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

Optimal. Leaf size=43 \[ -\frac{3}{25 (1-4 x)}+\frac{1}{10 (1-4 x)^2}-\frac{9}{125} \log (1-4 x)+\frac{9}{125} \log (2-3 x) \]

[Out]

1/(10*(1 - 4*x)^2) - 3/(25*(1 - 4*x)) - (9*Log[1 - 4*x])/125 + (9*Log[2 - 3*x])/125

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Rubi [A]  time = 0.0194159, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {44} \[ -\frac{3}{25 (1-4 x)}+\frac{1}{10 (1-4 x)^2}-\frac{9}{125} \log (1-4 x)+\frac{9}{125} \log (2-3 x) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 4*x)^3*(2 - 3*x)),x]

[Out]

1/(10*(1 - 4*x)^2) - 3/(25*(1 - 4*x)) - (9*Log[1 - 4*x])/125 + (9*Log[2 - 3*x])/125

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(1-4 x)^3 (2-3 x)} \, dx &=\int \left (\frac{27}{125 (-2+3 x)}-\frac{4}{5 (-1+4 x)^3}-\frac{12}{25 (-1+4 x)^2}-\frac{36}{125 (-1+4 x)}\right ) \, dx\\ &=\frac{1}{10 (1-4 x)^2}-\frac{3}{25 (1-4 x)}-\frac{9}{125} \log (1-4 x)+\frac{9}{125} \log (2-3 x)\\ \end{align*}

Mathematica [A]  time = 0.0187331, size = 46, normalized size = 1.07 \[ \frac{120 x+18 (1-4 x)^2 \log (8-12 x)-18 (1-4 x)^2 \log (4 x-1)-5}{250 (1-4 x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5 + 120*x + 18*(1 - 4*x)^2*Log[8 - 12*x] - 18*(1 - 4*x)^2*Log[-1 + 4*x])/(250*(1 - 4*x)^2)

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Maple [A]  time = 0.01, size = 36, normalized size = 0.8 \begin{align*}{\frac{9\,\ln \left ( -2+3\,x \right ) }{125}}+{\frac{1}{10\, \left ( -1+4\,x \right ) ^{2}}}+{\frac{3}{-25+100\,x}}-{\frac{9\,\ln \left ( -1+4\,x \right ) }{125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/125*ln(-2+3*x)+1/10/(-1+4*x)^2+3/25/(-1+4*x)-9/125*ln(-1+4*x)

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Maxima [A]  time = 0.936639, size = 49, normalized size = 1.14 \begin{align*} \frac{24 \, x - 1}{50 \,{\left (16 \, x^{2} - 8 \, x + 1\right )}} - \frac{9}{125} \, \log \left (4 \, x - 1\right ) + \frac{9}{125} \, \log \left (3 \, x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(24*x - 1)/(16*x^2 - 8*x + 1) - 9/125*log(4*x - 1) + 9/125*log(3*x - 2)

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Fricas [A]  time = 1.8645, size = 153, normalized size = 3.56 \begin{align*} -\frac{18 \,{\left (16 \, x^{2} - 8 \, x + 1\right )} \log \left (4 \, x - 1\right ) - 18 \,{\left (16 \, x^{2} - 8 \, x + 1\right )} \log \left (3 \, x - 2\right ) - 120 \, x + 5}{250 \,{\left (16 \, x^{2} - 8 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/250*(18*(16*x^2 - 8*x + 1)*log(4*x - 1) - 18*(16*x^2 - 8*x + 1)*log(3*x - 2) - 120*x + 5)/(16*x^2 - 8*x + 1
)

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Sympy [A]  time = 0.136271, size = 34, normalized size = 0.79 \begin{align*} \frac{24 x - 1}{800 x^{2} - 400 x + 50} + \frac{9 \log{\left (x - \frac{2}{3} \right )}}{125} - \frac{9 \log{\left (x - \frac{1}{4} \right )}}{125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(24*x - 1)/(800*x**2 - 400*x + 50) + 9*log(x - 2/3)/125 - 9*log(x - 1/4)/125

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Giac [A]  time = 1.06351, size = 45, normalized size = 1.05 \begin{align*} \frac{24 \, x - 1}{50 \,{\left (4 \, x - 1\right )}^{2}} - \frac{9}{125} \, \log \left ({\left | 4 \, x - 1 \right |}\right ) + \frac{9}{125} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(24*x - 1)/(4*x - 1)^2 - 9/125*log(abs(4*x - 1)) + 9/125*log(abs(3*x - 2))