3.101.20 \(\int \frac {-5 x^2+2 x^3+(-4 x+2 x^2) \log (-2+x)}{-8 x^2+4 x^3+(-8 x+4 x^2) \log (-2+x)+(-2+x) \log ^2(-2+x)} \, dx\)

Optimal. Leaf size=20 \[ 5+\log \left (\frac {9}{4}\right )+\frac {x^2}{2 x+\log (-2+x)} \]

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Rubi [F]  time = 0.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-5 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (-2+x)}{-8 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (-2+x)+(-2+x) \log ^2(-2+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*Defer[Int][(2*x + Log[-2 + x])^(-2), x] - 4*Defer[Int][1/((-2 + x)*(2*x + Log[-2 + x])^2), x] - Defer[Int][
x/(2*x + Log[-2 + x])^2, x] - 2*Defer[Int][x^2/(2*x + Log[-2 + x])^2, x] + 2*Defer[Int][x/(2*x + Log[-2 + x]),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x (-x (-5+2 x)-2 (-2+x) \log (-2+x))}{(2-x) (2 x+\log (-2+x))^2} \, dx\\ &=\int \left (-\frac {x^2 (-3+2 x)}{(-2+x) (2 x+\log (-2+x))^2}+\frac {2 x}{2 x+\log (-2+x)}\right ) \, dx\\ &=2 \int \frac {x}{2 x+\log (-2+x)} \, dx-\int \frac {x^2 (-3+2 x)}{(-2+x) (2 x+\log (-2+x))^2} \, dx\\ &=2 \int \frac {x}{2 x+\log (-2+x)} \, dx-\int \left (\frac {2}{(2 x+\log (-2+x))^2}+\frac {4}{(-2+x) (2 x+\log (-2+x))^2}+\frac {x}{(2 x+\log (-2+x))^2}+\frac {2 x^2}{(2 x+\log (-2+x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{(2 x+\log (-2+x))^2} \, dx\right )-2 \int \frac {x^2}{(2 x+\log (-2+x))^2} \, dx+2 \int \frac {x}{2 x+\log (-2+x)} \, dx-4 \int \frac {1}{(-2+x) (2 x+\log (-2+x))^2} \, dx-\int \frac {x}{(2 x+\log (-2+x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^2}{2 x+\log (-2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(2*x + Log[-2 + x])

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fricas [A]  time = 1.91, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{2}}{2 \, x + \log \left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-4*x)*log(x-2)+2*x^3-5*x^2)/((x-2)*log(x-2)^2+(4*x^2-8*x)*log(x-2)+4*x^3-8*x^2),x, algorithm=
"fricas")

[Out]

x^2/(2*x + log(x - 2))

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giac [A]  time = 0.14, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{2}}{2 \, x + \log \left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-4*x)*log(x-2)+2*x^3-5*x^2)/((x-2)*log(x-2)^2+(4*x^2-8*x)*log(x-2)+4*x^3-8*x^2),x, algorithm=
"giac")

[Out]

x^2/(2*x + log(x - 2))

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maple [A]  time = 0.04, size = 15, normalized size = 0.75

method	result	size
norman	\(\frac {x^{2}}{2 x +\ln \left (x -2\right )}\)	\(15\)
risch	\(\frac {x^{2}}{2 x +\ln \left (x -2\right )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-4*x)*ln(x-2)+2*x^3-5*x^2)/((x-2)*ln(x-2)^2+(4*x^2-8*x)*ln(x-2)+4*x^3-8*x^2),x,method=_RETURNVERBOS
E)

[Out]

x^2/(2*x+ln(x-2))

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maxima [A]  time = 0.48, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^{2}}{2 \, x + \log \left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-4*x)*log(x-2)+2*x^3-5*x^2)/((x-2)*log(x-2)^2+(4*x^2-8*x)*log(x-2)+4*x^3-8*x^2),x, algorithm=
"maxima")

[Out]

x^2/(2*x + log(x - 2))

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mupad [B]  time = 7.18, size = 14, normalized size = 0.70 \begin {gather*} \frac {x^2}{2\,x+\ln \left (x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x - 2)*(4*x - 2*x^2) + 5*x^2 - 2*x^3)/(log(x - 2)*(8*x - 4*x^2) + 8*x^2 - 4*x^3 - log(x - 2)^2*(x - 2
)),x)

[Out]

x^2/(2*x + log(x - 2))

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sympy [A]  time = 0.12, size = 10, normalized size = 0.50 \begin {gather*} \frac {x^{2}}{2 x + \log {\left (x - 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-4*x)*ln(x-2)+2*x**3-5*x**2)/((x-2)*ln(x-2)**2+(4*x**2-8*x)*ln(x-2)+4*x**3-8*x**2),x)

[Out]

x**2/(2*x + log(x - 2))

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