3.16.14 \(\int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx\) [1514]

3.16.14.1 Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {9}{28 (2+3 x)^4}-\frac {216}{49 (2+3 x)^3}-\frac {34371}{686 (2+3 x)^2}-\frac {1612242}{2401 (2+3 x)}-\frac {3125}{11 (3+5 x)}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (2+3 x)}{16807}-\frac {509375}{121} \log (3+5 x) \]

output

-9/28/(2+3*x)^4-216/49/(2+3*x)^3-34371/686/(2+3*x)^2-1612242/2401/(2+3*x)- 
3125/11/(3+5*x)-64/2033647*ln(1-2*x)[phone]/16807*ln(2+3*x)-509375/121*l 
n(3+5*x)
 

3.16.14.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {9}{28 (2+3 x)^4}-\frac {216}{49 (2+3 x)^3}-\frac {34371}{686 (2+3 x)^2}-\frac {1612242}{2401 (2+3 x)}-\frac {3125}{33+55 x}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (4+6 x)}{16807}-\frac {509375}{121} \log (6+10 x) \]

input

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

output

-9/(28*(2 + 3*x)^4) - 216/(49*(2 + 3*x)^3) - 34371/(686*(2 + 3*x)^2) - 161 
2242/(2401*(2 + 3*x)) - 3125/(33 + 55*x) - (64*Log[1 - 2*x])/2033647 + (70 
752609*Log[4 + 6*x])/16807 - (509375*Log[6 + 10*x])/121
 

3.16.14.3 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {212257827}{16807 (3 x+2)}-\frac {2546875}{121 (5 x+3)}+\frac {4836726}{2401 (3 x+2)^2}+\frac {15625}{11 (5 x+3)^2}+\frac {103113}{343 (3 x+2)^3}+\frac {1944}{49 (3 x+2)^4}+\frac {27}{7 (3 x+2)^5}-\frac {128}{2033647 (2 x-1)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1612242}{2401 (3 x+2)}-\frac {3125}{11 (5 x+3)}-\frac {34371}{686 (3 x+2)^2}-\frac {216}{49 (3 x+2)^3}-\frac {9}{28 (3 x+2)^4}-\frac {64 \log (1-2 x)}{2033647}+\frac {70752609 \log (3 x+2)}{16807}-\frac {509375}{121} \log (5 x+3)\)

input

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

output

-9/(28*(2 + 3*x)^4) - 216/(49*(2 + 3*x)^3) - 34371/(686*(2 + 3*x)^2) - 161 
2242/(2401*(2 + 3*x)) - 3125/(11*(3 + 5*x)) - (64*Log[1 - 2*x])/2033647 + 
(70752609*Log[2 + 3*x])/16807 - (509375*Log[3 + 5*x])/121
 

3.16.14.3.1 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]
 

3.16.14.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.71

method	result	size
norman	\(\frac {-\frac {15810178239}{52822} x^{3}-\frac {15605602857}{52822} x^{2}-\frac {13685553417}{105644} x -\frac {3001932495}{26411} x^{4}-\frac {2249141207}{105644}}{\left (2+3 x \right )^{4} \left (3+5 x \right )}-\frac {64 \ln \left (-1+2 x \right )}{2033647}+\frac {70752609 \ln \left (2+3 x \right )}{16807}-\frac {509375 \ln \left (3+5 x \right )}{121}\)	\(61\)
risch	\(\frac {-\frac {15810178239}{52822} x^{3}-\frac {15605602857}{52822} x^{2}-\frac {13685553417}{105644} x -\frac {3001932495}{26411} x^{4}-\frac {2249141207}{105644}}{\left (2+3 x \right )^{4} \left (3+5 x \right )}-\frac {64 \ln \left (-1+2 x \right )}{2033647}+\frac {70752609 \ln \left (2+3 x \right )}{16807}-\frac {509375 \ln \left (3+5 x \right )}{121}\)	\(62\)
default	\(-\frac {3125}{11 \left (3+5 x \right )}-\frac {509375 \ln \left (3+5 x \right )}{121}-\frac {64 \ln \left (-1+2 x \right )}{2033647}-\frac {9}{28 \left (2+3 x \right )^{4}}-\frac {216}{49 \left (2+3 x \right )^{3}}-\frac {34371}{686 \left (2+3 x \right )^{2}}-\frac {1612242}{2401 \left (2+3 x \right )}+\frac {70752609 \ln \left (2+3 x \right )}{16807}\)	\(71\)
parallelrisch	\(\frac {13149859812320 x -1854121348800000 \ln \left (x +\frac {3}{5}\right ) x^{2}+2840356130033664 \ln \left (\frac {2}{3}+x \right ) x^{3}-604890652800000 \ln \left (x +\frac {3}{5}\right ) x +1854121362660864 \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{3}[phone] x^{4}[phone] x^{2}-16257024 \ln \left (x -\frac {1}{2}\right ) x^{4}+2174647662057024 \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-21233664 \ln \left (x -\frac {1}{2}\right ) x^{3}-13860864 \ln \left (x -\frac {1}{2}\right ) x^{2}-4521984 \ln \left (x -\frac {1}{2}\right ) x -78898780800000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-2840356108800000 \ln \left (x +\frac {3}{5}\right ) x^{3}-665708463000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-2174647645800000 \ln \left (x +\frac {3}{5}\right ) x^{4}-589824 \ln \left (x -\frac {1}{2}\right )-4976640 \ln \left (x -\frac {1}{2}\right ) x^{5}}{390460224 \left (2+3 x \right )^{4} \left (3+5 x \right )}\)	\(188\)

input

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

output

(-15810178239/52822*x^3-15605602857/52822*x^2-13685553417/105644*x-3001932 
495/26411*x^4-2249141207/105644)/(2+3*x)^4/(3+5*x)-64/2033647*ln(-1+2*x)+7 
0752609/16807*ln(2+3*x)-509375/121*ln(3+5*x)
 

