Integrand size = 22, antiderivative size = 90 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {7672950131}{4096 (1-2 x)}+\frac {7277894263 x}{512}+\frac {21573106793 x^2}{2048}+\frac {2416569641 x^3}{256}+\frac {8502681987 x^4}{1024}+\frac {260574273 x^5}{40}+\frac {544462047 x^6}{128}+\frac {242570133 x^7}{112}+\frac {101721015 x^8}{128}+\frac {370575 x^9}{2}+\frac {164025 x^{10}}{8}+\frac {36770371407 \log (1-2 x)}{4096} \]
7672950131/4096/(1-2*x)+7277894263/512*x+21573106793/2048*x^2+2416569641/2 56*x^3+8502681987/1024*x^4+260574273/40*x^5+544462047/128*x^6+242570133/11 2*x^7+101721015/128*x^8+370575/2*x^9+164025/8*x^10+36770371407/4096*ln(1-2 *x)
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {11304620315803-43208575854086 x+20524026494160 x^2+13335647616480 x^3+12129460157920 x^4+11574822095424 x^5+10063991169792 x^6+7272841720320 x^7+4056416029440 x^8+1610338060800 x^9+401490432000 x^{10}+47029248000 x^{11}+10295703993960 (-1+2 x) \log (1-2 x)}{1146880 (-1+2 x)} \]
(11304620315803 - 43208575854086*x + 20524026494160*x^2 + 13335647616480*x ^3 + 12129460157920*x^4 + 11574822095424*x^5 + 10063991169792*x^6 + 727284 1720320*x^7 + 4056416029440*x^8 + 1610338060800*x^9 + 401490432000*x^10 + 47029248000*x^11 + 10295703993960*(-1 + 2*x)*Log[1 - 2*x])/(1146880*(-1 + 2*x))
Time = 0.21 (sec) , antiderivative size = 90, 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.
| \(\displaystyle \int \frac {(3 x+2)^8 (5 x+3)^3}{(1-2 x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {820125 x^9}{4}+\frac {3335175 x^8}{2}+\frac {101721015 x^7}{16}+\frac {242570133 x^6}{16}+\frac {1633386141 x^5}{64}+\frac {260574273 x^4}{8}+\frac {8502681987 x^3}{256}+\frac {7249708923 x^2}{256}+\frac {21573106793 x}{1024}+\frac {36770371407}{2048 (2 x-1)}+\frac {7672950131}{2048 (2 x-1)^2}+\frac {7277894263}{512}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {164025 x^{10}}{8}+\frac {370575 x^9}{2}+\frac {101721015 x^8}{128}+\frac {242570133 x^7}{112}+\frac {544462047 x^6}{128}+\frac {260574273 x^5}{40}+\frac {8502681987 x^4}{1024}+\frac {2416569641 x^3}{256}+\frac {21573106793 x^2}{2048}+\frac {7277894263 x}{512}+\frac {7672950131}{4096 (1-2 x)}+\frac {36770371407 \log (1-2 x)}{4096}\) |
7672950131/(4096*(1 - 2*x)) + (7277894263*x)/512 + (21573106793*x^2)/2048 + (2416569641*x^3)/256 + (8502681987*x^4)/1024 + (260574273*x^5)/40 + (544 462047*x^6)/128 + (242570133*x^7)/112 + (101721015*x^8)/128 + (370575*x^9) /2 + (164025*x^10)/8 + (36770371407*Log[1 - 2*x])/4096
3.16.71.3.1 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 = 2.67 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {164025 x^{10}}{8}+\frac {370575 x^{9}}{2}+\frac {101721015 x^{8}}{128}+\frac {242570133 x^{7}}{112}+\frac {544462047 x^{6}}{128}+\frac {260574273 x^{5}}{40}+\frac {8502681987 x^{4}}{1024}+\frac {2416569641 x^{3}}{256}+\frac {21573106793 x^{2}}{2048}+\frac {7277894263 x}{512}-\frac {7672950131}{8192 \left (x -\frac {1}{2}\right )}+\frac {36770371407 \ln \left (-1+2 x \right )}{4096}\) | \(65\) |
