3.74.18 \(\int \frac {-14-4 x}{4+7 x+x^2} \, dx\)

Optimal. Leaf size=15 \[ 2-2 \left (5+\log \left (4+7 x+x^2\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.73, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {628} \begin {gather*} -2 \log \left (x^2+7 x+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-14 - 4*x)/(4 + 7*x + x^2),x]

[Out]

-2*Log[4 + 7*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} &=-2 \log \left (4+7 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.73 \begin {gather*} -2 \log \left (4+7 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14 - 4*x)/(4 + 7*x + x^2),x]

[Out]

-2*Log[4 + 7*x + x^2]

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fricas [A]  time = 0.91, size = 11, normalized size = 0.73 \begin {gather*} -2 \, \log \left (x^{2} + 7 \, x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-14)/(x^2+7*x+4),x, algorithm="fricas")

[Out]

-2*log(x^2 + 7*x + 4)

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giac [A]  time = 0.20, size = 12, normalized size = 0.80 \begin {gather*} -2 \, \log \left ({\left | x^{2} + 7 \, x + 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-14)/(x^2+7*x+4),x, algorithm="giac")

[Out]

-2*log(abs(x^2 + 7*x + 4))

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maple [A]  time = 0.17, size = 12, normalized size = 0.80

method	result	size
default	\(-2 \ln \left (x^{2}+7 x +4\right )\)	\(12\)
norman	\(-2 \ln \left (x^{2}+7 x +4\right )\)	\(12\)
risch	\(-2 \ln \left (x^{2}+7 x +4\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x-14)/(x^2+7*x+4),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x^2+7*x+4)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.73 \begin {gather*} -2 \, \log \left (x^{2} + 7 \, x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-14)/(x^2+7*x+4),x, algorithm="maxima")

[Out]

-2*log(x^2 + 7*x + 4)

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mupad [B]  time = 0.06, size = 11, normalized size = 0.73 \begin {gather*} -2\,\ln \left (x^2+7\,x+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 14)/(7*x + x^2 + 4),x)

[Out]

-2*log(7*x + x^2 + 4)

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sympy [A]  time = 0.08, size = 12, normalized size = 0.80 \begin {gather*} - 2 \log {\left (x^{2} + 7 x + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x-14)/(x**2+7*x+4),x)

[Out]

-2*log(x**2 + 7*x + 4)

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