3.17.95 \(\int \frac {1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx\) [1695]

3.17.95.1 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/(1-2*x)-81/686/(2+3*x)^2-6156/2401/(2+3*x)- 
3125/1331/(3+5*x)-274224/246071287*ln(1-2*x)+333639/16807*ln(2+3*x)-290625 
/14641*ln(3+5*x)
 

3.17.95.2 Mathematica [A] (verified)

Time = 0.10 (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
 

3.17.95.3 Rubi [A] (verified)

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.

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

\(\Big \downarrow \) 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
 

3.17.95.3.1 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]
 

3.17.95.4 Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79

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

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)/(-1+2*x)^2/(2+3*x)^2/(3+5*x)-274224/24607 
1287*ln(-1+2*x)+333639/16807*ln(2+3*x)-290625/14641*ln(3+5*x)
 

3.17.95.5 Fricas [B] (verification not implemented)

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

Time = 0.23 (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)
 

3.17.95.6 Sympy [A] (verification not implemented)

Time = 0.12 (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
 

3.17.95.7 Maxima [A] (verification not implemented)

Time = 0.21 (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)
 

3.17.95.8 Giac [A] (verification not implemented)

Time = 0.28 (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))
 

3.17.95.9 Mupad [B] (verification not implemented)

Time = 1.37 (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)