Integrand size = 20, antiderivative size = 98 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=\frac {88}{117649 (1-2 x)^2}+\frac {2608}{823543 (1-2 x)}+\frac {3}{1715 (2+3 x)^5}-\frac {87}{9604 (2+3 x)^4}-\frac {186}{16807 (2+3 x)^3}-\frac {1140}{117649 (2+3 x)^2}-\frac {7680}{823543 (2+3 x)}-\frac {3312 \log (1-2 x)}{823543}+\frac {3312 \log (2+3 x)}{823543} \]
88/117649/(1-2*x)^2+2608/823543/(1-2*x)+3/1715/(2+3*x)^5-87/9604/(2+3*x)^4 -186/16807/(2+3*x)^3-1140/117649/(2+3*x)^2-7680/823543/(2+3*x)-3312/823543 *ln(1-2*x)+3312/823543*ln(2+3*x)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=\frac {3 \left (-\frac {7 \left (381394-1134751 x-7360644 x^2-5468940 x^3+13811040 x^4+24144480 x^5+10730880 x^6\right )}{3 (1-2 x)^2 (2+3 x)^5}-22080 \log (3-6 x)+22080 \log (2+3 x)\right )}{16470860} \]
(3*((-7*(381394 - 1134751*x - 7360644*x^2 - 5468940*x^3 + 13811040*x^4 + 2 4144480*x^5 + 10730880*x^6))/(3*(1 - 2*x)^2*(2 + 3*x)^5) - 22080*Log[3 - 6 *x] + 22080*Log[2 + 3*x]))/16470860
Time = 0.23 (sec) , antiderivative size = 98, 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.
\(\displaystyle \int \frac {5 x+3}{(1-2 x)^3 (3 x+2)^6} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {9936}{823543 (3 x+2)}+\frac {23040}{823543 (3 x+2)^2}+\frac {6840}{117649 (3 x+2)^3}+\frac {1674}{16807 (3 x+2)^4}+\frac {261}{2401 (3 x+2)^5}-\frac {9}{343 (3 x+2)^6}-\frac {6624}{823543 (2 x-1)}+\frac {5216}{823543 (2 x-1)^2}-\frac {352}{117649 (2 x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2608}{823543 (1-2 x)}-\frac {7680}{823543 (3 x+2)}+\frac {88}{117649 (1-2 x)^2}-\frac {1140}{117649 (3 x+2)^2}-\frac {186}{16807 (3 x+2)^3}-\frac {87}{9604 (3 x+2)^4}+\frac {3}{1715 (3 x+2)^5}-\frac {3312 \log (1-2 x)}{823543}+\frac {3312 \log (3 x+2)}{823543}\) |
88/(117649*(1 - 2*x)^2) + 2608/(823543*(1 - 2*x)) + 3/(1715*(2 + 3*x)^5) - 87/(9604*(2 + 3*x)^4) - 186/(16807*(2 + 3*x)^3) - 1140/(117649*(2 + 3*x)^ 2) - 7680/(823543*(2 + 3*x)) - (3312*Log[1 - 2*x])/823543 + (3312*Log[2 + 3*x])/823543
3.17.44.3.1 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 = 2.73 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.64
method | result | size |
norman | \(\frac {-\frac {1207224}{117649} x^{5}-\frac {690552}{117649} x^{4}-\frac {536544}{117649} x^{6}+\frac {273447}{117649} x^{3}+\frac {1134751}{2352980} x +\frac {1840161}{588245} x^{2}-\frac {190697}{1176490}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{5}}-\frac {3312 \ln \left (-1+2 x \right )}{823543}+\frac {3312 \ln \left (2+3 x \right )}{823543}\) | \(63\) |
risch | \(\frac {-\frac {1207224}{117649} x^{5}-\frac {690552}{117649} x^{4}-\frac {536544}{117649} x^{6}+\frac {273447}{117649} x^{3}+\frac {1134751}{2352980} x +\frac {1840161}{588245} x^{2}-\frac {190697}{1176490}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{5}}-\frac {3312 \ln \left (-1+2 x \right )}{823543}+\frac {3312 \ln \left (2+3 x \right )}{823543}\) | \(64\) |
default | \(\frac {88}{117649 \left (-1+2 x \right )^{2}}-\frac {2608}{823543 \left (-1+2 x \right )}-\frac {3312 \ln \left (-1+2 x \right )}{823543}+\frac {3}{1715 \left (2+3 x \right )^{5}}-\frac {87}{9604 \left (2+3 x \right )^{4}}-\frac {186}{16807 \left (2+3 x \right )^{3}}-\frac {1140}{117649 \left (2+3 x \right )^{2}}-\frac {7680}{823543 \left (2+3 x \right )}+\frac {3312 \ln \left (2+3 x \right )}{823543}\) | \(81\) |