| default | \(\frac {164025 x^{10}}{8}+\frac {370575 x^{9}}{2}+\frac {101721015 x^{8}}{128}+\frac {242570133 x^{7}}{112}+\frac {544462047 x^{6}}{128}+\frac {260574273 x^{5}}{40}+\frac {8502681987 x^{4}}{1024}+\frac {2416569641 x^{3}}{256}+\frac {21573106793 x^{2}}{2048}+\frac {7277894263 x}{512}+\frac {36770371407 \ln \left (-1+2 x \right )}{4096}-\frac {7672950131}{4096 \left (-1+2 x \right )}\) | \(67\) |
| norman | \(\frac {-\frac {36784527183}{2048} x +\frac {36650047311}{2048} x^{2}+\frac {11906828229}{1024} x^{3}+\frac {10829875141}{1024} x^{4}+\frac {25836656463}{2560} x^{5}+\frac {5616066501}{640} x^{6}+\frac {2840953797}{448} x^{7}+\frac {3169075023}{896} x^{8}+\frac {89862615}{64} x^{9}+\frac {2800575}{8} x^{10}+\frac {164025}{4} x^{11}}{-1+2 x}+\frac {36770371407 \ln \left (-1+2 x \right )}{4096}\) | \(72\) |
| parallelrisch | \(\frac {5878656000 x^{11}+50186304000 x^{10}+201292257600 x^{9}+507052003680 x^{8}+909105215040 x^{7}+1257998896224 x^{6}+1446852761928 x^{5}+1516182519740 x^{4}+1666955952060 x^{3}+2573925998490 \ln \left (x -\frac {1}{2}\right ) x +2565503311770 x^{2}-1286962999245 \ln \left (x -\frac {1}{2}\right )-2574916902810 x}{-143360+286720 x}\) | \(77\) |
| meijerg | \(\frac {65664 x}{1-2 x}+\frac {131464 x \left (-8 x^{2}-12 x +12\right )}{1-2 x}+\frac {36770371407 \ln \left (1-2 x \right )}{4096}+\frac {54142 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{1-2 x}+\frac {292593 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{4 \left (1-2 x \right )}+\frac {387153 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{40 \left (1-2 x \right )}+\frac {248967 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{80 \left (1-2 x \right )}+\frac {3792987 x \left (-8960 x^{8}-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{28672 \left (1-2 x \right )}+\frac {4097763 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{8960 \left (1-2 x \right )}+\frac {75648 x \left (-6 x +6\right )}{1-2 x}+\frac {650025 x \left (-157696 x^{9}-98560 x^{8}-63360 x^{7}-42240 x^{6}-29568 x^{5}-22176 x^{4}-18480 x^{3}-18480 x^{2}-27720 x +27720\right )}{315392 \left (1-2 x \right )}+\frac {18225 x \left (-258048 x^{10}-157696 x^{9}-98560 x^{8}-63360 x^{7}-42240 x^{6}-29568 x^{5}-22176 x^{4}-18480 x^{3}-18480 x^{2}-27720 x +27720\right )}{114688 \left (1-2 x \right )}\) | \(395\) |
164025/8*x^10+370575/2*x^9+101721015/128*x^8+242570133/112*x^7+544462047/1 28*x^6+260574273/40*x^5+8502681987/1024*x^4+2416569641/256*x^3+21573106793 /2048*x^2+7277894263/512*x-7672950131/8192/(x-1/2)+36770371407/4096*ln(-1+ 2*x)
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {5878656000 \, x^{11} + 50186304000 \, x^{10} + 201292257600 \, x^{9} + 507052003680 \, x^{8} + 909105215040 \, x^{7} + 1257998896224 \, x^{6} + 1446852761928 \, x^{5} + 1516182519740 \, x^{4} + 1666955952060 \, x^{3} + 2565503311770 \, x^{2} + 1286962999245 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 2037810393640 \, x - 268553254585}{143360 \, {\left (2 \, x - 1\right )}} \]
1/143360*(5878656000*x^11 + 50186304000*x^10 + 201292257600*x^9 + 50705200 3680*x^8 + 909105215040*x^7 + 1257998896224*x^6 + 1446852761928*x^5 + 1516 182519740*x^4 + 1666955952060*x^3 + 2565503311770*x^2 + 1286962999245*(2*x - 1)*log(2*x - 1) - 2037810393640*x - 268553254585)/(2*x - 1)
