\(\int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx\) [459]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=\frac {8}{41503 (1-2 x)^2}+\frac {2704}{3195731 (1-2 x)}-\frac {81}{686 (2+3 x)^2}-\frac {6156}{2401 (2+3 x)}-\frac {3125}{1331 (3+5 x)}-\frac {274224 \log (1-2 x)}{246071287}+\frac {333639 \log (2+3 x)}{16807}-\frac {290625 \log (3+5 x)}{14641} \] Output:

8/41503/(1-2*x)^2+2704/(3195731-6391462*x)-81/686/(2+3*x)^2-6156/(4802+720 
3*x)-3125/(3993+6655*x)-274224/246071287*ln(1-2*x)+333639/16807*ln(2+3*x)- 
290625/14641*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=\frac {-\frac {1155481250}{3+5 x}+\frac {41503 (-3251+6558 x)}{\left (-2+x+6 x^2\right )^2}-\frac {154 (-7937593+16395384 x)}{-2+x+6 x^2}-548448 \log (5-10 x)[phone] \log (5 (2+3 x))-9769068750 \log (3+5 x)}{492142574} \] Input:

Integrate[1/((1 - 2*x)^3*(2 + 3*x)^3*(3 + 5*x)^2),x]
 

Output:

(-1155481250/(3 + 5*x) + (41503*(-3251 + 6558*x))/(-2 + x + 6*x^2)^2 - (15 
4*(-7937593 + 16395384*x))/(-2 + x + 6*x^2) - 548448*Log[5 - 10*x] + 97696 
17198*Log[5*(2 + 3*x)] - 9769068750*Log[3 + 5*x])/492142574
 

Rubi [A] (verified)

Time = 0.26 (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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {1000917}{16807 (3 x+2)}-\frac {1453125}{14641 (5 x+3)}+\frac {18468}{2401 (3 x+2)^2}+\frac {15625}{1331 (5 x+3)^2}+\frac {243}{343 (3 x+2)^3}-\frac {548448}{246071287 (2 x-1)}+\frac {5408}{3195731 (2 x-1)^2}-\frac {32}{41503 (2 x-1)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2704}{3195731 (1-2 x)}-\frac {6156}{2401 (3 x+2)}-\frac {3125}{1331 (5 x+3)}+\frac {8}{41503 (1-2 x)^2}-\frac {81}{686 (3 x+2)^2}-\frac {274224 \log (1-2 x)}{246071287}+\frac {333639 \log (3 x+2)}{16807}-\frac {290625 \log (5 x+3)}{14641}\)

Input:

Int[1/((1 - 2*x)^3*(2 + 3*x)^3*(3 + 5*x)^2),x]
 

Output:

8/(41503*(1 - 2*x)^2) + 2704/(3195731*(1 - 2*x)) - 81/(686*(2 + 3*x)^2) - 
6156/(2401*(2 + 3*x)) - 3125/(1331*(3 + 5*x)) - (274224*Log[1 - 2*x])/2460 
71287 + (333639*Log[2 + 3*x])/16807 - (290625*Log[3 + 5*x])/14641
 

Defintions of rubi rules used

rule 99

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]))
 

