Integrand size = 20, antiderivative size = 76 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {63412811}{1024 (1-2 x)}+\frac {91609881 x}{256}+\frac {122887143 x^2}{512}+\frac {5892813 x^3}{32}+\frac {32991057 x^4}{256}+\frac {5859459 x^5}{80}+\frac {976617 x^6}{32}+\frac {56862 x^7}{7}+\frac {32805 x^8}{32}+\frac {246239357 \log (1-2 x)}{1024} \]
63412811/1024/(1-2*x)+91609881/256*x+122887143/512*x^2+5892813/32*x^3+3299 1057/256*x^4+5859459/80*x^5+976617/32*x^6+56862/7*x^7+32805/32*x^8+2462393 57/1024*ln(1-2*x)
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {60471800653-259057842186 x+136389333360 x^2+84833995680 x^3+68649225120 x^4+52899666624 x^5+33250113792 x^6+15171909120 x^7+4364202240 x^8+587865600 x^9+68947019960 (-1+2 x) \log (1-2 x)}{286720 (-1+2 x)} \]
(60471800653 - 259057842186*x + 136389333360*x^2 + 84833995680*x^3 + 68649 225120*x^4 + 52899666624*x^5 + 33250113792*x^6 + 15171909120*x^7 + 4364202 240*x^8 + 587865600*x^9 + 68947019960*(-1 + 2*x)*Log[1 - 2*x])/(286720*(-1 + 2*x))
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
| \(\displaystyle \int \frac {(3 x+2)^8 (5 x+3)}{(1-2 x)^2} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {32805 x^7}{4}+56862 x^6+\frac {2929851 x^5}{16}+\frac {5859459 x^4}{16}+\frac {32991057 x^3}{64}+\frac {17678439 x^2}{32}+\frac {122887143 x}{256}+\frac {246239357}{512 (2 x-1)}+\frac {63412811}{512 (2 x-1)^2}+\frac {91609881}{256}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {32805 x^8}{32}+\frac {56862 x^7}{7}+\frac {976617 x^6}{32}+\frac {5859459 x^5}{80}+\frac {32991057 x^4}{256}+\frac {5892813 x^3}{32}+\frac {122887143 x^2}{512}+\frac {91609881 x}{256}+\frac {63412811}{1024 (1-2 x)}+\frac {246239357 \log (1-2 x)}{1024}\) |
63412811/(1024*(1 - 2*x)) + (91609881*x)/256 + (122887143*x^2)/512 + (5892 813*x^3)/32 + (32991057*x^4)/256 + (5859459*x^5)/80 + (976617*x^6)/32 + (5 6862*x^7)/7 + (32805*x^8)/32 + (246239357*Log[1 - 2*x])/1024
3.16.38.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.87 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {32805 x^{8}}{32}+\frac {56862 x^{7}}{7}+\frac {976617 x^{6}}{32}+\frac {5859459 x^{5}}{80}+\frac {32991057 x^{4}}{256}+\frac {5892813 x^{3}}{32}+\frac {122887143 x^{2}}{512}+\frac {91609881 x}{256}-\frac {63412811}{2048 \left (x -\frac {1}{2}\right )}+\frac {246239357 \ln \left (-1+2 x \right )}{1024}\) | \(55\) |
| default | \(\frac {32805 x^{8}}{32}+\frac {56862 x^{7}}{7}+\frac {976617 x^{6}}{32}+\frac {5859459 x^{5}}{80}+\frac {32991057 x^{4}}{256}+\frac {5892813 x^{3}}{32}+\frac {122887143 x^{2}}{512}+\frac {91609881 x}{256}+\frac {246239357 \ln \left (-1+2 x \right )}{1024}-\frac {63412811}{1024 \left (-1+2 x \right )}\) | \(57\) |
| norman | \(\frac {-\frac {246632573}{512} x +\frac {243552381}{512} x^{2}+\frac {75744639}{256} x^{3}+\frac {61293951}{256} x^{4}+\frac {118079613}{640} x^{5}+\frac {18554751}{160} x^{6}+\frac {5926527}{112} x^{7}+\frac {3409533}{224} x^{8}+\frac {32805}{16} x^{9}}{-1+2 x}+\frac {246239357 \ln \left (-1+2 x \right )}{1024}\) | \(62\) |
| parallelrisch | \(\frac {73483200 x^{9}+545525280 x^{8}+1896488640 x^{7}+4156264224 x^{6}+6612458328 x^{5}+8581153140 x^{4}+10604249460 x^{3}+17236754990 \ln \left (x -\frac {1}{2}\right ) x +17048666670 x^{2}-8618377495 \ln \left (x -\frac {1}{2}\right )-17264280110 x}{-35840+71680 x}\) | \(67\) |
