3.34.96 \(\int \frac {3+x}{4+x} \, dx\)

Optimal. Leaf size=18 \[ -76-e^6+x-\log \left (\frac {4+x}{4}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} x-\log (x+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + x)/(4 + x),x]

[Out]

x - Log[4 + 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 {1}{-4-x}\right ) \, dx\\ &=x-\log (4+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.44 \begin {gather*} x-\log (4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/(4 + x),x]

[Out]

x - Log[4 + x]

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fricas [A]  time = 0.91, size = 8, normalized size = 0.44 \begin {gather*} x - \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(4+x),x, algorithm="fricas")

[Out]

x - log(x + 4)

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giac [A]  time = 0.22, size = 9, normalized size = 0.50 \begin {gather*} x - \log \left ({\left | x + 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(4+x),x, algorithm="giac")

[Out]

x - log(abs(x + 4))

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maple [A]  time = 0.47, size = 9, normalized size = 0.50

method	result	size
default	\(x -\ln \left (4+x \right )\)	\(9\)
norman	\(x -\ln \left (4+x \right )\)	\(9\)
risch	\(x -\ln \left (4+x \right )\)	\(9\)
meijerg	\(-\ln \left (1+\frac {x}{4}\right )+x\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+x)/(4+x),x,method=_RETURNVERBOSE)

[Out]

x-ln(4+x)

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maxima [A]  time = 0.38, size = 8, normalized size = 0.44 \begin {gather*} x - \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(4+x),x, algorithm="maxima")

[Out]

x - log(x + 4)

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mupad [B]  time = 0.03, size = 8, normalized size = 0.44 \begin {gather*} x-\ln \left (x+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3)/(x + 4),x)

[Out]

x - log(x + 4)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.28 \begin {gather*} x - \log {\left (x + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(4+x),x)

[Out]

x - log(x + 4)

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