250) + 166749*log(5*x + 3)/97656 25
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=-\frac {5832}{875} \, x^{7} - \frac {3402}{625} \, x^{6} + \frac {134622}{15625} \, x^{5} + \frac {74223}{12500} \, x^{4} - \frac {81747}{15625} \, x^{3} - \frac {915777}{390625} \, x^{2} + \frac {4571416}{1953125} \, x - \frac {121 \, {\left (1920 \, x + 1163\right )}}{19531250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {166749}{9765625} \, \log \left (5 \, x + 3\right ) \]
-5832/875*x^7 - 3402/625*x^6 + 134622/15625*x^5 + 74223/12500*x^4 - 81747/ 15625*x^3 - 915777/390625*x^2 + 4571416/1953125*x - 121/19531250*(1920*x + 1163)/(25*x^2 + 30*x + 9) + 166749/9765625*log(5*x + 3)
Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=-\frac {5832}{875} \, x^{7} - \frac {3402}{625} \, x^{6} + \frac {134622}{15625} \, x^{5} + \frac {74223}{12500} \, x^{4} - \frac {81747}{15625} \, x^{3} - \frac {915777}{390625} \, x^{2} + \frac {4571416}{1953125} \, x - \frac {121 \, {\left (1920 \, x + 1163\right )}}{19531250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {166749}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]
-5832/875*x^7 - 3402/625*x^6 + 134622/15625*x^5 + 74223/12500*x^4 - 81747/ 15625*x^3 - 915777/390625*x^2 + 4571416/1953125*x - 121/19531250*(1920*x + 1163)/(5*x + 3)^2 + 166749/9765625*log(abs(5*x + 3))
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^6}{(3+5 x)^3} \, dx=\frac {4571416\,x}{1953125}+\frac {166749\,\ln \left (x+\frac {3}{5}\right )}{9765625}-\frac {\frac {23232\,x}{48828125}+\frac {140723}{488281250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {915777\,x^2}{390625}-\frac {81747\,x^3}{15625}+\frac {74223\,x^4}{12500}+\frac {134622\,x^5}{15625}-\frac {3402\,x^6}{625}-\frac {5832\,x^7}{875} \]