3.66.48 \(\int \frac {-5+2 x}{-5 x+x^2} \, dx\)

Optimal. Leaf size=8 \[ \log ((5-x) x) \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.25, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {628} \begin {gather*} \log \left (5 x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + 2*x)/(-5*x + x^2),x]

[Out]

Log[5*x - x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (5 x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.12 \begin {gather*} \log (5-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 2*x)/(-5*x + x^2),x]

[Out]

Log[5 - x] + Log[x]

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fricas [A]  time = 1.02, size = 8, normalized size = 1.00 \begin {gather*} \log \left (x^{2} - 5 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 5*x)

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giac [A]  time = 0.20, size = 9, normalized size = 1.12 \begin {gather*} \log \left ({\left | x^{2} - 5 \, x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x^2 - 5*x))

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maple [A]  time = 0.42, size = 7, normalized size = 0.88

method	result	size
default	\(\ln \left (\left (x -5\right ) x \right )\)	\(7\)
norman	\(\ln \relax (x )+\ln \left (x -5\right )\)	\(8\)
derivativedivides	\(\ln \left (x^{2}-5 x \right )\)	\(9\)
risch	\(\ln \left (x^{2}-5 x \right )\)	\(9\)
meijerg	\(\ln \left (1-\frac {x}{5}\right )+\ln \relax (x )-\ln \relax (5)+i \pi \)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x-5)/(x^2-5*x),x,method=_RETURNVERBOSE)

[Out]

ln((x-5)*x)

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maxima [A]  time = 0.37, size = 8, normalized size = 1.00 \begin {gather*} \log \left (x^{2} - 5 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 5*x)

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mupad [B]  time = 0.05, size = 6, normalized size = 0.75 \begin {gather*} \ln \left (x\,\left (x-5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 5)/(5*x - x^2),x)

[Out]

log(x*(x - 5))

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sympy [A]  time = 0.07, size = 7, normalized size = 0.88 \begin {gather*} \log {\left (x^{2} - 5 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 - 5*x)

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