| meijerg | \(\frac {6016 x}{1-2 x}+\frac {246239357 \ln \left (1-2 x \right )}{1024}+\frac {21627 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{8960 \left (1-2 x \right )}+\frac {6561 x \left (-8960 x^{8}-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{28672 \left (1-2 x \right )}+\frac {7056 x \left (-8 x^{2}-12 x +12\right )}{1-2 x}+\frac {2142 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{1-2 x}+\frac {8127 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{4 \left (1-2 x \right )}+\frac {5312 x \left (-6 x +6\right )}{1-2 x}+\frac {7047 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{40 \left (1-2 x \right )}+\frac {2673 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{80 \left (1-2 x \right )}\) | \(280\) |
32805/32*x^8+56862/7*x^7+976617/32*x^6+5859459/80*x^5+32991057/256*x^4+589 2813/32*x^3+122887143/512*x^2+91609881/256*x-63412811/2048/(x-1/2)+2462393 57/1024*ln(-1+2*x)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {73483200 \, x^{9} + 545525280 \, x^{8} + 1896488640 \, x^{7} + 4156264224 \, x^{6} + 6612458328 \, x^{5} + 8581153140 \, x^{4} + 10604249460 \, x^{3} + 17048666670 \, x^{2} + 8618377495 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 12825383340 \, x - 2219448385}{35840 \, {\left (2 \, x - 1\right )}} \]
1/35840*(73483200*x^9 + 545525280*x^8 + 1896488640*x^7 + 4156264224*x^6 + 6612458328*x^5 + 8581153140*x^4 + 10604249460*x^3 + 17048666670*x^2 + 8618 377495*(2*x - 1)*log(2*x - 1) - 12825383340*x - 2219448385)/(2*x - 1)
Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {32805 x^{8}}{32} + \frac {56862 x^{7}}{7} + \frac {976617 x^{6}}{32} + \frac {5859459 x^{5}}{80} + \frac {32991057 x^{4}}{256} + \frac {5892813 x^{3}}{32} + \frac {122887143 x^{2}}{512} + \frac {91609881 x}{256} + \frac {246239357 \log {\left (2 x - 1 \right )}}{1024} - \frac {63412811}{2048 x - 1024} \]
32805*x**8/32 + 56862*x**7/7 + 976617*x**6/32 + 5859459*x**5/80 + 32991057 *x**4/256 + 5892813*x**3/32 + 122887143*x**2/512 + 91609881*x/256 + 246239 357*log(2*x - 1)/1024 - 63412811/(2048*x - 1024)
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {32805}{32} \, x^{8} + \frac {56862}{7} \, x^{7} + \frac {976617}{32} \, x^{6} + \frac {5859459}{80} \, x^{5} + \frac {32991057}{256} \, x^{4} + \frac {5892813}{32} \, x^{3} + \frac {122887143}{512} \, x^{2} + \frac {91609881}{256} \, x - \frac {63412811}{1024 \, {\left (2 \, x - 1\right )}} + \frac {246239357}{1024} \, \log \left (2 \, x - 1\right ) \]
32805/32*x^8 + 56862/7*x^7 + 976617/32*x^6 + 5859459/80*x^5 + 32991057/256 *x^4 + 5892813/32*x^3 + 122887143/512*x^2 + 91609881/256*x - 63412811/1024 /(2*x - 1) + 246239357/1024*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {3}{286720} \, {\left (2 \, x - 1\right )}^{8} {\left (\frac {9127080}{2 \, x - 1} + \frac {98748720}{{\left (2 \, x - 1\right )}^{2}} + \frac {641009376}{{\left (2 \, x - 1\right )}^{3}} + \frac {2786264460}{{\left (2 \, x - 1\right )}^{4}} + \frac {8611906800}{{\left (2 \, x - 1\right )}^{5}} + \frac {19962682320}{{\left (2 \, x - 1\right )}^{6}} + \frac {39661830880}{{\left (2 \, x - 1\right )}^{7}} + 382725\right )} - \frac {63412811}{1024 \, {\left (2 \, x - 1\right )}} - \frac {246239357}{1024} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]
3/286720*(2*x - 1)^8*(9127080/(2*x - 1) + 98748720/(2*x - 1)^2 + 641009376 /(2*x - 1)^3 + 2786264460/(2*x - 1)^4 + 8611906800/(2*x - 1)^5 + 199626823 20/(2*x - 1)^6 + 39661830880/(2*x - 1)^7 + 382725) - 63412811/1024/(2*x - 1) - 246239357/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
Time = 1.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^8 (3+5 x)}{(1-2 x)^2} \, dx=\frac {91609881\,x}{256}+\frac {246239357\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {63412811}{2048\,\left (x-\frac {1}{2}\right )}+\frac {122887143\,x^2}{512}+\frac {5892813\,x^3}{32}+\frac {32991057\,x^4}{256}+\frac {5859459\,x^5}{80}+\frac {976617\,x^6}{32}+\frac {56862\,x^7}{7}+\frac {32805\,x^8}{32} \]