3.83.44 \(\int \frac {-9-x}{7+x} \, dx\)

Optimal. Leaf size=19 \[ -7-x-\log \left (\frac {1}{25} (-7-x)^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.53, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} -x-2 \log (x+7) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - x)/(7 + x),x]

[Out]

-x - 2*Log[7 + x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {2}{7+x}\right ) \, dx\\ &=-x-2 \log (7+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.53 \begin {gather*} -x-2 \log (7+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - x)/(7 + x),x]

[Out]

-x - 2*Log[7 + x]

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fricas [A]  time = 0.49, size = 10, normalized size = 0.53 \begin {gather*} -x - 2 \, \log \left (x + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x-9)/(x+7),x, algorithm="fricas")

[Out]

-x - 2*log(x + 7)

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giac [A]  time = 0.20, size = 11, normalized size = 0.58 \begin {gather*} -x - 2 \, \log \left ({\left | x + 7 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x-9)/(x+7),x, algorithm="giac")

[Out]

-x - 2*log(abs(x + 7))

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maple [A]  time = 0.32, size = 11, normalized size = 0.58

method	result	size
default	\(-x -2 \ln \left (x +7\right )\)	\(11\)
norman	\(-x -2 \ln \left (x +7\right )\)	\(11\)
risch	\(-x -2 \ln \left (x +7\right )\)	\(11\)
meijerg	\(-2 \ln \left (1+\frac {x}{7}\right )-x\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x-9)/(x+7),x,method=_RETURNVERBOSE)

[Out]

-x-2*ln(x+7)

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maxima [A]  time = 0.36, size = 10, normalized size = 0.53 \begin {gather*} -x - 2 \, \log \left (x + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x-9)/(x+7),x, algorithm="maxima")

[Out]

-x - 2*log(x + 7)

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mupad [B]  time = 0.03, size = 10, normalized size = 0.53 \begin {gather*} -x-2\,\ln \left (x+7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + 9)/(x + 7),x)

[Out]

- x - 2*log(x + 7)

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sympy [A]  time = 0.07, size = 8, normalized size = 0.42 \begin {gather*} - x - 2 \log {\left (x + 7 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x-9)/(x+7),x)

[Out]

-x - 2*log(x + 7)

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