rule 2009

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79

method	result	size
norman	\(\frac {-\frac {761974020}{3195731} x^{4}-\frac {229003542}{3195731} x^{3}+\frac {147486147}{6391462} x +\frac {478741107}{3195731} x^{2}-\frac {160532983}{6391462}}{\left (3+5 x \right ) \left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}-\frac {274224 \ln \left (-1+2 x \right )}{246071287}+\frac {333639 \ln \left (2+3 x \right )}{16807}-\frac {290625 \ln \left (3+5 x \right )}{14641}\)	\(68\)
risch	\(\frac {-\frac {761974020}{3195731} x^{4}-\frac {229003542}{3195731} x^{3}+\frac {147486147}{6391462} x +\frac {478741107}{3195731} x^{2}-\frac {160532983}{6391462}}{\left (3+5 x \right ) \left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}-\frac {274224 \ln \left (-1+2 x \right )}{246071287}+\frac {333639 \ln \left (2+3 x \right )}{16807}-\frac {290625 \ln \left (3+5 x \right )}{14641}\)	\(69\)
default	\(-\frac {3125}{1331 \left (3+5 x \right )}-\frac {290625 \ln \left (3+5 x \right )}{14641}-\frac {81}{686 \left (2+3 x \right )^{2}}-\frac {6156}{2401 \left (2+3 x \right )}+\frac {333639 \ln \left (2+3 x \right )}{16807}+\frac {8}{41503 \left (-1+2 x \right )^{2}}-\frac {2704}{3195731 \left (-1+2 x \right )}-\frac {274224 \ln \left (-1+2 x \right )}{246071287}\)	\(71\)
parallelrisch	\(\frac {-235165517356+6955576950000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}-2846632790232 x^{4}[phone] x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{3}-52651008 \ln \left (x -\frac {1}{2}\right )-2044157997420 x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (x +\frac {3}{5}\right ) x^{3}-6174398069136 \ln \left (\frac {2}{3}+x \right ) x^{3}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x -6955967444976 \ln \left (\frac {2}{3}+x \right ) x^{2}-789765120 \ln \left (x -\frac {1}{2}\right ) x^{5}-625220400000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{4}-737114112 \ln \left (x -\frac {1}{2}\right ) x^{4}-13129628400000 \ln \left (x +\frac {3}{5}\right ) x^{4}-14067459000000 \ln \left (x +\frac {3}{5}\right ) x^{5}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}-937830600000 \ln \left (x +\frac {3}{5}\right )-35100672 \ln \left (x -\frac {1}{2}\right ) x}{3937140592 \left (3+5 x \right ) \left (-1+2 x \right )^{2} \left (2+3 x \right )^{2}}\)	\(193\)

Input:

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

Output:

(-761974020/3195731*x^4-229003542/3195731*x^3+147486147/6391462*x[phone] 
7/3195731*x^2-160532983/6391462)/(3+5*x)/(-1+2*x)^2/(2+3*x)^2-274224/24607 
1287*ln(-1+2*x)+333639/16807*ln(2+3*x)-290625/14641*ln(3+5*x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).

Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {117343999080 \, x^{4} + 35266545468 \, x^{3} - 73726130478 \, x^{2} + 9769068750 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 9769617198 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 548448 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (2 \, x - 1\right ) - 11356433319 \, x + 12361039691}{492142574 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \] Input:

integrate(1/(1-2*x)^3/(2+3*x)^3/(3+5*x)^2,x, algorithm="fricas")
 

Output:

-1/492142574*(117343999080*x^4 + 35266545468*x^3 - 73726130478*x^2 + 97690 
68750*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(5*x + 3) - 9769 
617198*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(3*x + 2) + 548 
448*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(2*x - 1) - 113564 
33319*x + 12361039691)/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=- \frac {1523948040 x^{4} + 458007084 x^{3} - 957482214 x^{2} - 147486147 x + 160532983}{1150463160 x^{5} + 1073765616 x^{4} - 504925498 x^{3} - 568840118 x^{2} + 51131696 x + 76697544} - \frac {274224 \log {\left (x - \frac {1}{2} \right )}}{246071287} - \frac {290625 \log {\left (x + \frac {3}{5} \right )}}{14641} + \frac {333639 \log {\left (x + \frac {2}{3} \right )}}{16807} \] Input:

integrate(1/(1-2*x)**3/(2+3*x)**3/(3+5*x)**2,x)
 

Output:

-(1523948040*x**4 + 458007084*x**3 - 957482214*x**2 - 147486147*x + 160532 
983)/(1150463160*x**5 + 1073765616*x**4 - 504925498*x**3 - 568840118*x**2 
+ 51131696*x + 76697544) - 274224*log(x - 1/2)/246071287 - 290625*log(x + 
3/5)/14641 + 333639*log(x + 2/3)/16807
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {1523948040 \, x^{4} + 458007084 \, x^{3} - 957482214 \, x^{2} - 147486147 \, x + 160532983}{6391462 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} - \frac {290625}{14641} \, \log \left (5 \, x + 3\right ) + \frac {333639}{16807} \, \log \left (3 \, x + 2\right ) - \frac {274224}{246071287} \, \log \left (2 \, x - 1\right ) \] Input:

integrate(1/(1-2*x)^3/(2+3*x)^3/(3+5*x)^2,x, algorithm="maxima")
 

Output:

-1/6391462*(1523948040*x^4 + 458007084*x^3 - 957482214*x^2 - 147486147*x + 
 160532983)/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12) - 290625/1464 
1*log(5*x + 3) + 333639/16807*log(3*x + 2) - 274224/246071287*log(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {3125}{1331 \, {\left (5 \, x + 3\right )}} - \frac {5 \, {\left (\frac {84659379036}{5 \, x + 3} - \frac {206753119043}{{\left (5 \, x + 3\right )}^{2}} - \frac {95568773322}{{\left (5 \, x + 3\right )}^{3}} - 7983405324\right )}}{70306082 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{2}} + \frac {333639}{16807} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {274224}{246071287} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \] Input:

integrate(1/(1-2*x)^3/(2+3*x)^3/(3+5*x)^2,x, algorithm="giac")
 

