### 3.2256 $$\int \frac{x}{6-5 x+x^2} \, dx$$

Optimal. Leaf size=17 $3 \log (3-x)-2 \log (2-x)$

[Out]

-2*Log[2 - x] + 3*Log[3 - x]

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Rubi [A]  time = 0.00531, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {632, 31} $3 \log (3-x)-2 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x/(6 - 5*x + x^2),x]

[Out]

-2*Log[2 - x] + 3*Log[3 - x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

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}{6-5 x+x^2} \, dx &=-\left (2 \int \frac{1}{-2+x} \, dx\right )+3 \int \frac{1}{-3+x} \, dx\\ &=-2 \log (2-x)+3 \log (3-x)\\ \end{align*}

Mathematica [A]  time = 0.003102, size = 17, normalized size = 1. $3 \log (3-x)-2 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/(6 - 5*x + x^2),x]

[Out]

-2*Log[2 - x] + 3*Log[3 - x]

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Maple [A]  time = 0.046, size = 14, normalized size = 0.8 \begin{align*} 3\,\ln \left ( -3+x \right ) -2\,\ln \left ( -2+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2-5*x+6),x)

[Out]

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

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Maxima [A]  time = 0.949461, size = 18, normalized size = 1.06 \begin{align*} -2 \, \log \left (x - 2\right ) + 3 \, \log \left (x - 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-5*x+6),x, algorithm="maxima")

[Out]

-2*log(x - 2) + 3*log(x - 3)

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Fricas [A]  time = 2.69446, size = 41, normalized size = 2.41 \begin{align*} -2 \, \log \left (x - 2\right ) + 3 \, \log \left (x - 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-5*x+6),x, algorithm="fricas")

[Out]

-2*log(x - 2) + 3*log(x - 3)

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Sympy [A]  time = 0.096094, size = 12, normalized size = 0.71 \begin{align*} 3 \log{\left (x - 3 \right )} - 2 \log{\left (x - 2 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2-5*x+6),x)

[Out]

3*log(x - 3) - 2*log(x - 2)

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Giac [A]  time = 1.09749, size = 20, normalized size = 1.18 \begin{align*} -2 \, \log \left ({\left | x - 2 \right |}\right ) + 3 \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-5*x+6),x, algorithm="giac")

[Out]

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