Integrand size = 22, antiderivative size = 83 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {1096135733}{2048 (1-2 x)}+\frac {3690540955 x}{1024}+\frac {1312685491 x^2}{512}+\frac {551942075 x^3}{256}+\frac {220950207 x^4}{128}+\frac {379446471 x^5}{320}+\frac {20626947 x^6}{32}+\frac {28463805 x^7}{112}+\frac {127575 x^8}{2}+\frac {30375 x^9}{4}+\frac {298946109}{128} \log (1-2 x) \]
1096135733/2048/(1-2*x)+3690540955/1024*x+1312685491/512*x^2+551942075/256 *x^3+220950207/128*x^4+379446471/320*x^5+20626947/32*x^6+28463805/112*x^7+ 127575/2*x^8+30375/4*x^9+298946109/128*ln(1-2*x)
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {167338715917-669744799994 x+332899764960 x^2+213008156480 x^3+185355446080 x^4+162468222336 x^5+123787657728 x^6+74191887360 x^7+31861382400 x^8+8600256000 x^9+1088640000 x^{10}+167409821040 (-1+2 x) \log (1-2 x)}{71680 (-1+2 x)} \]
(167338715917 - 669744799994*x + 332899764960*x^2 + 213008156480*x^3 + 185 355446080*x^4 + 162468222336*x^5 + 123787657728*x^6 + 74191887360*x^7 + 31 861382400*x^8 + 8600256000*x^9 + 1088640000*x^10 + 167409821040*(-1 + 2*x) *Log[1 - 2*x])/(71680*(-1 + 2*x))
Time = 0.20 (sec) , antiderivative size = 83, 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.
| \(\displaystyle \int \frac {(3 x+2)^7 (5 x+3)^3}{(1-2 x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {273375 x^8}{4}+510300 x^7+\frac {28463805 x^6}{16}+\frac {61880841 x^5}{16}+\frac {379446471 x^4}{64}+\frac {220950207 x^3}{32}+\frac {1655826225 x^2}{256}+\frac {1312685491 x}{256}+\frac {298946109}{64 (2 x-1)}+\frac {1096135733}{1024 (2 x-1)^2}+\frac {3690540955}{1024}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {30375 x^9}{4}+\frac {127575 x^8}{2}+\frac {28463805 x^7}{112}+\frac {20626947 x^6}{32}+\frac {379446471 x^5}{320}+\frac {220950207 x^4}{128}+\frac {551942075 x^3}{256}+\frac {1312685491 x^2}{512}+\frac {3690540955 x}{1024}+\frac {1096135733}{2048 (1-2 x)}+\frac {298946109}{128} \log (1-2 x)\) |
1096135733/(2048*(1 - 2*x)) + (3690540955*x)/1024 + (1312685491*x^2)/512 + (551942075*x^3)/256 + (220950207*x^4)/128 + (379446471*x^5)/320 + (206269 47*x^6)/32 + (28463805*x^7)/112 + (127575*x^8)/2 + (30375*x^9)/4 + (298946 109*Log[1 - 2*x])/128
3.16.72.3.1 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 = 0.87 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {30375 x^{9}}{4}+\frac {127575 x^{8}}{2}+\frac {28463805 x^{7}}{112}+\frac {20626947 x^{6}}{32}+\frac {379446471 x^{5}}{320}+\frac {220950207 x^{4}}{128}+\frac {551942075 x^{3}}{256}+\frac {1312685491 x^{2}}{512}+\frac {3690540955 x}{1024}-\frac {1096135733}{4096 \left (x -\frac {1}{2}\right )}+\frac {298946109 \ln \left (-1+2 x \right )}{128}\) | \(60\) |
