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

Optimal. Leaf size=6 \[ \log (2-x) \]

[Out]

Log[2 - x]

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Rubi [A]  time = 0.0079106, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1586, 31} \[ \log (2-x) \]

Antiderivative was successfully verified.

[In]

Int[(x + x^2)/(-2*x - x^2 + x^3),x]

[Out]

Log[2 - x]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0007617, size = 4, normalized size = 0.67 \[ \log (x-2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-2 + x]

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Maple [A]  time = 0., size = 5, normalized size = 0.8 \begin{align*} \ln \left ( -2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(-2+x)

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Maxima [A]  time = 1.00519, size = 5, normalized size = 0.83 \begin{align*} \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 2)

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Fricas [A]  time = 0.954969, size = 16, normalized size = 2.67 \begin{align*} \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 2)

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Sympy [A]  time = 0.058289, size = 3, normalized size = 0.5 \begin{align*} \log{\left (x - 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 2)

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Giac [A]  time = 1.104, size = 7, normalized size = 1.17 \begin{align*} \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x - 2))