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*Log[2 - x] - 6*Log[x] + Log[2 + x]

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Rubi [A]  time = 0.0128238, antiderivative size = 17, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0044864, 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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Maple [A]  time = 0.007, size = 16, normalized size = 0.9 \begin{align*} -6\,\ln \left ( x \right ) +\ln \left ( 2+x \right ) +5\,\ln \left ( -2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.973539, size = 20, normalized size = 1.18 \begin{align*} \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) - 6 \, \log \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.01461, size = 51, normalized size = 3. \begin{align*} \log \left (x + 2\right ) + 5 \, \log \left (x - 2\right ) - 6 \, \log \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.117198, size = 15, normalized size = 0.88 \begin{align*} - 6 \log{\left (x \right )} + 5 \log{\left (x - 2 \right )} + \log{\left (x + 2 \right )} \end{align*}

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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Giac [A]  time = 1.10588, size = 24, normalized size = 1.41 \begin{align*} \log \left ({\left | x + 2 \right |}\right ) + 5 \, \log \left ({\left | x - 2 \right |}\right ) - 6 \, \log \left ({\left | x \right |}\right ) \end{align*}

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))