| default | \(\frac {30375 x^{9}}{4}+\frac {127575 x^{8}}{2}+\frac {28463805 x^{7}}{112}+\frac {20626947 x^{6}}{32}+\frac {379446471 x^{5}}{320}+\frac {220950207 x^{4}}{128}+\frac {551942075 x^{3}}{256}+\frac {1312685491 x^{2}}{512}+\frac {3690540955 x}{1024}+\frac {298946109 \ln \left (-1+2 x \right )}{128}-\frac {1096135733}{2048 \left (-1+2 x \right )}\) | \(62\) |
| norman | \(\frac {-\frac {299167293}{64} x +\frac {297231933}{64} x^{2}+\frac {95092927}{32} x^{3}+\frac {82747967}{32} x^{4}+\frac {181326141}{80} x^{5}+\frac {34538967}{20} x^{6}+\frac {14490603}{14} x^{7}+\frac {24891705}{56} x^{8}+\frac {479925}{4} x^{9}+\frac {30375}{2} x^{10}}{-1+2 x}+\frac {298946109 \ln \left (-1+2 x \right )}{128}\) | \(67\) |
| parallelrisch | \(\frac {68040000 x^{10}+537516000 x^{9}+1991336400 x^{8}+4636992960 x^{7}+7736728608 x^{6}+10154263896 x^{5}+11584715380 x^{4}+13313009780 x^{3}+20926227630 \ln \left (x -\frac {1}{2}\right ) x +20806235310 x^{2}-10463113815 \ln \left (x -\frac {1}{2}\right )-20941710510 x}{-4480+8960 x}\) | \(72\) |
| meijerg | \(\frac {30240 x}{1-2 x}+\frac {31128 x \left (-6 x +6\right )}{1-2 x}+\frac {298946109 \ln \left (1-2 x \right )}{128}+\frac {647577 x \left (-48 x^{4}-40 x^{3}-40 x^{2}-60 x +60\right )}{32 \left (1-2 x \right )}+\frac {716013 x \left (-448 x^{5}-336 x^{4}-280 x^{3}-280 x^{2}-420 x +420\right )}{320 \left (1-2 x \right )}+\frac {2954853 x \left (-1280 x^{6}-896 x^{5}-672 x^{4}-560 x^{3}-560 x^{2}-840 x +840\right )}{5120 \left (1-2 x \right )}+\frac {114291 x \left (-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{1792 \left (1-2 x \right )}+\frac {353565 x \left (-8960 x^{8}-5760 x^{7}-3840 x^{6}-2688 x^{5}-2016 x^{4}-1680 x^{3}-1680 x^{2}-2520 x +2520\right )}{28672 \left (1-2 x \right )}+\frac {34853 x \left (-40 x^{3}-40 x^{2}-60 x +60\right )}{2 \left (1-2 x \right )}+\frac {30375 x \left (-157696 x^{9}-98560 x^{8}-63360 x^{7}-42240 x^{6}-29568 x^{5}-22176 x^{4}-18480 x^{3}-18480 x^{2}-27720 x +27720\right )}{315392 \left (1-2 x \right )}+\frac {96445 x \left (-8 x^{2}-12 x +12\right )}{2 \left (1-2 x \right )}\) | \(335\) |
30375/4*x^9+127575/2*x^8+28463805/112*x^7+20626947/32*x^6+379446471/320*x^ 5+220950207/128*x^4+551942075/256*x^3+1312685491/512*x^2+3690540955/1024*x -1096135733/4096/(x-1/2)+298946109/128*ln(-1+2*x)
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {1088640000 \, x^{10} + 8600256000 \, x^{9} + 31861382400 \, x^{8} + 74191887360 \, x^{7} + 123787657728 \, x^{6} + 162468222336 \, x^{5} + 185355446080 \, x^{4} + 213008156480 \, x^{3} + 332899764960 \, x^{2} + 167409821040 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 258337866850 \, x - 38364750655}{71680 \, {\left (2 \, x - 1\right )}} \]
1/71680*(1088640000*x^10 + 8600256000*x^9 + 31861382400*x^8 + 74191887360* x^7 + 123787657728*x^6 + 162468222336*x^5 + 185355446080*x^4 + 21300815648 0*x^3 + 332899764960*x^2 + 167409821040*(2*x - 1)*log(2*x - 1) - 258337866 850*x - 38364750655)/(2*x - 1)
