Integrand size = 22, antiderivative size = 76 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {2662}{16807 (1-2 x)}+\frac {1}{1764 (2+3 x)^4}-\frac {101}{9261 (2+3 x)^3}+\frac {363}{4802 (2+3 x)^2}-\frac {3267}{16807 (2+3 x)}-\frac {14520 \log (1-2 x)}{117649}+\frac {14520 \log (2+3 x)}{117649} \]
2662/16807/(1-2*x)+1/1764/(2+3*x)^4-101/9261/(2+3*x)^3+363/4802/(2+3*x)^2- 3267/16807/(2+3*x)-14520/117649*ln(1-2*x)+14520/117649*ln(2+3*x)
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {2 \left (-\frac {7 \left (2287541+21109490 x+66510750 x^2+88209000 x^3+42340320 x^4\right )}{8 (-1+2 x) (2+3 x)^4}-196020 \log (1-2 x)+196020 \log (4+6 x)\right )}{3176523} \]
(2*((-7*(2287541 + 21109490*x + 66510750*x^2 + 88209000*x^3 + 42340320*x^4 ))/(8*(-1 + 2*x)*(2 + 3*x)^4) - 196020*Log[1 - 2*x] + 196020*Log[4 + 6*x]) )/3176523
Time = 0.20 (sec) , antiderivative size = 76, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^2 (3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {43560}{117649 (3 x+2)}+\frac {9801}{16807 (3 x+2)^2}-\frac {1089}{2401 (3 x+2)^3}+\frac {101}{1029 (3 x+2)^4}-\frac {1}{147 (3 x+2)^5}-\frac {29040}{117649 (2 x-1)}+\frac {5324}{16807 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2662}{16807 (1-2 x)}-\frac {3267}{16807 (3 x+2)}+\frac {363}{4802 (3 x+2)^2}-\frac {101}{9261 (3 x+2)^3}+\frac {1}{1764 (3 x+2)^4}-\frac {14520 \log (1-2 x)}{117649}+\frac {14520 \log (3 x+2)}{117649}\) |
2662/(16807*(1 - 2*x)) + 1/(1764*(2 + 3*x)^4) - 101/(9261*(2 + 3*x)^3) + 3 63/(4802*(2 + 3*x)^2) - 3267/(16807*(2 + 3*x)) - (14520*Log[1 - 2*x])/1176 49 + (14520*Log[2 + 3*x])/117649
3.16.84.3.1 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 = 2.68 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
norman | \(\frac {-\frac {11085125}{302526} x^{2}-\frac {10554745}{907578} x -\frac {816750}{16807} x^{3}-\frac {392040}{16807} x^{4}-\frac {2287541}{1815156}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4}}-\frac {14520 \ln \left (-1+2 x \right )}{117649}+\frac {14520 \ln \left (2+3 x \right )}{117649}\) | \(53\) |
risch | \(\frac {-\frac {11085125}{302526} x^{2}-\frac {10554745}{907578} x -\frac {816750}{16807} x^{3}-\frac {392040}{16807} x^{4}-\frac {2287541}{1815156}}{\left (-1+2 x \right ) \left (2+3 x \right )^{4}}-\frac {14520 \ln \left (-1+2 x \right )}{117649}+\frac {14520 \ln \left (2+3 x \right )}{117649}\) | \(54\) |
default | \(-\frac {2662}{16807 \left (-1+2 x \right )}-\frac {14520 \ln \left (-1+2 x \right )}{117649}+\frac {1}{1764 \left (2+3 x \right )^{4}}-\frac {101}{9261 \left (2+3 x \right )^{3}}+\frac {363}{4802 \left (2+3 x \right )^{2}}-\frac {3267}{16807 \left (2+3 x \right )}+\frac {14520 \ln \left (2+3 x \right )}{117649}\) | \(63\) |
parallelrisch | \(\frac {150543360 \ln \left (\frac {2}{3}+x \right ) x^{5}-150543360 \ln \left (x -\frac {1}{2}\right ) x^{5}+326177280 \ln \left (\frac {2}{3}+x \right ) x^{4}-326177280 \ln \left (x -\frac {1}{2}\right ) x^{4}-96076722 x^{5}+200724480 \ln \left (\frac {2}{3}+x \right ) x^{3}-200724480 \ln \left (x -\frac {1}{2}\right ) x^{3}-383800151 x^{4}-22302720 \ln \left (\frac {2}{3}+x \right ) x^{2}+22302720 \ln \left (x -\frac {1}{2}\right ) x^{2}-494006296 x^{3}-59473920 \ln \left (\frac {2}{3}+x \right ) x +59473920 \ln \left (x -\frac {1}{2}\right ) x -261662856 x^{2}-14868480 \ln \left (\frac {2}{3}+x \right )+14868480 \ln \left (x -\frac {1}{2}\right )-49609056 x}{7529536 \left (-1+2 x \right ) \left (2+3 x \right )^{4}}\) | \(139\) |
