3.2.78 \(\int \frac {-1+4 x}{(-1+x) (2+x)} \, dx\) [178]

Optimal. Leaf size=13 \[ \log (1-x)+3 \log (2+x) \]

[Out]

ln(1-x)+3*ln(2+x)

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = {78} \begin {gather*} \log (1-x)+3 \log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 4*x)/((-1 + x)*(2 + x)),x]

[Out]

Log[1 - x] + 3*Log[2 + x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {-1+4 x}{(-1+x) (2+x)} \, dx &=\int \left (\frac {1}{-1+x}+\frac {3}{2+x}\right ) \, dx\\ &=\log (1-x)+3 \log (2+x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \log (1-x)+3 \log (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 4*x)/((-1 + x)*(2 + x)),x]

[Out]

Log[1 - x] + 3*Log[2 + x]

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Mathics [A]
time = 1.79, size = 11, normalized size = 0.85 \begin {gather*} \text {Log}\left [-1+x\right ]+3 \text {Log}\left [2+x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(-1 + 4*x)/((-1 + x)*(2 + x)),x]')

[Out]

Log[-1 + x] + 3 Log[2 + x]

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Maple [A]
time = 0.05, size = 12, normalized size = 0.92

method	result	size
default	\(\ln \left (-1+x \right )+3 \ln \left (2+x \right )\)	\(12\)
norman	\(\ln \left (-1+x \right )+3 \ln \left (2+x \right )\)	\(12\)
risch	\(\ln \left (-1+x \right )+3 \ln \left (2+x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+4*x)/(-1+x)/(2+x),x,method=_RETURNVERBOSE)

[Out]

ln(-1+x)+3*ln(2+x)

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.85 \begin {gather*} 3 \, \log \left (x + 2\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*x)/(-1+x)/(2+x),x, algorithm="maxima")

[Out]

3*log(x + 2) + log(x - 1)

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Fricas [A]
time = 0.32, size = 11, normalized size = 0.85 \begin {gather*} 3 \, \log \left (x + 2\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*x)/(-1+x)/(2+x),x, algorithm="fricas")

[Out]

3*log(x + 2) + log(x - 1)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (x - 1 \right )} + 3 \log {\left (x + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*x)/(-1+x)/(2+x),x)

[Out]

log(x - 1) + 3*log(x + 2)

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Giac [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \ln \left |x-1\right |+3 \ln \left |x+2\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*x)/(-1+x)/(2+x),x)

[Out]

3*log(abs(x + 2)) + log(abs(x - 1))

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Mupad [B]
time = 0.17, size = 11, normalized size = 0.85 \begin {gather*} \ln \left (x-1\right )+3\,\ln \left (x+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 1)/((x - 1)*(x + 2)),x)

[Out]

log(x - 1) + 3*log(x + 2)

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