3.16.14.5 Fricas [B] (verification not implemented)

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

Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {924595208460 \, x^{4} + 2434767448806 \, x^{3} + 2403262839978 \, x^{2} + 34244262500 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (5 \, x + 3\right ) - 34244262756 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (3 \, x + 2\right ) + 256 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (2 \, x - 1\right ) + 1053787613109 \, x + 173183872939}{8134588 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

input

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

output

-1/8134588*(924595208460*x^4 + 2434767448806*x^3 + 2403262839978*x^2 + 342 
44262500*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(5*x + 
 3) - 34244262756*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)* 
log(3*x + 2) + 256*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48) 
*log(2*x - 1) + 1053787613109*x + 173183872939)/(405*x^5 + 1323*x^4 + 1728 
*x^3 + 1128*x^2 + 368*x + 48)
 

3.16.14.6 Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=- \frac {12007729980 x^{4} + 31620356478 x^{3} + 31211205714 x^{2} + 13685553417 x + 2249141207}{42785820 x^{5} + 139767012 x^{4} + 182552832 x^{3} + 119166432 x^{2} + 38876992 x + 5070912} - \frac {64 \log {\left (x - \frac {1}{2} \right )}}{2033647} - \frac {509375 \log {\left (x + \frac {3}{5} \right )}}{121} + \frac {70752609 \log {\left (x + \frac {2}{3} \right )}}{16807} \]

input

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

output

-(12007729980*x**4 + 31620356478*x**3 + 31211205714*x**2 + 13685553417*x + 
 2249141207)/(42785820*x**5 + 139767012*x**4 + 182552832*x**3 + 119166432* 
x**2 + 38876992*x + 5070912) - 64*log(x - 1/2)/2033647 - 509375*log(x + 3/ 
5)/121 + 70752609*log(x + 2/3)/16807
 

3.16.14.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {12007729980 \, x^{4} + 31620356478 \, x^{3} + 31211205714 \, x^{2} + 13685553417 \, x + 2249141207}{105644 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} - \frac {509375}{121} \, \log \left (5 \, x + 3\right ) + \frac {70752609}{16807} \, \log \left (3 \, x + 2\right ) - \frac {64}{2033647} \, \log \left (2 \, x - 1\right ) \]

input

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

output

-1/105644*(12007729980*x^4 + 31620356478*x^3 + 31211205714*x^2 + 136855534 
17*x + 2249141207)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48) 
 - 509375/121*log(5*x + 3) + 70752609/16807*log(3*x + 2) - 64/2033647*log( 
2*x - 1)
 

3.16.14.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=-\frac {3125}{11 \, {\left (5 \, x + 3\right )}} + \frac {135 \, {\left (\frac {34747884}{5 \, x + 3} + \frac {13347468}{{\left (5 \, x + 3\right )}^{2}} + \frac {1775512}{{\left (5 \, x + 3\right )}^{3}} + 30897639\right )}}{9604 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{4}} + \frac {70752609}{16807} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {64}{2033647} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

input

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

output

-3125/11/(5*x + 3) + 135/9604*(34747884/(5*x + 3) + 13347468/(5*x + 3)^2 + 
 1775512/(5*x + 3)^3 + 30897639)/(1/(5*x + 3) + 3)^4 + 70752609/16807*log( 
abs(-1/(5*x + 3) - 3)) - 64/2033647*log(abs(-11/(5*x + 3) + 2))
 

3.16.14.9 Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x) (2+3 x)^5 (3+5 x)^2} \, dx=\frac {70752609\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {64\,\ln \left (x-\frac {1}{2}\right )}{2033647}-\frac {509375\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {\frac {7412179\,x^4}{26411}+\frac {585562157\,x^3}{792330}+\frac {577985291\,x^2}{792330}+\frac {4561851139\,x}{14261940}+\frac {2249141207}{42785820}}{x^5+\frac {49\,x^4}{15}+\frac {64\,x^3}{15}+\frac {376\,x^2}{135}+\frac {368\,x}{405}+\frac {16}{135}} \]

input

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

output

(70752609*log(x + 2/3))/16807 - (64*log(x - 1/2))/2033647 - (509375*log(x 
+ 3/5))/121 - ((4561851139*x)/14261940 + (577985291*x^2)/792330 + (5855621 
57*x^3)/792330 + (7412179*x^4)/26411 + 2249141207/42785820)/((368*x)/405 + 
 (376*x^2)/135 + (64*x^3)/15 + (49*x^4)/15 + x^5 + 16/135)