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

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

[Out]

ln(2-x)

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Rubi [A]  time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 31

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

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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.67, size = 4, normalized size = 0.67 \[ \log \left (x - 2\right ) \]

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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giac [A]  time = 0.28, size = 5, normalized size = 0.83 \[ \log \left ({\left | x - 2 \right |}\right ) \]

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))

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maple [A]  time = 0.00, size = 5, normalized size = 0.83 \[ \ln \left (x -2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x-2)

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maxima [A]  time = 0.57, size = 4, normalized size = 0.67 \[ \log \left (x - 2\right ) \]

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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mupad [B]  time = 0.02, size = 4, normalized size = 0.67 \[ \ln \left (x-2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + x^2)/(2*x + x^2 - x^3),x)

[Out]

log(x - 2)

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sympy [A]  time = 0.06, size = 3, normalized size = 0.50 \[ \log {\left (x - 2 \right )} \]

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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