3.43.72 \(\int \frac {24-4 x+x^2}{32-6 x^2+x^3} \, dx\)

Optimal. Leaf size=15 \[ -2+\frac {x}{4-x}+\log (2+x) \]

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Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2074} \begin {gather*} \frac {4}{4-x}+\log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24 - 4*x + x^2)/(32 - 6*x^2 + x^3),x]

[Out]

4/(4 - x) + Log[2 + x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.80 \begin {gather*} -\frac {4}{-4+x}+\log (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 - 4*x + x^2)/(32 - 6*x^2 + x^3),x]

[Out]

-4/(-4 + x) + Log[2 + x]

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fricas [A]  time = 0.66, size = 16, normalized size = 1.07 \begin {gather*} \frac {{\left (x - 4\right )} \log \left (x + 2\right ) - 4}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+24)/(x^3-6*x^2+32),x, algorithm="fricas")

[Out]

((x - 4)*log(x + 2) - 4)/(x - 4)

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giac [A]  time = 0.15, size = 13, normalized size = 0.87 \begin {gather*} -\frac {4}{x - 4} + \log \left ({\left | x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+24)/(x^3-6*x^2+32),x, algorithm="giac")

[Out]

-4/(x - 4) + log(abs(x + 2))

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maple [A]  time = 0.02, size = 13, normalized size = 0.87

method	result	size
default	\(\ln \left (2+x \right )-\frac {4}{x -4}\)	\(13\)
norman	\(\ln \left (2+x \right )-\frac {4}{x -4}\)	\(13\)
risch	\(\ln \left (2+x \right )-\frac {4}{x -4}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-4*x+24)/(x^3-6*x^2+32),x,method=_RETURNVERBOSE)

[Out]

ln(2+x)-4/(x-4)

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maxima [A]  time = 0.34, size = 12, normalized size = 0.80 \begin {gather*} -\frac {4}{x - 4} + \log \left (x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+24)/(x^3-6*x^2+32),x, algorithm="maxima")

[Out]

-4/(x - 4) + log(x + 2)

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mupad [B]  time = 0.05, size = 12, normalized size = 0.80 \begin {gather*} \ln \left (x+2\right )-\frac {4}{x-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 4*x + 24)/(x^3 - 6*x^2 + 32),x)

[Out]

log(x + 2) - 4/(x - 4)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.53 \begin {gather*} \log {\left (x + 2 \right )} - \frac {4}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-4*x+24)/(x**3-6*x**2+32),x)

[Out]

log(x + 2) - 4/(x - 4)

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