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {30375 x^{9}}{4} + \frac {127575 x^{8}}{2} + \frac {28463805 x^{7}}{112} + \frac {20626947 x^{6}}{32} + \frac {379446471 x^{5}}{320} + \frac {220950207 x^{4}}{128} + \frac {551942075 x^{3}}{256} + \frac {1312685491 x^{2}}{512} + \frac {3690540955 x}{1024} + \frac {298946109 \log {\left (2 x - 1 \right )}}{128} - \frac {1096135733}{4096 x - 2048} \]
30375*x**9/4 + 127575*x**8/2 + 28463805*x**7/112 + 20626947*x**6/32 + 3794 46471*x**5/320 + 220950207*x**4/128 + 551942075*x**3/256 + 1312685491*x**2 /512 + 3690540955*x/1024 + 298946109*log(2*x - 1)/128 - 1096135733/(4096*x - 2048)
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {30375}{4} \, x^{9} + \frac {127575}{2} \, x^{8} + \frac {28463805}{112} \, x^{7} + \frac {20626947}{32} \, x^{6} + \frac {379446471}{320} \, x^{5} + \frac {220950207}{128} \, x^{4} + \frac {551942075}{256} \, x^{3} + \frac {1312685491}{512} \, x^{2} + \frac {3690540955}{1024} \, x - \frac {1096135733}{2048 \, {\left (2 \, x - 1\right )}} + \frac {298946109}{128} \, \log \left (2 \, x - 1\right ) \]
30375/4*x^9 + 127575/2*x^8 + 28463805/112*x^7 + 20626947/32*x^6 + 37944647 1/320*x^5 + 220950207/128*x^4 + 551942075/256*x^3 + 1312685491/512*x^2 + 3 690540955/1024*x - 1096135733/2048/(2*x - 1) + 298946109/128*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.34 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {1}{71680} \, {\left (2 \, x - 1\right )}^{9} {\left (\frac {27428625}{2 \, x - 1} + \frac {323475525}{{\left (2 \, x - 1\right )}^{2}} + \frac {2307572820}{{\left (2 \, x - 1\right )}^{3}} + \frac {11110625442}{{\left (2 \, x - 1\right )}^{4}} + \frac {38208385530}{{\left (2 \, x - 1\right )}^{5}} + \frac {97321773850}{{\left (2 \, x - 1\right )}^{6}} + \frac {191214919700}{{\left (2 \, x - 1\right )}^{7}} + \frac {328704835305}{{\left (2 \, x - 1\right )}^{8}} + 1063125\right )} - \frac {1096135733}{2048 \, {\left (2 \, x - 1\right )}} - \frac {298946109}{128} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \]
1/71680*(2*x - 1)^9*(27428625/(2*x - 1) + 323475525/(2*x - 1)^2 + 23075728 20/(2*x - 1)^3 + 11110625442/(2*x - 1)^4 + 38208385530/(2*x - 1)^5 + 97321 773850/(2*x - 1)^6 + 191214919700/(2*x - 1)^7 + 328704835305/(2*x - 1)^8 + 1063125) - 1096135733/2048/(2*x - 1) - 298946109/128*log(1/2*abs(2*x - 1) /(2*x - 1)^2)
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^7 (3+5 x)^3}{(1-2 x)^2} \, dx=\frac {3690540955\,x}{1024}+\frac {298946109\,\ln \left (x-\frac {1}{2}\right )}{128}-\frac {1096135733}{4096\,\left (x-\frac {1}{2}\right )}+\frac {1312685491\,x^2}{512}+\frac {551942075\,x^3}{256}+\frac {220950207\,x^4}{128}+\frac {379446471\,x^5}{320}+\frac {20626947\,x^6}{32}+\frac {28463805\,x^7}{112}+\frac {127575\,x^8}{2}+\frac {30375\,x^9}{4} \]