Integrand size = 22, antiderivative size = 72 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=-\frac {155706083 x}{512}-\frac {149512931 x^2}{512}-\frac {130251491 x^3}{384}-\frac {95317731 x^4}{256}-\frac {54600291 x^5}{160}-\frac {7656993 x^6}{32}-\frac {6596235 x^7}{56}-\frac {1148175 x^8}{32}-\frac {10125 x^9}{2}-\frac {156590819 \log (1-2 x)}{1024} \]
-155706083/512*x-149512931/512*x^2-130251491/384*x^3-95317731/256*x^4-5460 0291/160*x^5-7656993/32*x^6-6596235/56*x^7-1148175/32*x^8-10125/2*x^9-1565 90819/1024*ln(1-2*x)
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=\frac {263385079253}{860160}-\frac {155706083 x}{512}-\frac {149512931 x^2}{512}-\frac {130251491 x^3}{384}-\frac {95317731 x^4}{256}-\frac {54600291 x^5}{160}-\frac {7656993 x^6}{32}-\frac {6596235 x^7}{56}-\frac {1148175 x^8}{32}-\frac {10125 x^9}{2}-\frac {156590819 \log (1-2 x)}{1024} \]
263385079253/860160 - (155706083*x)/512 - (149512931*x^2)/512 - (130251491 *x^3)/384 - (95317731*x^4)/256 - (54600291*x^5)/160 - (7656993*x^6)/32 - ( 6596235*x^7)/56 - (1148175*x^8)/32 - (10125*x^9)/2 - (156590819*Log[1 - 2* x])/1024
Time = 0.19 (sec) , antiderivative size = 72, 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.
| \(\displaystyle \int \frac {(3 x+2)^6 (5 x+3)^3}{1-2 x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {91125 x^8}{2}-\frac {1148175 x^7}{4}-\frac {6596235 x^6}{8}-\frac {22970979 x^5}{16}-\frac {54600291 x^4}{32}-\frac {95317731 x^3}{64}-\frac {130251491 x^2}{128}-\frac {149512931 x}{256}-\frac {156590819}{512 (2 x-1)}-\frac {155706083}{512}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10125 x^9}{2}-\frac {1148175 x^8}{32}-\frac {6596235 x^7}{56}-\frac {7656993 x^6}{32}-\frac {54600291 x^5}{160}-\frac {95317731 x^4}{256}-\frac {130251491 x^3}{384}-\frac {149512931 x^2}{512}-\frac {155706083 x}{512}-\frac {156590819 \log (1-2 x)}{1024}\) |
(-155706083*x)/512 - (149512931*x^2)/512 - (130251491*x^3)/384 - (95317731 *x^4)/256 - (54600291*x^5)/160 - (7656993*x^6)/32 - (6596235*x^7)/56 - (11 48175*x^8)/32 - (10125*x^9)/2 - (156590819*Log[1 - 2*x])/1024
3.15.69.3.1 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 = 0.85 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(-\frac {10125 x^{9}}{2}-\frac {1148175 x^{8}}{32}-\frac {6596235 x^{7}}{56}-\frac {7656993 x^{6}}{32}-\frac {54600291 x^{5}}{160}-\frac {95317731 x^{4}}{256}-\frac {130251491 x^{3}}{384}-\frac {149512931 x^{2}}{512}-\frac {155706083 x}{512}-\frac {156590819 \ln \left (x -\frac {1}{2}\right )}{1024}\) | \(51\) |
| default | \(-\frac {10125 x^{9}}{2}-\frac {1148175 x^{8}}{32}-\frac {6596235 x^{7}}{56}-\frac {7656993 x^{6}}{32}-\frac {54600291 x^{5}}{160}-\frac {95317731 x^{4}}{256}-\frac {130251491 x^{3}}{384}-\frac {149512931 x^{2}}{512}-\frac {155706083 x}{512}-\frac {156590819 \ln \left (-1+2 x \right )}{1024}\) | \(53\) |
| norman | \(-\frac {10125 x^{9}}{2}-\frac {1148175 x^{8}}{32}-\frac {6596235 x^{7}}{56}-\frac {7656993 x^{6}}{32}-\frac {54600291 x^{5}}{160}-\frac {95317731 x^{4}}{256}-\frac {130251491 x^{3}}{384}-\frac {149512931 x^{2}}{512}-\frac {155706083 x}{512}-\frac {156590819 \ln \left (-1+2 x \right )}{1024}\) | \(53\) |
| risch | \(-\frac {10125 x^{9}}{2}-\frac {1148175 x^{8}}{32}-\frac {6596235 x^{7}}{56}-\frac {7656993 x^{6}}{32}-\frac {54600291 x^{5}}{160}-\frac {95317731 x^{4}}{256}-\frac {130251491 x^{3}}{384}-\frac {149512931 x^{2}}{512}-\frac {155706083 x}{512}-\frac {156590819 \ln \left (-1+2 x \right )}{1024}\) | \(53\) |
