Integrand size = 22, antiderivative size = 75 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {3}{28 (2+3 x)^4}+\frac {37}{49 (2+3 x)^3}+\frac {3897}{686 (2+3 x)^2}+\frac {136419}{2401 (2+3 x)}-\frac {32 \log (1-2 x)}{184877}-\frac {4774713 \log (2+3 x)}{16807}+\frac {3125}{11} \log (3+5 x) \] Output:
3/28/(2+3*x)^4+37/49/(2+3*x)^3+3897/686/(2+3*x)^2+136419/(4802+7203*x)-32/ 184877*ln(1-2*x)-4774713/16807*ln(2+3*x)+3125/11*ln(3+5*x)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {\frac {77 \left (4599173+20320788 x+29957526 x^2+14733252 x^3\right )}{4 (2+3 x)^4}-32 \log (1-2 x)-52521843 \log (4+6 x)+52521875 \log (6+10 x)}{184877} \] Input:
Integrate[1/((1 - 2*x)*(2 + 3*x)^5*(3 + 5*x)),x]
Output:
((77*(4599173 + 20320788*x + 29957526*x^2 + 14733252*x^3))/(4*(2 + 3*x)^4) - 32*Log[1 - 2*x] - 52521843*Log[4 + 6*x] + 52521875*Log[6 + 10*x])/18487 7
Time = 0.21 (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 = {93, 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.
\(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^5 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \int \left (-\frac {14324139}{16807 (3 x+2)}+\frac {15625}{11 (5 x+3)}-\frac {409257}{2401 (3 x+2)^2}-\frac {11691}{343 (3 x+2)^3}-\frac {333}{49 (3 x+2)^4}-\frac {9}{7 (3 x+2)^5}-\frac {64}{184877 (2 x-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {136419}{2401 (3 x+2)}+\frac {3897}{686 (3 x+2)^2}+\frac {37}{49 (3 x+2)^3}+\frac {3}{28 (3 x+2)^4}-\frac {32 \log (1-2 x)}{184877}-\frac {4774713 \log (3 x+2)}{16807}+\frac {3125}{11} \log (5 x+3)\) |
Input:
Int[1/((1 - 2*x)*(2 + 3*x)^5*(3 + 5*x)),x]
Output:
3/(28*(2 + 3*x)^4) + 37/(49*(2 + 3*x)^3) + 3897/(686*(2 + 3*x)^2) + 136419 /(2401*(2 + 3*x)) - (32*Log[1 - 2*x])/184877 - (4774713*Log[2 + 3*x])/1680 7 + (3125*Log[3 + 5*x])/11
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.65
method | result | size |
norman | \(\frac {\frac {3683313}{2401} x^{3}+\frac {5080197}{2401} x +\frac {14978763}{4802} x^{2}+\frac {4599173}{9604}}{\left (2+3 x \right )^{4}}-\frac {32 \ln \left (-1+2 x \right )}{184877}-\frac {4774713 \ln \left (2+3 x \right )}{16807}+\frac {3125 \ln \left (3+5 x \right )}{11}\) | \(49\) |
risch | \(\frac {\frac {3683313}{2401} x^{3}+\frac {5080197}{2401} x +\frac {14978763}{4802} x^{2}+\frac {4599173}{9604}}{\left (2+3 x \right )^{4}}-\frac {32 \ln \left (-1+2 x \right )}{184877}-\frac {4774713 \ln \left (2+3 x \right )}{16807}+\frac {3125 \ln \left (3+5 x \right )}{11}\) | \(50\) |
default | \(\frac {3125 \ln \left (3+5 x \right )}{11}+\frac {3}{28 \left (2+3 x \right )^{4}}+\frac {37}{49 \left (2+3 x \right )^{3}}+\frac {3897}{686 \left (2+3 x \right )^{2}}+\frac {136419}{2401 \left (2+3 x \right )}-\frac {4774713 \ln \left (2+3 x \right )}{16807}-\frac {32 \ln \left (-1+2 x \right )}{184877}\) | \(62\) |
parallelrisch | \(-\frac {272273234112 \ln \left (\frac {2}{3}+x \right ) x^{4}+165888 \ln \left (x -\frac {1}{2}\right ) x^{4}-272273400000 \ln \left (x +\frac {3}{5}\right ) x^{4}+726061957632 \ln \left (\frac {2}{3}+x \right ) x^{3}+442368 \ln \left (x -\frac {1}{2}\right ) x^{3}-726062400000 \ln \left (x +\frac {3}{5}\right ) x^{3}+28685042001 x^{4}+726061957632 \ln \left (\frac {2}{3}+x \right ) x^{2}+442368 \ln \left (x -\frac {1}{2}\right ) x^{2}-726062400000 \ln \left (x +\frac {3}{5}\right ) x^{2}+58342078872 x^{3}+322694203392 \ln \left (\frac {2}{3}+x \right ) x +196608 \ln \left (x -\frac {1}{2}\right ) x -322694400000 \ln \left (x +\frac {3}{5}\right ) x +39585773304 x^{2}+53782367232 \ln \left (\frac {2}{3}+x \right )+32768 \ln \left (x -\frac {1}{2}\right )-53782400000 \ln \left (x +\frac {3}{5}\right )+8961876000 x}{11832128 \left (2+3 x \right )^{4}}\) | \(149\) |
Input:
int(1/(1-2*x)/(2+3*x)^5/(3+5*x),x,method=_RETURNVERBOSE)
Output:
(3683313/2401*x^3+5080197/2401*x+14978763/4802*x^2+4599173/9604)/(2+3*x)^4 -32/184877*ln(-1+2*x)-4774713/16807*ln(2+3*x)+3125/11*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (61) = 122\).
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.64 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {1134460404 \, x^{3} + 2306729502 \, x^{2} + 210087500 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (5 \, x + 3\right ) - 210087372 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) - 128 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 1564700676 \, x + 354136321}{739508 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:
integrate(1/(1-2*x)/(2+3*x)^5/(3+5*x),x, algorithm="fricas")
Output:
1/739508*(1134460404*x^3 + 2306729502*x^2 + 210087500*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(5*x + 3) - 210087372*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) - 128*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*l og(2*x - 1) + 1564700676*x + 354136321)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=- \frac {- 14733252 x^{3} - 29957526 x^{2} - 20320788 x - 4599173}{777924 x^{4} + 2074464 x^{3} + 2074464 x^{2} + 921984 x + 153664} - \frac {32 \log {\left (x - \frac {1}{2} \right )}}{184877} + \frac {3125 \log {\left (x + \frac {3}{5} \right )}}{11} - \frac {4774713 \log {\left (x + \frac {2}{3} \right )}}{16807} \] Input:
integrate(1/(1-2*x)/(2+3*x)**5/(3+5*x),x)
Output:
-(-14733252*x**3 - 29957526*x**2 - 20320788*x - 4599173)/(777924*x**4 + 20 74464*x**3 + 2074464*x**2 + 921984*x + 153664) - 32*log(x - 1/2)/184877 + 3125*log(x + 3/5)/11 - 4774713*log(x + 2/3)/16807
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {14733252 \, x^{3} + 29957526 \, x^{2} + 20320788 \, x + 4599173}{9604 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {3125}{11} \, \log \left (5 \, x + 3\right ) - \frac {4774713}{16807} \, \log \left (3 \, x + 2\right ) - \frac {32}{184877} \, \log \left (2 \, x - 1\right ) \] Input:
integrate(1/(1-2*x)/(2+3*x)^5/(3+5*x),x, algorithm="maxima")
Output:
1/9604*(14733252*x^3 + 29957526*x^2 + 20320788*x + 4599173)/(81*x^4 + 216* x^3 + 216*x^2 + 96*x + 16) + 3125/11*log(5*x + 3) - 4774713/16807*log(3*x + 2) - 32/184877*log(2*x - 1)
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {136419}{2401 \, {\left (3 \, x + 2\right )}} + \frac {3897}{686 \, {\left (3 \, x + 2\right )}^{2}} + \frac {37}{49 \, {\left (3 \, x + 2\right )}^{3}} + \frac {3}{28 \, {\left (3 \, x + 2\right )}^{4}} + \frac {3125}{11} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {32}{184877} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \] Input:
integrate(1/(1-2*x)/(2+3*x)^5/(3+5*x),x, algorithm="giac")
Output:
136419/2401/(3*x + 2) + 3897/686/(3*x + 2)^2 + 37/49/(3*x + 2)^3 + 3/28/(3 *x + 2)^4 + 3125/11*log(abs(-1/(3*x + 2) + 5)) - 32/184877*log(abs(-7/(3*x + 2) + 2))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {3125\,\ln \left (x+\frac {3}{5}\right )}{11}-\frac {4774713\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {32\,\ln \left (x-\frac {1}{2}\right )}{184877}+\frac {\frac {45473\,x^3}{2401}+\frac {184923\,x^2}{4802}+\frac {1693399\,x}{64827}+\frac {4599173}{777924}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \] Input:
int(-1/((2*x - 1)*(3*x + 2)^5*(5*x + 3)),x)
Output:
(3125*log(x + 3/5))/11 - (4774713*log(x + 2/3))/16807 - (32*log(x - 1/2))/ 184877 + ((1693399*x)/64827 + (184923*x^2)/4802 + (45473*x^3)/2401 + 45991 73/777924)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81)
Time = 0.16 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.51 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {34034175000 \,\mathrm {log}\left (5 x +3\right ) x^{4}+90757800000 \,\mathrm {log}\left (5 x +3\right ) x^{3}+90757800000 \,\mathrm {log}\left (5 x +3\right ) x^{2}+40336800000 \,\mathrm {log}\left (5 x +3\right ) x +6722800000 \,\mathrm {log}\left (5 x +3\right )-34034154264 \,\mathrm {log}\left (3 x +2\right ) x^{4}-90757744704 \,\mathrm {log}\left (3 x +2\right ) x^{3}-90757744704 \,\mathrm {log}\left (3 x +2\right ) x^{2}-40336775424 \,\mathrm {log}\left (3 x +2\right ) x -6722795904 \,\mathrm {log}\left (3 x +2\right )-20736 \,\mathrm {log}\left (2 x -1\right ) x^{4}-55296 \,\mathrm {log}\left (2 x -1\right ) x^{3}-55296 \,\mathrm {log}\left (2 x -1\right ) x^{2}-24576 \,\mathrm {log}\left (2 x -1\right ) x -4096 \,\mathrm {log}\left (2 x -1\right )-850845303 x^{4}+2344538196 x^{2}+2120992104 x +540204434}{119800296 x^{4}+319467456 x^{3}+319467456 x^{2}+141985536 x +23664256} \] Input:
int(1/(1-2*x)/(2+3*x)^5/(3+5*x),x)
Output:
(34034175000*log(5*x + 3)*x**4 + 90757800000*log(5*x + 3)*x**3 + 907578000 00*log(5*x + 3)*x**2 + 40336800000*log(5*x + 3)*x + 6722800000*log(5*x + 3 ) - 34034154264*log(3*x + 2)*x**4 - 90757744704*log(3*x + 2)*x**3 - 907577 44704*log(3*x + 2)*x**2 - 40336775424*log(3*x + 2)*x - 6722795904*log(3*x + 2) - 20736*log(2*x - 1)*x**4 - 55296*log(2*x - 1)*x**3 - 55296*log(2*x - 1)*x**2 - 24576*log(2*x - 1)*x - 4096*log(2*x - 1) - 850845303*x**4 + 234 4538196*x**2 + 2120992104*x + 540204434)/(1479016*(81*x**4 + 216*x**3 + 21 6*x**2 + 96*x + 16))