Integrand size = 22, antiderivative size = 75 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {16}{65219 (1-2 x)}+\frac {81}{49 (2+3 x)}-\frac {125}{242 (3+5 x)^2}+\frac {7750}{1331 (3+5 x)}-\frac {2736 \log (1-2 x)}{5021863}-\frac {8829}{343} \log (2+3 x)+\frac {376875 \log (3+5 x)}{14641} \] Output:
16/(65219-130438*x)+81/(98+147*x)-125/242/(3+5*x)^2+7750/(3993+6655*x)-273 6/5021863*ln(1-2*x)-8829/343*ln(2+3*x)+376875/14641*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {\frac {77 \left (-6363424-7974123 x+24606540 x^2+33563700 x^3\right )}{(3+5 x)^2 \left (-2+x+6 x^2\right )}-5472 \log (5-10 x)-258530778 \log (5 (2+3 x))+258536250 \log (3+5 x)}{10043726} \] Input:
Integrate[1/((1 - 2*x)^2*(2 + 3*x)^2*(3 + 5*x)^3),x]
Output:
((77*(-6363424 - 7974123*x + 24606540*x^2 + 33563700*x^3))/((3 + 5*x)^2*(- 2 + x + 6*x^2)) - 5472*Log[5 - 10*x] - 258530778*Log[5*(2 + 3*x)] + 258536 250*Log[3 + 5*x])/10043726
Time = 0.25 (sec) , antiderivative size = 75, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^2 (3 x+2)^2 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {26487}{343 (3 x+2)}+\frac {1884375}{14641 (5 x+3)}-\frac {243}{49 (3 x+2)^2}-\frac {38750}{1331 (5 x+3)^2}+\frac {625}{121 (5 x+3)^3}-\frac {5472}{5021863 (2 x-1)}+\frac {32}{65219 (2 x-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {16}{65219 (1-2 x)}+\frac {81}{49 (3 x+2)}+\frac {7750}{1331 (5 x+3)}-\frac {125}{242 (5 x+3)^2}-\frac {2736 \log (1-2 x)}{5021863}-\frac {8829}{343} \log (3 x+2)+\frac {376875 \log (5 x+3)}{14641}\) |
Input:
Int[1/((1 - 2*x)^2*(2 + 3*x)^2*(3 + 5*x)^3),x]
Output:
16/(65219*(1 - 2*x)) + 81/(49*(2 + 3*x)) - 125/(242*(3 + 5*x)^2) + 7750/(1 331*(3 + 5*x)) - (2736*Log[1 - 2*x])/5021863 - (8829*Log[2 + 3*x])/343 + ( 376875*Log[3 + 5*x])/14641
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 = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {125}{242 \left (3+5 x \right )^{2}}+\frac {7750}{1331 \left (3+5 x \right )}+\frac {376875 \ln \left (3+5 x \right )}{14641}+\frac {81}{49 \left (2+3 x \right )}-\frac {8829 \ln \left (2+3 x \right )}{343}-\frac {16}{65219 \left (-1+2 x \right )}-\frac {2736 \ln \left (-1+2 x \right )}{5021863}\) | \(62\) |
risch | \(\frac {\frac {16781850}{65219} x^{3}+\frac {1757610}{9317} x^{2}-\frac {7974123}{130438} x -\frac {3181712}{65219}}{\left (3+5 x \right )^{2} \left (-1+2 x \right ) \left (2+3 x \right )}-\frac {2736 \ln \left (-1+2 x \right )}{5021863}-\frac {8829 \ln \left (2+3 x \right )}{343}+\frac {376875 \ln \left (3+5 x \right )}{14641}\) | \(64\) |
norman | \(\frac {-\frac {25012690}{83853} x^{3}-\frac {79542800}{195657} x^{4}+\frac {56640326}{586971} x^{2}+\frac {30166735}{391314} x}{\left (3+5 x \right )^{2} \left (-1+2 x \right ) \left (2+3 x \right )}-\frac {2736 \ln \left (-1+2 x \right )}{5021863}-\frac {8829 \ln \left (2+3 x \right )}{343}+\frac {376875 \ln \left (3+5 x \right )}{14641}\) | \(67\) |
