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

Optimal. Leaf size=64 \[ \frac {404}{41503 (1-2 x)}+\frac {27}{343 (3 x+2)}+\frac {2}{539 (1-2 x)^2}-\frac {27208 \log (1-2 x)}{3195731}-\frac {1107 \log (3 x+2)}{2401}+\frac {625 \log (5 x+3)}{1331} \]

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Rubi [A]  time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {404}{41503 (1-2 x)}+\frac {27}{343 (3 x+2)}+\frac {2}{539 (1-2 x)^2}-\frac {27208 \log (1-2 x)}{3195731}-\frac {1107 \log (3 x+2)}{2401}+\frac {625 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/(539*(1 - 2*x)^2) + 404/(41503*(1 - 2*x)) + 27/(343*(2 + 3*x)) - (27208*Log[1 - 2*x])/3195731 - (1107*Log[2
+ 3*x])/2401 + (625*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)^2 (3+5 x)} \, dx &=\int \left (-\frac {8}{539 (-1+2 x)^3}+\frac {808}{41503 (-1+2 x)^2}-\frac {54416}{3195731 (-1+2 x)}-\frac {81}{343 (2+3 x)^2}-\frac {3321}{2401 (2+3 x)}+\frac {3125}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {2}{539 (1-2 x)^2}+\frac {404}{41503 (1-2 x)}+\frac {27}{343 (2+3 x)}-\frac {27208 \log (1-2 x)}{3195731}-\frac {1107 \log (2+3 x)}{2401}+\frac {625 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 57, normalized size = 0.89 \begin {gather*} \frac {\frac {77 \left (10644 x^2-13010 x+4383\right )}{(1-2 x)^2 (3 x+2)}-27208 \log (5-10 x)-1473417 \log (5 (3 x+2))+1500625 \log (5 x+3)}{3195731} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(4383 - 13010*x + 10644*x^2))/((1 - 2*x)^2*(2 + 3*x)) - 27208*Log[5 - 10*x] - 1473417*Log[5*(2 + 3*x)] +
1500625*Log[3 + 5*x])/3195731

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.59, size = 98, normalized size = 1.53 \begin {gather*} \frac {819588 \, x^{2} + 1500625 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (5 \, x + 3\right ) - 1473417 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 27208 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (2 \, x - 1\right ) - 1001770 \, x + 337491}{3195731 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3195731*(819588*x^2 + 1500625*(12*x^3 - 4*x^2 - 5*x + 2)*log(5*x + 3) - 1473417*(12*x^3 - 4*x^2 - 5*x + 2)*l
og(3*x + 2) - 27208*(12*x^3 - 4*x^2 - 5*x + 2)*log(2*x - 1) - 1001770*x + 337491)/(12*x^3 - 4*x^2 - 5*x + 2)

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giac [A]  time = 1.17, size = 66, normalized size = 1.03 \begin {gather*} \frac {27}{343 \, {\left (3 \, x + 2\right )}} + \frac {24 \, {\left (\frac {938}{3 \, x + 2} - 235\right )}}{290521 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} + \frac {625}{1331} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {27208}{3195731} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/343/(3*x + 2) + 24/290521*(938/(3*x + 2) - 235)/(7/(3*x + 2) - 2)^2 + 625/1331*log(abs(-1/(3*x + 2) + 5)) -
27208/3195731*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 53, normalized size = 0.83 \begin {gather*} -\frac {27208 \ln \left (2 x -1\right )}{3195731}-\frac {1107 \ln \left (3 x +2\right )}{2401}+\frac {625 \ln \left (5 x +3\right )}{1331}+\frac {27}{343 \left (3 x +2\right )}+\frac {2}{539 \left (2 x -1\right )^{2}}-\frac {404}{41503 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/1331*ln(5*x+3)+27/343/(3*x+2)-1107/2401*ln(3*x+2)+2/539/(2*x-1)^2-404/41503/(2*x-1)-27208/3195731*ln(2*x-1
)

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maxima [A]  time = 0.54, size = 54, normalized size = 0.84 \begin {gather*} \frac {10644 \, x^{2} - 13010 \, x + 4383}{41503 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} + \frac {625}{1331} \, \log \left (5 \, x + 3\right ) - \frac {1107}{2401} \, \log \left (3 \, x + 2\right ) - \frac {27208}{3195731} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/41503*(10644*x^2 - 13010*x + 4383)/(12*x^3 - 4*x^2 - 5*x + 2) + 625/1331*log(5*x + 3) - 1107/2401*log(3*x +
2) - 27208/3195731*log(2*x - 1)

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mupad [B]  time = 0.04, size = 48, normalized size = 0.75 \begin {gather*} \frac {625\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {1107\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {27208\,\ln \left (x-\frac {1}{2}\right )}{3195731}-\frac {\frac {887\,x^2}{41503}-\frac {6505\,x}{249018}+\frac {1461}{166012}}{-x^3+\frac {x^2}{3}+\frac {5\,x}{12}-\frac {1}{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(625*log(x + 3/5))/1331 - (1107*log(x + 2/3))/2401 - (27208*log(x - 1/2))/3195731 - ((887*x^2)/41503 - (6505*x
)/249018 + 1461/166012)/((5*x)/12 + x^2/3 - x^3 - 1/6)

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sympy [A]  time = 0.24, size = 54, normalized size = 0.84 \begin {gather*} - \frac {- 10644 x^{2} + 13010 x - 4383}{498036 x^{3} - 166012 x^{2} - 207515 x + 83006} - \frac {27208 \log {\left (x - \frac {1}{2} \right )}}{3195731} + \frac {625 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {1107 \log {\left (x + \frac {2}{3} \right )}}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-10644*x**2 + 13010*x - 4383)/(498036*x**3 - 166012*x**2 - 207515*x + 83006) - 27208*log(x - 1/2)/3195731 +
625*log(x + 3/5)/1331 - 1107*log(x + 2/3)/2401

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