\(\int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx\) [1499]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {3}{35 (2+3 x)^5}+\frac {111}{196 (2+3 x)^4}+\frac {1299}{343 (2+3 x)^3}+\frac {136419}{4802 (2+3 x)^2}+\frac {4774713}{16807 (2+3 x)}-\frac {64 \log (1-2 x)}{1294139}-\frac {167115051 \log (2+3 x)}{117649}+\frac {15625}{11} \log (3+5 x) \]

[Out]

3/35/(2+3*x)^5+111/196/(2+3*x)^4+1299/343/(2+3*x)^3+136419/4802/(2+3*x)^2+4774713/16807/(2+3*x)-64/1294139*ln(
1-2*x)-167115051/117649*ln(2+3*x)+15625/11*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {84} \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {4774713}{16807 (3 x+2)}+\frac {136419}{4802 (3 x+2)^2}+\frac {1299}{343 (3 x+2)^3}+\frac {111}{196 (3 x+2)^4}+\frac {3}{35 (3 x+2)^5}-\frac {64 \log (1-2 x)}{1294139}-\frac {167115051 \log (3 x+2)}{117649}+\frac {15625}{11} \log (5 x+3) \]

[In]

Int[1/((1 - 2*x)*(2 + 3*x)^6*(3 + 5*x)),x]

[Out]

3/(35*(2 + 3*x)^5) + 111/(196*(2 + 3*x)^4) + 1299/(343*(2 + 3*x)^3) + 136419/(4802*(2 + 3*x)^2) + 4774713/(168
07*(2 + 3*x)) - (64*Log[1 - 2*x])/1294139 - (167115051*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/11

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {128}{1294139 (-1+2 x)}-\frac {9}{7 (2+3 x)^6}-\frac {333}{49 (2+3 x)^5}-\frac {11691}{343 (2+3 x)^4}-\frac {409257}{2401 (2+3 x)^3}-\frac {14324139}{16807 (2+3 x)^2}-\frac {501345153}{117649 (2+3 x)}+\frac {78125}{11 (3+5 x)}\right ) \, dx \\ & = \frac {3}{35 (2+3 x)^5}+\frac {111}{196 (2+3 x)^4}+\frac {1299}{343 (2+3 x)^3}+\frac {136419}{4802 (2+3 x)^2}+\frac {4774713}{16807 (2+3 x)}-\frac {64 \log (1-2 x)}{1294139}-\frac {167115051 \log (2+3 x)}{117649}+\frac {15625}{11} \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {\frac {2079 \left (59622386+352854525 x+783477080 x^2+773503410 x^3+286482780 x^4\right )}{4 (2+3 x)^5}-320 \log (1-2 x)-9191327805 \log (4+6 x)[phone] \log (6+10 x)}{6470695} \]

[In]

Integrate[1/((1 - 2*x)*(2 + 3*x)^6*(3 + 5*x)),x]

[Out]

((2079*(59622386 + 352854525*x + 783477080*x^2 + 773503410*x^3 + 286482780*x^4))/(4*(2 + 3*x)^5) - 320*Log[1 -
2*x] - 9191327805*Log[4 + 6*x] + 9191328125*Log[6 + 10*x])/6470695

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.63

method	result	size
norman	\(\frac {\frac {386751753}{16807} x^{4}+\frac {1057694058}{16807} x^{2}+\frac {1905414435}{67228} x +\frac {2088459207}{33614} x^{3}+\frac {804902211}{168070}}{\left (2+3 x \right )^{5}}-\frac {64 \ln \left (-1+2 x \right )}{1294139}-\frac {167115051 \ln \left (2+3 x \right )}{117649}+\frac {15625 \ln \left (3+5 x \right )}{11}\)	\(54\)
risch	\(\frac {\frac {386751753}{16807} x^{4}+\frac {1057694058}{16807} x^{2}+\frac {1905414435}{67228} x +\frac {2088459207}{33614} x^{3}+\frac {804902211}{168070}}{\left (2+3 x \right )^{5}}-\frac {64 \ln \left (-1+2 x \right )}{1294139}-\frac {167115051 \ln \left (2+3 x \right )}{117649}+\frac {15625 \ln \left (3+5 x \right )}{11}\)	\(55\)
default	\(\frac {15625 \ln \left (3+5 x \right )}{11}-\frac {64 \ln \left (-1+2 x \right )}{1294139}+\frac {3}{35 \left (2+3 x \right )^{5}}+\frac {111}{196 \left (2+3 x \right )^{4}}+\frac {1299}{343 \left (2+3 x \right )^{3}}+\frac {136419}{4802 \left (2+3 x \right )^{2}}+\frac {4774713}{16807 \left (2+3 x \right )}-\frac {167115051 \ln \left (2+3 x \right )}{117649}\)	\(71\)
parallelrisch	\(-\frac {3137239939680 x -423536400000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-141178800000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (x -\frac {1}{2}\right ) x^{3}[phone] \ln \left (x -\frac {1}{2}\right ) x^{2}+4915200 \ln \left (x -\frac {1}{2}\right ) x -18823840000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-635304600000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-142943535000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-476478450000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+655360 \ln \left (x -\frac {1}{2}\right )+4976640 \ln \left (x -\frac {1}{2}\right ) x^{5}}{414124480 \left (2+3 x \right )^{5}}\)	\(181\)

