3.1499 \(\int \frac{1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=86 \[ \frac{4774713}{16807 (3 x+2)}+\frac{136419}{4802 (3 x+2)^2}+\frac{1299}{343 (3 x+2)^3}+\frac{111}{196 (3 x+2)^4}+\frac{3}{35 (3 x+2)^5}-\frac{64 \log (1-2 x)}{1294139}-\frac{167115051 \log (3 x+2)}{117649}+\frac{15625}{11} \log (5 x+3) \]

[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

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Rubi [A]  time = 0.0398258, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {72} \[ \frac{4774713}{16807 (3 x+2)}+\frac{136419}{4802 (3 x+2)^2}+\frac{1299}{343 (3 x+2)^3}+\frac{111}{196 (3 x+2)^4}+\frac{3}{35 (3 x+2)^5}-\frac{64 \log (1-2 x)}{1294139}-\frac{167115051 \log (3 x+2)}{117649}+\frac{15625}{11} \log (5 x+3) \]

Antiderivative was successfully verified.

[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 72

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*} \int \frac{1}{(1-2 x) (2+3 x)^6 (3+5 x)} \, dx &=\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]  time = 0.0653973, size = 60, normalized size = 0.7 \[ \frac{\frac{2079 \left (286482780 x^4+773503410 x^3+783477080 x^2+352854525 x+59622386\right )}{4 (3 x+2)^5}-320 \log (1-2 x)-9191327805 \log (6 x+4)[phone] \log (10 x+6)}{6470695} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.008, size = 71, normalized size = 0.8 \begin{align*} -{\frac{64\,\ln \left ( 2\,x-1 \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}{33614+50421\,x}}-{\frac{167115051\,\ln \left ( 2+3\,x \right ) }{117649}}+{\frac{15625\,\ln \left ( 3+5\,x \right ) }{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-64/1294139*ln(2*x-1)+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)-167115051/117649*ln(2+3*x)+15625/11*ln(3+5*x)

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Maxima [A]  time = 2.50955, size = 100, normalized size = 1.16 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.23343, size = 531, normalized size = 6.17 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.236017, size = 75, normalized size = 0.87 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 272273400
*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

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Giac [A]  time = 3.08002, size = 77, normalized size = 0.9 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))