3.46.72 \(\int \frac {-36+8 x-x^2}{16-8 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ -2+2^{2/3}-x-\frac {5 x}{4-x}+\log (3) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {27, 683} \begin {gather*} -x-\frac {20}{4-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36 + 8*x - x^2)/(16 - 8*x + x^2),x]

[Out]

-20/(4 - x) - x

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-36+8 x-x^2}{(-4+x)^2} \, dx\\ &=\int \left (-1-\frac {20}{(-4+x)^2}\right ) \, dx\\ &=-\frac {20}{4-x}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.50 \begin {gather*} \frac {20}{-4+x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36 + 8*x - x^2)/(16 - 8*x + x^2),x]

[Out]

20/(-4 + x) - x

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fricas [A]  time = 0.57, size = 15, normalized size = 0.68 \begin {gather*} -\frac {x^{2} - 4 \, x - 20}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+8*x-36)/(x^2-8*x+16),x, algorithm="fricas")

[Out]

-(x^2 - 4*x - 20)/(x - 4)

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giac [A]  time = 0.12, size = 11, normalized size = 0.50 \begin {gather*} -x + \frac {20}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+8*x-36)/(x^2-8*x+16),x, algorithm="giac")

[Out]

-x + 20/(x - 4)

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maple [A]  time = 0.19, size = 12, normalized size = 0.55

method	result	size
default	\(-x +\frac {20}{x -4}\)	\(12\)
risch	\(-x +\frac {20}{x -4}\)	\(12\)
gosper	\(-\frac {x^{2}-36}{x -4}\)	\(13\)
meijerg	\(-\frac {x}{4 \left (-\frac {x}{4}+1\right )}-\frac {x \left (-\frac {3 x}{4}+6\right )}{3 \left (-\frac {x}{4}+1\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+8*x-36)/(x^2-8*x+16),x,method=_RETURNVERBOSE)

[Out]

-x+20/(x-4)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.50 \begin {gather*} -x + \frac {20}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+8*x-36)/(x^2-8*x+16),x, algorithm="maxima")

[Out]

-x + 20/(x - 4)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.50 \begin {gather*} \frac {20}{x-4}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 8*x + 36)/(x^2 - 8*x + 16),x)

[Out]

20/(x - 4) - x

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sympy [A]  time = 0.07, size = 5, normalized size = 0.23 \begin {gather*} - x + \frac {20}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+8*x-36)/(x**2-8*x+16),x)

[Out]

-x + 20/(x - 4)

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