3.1683 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=86 \[ \frac{2672}{2033647 (1-2 x)}+\frac{39393}{16807 (3 x+2)}+\frac{8}{26411 (1-2 x)^2}+\frac{1107}{4802 (3 x+2)^2}+\frac{9}{343 (3 x+2)^3}-\frac{267760 \log (1-2 x)}{156590819}-\frac{1380915 \log (3 x+2)}{117649}+\frac{15625 \log (5 x+3)}{1331} \]

[Out]

8/(26411*(1 - 2*x)^2) + 2672/(2033647*(1 - 2*x)) + 9/(343*(2 + 3*x)^3) + 1107/(4802*(2 + 3*x)^2) + 39393/(1680
7*(2 + 3*x)) - (267760*Log[1 - 2*x])/156590819 - (1380915*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0450109, 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 = {88} \[ \frac{2672}{2033647 (1-2 x)}+\frac{39393}{16807 (3 x+2)}+\frac{8}{26411 (1-2 x)^2}+\frac{1107}{4802 (3 x+2)^2}+\frac{9}{343 (3 x+2)^3}-\frac{267760 \log (1-2 x)}{156590819}-\frac{1380915 \log (3 x+2)}{117649}+\frac{15625 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

8/(26411*(1 - 2*x)^2) + 2672/(2033647*(1 - 2*x)) + 9/(343*(2 + 3*x)^3) + 1107/(4802*(2 + 3*x)^2) + 39393/(1680
7*(2 + 3*x)) - (267760*Log[1 - 2*x])/156590819 - (1380915*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/1331

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^3 (2+3 x)^4 (3+5 x)} \, dx &=\int \left (-\frac{32}{26411 (-1+2 x)^3}+\frac{5344}{2033647 (-1+2 x)^2}-\frac{535520}{156590819 (-1+2 x)}-\frac{81}{343 (2+3 x)^4}-\frac{3321}{2401 (2+3 x)^3}-\frac{118179}{16807 (2+3 x)^2}-\frac{4142745}{117649 (2+3 x)}+\frac{78125}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{8}{26411 (1-2 x)^2}+\frac{2672}{2033647 (1-2 x)}+\frac{9}{343 (2+3 x)^3}+\frac{1107}{4802 (2+3 x)^2}+\frac{39393}{16807 (2+3 x)}-\frac{267760 \log (1-2 x)}{156590819}-\frac{1380915 \log (2+3 x)}{117649}+\frac{15625 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.083036, size = 69, normalized size = 0.8 \[ \frac{5 \left (\frac{77 \left (342903240 x^4+125249220 x^3-222614730 x^2-43096225 x+40167012\right )}{5 (1-2 x)^2 (3 x+2)^3}-107104 \log (5-10 x)-735199146 \log (5 (3 x+2))[phone] \log (5 x+3)\right )}{313181638} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^3*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(5*((77*(40167012 - 43096225*x - 222614730*x^2 + 125249220*x^3 + 342903240*x^4))/(5*(1 - 2*x)^2*(2 + 3*x)^3) -
107104*Log[5 - 10*x] - 735199146*Log[5*(2 + 3*x)] + 735306250*Log[3 + 5*x]))/313181638

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Maple [A]  time = 0.008, size = 71, normalized size = 0.8 \begin{align*}{\frac{8}{26411\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{2672}{4067294\,x-2033647}}-{\frac{267760\,\ln \left ( 2\,x-1 \right ) }{156590819}}+{\frac{9}{343\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1107}{4802\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{39393}{33614+50421\,x}}-{\frac{1380915\,\ln \left ( 2+3\,x \right ) }{117649}}+{\frac{15625\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^3/(2+3*x)^4/(3+5*x),x)

[Out]

8/26411/(2*x-1)^2-2672/2033647/(2*x-1)-267760/156590819*ln(2*x-1)+9/343/(2+3*x)^3+1107/4802/(2+3*x)^2+39393/16
807/(2+3*x)-1380915/117649*ln(2+3*x)+15625/1331*ln(3+5*x)

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Maxima [A]  time = 2.39449, size = 100, normalized size = 1.16 \begin{align*} \frac{342903240 \, x^{4} + 125249220 \, x^{3} - 222614730 \, x^{2} - 43096225 \, x + 40167012}{4067294 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac{15625}{1331} \, \log \left (5 \, x + 3\right ) - \frac{1380915}{117649} \, \log \left (3 \, x + 2\right ) - \frac{267760}{156590819} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^3/(2+3*x)^4/(3+5*x),x, algorithm="maxima")

[Out]

1/4067294*(342903240*x^4 + 125249220*x^3 - 222614730*x^2 - 43096225*x + 40167012)/(108*x^5 + 108*x^4 - 45*x^3
- 58*x^2 + 4*x + 8) + 15625/1331*log(5*x + 3) - 1380915/117649*log(3*x + 2) - 267760/156590819*log(2*x - 1)

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Fricas [B]  time = 1.45376, size = 486, normalized size = 5.65 \begin{align*} \frac{26403549480 \, x^{4} + 9644189940 \, x^{3} - 17141334210 \, x^{2} + 3676531250 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 3675995730 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 535520 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 3318409325 \, x + 3092859924}{313181638 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^3/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/313181638*(26403549480*x^4 + 9644189940*x^3 - 17141334210*x^2 + 3676531250*(108*x^5 + 108*x^4 - 45*x^3 - 58*
x^2 + 4*x + 8)*log(5*x + 3) - 3675995730*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(3*x + 2) - 535520
*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(2*x - 1) - 3318409325*x + 3092859924)/(108*x^5 + 108*x^4
- 45*x^3 - 58*x^2 + 4*x + 8)

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Sympy [A]  time = 0.232034, size = 75, normalized size = 0.87 \begin{align*} \frac{342903240 x^{4} + 125249220 x^{3} - 222614730 x^{2} - 43096225 x + 40167012}{439267752 x^{5} + 439267752 x^{4} - 183028230 x^{3} - 235903052 x^{2} + 16269176 x + 32538352} - \frac{267760 \log{\left (x - \frac{1}{2} \right )}}{156590819} + \frac{15625 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{1380915 \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)**3/(2+3*x)**4/(3+5*x),x)

[Out]

(342903240*x**4 + 125249220*x**3 - 222614730*x**2 - 43096225*x + 40167012)/(439267752*x**5 + 439267752*x**4 -
183028230*x**3 - 235903052*x**2 + 16269176*x + 32538352) - 267760*log(x - 1/2)/156590819 + 15625*log(x + 3/5)/
1331 - 1380915*log(x + 2/3)/117649

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Giac [A]  time = 2.88661, size = 86, normalized size = 1. \begin{align*} \frac{342903240 \, x^{4} + 125249220 \, x^{3} - 222614730 \, x^{2} - 43096225 \, x + 40167012}{4067294 \,{\left (3 \, x + 2\right )}^{3}{\left (2 \, x - 1\right )}^{2}} + \frac{15625}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{1380915}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{267760}{156590819} \, \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)^3/(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

1/4067294*(342903240*x^4 + 125249220*x^3 - 222614730*x^2 - 43096225*x + 40167012)/((3*x + 2)^3*(2*x - 1)^2) +
15625/1331*log(abs(5*x + 3)) - 1380915/117649*log(abs(3*x + 2)) - 267760/156590819*log(abs(2*x - 1))