parallelrisch | \(-\frac {349016550300 \ln \left (\frac {2}{3}+x \right ) x^{4}+7387200 \ln \left (x -\frac {1}{2}\right ) x^{4}-349023937500 \ln \left (x +\frac {3}{5}\right ) x^{4}+476989285410 \ln \left (\frac {2}{3}+x \right ) x^{3}+10095840 \ln \left (x -\frac {1}{2}\right ) x^{3}-476999381250 \ln \left (x +\frac {3}{5}\right ) x^{3}+36748773600 x^{4}+79110418068 \ln \left (\frac {2}{3}+x \right ) x^{2}+1674432 \ln \left (x -\frac {1}{2}\right ) x^{2}-79112092500 \ln \left (x +\frac {3}{5}\right ) x^{2}+26963679820 x^{3}-118665627102 \ln \left (\frac {2}{3}+x \right ) x -2511648 \ln \left (x -\frac {1}{2}\right ) x +118668138750 \ln \left (x +\frac {3}{5}\right ) x -8722610204 x^{2}-41881986036 \ln \left (\frac {2}{3}+x \right )-886464 \ln \left (x -\frac {1}{2}\right )+41882872500 \ln \left (x +\frac {3}{5}\right )-6968515785 x}{90393534 \left (3+5 x \right )^{2} \left (-1+2 x \right ) \left (2+3 x \right )}\) | \(163\) |
Input:
int(1/(1-2*x)^2/(2+3*x)^2/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-125/242/(3+5*x)^2+7750/1331/(3+5*x)+376875/14641*ln(3+5*x)+81/49/(2+3*x)- 8829/343*ln(2+3*x)-16/65219/(-1+2*x)-2736/5021863*ln(-1+2*x)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (61) = 122\).
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {2584404900 \, x^{3} + 1894703580 \, x^{2} + 258536250 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (5 \, x + 3\right ) - 258530778 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (3 \, x + 2\right ) - 5472 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (2 \, x - 1\right ) - 614007471 \, x - 489983648}{10043726 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \] Input:
integrate(1/(1-2*x)^2/(2+3*x)^2/(3+5*x)^3,x, algorithm="fricas")
Output:
1/10043726*(2584404900*x^3 + 1894703580*x^2 + 258536250*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(5*x + 3) - 258530778*(150*x^4 + 205*x^3 + 34*x^ 2 - 51*x - 18)*log(3*x + 2) - 5472*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18 )*log(2*x - 1) - 614007471*x - 489983648)/(150*x^4 + 205*x^3 + 34*x^2 - 51 *x - 18)
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {33563700 x^{3} + 24606540 x^{2} - 7974123 x - 6363424}{19565700 x^{4} + 26739790 x^{3} + 4434892 x^{2} - 6652338 x - 2347884} - \frac {2736 \log {\left (x - \frac {1}{2} \right )}}{5021863} + \frac {376875 \log {\left (x + \frac {3}{5} \right )}}{14641} - \frac {8829 \log {\left (x + \frac {2}{3} \right )}}{343} \] Input:
integrate(1/(1-2*x)**2/(2+3*x)**2/(3+5*x)**3,x)
Output:
(33563700*x**3 + 24606540*x**2 - 7974123*x - 6363424)/(19565700*x**4 + 267 39790*x**3 + 4434892*x**2 - 6652338*x - 2347884) - 2736*log(x - 1/2)/50218 63 + 376875*log(x + 3/5)/14641 - 8829*log(x + 2/3)/343
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {33563700 \, x^{3} + 24606540 \, x^{2} - 7974123 \, x - 6363424}{130438 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} + \frac {376875}{14641} \, \log \left (5 \, x + 3\right ) - \frac {8829}{343} \, \log \left (3 \, x + 2\right ) - \frac {2736}{5021863} \, \log \left (2 \, x - 1\right ) \] Input:
integrate(1/(1-2*x)^2/(2+3*x)^2/(3+5*x)^3,x, algorithm="maxima")
Output:
1/130438*(33563700*x^3 + 24606540*x^2 - 7974123*x - 6363424)/(150*x^4 + 20 5*x^3 + 34*x^2 - 51*x - 18) + 376875/14641*log(5*x + 3) - 8829/343*log(3*x + 2) - 2736/5021863*log(2*x - 1)
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {81}{49 \, {\left (3 \, x + 2\right )}} + \frac {27 \, {\left (\frac {139939165}{3 \, x + 2} - \frac {31679854}{{\left (3 \, x + 2\right )}^{2}} - 37396350\right )}}{913066 \, {\left (\frac {7}{3 \, x + 2} - 2\right )} {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {376875}{14641} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {2736}{5021863} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \] Input:
integrate(1/(1-2*x)^2/(2+3*x)^2/(3+5*x)^3,x, algorithm="giac")
Output:
81/49/(3*x + 2) + 27/913066*(139939165/(3*x + 2) - 31679854/(3*x + 2)^2 - 37396350)/((7/(3*x + 2) - 2)*(1/(3*x + 2) - 5)^2) + 376875/14641*log(abs(- 1/(3*x + 2) + 5)) - 2736/5021863*log(abs(-7/(3*x + 2) + 2))
Time = 1.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {376875\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {8829\,\ln \left (x+\frac {2}{3}\right )}{343}-\frac {2736\,\ln \left (x-\frac {1}{2}\right )}{5021863}-\frac {-\frac {111879\,x^3}{65219}-\frac {58587\,x^2}{46585}+\frac {2658041\,x}{6521900}+\frac {1590856}{4891425}}{x^4+\frac {41\,x^3}{30}+\frac {17\,x^2}{75}-\frac {17\,x}{50}-\frac {3}{25}} \] Input:
int(1/((2*x - 1)^2*(3*x + 2)^2*(5*x + 3)^3),x)
Output:
(376875*log(x + 3/5))/14641 - (8829*log(x + 2/3))/343 - (2736*log(x - 1/2) )/5021863 - ((2658041*x)/6521900 - (58587*x^2)/46585 - (111879*x^3)/65219 + 1590856/4891425)/((17*x^2)/75 - (17*x)/50 + (41*x^3)/30 + x^4 - 3/25)
Time = 0.16 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.51 \[ \int \frac {1}{(1-2 x)^2 (2+3 x)^2 (3+5 x)^3} \, dx=\frac {1589997937500 \,\mathrm {log}\left (5 x +3\right ) x^{4}+2172997181250 \,\mathrm {log}\left (5 x +3\right ) x^{3}+360399532500 \,\mathrm {log}\left (5 x +3\right ) x^{2}-540599298750 \,\mathrm {log}\left (5 x +3\right ) x -190799752500 \,\mathrm {log}\left (5 x +3\right )-1589964284700 \,\mathrm {log}\left (3 x +2\right ) x^{4}-2172951189090 \,\mathrm {log}\left (3 x +2\right ) x^{3}-360391904532 \,\mathrm {log}\left (3 x +2\right ) x^{2}+540587856798 \,\mathrm {log}\left (3 x +2\right ) x +190795714164 \,\mathrm {log}\left (3 x +2\right )-33652800 \,\mathrm {log}\left (2 x -1\right ) x^{4}-45992160 \,\mathrm {log}\left (2 x -1\right ) x^{3}-7627968 \,\mathrm {log}\left (2 x -1\right ) x^{2}+11441952 \,\mathrm {log}\left (2 x -1\right ) x +4038336 \,\mathrm {log}\left (2 x -1\right )-77532147000 x^{4}+60108893460 x^{2}+1186623669 x -10785471928}{61768914900 x^{4}+84417517030 x^{3}+14000954044 x^{2}-21001431066 x -7412269788} \] Input:
int(1/(1-2*x)^2/(2+3*x)^2/(3+5*x)^3,x)
Output:
(1589997937500*log(5*x + 3)*x**4 + 2172997181250*log(5*x + 3)*x**3 + 36039 9532500*log(5*x + 3)*x**2 - 540599298750*log(5*x + 3)*x - 190799752500*log (5*x + 3) - 1589964284700*log(3*x + 2)*x**4 - 2172951189090*log(3*x + 2)*x **3 - 360391904532*log(3*x + 2)*x**2 + 540587856798*log(3*x + 2)*x + 19079 5714164*log(3*x + 2) - 33652800*log(2*x - 1)*x**4 - 45992160*log(2*x - 1)* x**3 - 7627968*log(2*x - 1)*x**2 + 11441952*log(2*x - 1)*x + 4038336*log(2 *x - 1) - 77532147000*x**4 + 60108893460*x**2 + 1186623669*x - 10785471928 )/(411792766*(150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18))