(-11085125/302526*x^2-10554745/907578*x-816750/16807*x^3-392040/16807*x^4- 2287541/1815156)/(-1+2*x)/(2+3*x)^4-14520/117649*ln(-1+2*x)+14520/117649*l n(2+3*x)
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.51 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=-\frac {296382240 \, x^{4} + 617463000 \, x^{3} + 465575250 \, x^{2} - 1568160 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 1568160 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) + 147766430 \, x + 16012787}{12706092 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
-1/12706092*(296382240*x^4 + 617463000*x^3 + 465575250*x^2 - 1568160*(162* x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(3*x + 2) + 1568160*(162* x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) + 147766430*x + 16012787)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {- 42340320 x^{4} - 88209000 x^{3} - 66510750 x^{2} - 21109490 x - 2287541}{294055272 x^{5} + 637119756 x^{4} + 392073696 x^{3} - 43563744 x^{2} - 116169984 x - 29042496} - \frac {14520 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {14520 \log {\left (x + \frac {2}{3} \right )}}{117649} \]
(-42340320*x**4 - 88209000*x**3 - 66510750*x**2 - 21109490*x - 2287541)/(2 94055272*x**5 + 637119756*x**4 + 392073696*x**3 - 43563744*x**2 - 11616998 4*x - 29042496) - 14520*log(x - 1/2)/117649 + 14520*log(x + 2/3)/117649
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=-\frac {42340320 \, x^{4} + 88209000 \, x^{3} + 66510750 \, x^{2} + 21109490 \, x + 2287541}{1815156 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {14520}{117649} \, \log \left (3 \, x + 2\right ) - \frac {14520}{117649} \, \log \left (2 \, x - 1\right ) \]
-1/1815156*(42340320*x^4 + 88209000*x^3 + 66510750*x^2 + 21109490*x + 2287 541)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16) + 14520/117649*log (3*x + 2) - 14520/117649*log(2*x - 1)
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=-\frac {3267}{16807 \, {\left (3 \, x + 2\right )}} + \frac {15972}{117649 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} + \frac {363}{4802 \, {\left (3 \, x + 2\right )}^{2}} - \frac {101}{9261 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{1764 \, {\left (3 \, x + 2\right )}^{4}} - \frac {14520}{117649} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]
-3267/16807/(3*x + 2) + 15972/117649/(7/(3*x + 2) - 2) + 363/4802/(3*x + 2 )^2 - 101/9261/(3*x + 2)^3 + 1/1764/(3*x + 2)^4 - 14520/117649*log(abs(-7/ (3*x + 2) + 2))
Time = 1.62 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx=\frac {29040\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {\frac {2420\,x^4}{16807}+\frac {15125\,x^3}{50421}+\frac {11085125\,x^2}{49009212}+\frac {10554745\,x}{147027636}+\frac {2287541}{294055272}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \]