Integrand size = 22, antiderivative size = 87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {1331}{16807 (1-2 x)^2}+\frac {14520}{117649 (1-2 x)}+\frac {1}{4116 (2+3 x)^4}-\frac {11}{2401 (2+3 x)^3}+\frac {1023}{33614 (2+3 x)^2}-\frac {7755}{117649 (2+3 x)}-\frac {59070 \log (1-2 x)}{823543}+\frac {59070 \log (2+3 x)}{823543} \]
1331/16807/(1-2*x)^2+14520/117649/(1-2*x)+1/4116/(2+3*x)^4-11/2401/(2+3*x) ^3+1023/33614/(2+3*x)^2-7755/117649/(2+3*x)-59070/823543*ln(1-2*x)+59070/8 23543*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {-\frac {7 \left (-3991495-21371408 x-32767930 x^2+8860500 x^3+60605820 x^4+38277360 x^5\right )}{4 (1-2 x)^2 (2+3 x)^4}-177210 \log (1-2 x)+177210 \log (4+6 x)}{2470629} \]
((-7*(-3991495 - 21371408*x - 32767930*x^2 + 8860500*x^3 + 60605820*x^4 + 38277360*x^5))/(4*(1 - 2*x)^2*(2 + 3*x)^4) - 177210*Log[1 - 2*x] + 177210* Log[4 + 6*x])/2470629
Time = 0.22 (sec) , antiderivative size = 87, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^3 (3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {177210}{823543 (3 x+2)}+\frac {23265}{117649 (3 x+2)^2}-\frac {3069}{16807 (3 x+2)^3}+\frac {99}{2401 (3 x+2)^4}-\frac {1}{343 (3 x+2)^5}-\frac {118140}{823543 (2 x-1)}+\frac {29040}{117649 (2 x-1)^2}-\frac {5324}{16807 (2 x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {14520}{117649 (1-2 x)}-\frac {7755}{117649 (3 x+2)}+\frac {1331}{16807 (1-2 x)^2}+\frac {1023}{33614 (3 x+2)^2}-\frac {11}{2401 (3 x+2)^3}+\frac {1}{4116 (3 x+2)^4}-\frac {59070 \log (1-2 x)}{823543}+\frac {59070 \log (3 x+2)}{823543}\) |
1331/(16807*(1 - 2*x)^2) + 14520/(117649*(1 - 2*x)) + 1/(4116*(2 + 3*x)^4) - 11/(2401*(2 + 3*x)^3) + 1023/(33614*(2 + 3*x)^2) - 7755/(117649*(2 + 3* x)) - (59070*Log[1 - 2*x])/823543 + (59070*Log[2 + 3*x])/823543
3.17.70.3.1 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 = 0.90 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67
method | result | size |
norman | \(\frac {-\frac {5050485}{117649} x^{4}-\frac {3189780}{117649} x^{5}-\frac {738375}{117649} x^{3}+\frac {5342852}{352947} x +\frac {16383965}{705894} x^{2}+\frac {3991495}{1411788}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}-\frac {59070 \ln \left (-1+2 x \right )}{823543}+\frac {59070 \ln \left (2+3 x \right )}{823543}\) | \(58\) |
risch | \(\frac {-\frac {5050485}{117649} x^{4}-\frac {3189780}{117649} x^{5}-\frac {738375}{117649} x^{3}+\frac {5342852}{352947} x +\frac {16383965}{705894} x^{2}+\frac {3991495}{1411788}}{\left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}-\frac {59070 \ln \left (-1+2 x \right )}{823543}+\frac {59070 \ln \left (2+3 x \right )}{823543}\) | \(59\) |
default | \(\frac {1331}{16807 \left (-1+2 x \right )^{2}}-\frac {14520}{117649 \left (-1+2 x \right )}-\frac {59070 \ln \left (-1+2 x \right )}{823543}+\frac {1}{4116 \left (2+3 x \right )^{4}}-\frac {11}{2401 \left (2+3 x \right )^{3}}+\frac {1023}{33614 \left (2+3 x \right )^{2}}-\frac {7755}{117649 \left (2+3 x \right )}+\frac {59070 \ln \left (2+3 x \right )}{823543}\) | \(72\) |
parallelrisch | \(\frac {499834272 x -998046720 \ln \left (\frac {2}{3}+x \right ) x^{3}-393169920 \ln \left (\frac {2}{3}+x \right ) x^{2}+120975360 \ln \left (\frac {2}{3}+x \right ) x -6458305140 x^{5}-3017570220 x^{6}+2127968920 x^{3}-3017009835 x^{4}+2191938840 x^{2}-306218880 \ln \left (x -\frac {1}{2}\right ) x^{4}+306218880 \ln \left (\frac {2}{3}+x \right ) x^{4}+60487680 \ln \left (\frac {2}{3}+x \right )+998046720 \ln \left (x -\frac {1}{2}\right ) x^{3}+393169920 \ln \left (x -\frac {1}{2}\right ) x^{2}-120975360 \ln \left (x -\frac {1}{2}\right ) x +2041459200 \ln \left (\frac {2}{3}+x \right ) x^{5}+1224875520 \ln \left (\frac {2}{3}+x \right ) x^{6}-60487680 \ln \left (x -\frac {1}{2}\right )-1224875520 \ln \left (x -\frac {1}{2}\right ) x^{6}-2041459200 \ln \left (x -\frac {1}{2}\right ) x^{5}}{52706752 \left (-1+2 x \right )^{2} \left (2+3 x \right )^{4}}\) | \(162\) |
