3.1615 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=86 \[ \frac{32}{290521 (1-2 x)}-\frac{107109}{2401 (3 x+2)}-\frac{3125}{121 (5 x+3)}-\frac{999}{343 (3 x+2)^2}-\frac{9}{49 (3 x+2)^3}-\frac{6464 \log (1-2 x)}{22370117}+\frac{5050944 \log (3 x+2)}{16807}-\frac{400000 \log (5 x+3)}{1331} \]

[Out]

32/(290521*(1 - 2*x)) - 9/(49*(2 + 3*x)^3) - 999/(343*(2 + 3*x)^2) - 107109/(2401*(2 + 3*x)) - 3125/(121*(3 +
5*x)) - (6464*Log[1 - 2*x])/22370117 + (5050944*Log[2 + 3*x])/16807 - (400000*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0450506, 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} \[ \frac{32}{290521 (1-2 x)}-\frac{107109}{2401 (3 x+2)}-\frac{3125}{121 (5 x+3)}-\frac{999}{343 (3 x+2)^2}-\frac{9}{49 (3 x+2)^3}-\frac{6464 \log (1-2 x)}{22370117}+\frac{5050944 \log (3 x+2)}{16807}-\frac{400000 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

32/(290521*(1 - 2*x)) - 9/(49*(2 + 3*x)^3) - 999/(343*(2 + 3*x)^2) - 107109/(2401*(2 + 3*x)) - 3125/(121*(3 +
5*x)) - (6464*Log[1 - 2*x])/22370117 + (5050944*Log[2 + 3*x])/16807 - (400000*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)^2 (2+3 x)^4 (3+5 x)^2} \, dx &=\int \left (\frac{64}{290521 (-1+2 x)^2}-\frac{12928}{22370117 (-1+2 x)}+\frac{81}{49 (2+3 x)^4}+\frac{5994}{343 (2+3 x)^3}+\frac{321327}{2401 (2+3 x)^2}+\frac{15152832}{16807 (2+3 x)}+\frac{15625}{121 (3+5 x)^2}-\frac{2000000}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{32}{290521 (1-2 x)}-\frac{9}{49 (2+3 x)^3}-\frac{999}{343 (2+3 x)^2}-\frac{107109}{2401 (2+3 x)}-\frac{3125}{121 (3+5 x)}-\frac{6464 \log (1-2 x)}{22370117}+\frac{5050944 \log (2+3 x)}{16807}-\frac{400000 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0810015, size = 70, normalized size = 0.81 \[ \frac{-\frac{77 \left (1571590080 x^4+2305013328 x^3+479067048 x^2-570653522 x-220783501\right )}{(3 x+2)^3 \left (10 x^2+x-3\right )}-6464 \log (3-6 x)[phone] \log (3 x+2)-6722800000 \log (-3 (5 x+3))}{22370117} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-77*(-220783501 - 570653522*x + 479067048*x^2 + 2305013328*x^3 + 1571590080*x^4))/((2 + 3*x)^3*(-3 + x + 10*
x^2)) - 6464*Log[3 - 6*x] + 6722806464*Log[2 + 3*x] - 6722800000*Log[-3*(3 + 5*x)])/22370117

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Maple [A]  time = 0.011, size = 71, normalized size = 0.8 \begin{align*} -{\frac{32}{581042\,x-290521}}-{\frac{6464\,\ln \left ( 2\,x-1 \right ) }{22370117}}-{\frac{9}{49\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{999}{343\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{107109}{4802+7203\,x}}+{\frac{5050944\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{3125}{363+605\,x}}-{\frac{400000\,\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)^2/(2+3*x)^4/(3+5*x)^2,x)

[Out]

-32/290521/(2*x-1)-6464/22370117*ln(2*x-1)-9/49/(2+3*x)^3-999/343/(2+3*x)^2-107109/2401/(2+3*x)+5050944/16807*
ln(2+3*x)-3125/121/(3+5*x)-400000/1331*ln(3+5*x)

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Maxima [A]  time = 1.15007, size = 100, normalized size = 1.16 \begin{align*} -\frac{1571590080 \, x^{4} + 2305013328 \, x^{3} + 479067048 \, x^{2} - 570653522 \, x - 220783501}{290521 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} - \frac{400000}{1331} \, \log \left (5 \, x + 3\right ) + \frac{5050944}{16807} \, \log \left (3 \, x + 2\right ) - \frac{6464}{22370117} \, \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/(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/290521*(1571590080*x^4 + 2305013328*x^3 + 479067048*x^2 - 570653522*x - 220783501)/(270*x^5 + 567*x^4 + 333
*x^3 - 46*x^2 - 100*x - 24) - 400000/1331*log(5*x + 3) + 5050944/16807*log(3*x + 2) - 6464/22370117*log(2*x -
1)

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Fricas [B]  time = 1.53268, size = 512, normalized size = 5.95 \begin{align*} -\frac{121012436160 \, x^{4} + 177486026256 \, x^{3} + 36888162696 \, x^{2} + 6722800000 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (5 \, x + 3\right ) - 6722806464 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (3 \, x + 2\right ) + 6464 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \log \left (2 \, x - 1\right ) - 43940321194 \, x - 17000329577}{22370117 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/22370117*(121012436160*x^4 + 177486026256*x^3 + 36888162696*x^2 + 6722800000*(270*x^5 + 567*x^4 + 333*x^3 -
46*x^2 - 100*x - 24)*log(5*x + 3) - 6722806464*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(3*x +
2) + 6464*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*log(2*x - 1) - 43940321194*x - 17000329577)/(270
*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)

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Sympy [A]  time = 0.239733, size = 75, normalized size = 0.87 \begin{align*} - \frac{1571590080 x^{4} + 2305013328 x^{3} + 479067048 x^{2} - 570653522 x - 220783501}{78440670 x^{5} + 164725407 x^{4} + 96743493 x^{3} - 13363966 x^{2} - 29052100 x - 6972504} - \frac{6464 \log{\left (x - \frac{1}{2} \right )}}{22370117} - \frac{400000 \log{\left (x + \frac{3}{5} \right )}}{1331} + \frac{5050944 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1571590080*x**4 + 2305013328*x**3 + 479067048*x**2 - 570653522*x - 220783501)/(78440670*x**5 + 164725407*x**
4 + 96743493*x**3 - 13363966*x**2 - 29052100*x - 6972504) - 6464*log(x - 1/2)/22370117 - 400000*log(x + 3/5)/1
331 + 5050944*log(x + 2/3)/16807

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Giac [A]  time = 2.47352, size = 128, normalized size = 1.49 \begin{align*} -\frac{3125}{121 \,{\left (5 \, x + 3\right )}} + \frac{5 \,{\left (\frac{52083388017}{5 \, x + 3} + \frac{44729490744}{{\left (5 \, x + 3\right )}^{2}} + \frac{9228837286}{{\left (5 \, x + 3\right )}^{3}} - 11003835798\right )}}{3195731 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}{\left (\frac{1}{5 \, x + 3} + 3\right )}^{3}} + \frac{5050944}{16807} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{6464}{22370117} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3125/121/(5*x + 3) + 5/3195731*(52083388017/(5*x + 3) + 44729490744/(5*x + 3)^2 + 9228837286/(5*x + 3)^3 - 11
003835798)/((11/(5*x + 3) - 2)*(1/(5*x + 3) + 3)^3) + 5050944/16807*log(abs(-1/(5*x + 3) - 3)) - 6464/22370117
*log(abs(-11/(5*x + 3) + 2))