3.68.88 \(\int \frac {4+2 x}{4 x+x^2} \, dx\)

Optimal. Leaf size=15 \[ \log \left (\left (4+5 e^3\right )^2 x (4+x)\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.53, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {628} \begin {gather*} \log \left (x^2+4 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 2*x)/(4*x + x^2),x]

[Out]

Log[4*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 (4 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.13 \begin {gather*} 2 \left (\frac {\log (x)}{2}+\frac {1}{2} \log (4+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 2*x)/(4*x + x^2),x]

[Out]

2*(Log[x]/2 + Log[4 + x]/2)

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fricas [A]  time = 0.54, size = 8, normalized size = 0.53 \begin {gather*} \log \left (x^{2} + 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 4*x)

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giac [A]  time = 0.24, size = 13, normalized size = 0.87 \begin {gather*} \log \left (2 \, {\left | \frac {1}{2} \, x^{2} + 2 \, x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*abs(1/2*x^2 + 2*x))

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maple [A]  time = 0.07, size = 7, normalized size = 0.47

method	result	size
default	\(\ln \left (\left (4+x \right ) x \right )\)	\(7\)
norman	\(\ln \relax (x )+\ln \left (4+x \right )\)	\(8\)
derivativedivides	\(\ln \left (x^{2}+4 x \right )\)	\(9\)
risch	\(\ln \left (x^{2}+4 x \right )\)	\(9\)
meijerg	\(\ln \relax (x )-2 \ln \relax (2)+\ln \left (1+\frac {x}{4}\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln((4+x)*x)

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maxima [A]  time = 0.35, size = 8, normalized size = 0.53 \begin {gather*} \log \left (x^{2} + 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 + 4*x)

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mupad [B]  time = 0.05, size = 6, normalized size = 0.40 \begin {gather*} \ln \left (x\,\left (x+4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 4)/(4*x + x^2),x)

[Out]

log(x*(x + 4))

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sympy [A]  time = 0.08, size = 7, normalized size = 0.47 \begin {gather*} \log {\left (x^{2} + 4 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + 4*x)

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