| meijerg | \(-\frac {156590819 \ln \left (1-2 x \right )}{1024}-6270 x \left (6 x +6\right )-\frac {34115 x \left (16 x^{2}+12 x +12\right )}{6}-\frac {21207 x \left (120 x^{3}+80 x^{2}+60 x +60\right )}{16}-\frac {90801 x \left (7680 x^{6}+4480 x^{5}+2688 x^{4}+1680 x^{3}+1120 x^{2}+840 x +840\right )}{7168}-\frac {11745 x \left (40320 x^{7}+23040 x^{6}+13440 x^{5}+8064 x^{4}+5040 x^{3}+3360 x^{2}+2520 x +2520\right )}{14336}-\frac {2025 x \left (71680 x^{8}+40320 x^{7}+23040 x^{6}+13440 x^{5}+8064 x^{4}+5040 x^{3}+3360 x^{2}+2520 x +2520\right )}{28672}-\frac {5148 x \left (192 x^{4}+120 x^{3}+80 x^{2}+60 x +60\right )}{5}-\frac {682281 x \left (2240 x^{5}+1344 x^{4}+840 x^{3}+560 x^{2}+420 x +420\right )}{8960}-12096 x\) | \(217\) |
-10125/2*x^9-1148175/32*x^8-6596235/56*x^7-7656993/32*x^6-54600291/160*x^5 -95317731/256*x^4-130251491/384*x^3-149512931/512*x^2-155706083/512*x-1565 90819/1024*ln(x-1/2)
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=-\frac {10125}{2} \, x^{9} - \frac {1148175}{32} \, x^{8} - \frac {6596235}{56} \, x^{7} - \frac {7656993}{32} \, x^{6} - \frac {54600291}{160} \, x^{5} - \frac {95317731}{256} \, x^{4} - \frac {130251491}{384} \, x^{3} - \frac {149512931}{512} \, x^{2} - \frac {155706083}{512} \, x - \frac {156590819}{1024} \, \log \left (2 \, x - 1\right ) \]
-10125/2*x^9 - 1148175/32*x^8 - 6596235/56*x^7 - 7656993/32*x^6 - 54600291 /160*x^5 - 95317731/256*x^4 - 130251491/384*x^3 - 149512931/512*x^2 - 1557 06083/512*x - 156590819/1024*log(2*x - 1)
Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=- \frac {10125 x^{9}}{2} - \frac {1148175 x^{8}}{32} - \frac {6596235 x^{7}}{56} - \frac {7656993 x^{6}}{32} - \frac {54600291 x^{5}}{160} - \frac {95317731 x^{4}}{256} - \frac {130251491 x^{3}}{384} - \frac {149512931 x^{2}}{512} - \frac {155706083 x}{512} - \frac {156590819 \log {\left (2 x - 1 \right )}}{1024} \]
-10125*x**9/2 - 1148175*x**8/32 - 6596235*x**7/56 - 7656993*x**6/32 - 5460 0291*x**5/160 - 95317731*x**4/256 - 130251491*x**3/384 - 149512931*x**2/51 2 - 155706083*x/512 - 156590819*log(2*x - 1)/1024
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=-\frac {10125}{2} \, x^{9} - \frac {1148175}{32} \, x^{8} - \frac {6596235}{56} \, x^{7} - \frac {7656993}{32} \, x^{6} - \frac {54600291}{160} \, x^{5} - \frac {95317731}{256} \, x^{4} - \frac {130251491}{384} \, x^{3} - \frac {149512931}{512} \, x^{2} - \frac {155706083}{512} \, x - \frac {156590819}{1024} \, \log \left (2 \, x - 1\right ) \]
-10125/2*x^9 - 1148175/32*x^8 - 6596235/56*x^7 - 7656993/32*x^6 - 54600291 /160*x^5 - 95317731/256*x^4 - 130251491/384*x^3 - 149512931/512*x^2 - 1557 06083/512*x - 156590819/1024*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=-\frac {10125}{2} \, x^{9} - \frac {1148175}{32} \, x^{8} - \frac {6596235}{56} \, x^{7} - \frac {7656993}{32} \, x^{6} - \frac {54600291}{160} \, x^{5} - \frac {95317731}{256} \, x^{4} - \frac {130251491}{384} \, x^{3} - \frac {149512931}{512} \, x^{2} - \frac {155706083}{512} \, x - \frac {156590819}{1024} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-10125/2*x^9 - 1148175/32*x^8 - 6596235/56*x^7 - 7656993/32*x^6 - 54600291 /160*x^5 - 95317731/256*x^4 - 130251491/384*x^3 - 149512931/512*x^2 - 1557 06083/512*x - 156590819/1024*log(abs(2*x - 1))
Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^6 (3+5 x)^3}{1-2 x} \, dx=-\frac {155706083\,x}{512}-\frac {156590819\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {149512931\,x^2}{512}-\frac {130251491\,x^3}{384}-\frac {95317731\,x^4}{256}-\frac {54600291\,x^5}{160}-\frac {7656993\,x^6}{32}-\frac {6596235\,x^7}{56}-\frac {1148175\,x^8}{32}-\frac {10125\,x^9}{2} \]