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

Optimal. Leaf size=23 \[ 2 \log (1-x)+\frac{\log (x)}{2}-\frac{1}{2} \log (x+2) \]

[Out]

2*Log[1 - x] + Log[x]/2 - Log[2 + x]/2

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Rubi [A]  time = 0.033277, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1594, 1628} \[ 2 \log (1-x)+\frac{\log (x)}{2}-\frac{1}{2} \log (x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[1 - x] + Log[x]/2 - Log[2 + x]/2

Rule 1594

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

Rule 1628

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

Rubi steps

\begin{align*} \int \frac{-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx &=\int \frac{-1+5 x+2 x^2}{x \left (-2+x+x^2\right )} \, dx\\ &=\int \left (\frac{2}{-1+x}+\frac{1}{2 x}-\frac{1}{2 (2+x)}\right ) \, dx\\ &=2 \log (1-x)+\frac{\log (x)}{2}-\frac{1}{2} \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0049563, size = 23, normalized size = 1. \[ 2 \log (1-x)+\frac{\log (x)}{2}-\frac{1}{2} \log (x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[1 - x] + Log[x]/2 - Log[2 + x]/2

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Maple [A]  time = 0.007, size = 18, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 2+x \right ) }{2}}+{\frac{\ln \left ( x \right ) }{2}}+2\,\ln \left ( -1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*ln(2+x)+1/2*ln(x)+2*ln(-1+x)

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Maxima [A]  time = 0.929451, size = 23, normalized size = 1. \begin{align*} -\frac{1}{2} \, \log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x + 2) + 2*log(x - 1) + 1/2*log(x)

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Fricas [A]  time = 0.654882, size = 61, normalized size = 2.65 \begin{align*} -\frac{1}{2} \, \log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(x + 2) + 2*log(x - 1) + 1/2*log(x)

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Sympy [A]  time = 0.126741, size = 17, normalized size = 0.74 \begin{align*} \frac{\log{\left (x \right )}}{2} + 2 \log{\left (x - 1 \right )} - \frac{\log{\left (x + 2 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/2 + 2*log(x - 1) - log(x + 2)/2

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Giac [A]  time = 1.09716, size = 27, normalized size = 1.17 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x + 2 \right |}\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(x + 2)) + 2*log(abs(x - 1)) + 1/2*log(abs(x))