[In]

int(1/(1-2*x)/(2+3*x)^6/(3+5*x),x,method=_RETURNVERBOSE)

[Out]

(386751753/16807*x^4+1057694058/16807*x^2+1905414435/67228*x+2088459207/33614*x^3+804902211/168070)/(2+3*x)^5-
64/1294139*ln(-1+2*x)-167115051/117649*ln(2+3*x)+15625/11*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 = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {595597699620 \, x^{4} + 1608113589390 \, x^{3} + 1628848849320 \, x^{2} + 36765312500 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (5 \, x + 3\right ) - 36765311220 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) - 1280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 733584557475 \, x + 123954940494}{25882780 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

[In]

integrate(1/(1-2*x)/(2+3*x)^6/(3+5*x),x, algorithm="fricas")

[Out]

1/25882780*(595597699620*x^4 + 1608113589390*x^3 + 1628848849320*x^2 + 36765312500*(243*x^5 + 810*x^4 + 1080*x
^3 + 720*x^2 + 240*x + 32)*log(5*x + 3) - 36765311220*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*lo
g(3*x + 2) - 1280*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 733584557475*x + 123954
940494)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=- \frac {- 7735035060 x^{4} - 20884592070 x^{3} - 21153881160 x^{2} - 9527072175 x - 1609804422}{81682020 x^{5} + 272273400 x^{4} + 363031200 x^{3} + 242020800 x^{2} + 80673600 x + 10756480} - \frac {64 \log {\left (x - \frac {1}{2} \right )}}{1294139} + \frac {15625 \log {\left (x + \frac {3}{5} \right )}}{11} - \frac {167115051 \log {\left (x + \frac {2}{3} \right )}}{117649} \]

[In]

integrate(1/(1-2*x)/(2+3*x)**6/(3+5*x),x)

[Out]

-(-7735035060*x**4 - 20884592070*x**3 - 21153881160*x**2 - 9527072175*x - 1609804422)/(81682020*x**5 + 2722734
00*x**4 + 363031200*x**3 + 242020800*x**2 + 80673600*x + 10756480) - 64*log(x - 1/2)/1294139 + 15625*log(x + 3
/5)/11 - 167115051*log(x + 2/3)/117649

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {27 \, {\left (286482780 \, x^{4} + 773503410 \, x^{3} + 783477080 \, x^{2} + 352854525 \, x + 59622386\right )}}{336140 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {15625}{11} \, \log \left (5 \, x + 3\right ) - \frac {167115051}{117649} \, \log \left (3 \, x + 2\right ) - \frac {64}{1294139} \, \log \left (2 \, x - 1\right ) \]

[In]

integrate(1/(1-2*x)/(2+3*x)^6/(3+5*x),x, algorithm="maxima")

[Out]

27/336140*(286482780*x^4 + 773503410*x^3 + 783477080*x^2 + 352854525*x + 59622386)/(243*x^5 + 810*x^4 + 1080*x
^3 + 720*x^2 + 240*x + 32) + 15625/11*log(5*x + 3) - 167115051/117649*log(3*x + 2) - 64/1294139*log(2*x - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {27 \, {\left (286482780 \, x^{4} + 773503410 \, x^{3} + 783477080 \, x^{2} + 352854525 \, x + 59622386\right )}}{336140 \, {\left (3 \, x + 2\right )}^{5}} + \frac {15625}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {167115051}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {64}{1294139} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[In]

integrate(1/(1-2*x)/(2+3*x)^6/(3+5*x),x, algorithm="giac")

[Out]

27/336140*(286482780*x^4 + 773503410*x^3 + 783477080*x^2 + 352854525*x + 59622386)/(3*x + 2)^5 + 15625/11*log(
abs(5*x + 3)) - 167115051/117649*log(abs(3*x + 2)) - 64/1294139*log(abs(2*x - 1))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx=\frac {15625\,\ln \left (x+\frac {3}{5}\right )}{11}-\frac {167115051\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {64\,\ln \left (x-\frac {1}{2}\right )}{1294139}+\frac {\frac {1591571\,x^4}{16807}+\frac {25783447\,x^3}{100842}+\frac {39173854\,x^2}{151263}+\frac {23523635\,x}{201684}+\frac {29811193}{1512630}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \]

[In]

int(-1/((2*x - 1)*(3*x + 2)^6*(5*x + 3)),x)

[Out]

(15625*log(x + 3/5))/11 - (167115051*log(x + 2/3))/117649 - (64*log(x - 1/2))/1294139 + ((23523635*x)/201684 +
(39173854*x^2)/151263 + (25783447*x^3)/100842 + (1591571*x^4)/16807 + 29811193/1512630)/((80*x)/81 + (80*x^2)
/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)