3.281 \(\int \frac{-3+2 x}{-x^2+x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac{3}{x}-\log (1-x)+\log (x) \]

[Out]

-3/x - Log[1 - x] + Log[x]

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Rubi [A]  time = 0.0136957, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1593, 77} \[ -\frac{3}{x}-\log (1-x)+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*x)/(-x^2 + x^3),x]

[Out]

-3/x - Log[1 - x] + Log[x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{-3+2 x}{-x^2+x^3} \, dx &=\int \frac{-3+2 x}{(-1+x) x^2} \, dx\\ &=\int \left (\frac{1}{1-x}+\frac{3}{x^2}+\frac{1}{x}\right ) \, dx\\ &=-\frac{3}{x}-\log (1-x)+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.0032564, size = 16, normalized size = 1. \[ -\frac{3}{x}-\log (1-x)+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*x)/(-x^2 + x^3),x]

[Out]

-3/x - Log[1 - x] + Log[x]

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Maple [A]  time = 0.051, size = 15, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -3\,{x}^{-1}-\ln \left ( -1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)-3/x-ln(-1+x)

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Maxima [A]  time = 1.12229, size = 19, normalized size = 1.19 \begin{align*} -\frac{3}{x} - \log \left (x - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/x - log(x - 1) + log(x)

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Fricas [A]  time = 2.06159, size = 46, normalized size = 2.88 \begin{align*} -\frac{x \log \left (x - 1\right ) - x \log \left (x\right ) + 3}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*log(x - 1) - x*log(x) + 3)/x

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Sympy [A]  time = 0.147027, size = 10, normalized size = 0.62 \begin{align*} \log{\left (x \right )} - \log{\left (x - 1 \right )} - \frac{3}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x - 1) - 3/x

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Giac [A]  time = 1.24333, size = 22, normalized size = 1.38 \begin{align*} -\frac{3}{x} - \log \left ({\left | x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/x - log(abs(x - 1)) + log(abs(x))