3.432 \(\int \frac {24+8 x}{x (-4+x^2)} \, dx\)

Optimal. Leaf size=17 \[ 5 \log (2-x)-6 \log (x)+\log (x+2) \]

[Out]

5*ln(2-x)-6*ln(x)+ln(2+x)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {801} \[ 5 \log (2-x)-6 \log (x)+\log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(24 + 8*x)/(x*(-4 + x^2)),x]

[Out]

5*Log[2 - x] - 6*Log[x] + Log[2 + x]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

\begin {align*} \int \frac {24+8 x}{x \left (-4+x^2\right )} \, dx &=\int \left (\frac {5}{-2+x}-\frac {6}{x}+\frac {1}{2+x}\right ) \, dx\\ &=5 \log (2-x)-6 \log (x)+\log (2+x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.59 \[ 8 \left (\frac {5}{8} \log (2-x)-\frac {3 \log (x)}{4}+\frac {1}{8} \log (x+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(24 + 8*x)/(x*(-4 + x^2)),x]

[Out]

8*((5*Log[2 - x])/8 - (3*Log[x])/4 + Log[2 + x]/8)

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fricas [A]  time = 0.71, size = 15, normalized size = 0.88 \[ \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) - 6 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24+8*x)/x/(x^2-4),x, algorithm="fricas")

[Out]

log(x + 2) + 5*log(x - 2) - 6*log(x)

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giac [A]  time = 0.33, size = 18, normalized size = 1.06 \[ \log \left ({\left | x + 2 \right |}\right ) + 5 \, \log \left ({\left | x - 2 \right |}\right ) - 6 \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24+8*x)/x/(x^2-4),x, algorithm="giac")

[Out]

log(abs(x + 2)) + 5*log(abs(x - 2)) - 6*log(abs(x))

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maple [A]  time = 0.01, size = 16, normalized size = 0.94 \[ -6 \ln \relax (x )+5 \ln \left (x -2\right )+\ln \left (x +2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24+8*x)/x/(x^2-4),x)

[Out]

5*ln(x-2)+ln(x+2)-6*ln(x)

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maxima [A]  time = 0.75, size = 15, normalized size = 0.88 \[ \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) - 6 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24+8*x)/x/(x^2-4),x, algorithm="maxima")

[Out]

log(x + 2) + 5*log(x - 2) - 6*log(x)

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mupad [B]  time = 0.06, size = 15, normalized size = 0.88 \[ 5\,\ln \left (x-2\right )+\ln \left (x+2\right )-6\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 24)/(x*(x^2 - 4)),x)

[Out]

5*log(x - 2) + log(x + 2) - 6*log(x)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.88 \[ - 6 \log {\relax (x )} + 5 \log {\left (x - 2 \right )} + \log {\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24+8*x)/x/(x**2-4),x)

[Out]

-6*log(x) + 5*log(x - 2) + log(x + 2)

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