3.118 \(\int \frac{-6+2 x+x^4}{-2 x+x^2+x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]

[Out]

-x + x^2/2 - Log[1 - x] + 3*Log[x] + Log[2 + x]

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Rubi [A]  time = 0.0307892, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1594, 1628} \[ \frac{x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(-6 + 2*x + x^4)/(-2*x + x^2 + x^3),x]

[Out]

-x + x^2/2 - Log[1 - x] + 3*Log[x] + Log[2 + x]

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{-6+2 x+x^4}{-2 x+x^2+x^3} \, dx &=\int \frac{-6+2 x+x^4}{x \left (-2+x+x^2\right )} \, dx\\ &=\int \left (-1+\frac{1}{1-x}+\frac{3}{x}+x+\frac{1}{2+x}\right ) \, dx\\ &=-x+\frac{x^2}{2}-\log (1-x)+3 \log (x)+\log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0051538, size = 27, normalized size = 1. \[ \frac{x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 2*x + x^4)/(-2*x + x^2 + x^3),x]

[Out]

-x + x^2/2 - Log[1 - x] + 3*Log[x] + Log[2 + x]

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Maple [A]  time = 0.006, size = 24, normalized size = 0.9 \begin{align*} -x+{\frac{{x}^{2}}{2}}+\ln \left ( 2+x \right ) +3\,\ln \left ( x \right ) -\ln \left ( -1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+2*x-6)/(x^3+x^2-2*x),x)

[Out]

-x+1/2*x^2+ln(2+x)+3*ln(x)-ln(-1+x)

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Maxima [A]  time = 0.944474, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left (x + 2\right ) - \log \left (x - 1\right ) + 3 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+2*x-6)/(x^3+x^2-2*x),x, algorithm="maxima")

[Out]

1/2*x^2 - x + log(x + 2) - log(x - 1) + 3*log(x)

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Fricas [A]  time = 0.660478, size = 68, normalized size = 2.52 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left (x + 2\right ) - \log \left (x - 1\right ) + 3 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+2*x-6)/(x^3+x^2-2*x),x, algorithm="fricas")

[Out]

1/2*x^2 - x + log(x + 2) - log(x - 1) + 3*log(x)

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Sympy [A]  time = 0.114105, size = 20, normalized size = 0.74 \begin{align*} \frac{x^{2}}{2} - x + 3 \log{\left (x \right )} - \log{\left (x - 1 \right )} + \log{\left (x + 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+2*x-6)/(x**3+x**2-2*x),x)

[Out]

x**2/2 - x + 3*log(x) - log(x - 1) + log(x + 2)

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Giac [A]  time = 1.09188, size = 35, normalized size = 1.3 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x - 1 \right |}\right ) + 3 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+2*x-6)/(x^3+x^2-2*x),x, algorithm="giac")

[Out]

1/2*x^2 - x + log(abs(x + 2)) - log(abs(x - 1)) + 3*log(abs(x))