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {164025 x^{10}}{8} + \frac {370575 x^{9}}{2} + \frac {101721015 x^{8}}{128} + \frac {242570133 x^{7}}{112} + \frac {544462047 x^{6}}{128} + \frac {260574273 x^{5}}{40} + \frac {8502681987 x^{4}}{1024} + \frac {2416569641 x^{3}}{256} + \frac {21573106793 x^{2}}{2048} + \frac {7277894263 x}{512} + \frac {36770371407 \log {\left (2 x - 1 \right )}}{4096} - \frac {7672950131}{8192 x - 4096} \]
164025*x**10/8 + 370575*x**9/2 + 101721015*x**8/128 + 242570133*x**7/112 + 544462047*x**6/128 + 260574273*x**5/40 + 8502681987*x**4/1024 + 241656964 1*x**3/256 + 21573106793*x**2/2048 + 7277894263*x/512 + 36770371407*log(2* x - 1)/4096 - 7672950131/(8192*x - 4096)
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {164025}{8} \, x^{10} + \frac {370575}{2} \, x^{9} + \frac {101721015}{128} \, x^{8} + \frac {242570133}{112} \, x^{7} + \frac {544462047}{128} \, x^{6} + \frac {260574273}{40} \, x^{5} + \frac {8502681987}{1024} \, x^{4} + \frac {2416569641}{256} \, x^{3} + \frac {21573106793}{2048} \, x^{2} + \frac {7277894263}{512} \, x - \frac {7672950131}{4096 \, {\left (2 \, x - 1\right )}} + \frac {36770371407}{4096} \, \log \left (2 \, x - 1\right ) \]
164025/8*x^10 + 370575/2*x^9 + 101721015/128*x^8 + 242570133/112*x^7 + 544 462047/128*x^6 + 260574273/40*x^5 + 8502681987/1024*x^4 + 2416569641/256*x ^3 + 21573106793/2048*x^2 + 7277894263/512*x - 7672950131/4096/(2*x - 1) + 36770371407/4096*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.33 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {1}{1146880} \, {\left (2 \, x - 1\right )}^{10} {\left (\frac {644679000}{2 \, x - 1} + \frac {8328989025}{{\left (2 \, x - 1\right )}^{2}} + \frac {65584698840}{{\left (2 \, x - 1\right )}^{3}} + \frac {351436586760}{{\left (2 \, x - 1\right )}^{4}} + \frac {1355796026928}{{\left (2 \, x - 1\right )}^{5}} + \frac {3891461518980}{{\left (2 \, x - 1\right )}^{6}} + \frac {8509458050800}{{\left (2 \, x - 1\right )}^{7}} + \frac {14652493526860}{{\left (2 \, x - 1\right )}^{8}} + \frac {22425306482040}{{\left (2 \, x - 1\right )}^{9}} + 22963500\right )} - \frac {7672950131}{4096 \, {\left (2 \, x - 1\right )}} - \frac {36770371407}{4096} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]
1/1146880*(2*x - 1)^10*(644679000/(2*x - 1) + 8328989025/(2*x - 1)^2 + 655 84698840/(2*x - 1)^3 + 351436586760/(2*x - 1)^4 + 1355796026928/(2*x - 1)^ 5 + 3891461518980/(2*x - 1)^6 + 8509458050800/(2*x - 1)^7 + 14652493526860 /(2*x - 1)^8 + 22425306482040/(2*x - 1)^9 + 22963500) - 7672950131/4096/(2 *x - 1) - 36770371407/4096*log(1/2*abs(2*x - 1)/(2*x - 1)^2)
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^8 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {7277894263\,x}{512}+\frac {36770371407\,\ln \left (x-\frac {1}{2}\right )}{4096}-\frac {7672950131}{8192\,\left (x-\frac {1}{2}\right )}+\frac {21573106793\,x^2}{2048}+\frac {2416569641\,x^3}{256}+\frac {8502681987\,x^4}{1024}+\frac {260574273\,x^5}{40}+\frac {544462047\,x^6}{128}+\frac {242570133\,x^7}{112}+\frac {101721015\,x^8}{128}+\frac {370575\,x^9}{2}+\frac {164025\,x^{10}}{8} \]