3.259 \(\int \frac{-9-9 x+2 x^2}{-9 x+x^3} \, dx\)

Optimal. Leaf size=17 \[ -\log (3-x)+\log (x)+2 \log (x+3) \]

[Out]

-Log[3 - x] + Log[x] + 2*Log[3 + x]

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Rubi [A]  time = 0.0370496, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1593, 1802} \[ -\log (3-x)+\log (x)+2 \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[(-9 - 9*x + 2*x^2)/(-9*x + x^3),x]

[Out]

-Log[3 - x] + Log[x] + 2*Log[3 + 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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \frac{-9-9 x+2 x^2}{-9 x+x^3} \, dx &=\int \frac{-9-9 x+2 x^2}{x \left (-9+x^2\right )} \, dx\\ &=\int \left (\frac{1}{3-x}+\frac{1}{x}+\frac{2}{3+x}\right ) \, dx\\ &=-\log (3-x)+\log (x)+2 \log (3+x)\\ \end{align*}

Mathematica [A]  time = 0.0046504, size = 17, normalized size = 1. \[ -\log (3-x)+\log (x)+2 \log (x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 9*x + 2*x^2)/(-9*x + x^3),x]

[Out]

-Log[3 - x] + Log[x] + 2*Log[3 + x]

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Maple [A]  time = 0.006, size = 16, normalized size = 0.9 \begin{align*} -\ln \left ( -3+x \right ) +\ln \left ( x \right ) +2\,\ln \left ( 3+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-9*x-9)/(x^3-9*x),x)

[Out]

-ln(-3+x)+ln(x)+2*ln(3+x)

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Maxima [A]  time = 0.974869, size = 20, normalized size = 1.18 \begin{align*} 2 \, \log \left (x + 3\right ) - \log \left (x - 3\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-9*x-9)/(x^3-9*x),x, algorithm="maxima")

[Out]

2*log(x + 3) - log(x - 3) + log(x)

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Fricas [A]  time = 1.44348, size = 49, normalized size = 2.88 \begin{align*} 2 \, \log \left (x + 3\right ) - \log \left (x - 3\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-9*x-9)/(x^3-9*x),x, algorithm="fricas")

[Out]

2*log(x + 3) - log(x - 3) + log(x)

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Sympy [A]  time = 0.115618, size = 14, normalized size = 0.82 \begin{align*} \log{\left (x \right )} - \log{\left (x - 3 \right )} + 2 \log{\left (x + 3 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-9*x-9)/(x**3-9*x),x)

[Out]

log(x) - log(x - 3) + 2*log(x + 3)

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Giac [A]  time = 1.41245, size = 24, normalized size = 1.41 \begin{align*} 2 \, \log \left ({\left | x + 3 \right |}\right ) - \log \left ({\left | x - 3 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-9*x-9)/(x^3-9*x),x, algorithm="giac")

[Out]

2*log(abs(x + 3)) - log(abs(x - 3)) + log(abs(x))