3.5.56 \(\int \frac {-31-2 x}{253+31 x+x^2} \, dx\)

Optimal. Leaf size=13 \[ \log \left (\frac {1}{3+x-(16+x)^2}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.85, 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*} -\log \left (x^2+31 x+253\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-31 - 2*x)/(253 + 31*x + x^2),x]

[Out]

-Log[253 + 31*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 (253+31 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} -\log \left (253+31 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-31 - 2*x)/(253 + 31*x + x^2),x]

[Out]

-Log[253 + 31*x + x^2]

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fricas [A]  time = 0.69, size = 11, normalized size = 0.85 \begin {gather*} -\log \left (x^{2} + 31 \, x + 253\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-31)/(x^2+31*x+253),x, algorithm="fricas")

[Out]

-log(x^2 + 31*x + 253)

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giac [A]  time = 0.41, size = 11, normalized size = 0.85 \begin {gather*} -\log \left (x^{2} + 31 \, x + 253\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-31)/(x^2+31*x+253),x, algorithm="giac")

[Out]

-log(x^2 + 31*x + 253)

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maple [A]  time = 0.49, size = 12, normalized size = 0.92

method	result	size
default	\(-\ln \left (x^{2}+31 x +253\right )\)	\(12\)
norman	\(-\ln \left (x^{2}+31 x +253\right )\)	\(12\)
risch	\(-\ln \left (x^{2}+31 x +253\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x-31)/(x^2+31*x+253),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2+31*x+253)

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maxima [A]  time = 0.40, size = 11, normalized size = 0.85 \begin {gather*} -\log \left (x^{2} + 31 \, x + 253\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-31)/(x^2+31*x+253),x, algorithm="maxima")

[Out]

-log(x^2 + 31*x + 253)

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mupad [B]  time = 0.04, size = 11, normalized size = 0.85 \begin {gather*} -\ln \left (x^2+31\,x+253\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 31)/(31*x + x^2 + 253),x)

[Out]

-log(31*x + x^2 + 253)

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sympy [A]  time = 0.07, size = 10, normalized size = 0.77 \begin {gather*} - \log {\left (x^{2} + 31 x + 253 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x-31)/(x**2+31*x+253),x)

[Out]

-log(x**2 + 31*x + 253)

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