\(\int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx\) [301]

Optimal result

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)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {\frac {77 \left (4599173+20320788 x+29957526 x^2+14733252 x^3\right )}{4 (2+3 x)^4}-32 \log (1-2 x)-52521843 \log (4+6 x)[phone] \log (6+10 x)}{184877} \] 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
 

Rubi [A] (verified)

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
 

Defintions of rubi rules used

rule 93

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

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	\(-\frac {272273234112 \ln \left (\frac {2}{3}+x \right ) x^{4}+165888 \ln \left (x -\frac {1}{2}\right ) x^{4}-272273400000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}+442368 \ln \left (x -\frac {1}{2}\right ) x^{3}-726062400000 \ln \left (x +\frac {3}{5}\right ) x^{3}[phone] x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}+442368 \ln \left (x -\frac {1}{2}\right ) x^{2}-726062400000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x +196608 \ln \left (x -\frac {1}{2}\right ) x -322694400000 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right )+32768 \ln \left (x -\frac {1}{2}\right )-53782400000 \ln \left (x +\frac {3}{5}\right )[phone] x}{11832128 \left (2+3 x \right )^{4}}\)	\(149\)

Input:

int(1/(1-2*x)/(2+3*x)^5/(3+5*x),x,method=_RETURNVERBOSE)
 

Output:

(3683313/2401*x^3+5080197/2401*x[phone]/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)
 

Fricas [B] (verification not implemented)

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)
 

Sympy [A] (verification not implemented)

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
 

Maxima [A] (verification not implemented)

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)
 

Giac [A] (verification not implemented)

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))
 

Mupad [B] (verification not implemented)

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)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.51 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)} \, dx=\frac {34034175000 \,\mathrm {log}\left (5 x +3\right ) x^{4}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{3}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{2}[phone] \,\mathrm {log}\left (5 x +3\right ) x [phone] \,\mathrm {log}\left (5 x +3\right )-34034154264 \,\mathrm {log}\left (3 x +2\right ) x^{4}-90757744704 \,\mathrm {log}\left (3 x +2\right ) x^{3}-90757744704 \,\mathrm {log}\left (3 x +2\right ) x^{2}-40336775424 \,\mathrm {log}\left (3 x +2\right ) x -6722795904 \,\mathrm {log}\left (3 x +2\right )-20736 \,\mathrm {log}\left (2 x -1\right ) x^{4}-55296 \,\mathrm {log}\left (2 x -1\right ) x^{3}-55296 \,\mathrm {log}\left (2 x -1\right ) x^{2}-24576 \,\mathrm {log}\left (2 x -1\right ) x -4096 \,\mathrm {log}\left (2 x -1\right )-850845303 x^{4}[phone] x^{2}[phone] x [phone]}{119800296 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] 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))