3.92 \(\int \frac{1}{2-7 x+3 x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{5} \log (2-x)-\frac{1}{5} \log (1-3 x) \]

[Out]

-Log[1 - 3*x]/5 + Log[2 - x]/5

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Rubi [A]  time = 0.0055049, antiderivative size = 21, 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 = {616, 31} \[ \frac{1}{5} \log (2-x)-\frac{1}{5} \log (1-3 x) \]

Antiderivative was successfully verified.

[In]

Int[(2 - 7*x + 3*x^2)^(-1),x]

[Out]

-Log[1 - 3*x]/5 + Log[2 - x]/5

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[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{1}{2-7 x+3 x^2} \, dx &=\frac{3}{5} \int \frac{1}{-6+3 x} \, dx-\frac{3}{5} \int \frac{1}{-1+3 x} \, dx\\ &=-\frac{1}{5} \log (1-3 x)+\frac{1}{5} \log (2-x)\\ \end{align*}

Mathematica [A]  time = 0.0029235, size = 21, normalized size = 1. \[ \frac{1}{5} \log (2-x)-\frac{1}{5} \log (1-3 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 7*x + 3*x^2)^(-1),x]

[Out]

-Log[1 - 3*x]/5 + Log[2 - x]/5

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 3\,x-1 \right ) }{5}}+{\frac{\ln \left ( -2+x \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2-7*x+2),x)

[Out]

-1/5*ln(3*x-1)+1/5*ln(-2+x)

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Maxima [A]  time = 0.934454, size = 20, normalized size = 0.95 \begin{align*} -\frac{1}{5} \, \log \left (3 \, x - 1\right ) + \frac{1}{5} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-7*x+2),x, algorithm="maxima")

[Out]

-1/5*log(3*x - 1) + 1/5*log(x - 2)

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Fricas [A]  time = 1.77085, size = 49, normalized size = 2.33 \begin{align*} -\frac{1}{5} \, \log \left (3 \, x - 1\right ) + \frac{1}{5} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-7*x+2),x, algorithm="fricas")

[Out]

-1/5*log(3*x - 1) + 1/5*log(x - 2)

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Sympy [A]  time = 0.089842, size = 14, normalized size = 0.67 \begin{align*} \frac{\log{\left (x - 2 \right )}}{5} - \frac{\log{\left (x - \frac{1}{3} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2-7*x+2),x)

[Out]

log(x - 2)/5 - log(x - 1/3)/5

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Giac [A]  time = 1.04913, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{5} \, \log \left ({\left | 3 \, x - 1 \right |}\right ) + \frac{1}{5} \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-7*x+2),x, algorithm="giac")

[Out]

-1/5*log(abs(3*x - 1)) + 1/5*log(abs(x - 2))