Output:

-3125/1331/(5*x + 3) - 5/70306082*(84659379036/(5*x + 3) - 206753119043/(5 
*x + 3)^2 - 95568773322/(5*x + 3)^3 - 7983405324)/((11/(5*x + 3) - 2)^2*(1 
/(5*x + 3) + 3)^2) + 333639/16807*log(abs(-1/(5*x + 3) - 3)) - 274224/2460 
71287*log(abs(-11/(5*x + 3) + 2))
 

Mupad [B] (verification not implemented)

Time = 1.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=\frac {333639\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {274224\,\ln \left (x-\frac {1}{2}\right )}{246071287}-\frac {290625\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {\frac {4233189\,x^4}{3195731}+\frac {12722419\,x^3}{31957310}-\frac {159580369\,x^2}{191743860}-\frac {49162049\,x}{383487720}+\frac {160532983}{1150463160}}{x^5+\frac {14\,x^4}{15}-\frac {79\,x^3}{180}-\frac {89\,x^2}{180}+\frac {2\,x}{45}+\frac {1}{15}} \] Input:

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

Output:

(333639*log(x + 2/3))/16807 - (274224*log(x - 1/2))/246071287 - (290625*lo 
g(x + 3/5))/14641 - ((12722419*x^3)/31957310 - (159580369*x^2)/191743860 - 
 (49162049*x)/383487720 + (4233189*x^4)/3195731 + 160532983/1150463160)/(( 
2*x)/45 - (89*x^2)/180 - (79*x^3)/180 + (14*x^4)/15 + x^5 + 1/15)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.69 \[ \int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx=\frac {-1758432375000 \,\mathrm {log}\left (5 x +3\right ) x^{5}-1641203550000 \,\mathrm {log}\left (5 x +3\right ) x^{4}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{3}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{2}-78152550000 \,\mathrm {log}\left (5 x +3\right ) x -117228825000 \,\mathrm {log}\left (5 x +3\right )[phone] \,\mathrm {log}\left (3 x +2\right ) x^{5}[phone] \,\mathrm {log}\left (3 x +2\right ) x^{4}-771799758642 \,\mathrm {log}\left (3 x +2\right ) x^{3}-869495930622 \,\mathrm {log}\left (3 x +2\right ) x^{2}[phone] \,\mathrm {log}\left (3 x +2\right ) x [phone] \,\mathrm {log}\left (3 x +2\right )-98720640 \,\mathrm {log}\left (2 x -1\right ) x^{5}-92139264 \,\mathrm {log}\left (2 x -1\right ) x^{4}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{3}[phone] \,\mathrm {log}\left (2 x -1\right ) x^{2}-4387584 \,\mathrm {log}\left (2 x -1\right ) x -6581376 \,\mathrm {log}\left (2 x -1\right )[phone] x^{5}-90446164083 x^{3}[phone] x^{2}[phone] x -3979325471}{88585663320 x^{5}[phone] x^{4}-38879263346 x^{3}-43800689086 x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 1758432375000*log(5*x + 3)*x**5 - 1641203550000*log(5*x + 3)*x**4 + 77 
1756431250*log(5*x + 3)*x**3 + 869447118750*log(5*x + 3)*x**2 - 7815255000 
0*log(5*x + 3)*x - 117228825000*log(5*x + 3) + 1758531095640*log(3*x + 2)* 
x**5 + 1641295689264*log(3*x + 2)*x**4 - 771799758642*log(3*x + 2)*x**3 - 
869495930622*log(3*x + 2)*x**2 + 78156937584*log(3*x + 2)*x + 117235406376 
*log(3*x + 2) - 98720640*log(2*x - 1)*x**5 - 92139264*log(2*x - 1)*x**4 + 
43327392*log(2*x - 1)*x**3 + 48811872*log(2*x - 1)*x**2 - 4387584*log(2*x 
- 1)*x - 6581376*log(2*x - 1) + 125725713300*x**5 - 90446164083*x**3 + 115 
61750013*x**2 + 16944242799*x - 3979325471)/(492142574*(180*x**5 + 168*x** 
4 - 79*x**3 - 89*x**2 + 8*x + 12))