Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=\frac {9}{7 (2+3 x)^3}+\frac {2889}{98 (2+3 x)^2}+\frac {204228}{343 (2+3 x)}-\frac {625}{22 (3+5 x)^2}+\frac {81250}{121 (3+5 x)}-\frac {64 \log (1-2 x)}{3195731}-\frac {11984706 \log (2+3 x)}{2401}+\frac {6643750 \log (3+5 x)}{1331} \]
9/7/(2+3*x)^3+2889/98/(2+3*x)^2+204228/343/(2+3*x)-625/22/(3+5*x)^2+81250/ 121/(3+5*x)-64/3195731*ln(1-2*x)-11984706/2401*ln(2+3*x)+6643750/1331*ln(3 +5*x)
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=\frac {9}{7 (2+3 x)^3}+\frac {2889}{98 (2+3 x)^2}+\frac {204228}{343 (2+3 x)}-\frac {625}{22 (3+5 x)^2}+\frac {81250}{363+605 x}-\frac {64 \log (1-2 x)}{3195731}-\frac {11984706 \log (4+6 x)}{2401}+\frac {6643750 \log (6+10 x)}{1331} \]
9/(7*(2 + 3*x)^3) + 2889/(98*(2 + 3*x)^2) + 204228/(343*(2 + 3*x)) - 625/( 22*(3 + 5*x)^2) + 81250/(363 + 605*x) - (64*Log[1 - 2*x])/3195731 - (11984 706*Log[4 + 6*x])/2401 + (6643750*Log[6 + 10*x])/1331
Time = 0.21 (sec) , antiderivative size = 86, 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.
\(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^4 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {35954118}{2401 (3 x+2)}+\frac {33218750}{1331 (5 x+3)}-\frac {612684}{343 (3 x+2)^2}-\frac {406250}{121 (5 x+3)^2}-\frac {8667}{49 (3 x+2)^3}+\frac {3125}{11 (5 x+3)^3}-\frac {81}{7 (3 x+2)^4}-\frac {128}{3195731 (2 x-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {204228}{343 (3 x+2)}+\frac {81250}{121 (5 x+3)}+\frac {2889}{98 (3 x+2)^2}-\frac {625}{22 (5 x+3)^2}+\frac {9}{7 (3 x+2)^3}-\frac {64 \log (1-2 x)}{3195731}-\frac {11984706 \log (3 x+2)}{2401}+\frac {6643750 \log (5 x+3)}{1331}\) |
9/(7*(2 + 3*x)^3) + 2889/(98*(2 + 3*x)^2) + 204228/(343*(2 + 3*x)) - 625/( 22*(3 + 5*x)^2) + 81250/(121*(3 + 5*x)) - (64*Log[1 - 2*x])/3195731 - (119 84706*Log[2 + 3*x])/2401 + (6643750*Log[3 + 5*x])/1331
3.16.28.3.1 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 = 2.60 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.71
method | result | size |
norman | \(\frac {\frac {9322388550}{41503} x^{4}+\frac {19648830809}{83006} x +\frac {23009035068}{41503} x^{2}+\frac {23927463585}{41503} x^{3}+\frac {1571537764}{41503}}{\left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {64 \ln \left (-1+2 x \right )}{3195731}-\frac {11984706 \ln \left (2+3 x \right )}{2401}+\frac {6643750 \ln \left (3+5 x \right )}{1331}\) | \(61\) |
risch | \(\frac {\frac {9322388550}{41503} x^{4}+\frac {19648830809}{83006} x +\frac {23009035068}{41503} x^{2}+\frac {23927463585}{41503} x^{3}+\frac {1571537764}{41503}}{\left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {64 \ln \left (-1+2 x \right )}{3195731}-\frac {11984706 \ln \left (2+3 x \right )}{2401}+\frac {6643750 \ln \left (3+5 x \right )}{1331}\) | \(62\) |
default | \(-\frac {625}{22 \left (3+5 x \right )^{2}}+\frac {81250}{121 \left (3+5 x \right )}+\frac {6643750 \ln \left (3+5 x \right )}{1331}-\frac {64 \ln \left (-1+2 x \right )}{3195731}+\frac {9}{7 \left (2+3 x \right )^{3}}+\frac {2889}{98 \left (2+3 x \right )^{2}}+\frac {204228}{343 \left (2+3 x \right )}-\frac {11984706 \ln \left (2+3 x \right )}{2401}\) | \(71\) |
parallelrisch | \(-\frac {13782183012444 x -2028283406100000 \ln \left (x +\frac {3}{5}\right ) x^{2}+3173356188318096 \ln \left (\frac {2}{3}+x \right ) x^{3}-647764349400000 \ln \left (x +\frac {3}{5}\right ) x +2028283397962272 \ln \left (\frac {2}{3}+x \right ) x^{2}+647764346801088 \ln \left (\frac {2}{3}+x \right ) x +81680675283900 x^{5}+201692372713524 x^{3}+209694838787280 x^{4}+86138757807256 x^{2}+9953280 \ln \left (x -\frac {1}{2}\right ) x^{4}+2480799626046720 \ln \left (\frac {2}{3}+x \right ) x^{4}+82693320868224 \ln \left (\frac {2}{3}+x \right )+12731904 \ln \left (x -\frac {1}{2}\right ) x^{3}+8137728 \ln \left (x -\frac {1}{2}\right ) x^{2}+2598912 \ln \left (x -\frac {1}{2}\right ) x -82693321200000 \ln \left (x +\frac {3}{5}\right )+775249883139600 \ln \left (\frac {2}{3}+x \right ) x^{5}-3173356201050000 \ln \left (x +\frac {3}{5}\right ) x^{3}-775249886250000 \ln \left (x +\frac {3}{5}\right ) x^{5}-2480799636000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+331776 \ln \left (x -\frac {1}{2}\right )+3110400 \ln \left (x -\frac {1}{2}\right ) x^{5}}{230092632 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(188\) |
(9322388550/41503*x^4+19648830809/83006*x+23009035068/41503*x^2+2392746358 5/41503*x^3+1571537764/41503)/(2+3*x)^3/(3+5*x)^2-64/3195731*ln(-1+2*x)-11 984706/2401*ln(2+3*x)+6643750/1331*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).
Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=\frac {1435647836700 \, x^{4} + 3684829392090 \, x^{3} + 3543391400472 \, x^{2} + 31903287500 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 31903287372 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) - 128 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (2 \, x - 1\right ) + 1512959972293 \, x + 242016815656}{6391462 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
1/6391462*(1435647836700*x^4 + 3684829392090*x^3 + 3543391400472*x^2 + 319 03287500*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(5*x + 3) - 31903287372*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)* log(3*x + 2) - 128*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72) *log(2*x - 1) + 1512959972293*x + 242016815656)/(675*x^5 + 2160*x^4 + 2763 *x^3 + 1766*x^2 + 564*x + 72)
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=- \frac {- 18644777100 x^{4} - 47854927170 x^{3} - 46018070136 x^{2} - 19648830809 x - 3143075528}{56029050 x^{5} + 179292960 x^{4} + 229345578 x^{3} + 146588596 x^{2} + 46815384 x + 5976432} - \frac {64 \log {\left (x - \frac {1}{2} \right )}}{3195731} + \frac {6643750 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {11984706 \log {\left (x + \frac {2}{3} \right )}}{2401} \]
-(-18644777100*x**4 - 47854927170*x**3 - 46018070136*x**2 - 19648830809*x - 3143075528)/(56029050*x**5 + 179292960*x**4 + 229345578*x**3 + 146588596 *x**2 + 46815384*x + 5976432) - 64*log(x - 1/2)/3195731 + 6643750*log(x + 3/5)/1331 - 11984706*log(x + 2/3)/2401
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=\frac {18644777100 \, x^{4} + 47854927170 \, x^{3} + 46018070136 \, x^{2} + 19648830809 \, x + 3143075528}{83006 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + \frac {6643750}{1331} \, \log \left (5 \, x + 3\right ) - \frac {11984706}{2401} \, \log \left (3 \, x + 2\right ) - \frac {64}{3195731} \, \log \left (2 \, x - 1\right ) \]
1/83006*(18644777100*x^4 + 47854927170*x^3 + 46018070136*x^2 + 19648830809 *x + 3143075528)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72) + 6643750/1331*log(5*x + 3) - 11984706/2401*log(3*x + 2) - 64/3195731*log(2 *x - 1)
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=\frac {18644777100 \, x^{4} + 47854927170 \, x^{3} + 46018070136 \, x^{2} + 19648830809 \, x + 3143075528}{83006 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + \frac {6643750}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {11984706}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {64}{3195731} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
1/83006*(18644777100*x^4 + 47854927170*x^3 + 46018070136*x^2 + 19648830809 *x + 3143075528)/((5*x + 3)^2*(3*x + 2)^3) + 6643750/1331*log(abs(5*x + 3) ) - 11984706/2401*log(abs(3*x + 2)) - 64/3195731*log(abs(2*x - 1))
Time = 1.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1-2 x) (2+3 x)^4 (3+5 x)^3} \, dx=\frac {6643750\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {11984706\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {64\,\ln \left (x-\frac {1}{2}\right )}{3195731}+\frac {\frac {13810946\,x^4}{41503}+\frac {177240471\,x^3}{207515}+\frac {852186484\,x^2}{1037575}+\frac {19648830809\,x}{56029050}+\frac {1571537764}{28014525}}{x^5+\frac {16\,x^4}{5}+\frac {307\,x^3}{75}+\frac {1766\,x^2}{675}+\frac {188\,x}{225}+\frac {8}{75}} \]