Integrand size = 22, antiderivative size = 73 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=\frac {14235529}{1024 (1-2 x)^2}-\frac {12386759}{128 (1-2 x)}-\frac {39980457 x}{256}-\frac {17700255 x^2}{256}-\frac {1024389 x^3}{32}-\frac {770067 x^4}{64}-\frac {48843 x^5}{16}-\frac {6075 x^6}{16}-\frac {18859855}{128} \log (1-2 x) \]
14235529/1024/(1-2*x)^2-12386759/128/(1-2*x)-39980457/256*x-17700255/256*x ^2-1024389/32*x^3-770067/64*x^4-48843/16*x^5-6075/16*x^6-18859855/128*ln(1 -2*x)
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=-\frac {-8887005+186131948 x-489708252 x^2+194631840 x^3+82201680 x^4+42481728 x^5+18584640 x^6+5474304 x^7+777600 x^8+75439420 (1-2 x)^2 \log (1-2 x)}{512 (1-2 x)^2} \]
-1/512*(-8887005 + 186131948*x - 489708252*x^2 + 194631840*x^3 + 82201680* x^4 + 42481728*x^5 + 18584640*x^6 + 5474304*x^7 + 777600*x^8 + 75439420*(1 - 2*x)^2*Log[1 - 2*x])/(1 - 2*x)^2
Time = 0.20 (sec) , antiderivative size = 73, 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.
\(\displaystyle \int \frac {(3 x+2)^6 (5 x+3)^2}{(1-2 x)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {18225 x^5}{8}-\frac {244215 x^4}{16}-\frac {770067 x^3}{16}-\frac {3073167 x^2}{32}-\frac {17700255 x}{128}-\frac {18859855}{64 (2 x-1)}-\frac {12386759}{64 (2 x-1)^2}-\frac {14235529}{256 (2 x-1)^3}-\frac {39980457}{256}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {6075 x^6}{16}-\frac {48843 x^5}{16}-\frac {770067 x^4}{64}-\frac {1024389 x^3}{32}-\frac {17700255 x^2}{256}-\frac {39980457 x}{256}-\frac {12386759}{128 (1-2 x)}+\frac {14235529}{1024 (1-2 x)^2}-\frac {18859855}{128} \log (1-2 x)\) |
14235529/(1024*(1 - 2*x)^2) - 12386759/(128*(1 - 2*x)) - (39980457*x)/256 - (17700255*x^2)/256 - (1024389*x^3)/32 - (770067*x^4)/64 - (48843*x^5)/16 - (6075*x^6)/16 - (18859855*Log[1 - 2*x])/128
3.17.46.3.1 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 = 0.89 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {6075 x^{6}}{16}-\frac {48843 x^{5}}{16}-\frac {770067 x^{4}}{64}-\frac {1024389 x^{3}}{32}-\frac {17700255 x^{2}}{256}-\frac {39980457 x}{256}+\frac {\frac {12386759 x}{64}-\frac {84858543}{1024}}{\left (-1+2 x \right )^{2}}-\frac {18859855 \ln \left (-1+2 x \right )}{128}\) | \(52\) |
default | \(-\frac {6075 x^{6}}{16}-\frac {48843 x^{5}}{16}-\frac {770067 x^{4}}{64}-\frac {1024389 x^{3}}{32}-\frac {17700255 x^{2}}{256}-\frac {39980457 x}{256}-\frac {18859855 \ln \left (-1+2 x \right )}{128}+\frac {12386759}{128 \left (-1+2 x \right )}+\frac {14235529}{1024 \left (-1+2 x \right )^{2}}\) | \(56\) |
norman | \(\frac {-\frac {18822991}{64} x +\frac {56770029}{64} x^{2}-\frac {6082245}{16} x^{3}-\frac {5137605}{32} x^{4}-\frac {663777}{8} x^{5}-\frac {290385}{8} x^{6}-10692 x^{7}-\frac {6075}{4} x^{8}}{\left (-1+2 x \right )^{2}}-\frac {18859855 \ln \left (-1+2 x \right )}{128}\) | \(57\) |
parallelrisch | \(-\frac {194400 x^{8}+1368576 x^{7}+4646160 x^{6}+10620432 x^{5}+20550420 x^{4}+75439420 \ln \left (x -\frac {1}{2}\right ) x^{2}+48657960 x^{3}-75439420 \ln \left (x -\frac {1}{2}\right ) x -113540058 x^{2}+18859855 \ln \left (x -\frac {1}{2}\right )+37645982 x}{128 \left (-1+2 x \right )^{2}}\) | \(71\) |
meijerg | \(\frac {288 x \left (2-2 x \right )}{\left (1-2 x \right )^{2}}+\frac {3552 x^{2}}{\left (1-2 x \right )^{2}}-\frac {4790 x \left (-18 x +6\right )}{3 \left (1-2 x \right )^{2}}-\frac {18859855 \ln \left (1-2 x \right )}{128}-\frac {3690 x \left (16 x^{2}-36 x +12\right )}{\left (1-2 x \right )^{2}}-\frac {11367 x \left (40 x^{3}+80 x^{2}-180 x +60\right )}{8 \left (1-2 x \right )^{2}}-\frac {23337 x \left (32 x^{4}+40 x^{3}+80 x^{2}-180 x +60\right )}{16 \left (1-2 x \right )^{2}}-\frac {215541 x \left (224 x^{5}+224 x^{4}+280 x^{3}+560 x^{2}-1260 x +420\right )}{1792 \left (1-2 x \right )^{2}}-\frac {9477 x \left (512 x^{6}+448 x^{5}+448 x^{4}+560 x^{3}+1120 x^{2}-2520 x +840\right )}{512 \left (1-2 x \right )^{2}}-\frac {405 x \left (1920 x^{7}+1536 x^{6}+1344 x^{5}+1344 x^{4}+1680 x^{3}+3360 x^{2}-7560 x +2520\right )}{512 \left (1-2 x \right )^{2}}\) | \(247\) |
-6075/16*x^6-48843/16*x^5-770067/64*x^4-1024389/32*x^3-17700255/256*x^2-39 980457/256*x+4*(12386759/256*x-84858543/4096)/(-1+2*x)^2-18859855/128*ln(- 1+2*x)
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=-\frac {1555200 \, x^{8} + 10948608 \, x^{7} + 37169280 \, x^{6} + 84963456 \, x^{5} + 164403360 \, x^{4} + 389263680 \, x^{3} - 568886292 \, x^{2} + 150878840 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 38266316 \, x + 84858543}{1024 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/1024*(1555200*x^8 + 10948608*x^7 + 37169280*x^6 + 84963456*x^5 + 164403 360*x^4 + 389263680*x^3 - 568886292*x^2 + 150878840*(4*x^2 - 4*x + 1)*log( 2*x - 1) - 38266316*x + 84858543)/(4*x^2 - 4*x + 1)
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=- \frac {6075 x^{6}}{16} - \frac {48843 x^{5}}{16} - \frac {770067 x^{4}}{64} - \frac {1024389 x^{3}}{32} - \frac {17700255 x^{2}}{256} - \frac {39980457 x}{256} - \frac {84858543 - 198188144 x}{4096 x^{2} - 4096 x + 1024} - \frac {18859855 \log {\left (2 x - 1 \right )}}{128} \]
-6075*x**6/16 - 48843*x**5/16 - 770067*x**4/64 - 1024389*x**3/32 - 1770025 5*x**2/256 - 39980457*x/256 - (84858543 - 198188144*x)/(4096*x**2 - 4096*x + 1024) - 18859855*log(2*x - 1)/128
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=-\frac {6075}{16} \, x^{6} - \frac {48843}{16} \, x^{5} - \frac {770067}{64} \, x^{4} - \frac {1024389}{32} \, x^{3} - \frac {17700255}{256} \, x^{2} - \frac {39980457}{256} \, x + \frac {184877 \, {\left (1072 \, x - 459\right )}}{1024 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {18859855}{128} \, \log \left (2 \, x - 1\right ) \]
-6075/16*x^6 - 48843/16*x^5 - 770067/64*x^4 - 1024389/32*x^3 - 17700255/25 6*x^2 - 39980457/256*x + 184877/1024*(1072*x - 459)/(4*x^2 - 4*x + 1) - 18 859855/128*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=-\frac {6075}{16} \, x^{6} - \frac {48843}{16} \, x^{5} - \frac {770067}{64} \, x^{4} - \frac {1024389}{32} \, x^{3} - \frac {17700255}{256} \, x^{2} - \frac {39980457}{256} \, x + \frac {184877 \, {\left (1072 \, x - 459\right )}}{1024 \, {\left (2 \, x - 1\right )}^{2}} - \frac {18859855}{128} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-6075/16*x^6 - 48843/16*x^5 - 770067/64*x^4 - 1024389/32*x^3 - 17700255/25 6*x^2 - 39980457/256*x + 184877/1024*(1072*x - 459)/(2*x - 1)^2 - 18859855 /128*log(abs(2*x - 1))
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^6 (3+5 x)^2}{(1-2 x)^3} \, dx=\frac {\frac {12386759\,x}{256}-\frac {84858543}{4096}}{x^2-x+\frac {1}{4}}-\frac {18859855\,\ln \left (x-\frac {1}{2}\right )}{128}-\frac {39980457\,x}{256}-\frac {17700255\,x^2}{256}-\frac {1024389\,x^3}{32}-\frac {770067\,x^4}{64}-\frac {48843\,x^5}{16}-\frac {6075\,x^6}{16} \]