parallelrisch | \(\frac {276598560 x -890265600 \ln \left (\frac {2}{3}+x \right ) x^{3}-118702080 \ln \left (\frac {2}{3}+x \right ) x^{2}+118702080 \ln \left (\frac {2}{3}+x \right ) x -938136843 x^{5}+1825647012 x^{6}+1297502388 x^{7}-508777080 x^{3}-2387810250 x^{4}+674885680 x^{2}+667699200 \ln \left (x -\frac {1}{2}\right ) x^{4}-667699200 \ln \left (\frac {2}{3}+x \right ) x^{4}+33914880 \ln \left (\frac {2}{3}+x \right )+890265600 \ln \left (x -\frac {1}{2}\right ) x^{3}+1030164480 \ln \left (\frac {2}{3}+x \right ) x^{7}+118702080 \ln \left (x -\frac {1}{2}\right ) x^{2}-118702080 \ln \left (x -\frac {1}{2}\right ) x +1402168320 \ln \left (\frac {2}{3}+x \right ) x^{5}+2403717120 \ln \left (\frac {2}{3}+x \right ) x^{6}-33914880 \ln \left (x -\frac {1}{2}\right )-1030164480 \ln \left (x -\frac {1}{2}\right ) x^{7}-2403717120 \ln \left (x -\frac {1}{2}\right ) x^{6}-1402168320 \ln \left (x -\frac {1}{2}\right ) x^{5}}{263533760 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{5}}\) | \(185\) |
(-1207224/117649*x^5-690552/117649*x^4-536544/117649*x^6+273447/117649*x^3 +1134751/2352980*x+1840161/588245*x^2-190697/1176490)/(-1+2*x)^2/(2+3*x)^5 -3312/823543*ln(-1+2*x)+3312/823543*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.58 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=-\frac {75116160 \, x^{6} + 169011360 \, x^{5} + 96677280 \, x^{4} - 38282580 \, x^{3} - 51524508 \, x^{2} - 66240 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 66240 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (2 \, x - 1\right ) - 7943257 \, x + 2669758}{16470860 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} \]
-1/16470860*(75116160*x^6 + 169011360*x^5 + 96677280*x^4 - 38282580*x^3 - 51524508*x^2 - 66240*(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(3*x + 2) + 66240*(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log(2*x - 1) - 7943257*x + 266 9758)/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=- \frac {10730880 x^{6} + 24144480 x^{5} + 13811040 x^{4} - 5468940 x^{3} - 7360644 x^{2} - 1134751 x + 381394}{2287096560 x^{7} + 5336558640 x^{6} + 3112992540 x^{5} - 1482377400 x^{4} - 1976503200 x^{3} - 263533760 x^{2} + 263533760 x + 75295360} - \frac {3312 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {3312 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
-(10730880*x**6 + 24144480*x**5 + 13811040*x**4 - 5468940*x**3 - 7360644*x **2 - 1134751*x + 381394)/(2287096560*x**7 + 5336558640*x**6 + 3112992540* x**5 - 1482377400*x**4 - 1976503200*x**3 - 263533760*x**2 + 263533760*x + 75295360) - 3312*log(x - 1/2)/823543 + 3312*log(x + 2/3)/823543
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=-\frac {10730880 \, x^{6} + 24144480 \, x^{5} + 13811040 \, x^{4} - 5468940 \, x^{3} - 7360644 \, x^{2} - 1134751 \, x + 381394}{2352980 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} + \frac {3312}{823543} \, \log \left (3 \, x + 2\right ) - \frac {3312}{823543} \, \log \left (2 \, x - 1\right ) \]
-1/2352980*(10730880*x^6 + 24144480*x^5 + 13811040*x^4 - 5468940*x^3 - 736 0644*x^2 - 1134751*x + 381394)/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32) + 3312/823543*log(3*x + 2) - 3312/823543*l og(2*x - 1)
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.66 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=-\frac {10730880 \, x^{6} + 24144480 \, x^{5} + 13811040 \, x^{4} - 5468940 \, x^{3} - 7360644 \, x^{2} - 1134751 \, x + 381394}{2352980 \, {\left (3 \, x + 2\right )}^{5} {\left (2 \, x - 1\right )}^{2}} + \frac {3312}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3312}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
-1/2352980*(10730880*x^6 + 24144480*x^5 + 13811040*x^4 - 5468940*x^3 - 736 0644*x^2 - 1134751*x + 381394)/((3*x + 2)^5*(2*x - 1)^2) + 3312/823543*log (abs(3*x + 2)) - 3312/823543*log(abs(2*x - 1))
Time = 1.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^6} \, dx=\frac {6624\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {552\,x^6}{117649}+\frac {1242\,x^5}{117649}+\frac {6394\,x^4}{1058841}-\frac {30383\,x^3}{12706092}-\frac {613387\,x^2}{190591380}-\frac {1134751\,x}{2287096560}+\frac {190697}{1143548280}}{x^7+\frac {7\,x^6}{3}+\frac {49\,x^5}{36}-\frac {35\,x^4}{54}-\frac {70\,x^3}{81}-\frac {28\,x^2}{243}+\frac {28\,x}{243}+\frac {8}{243}} \]