(-5050485/117649*x^4-3189780/117649*x^5-738375/117649*x^3+5342852/352947*x +16383965/705894*x^2+3991495/1411788)/(-1+2*x)^2/(2+3*x)^4-59070/823543*ln (-1+2*x)+59070/823543*ln(2+3*x)
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {267941520 \, x^{5} + 424240740 \, x^{4} + 62023500 \, x^{3} - 229375510 \, x^{2} - 708840 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 708840 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 149599856 \, x - 27940465}{9882516 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
-1/9882516*(267941520*x^5 + 424240740*x^4 + 62023500*x^3 - 229375510*x^2 - 708840*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log(3 *x + 2) + 708840*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log(2*x - 1) - 149599856*x - 27940465)/(324*x^6 + 540*x^5 + 81*x^4 - 2 64*x^3 - 104*x^2 + 32*x + 16)
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=- \frac {38277360 x^{5} + 60605820 x^{4} + 8860500 x^{3} - 32767930 x^{2} - 21371408 x - 3991495}{457419312 x^{6} + 762365520 x^{5} + 114354828 x^{4} - 372712032 x^{3} - 146825952 x^{2} + 45177216 x + 22588608} - \frac {59070 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {59070 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
-(38277360*x**5 + 60605820*x**4 + 8860500*x**3 - 32767930*x**2 - 21371408* x - 3991495)/(457419312*x**6 + 762365520*x**5 + 114354828*x**4 - 372712032 *x**3 - 146825952*x**2 + 45177216*x + 22588608) - 59070*log(x - 1/2)/82354 3 + 59070*log(x + 2/3)/823543
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {38277360 \, x^{5} + 60605820 \, x^{4} + 8860500 \, x^{3} - 32767930 \, x^{2} - 21371408 \, x - 3991495}{1411788 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac {59070}{823543} \, \log \left (3 \, x + 2\right ) - \frac {59070}{823543} \, \log \left (2 \, x - 1\right ) \]
-1/1411788*(38277360*x^5 + 60605820*x^4 + 8860500*x^3 - 32767930*x^2 - 213 71408*x - 3991495)/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16) + 59070/823543*log(3*x + 2) - 59070/823543*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=-\frac {7755}{117649 \, {\left (3 \, x + 2\right )}} + \frac {4356 \, {\left (\frac {217}{3 \, x + 2} - 51\right )}}{823543 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} + \frac {1023}{33614 \, {\left (3 \, x + 2\right )}^{2}} - \frac {11}{2401 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{4116 \, {\left (3 \, x + 2\right )}^{4}} - \frac {59070}{823543} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
-7755/117649/(3*x + 2) + 4356/823543*(217/(3*x + 2) - 51)/(7/(3*x + 2) - 2 )^2 + 1023/33614/(3*x + 2)^2 - 11/2401/(3*x + 2)^3 + 1/4116/(3*x + 2)^4 - 59070/823543*log(abs(-7/(3*x + 2) + 2))
Time = 1.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^5} \, dx=\frac {118140\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}+\frac {-\frac {9845\,x^5}{117649}-\frac {187055\,x^4}{1411788}-\frac {246125\,x^3}{12706092}+\frac {16383965\,x^2}{228709656}+\frac {1335713\,x}{28588707}+\frac {3991495}{457419312}}{x^6+\frac {5\,x^5}{3}+\frac {x^4}{4}-\frac {22\,x^3}{27}-\frac {26\,x^2}{81}+\frac {8\,x}{81}+\frac {4}{81}} \]