3.57.26 \(\int \frac {8+3 x}{4 x+x^2} \, dx\)

Optimal. Leaf size=13 \[ \log (7 x (-x+x (5+x))) \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.69, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {631} \begin {gather*} 2 \log (x)+\log (x+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8 + 3*x)/(4*x + x^2),x]

[Out]

2*Log[x] + Log[4 + x]

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2}{x}+\frac {1}{4+x}\right ) \, dx\\ &=2 \log (x)+\log (4+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.69 \begin {gather*} 2 \log (x)+\log (4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 3*x)/(4*x + x^2),x]

[Out]

2*Log[x] + Log[4 + x]

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fricas [A]  time = 0.52, size = 9, normalized size = 0.69 \begin {gather*} \log \left (x + 4\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+8)/(x^2+4*x),x, algorithm="fricas")

[Out]

log(x + 4) + 2*log(x)

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giac [A]  time = 0.22, size = 11, normalized size = 0.85 \begin {gather*} \log \left ({\left | x + 4 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+8)/(x^2+4*x),x, algorithm="giac")

[Out]

log(abs(x + 4)) + 2*log(abs(x))

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maple [A]  time = 0.25, size = 10, normalized size = 0.77

method	result	size
default	\(\ln \left (4+x \right )+2 \ln \relax (x )\)	\(10\)
norman	\(\ln \left (4+x \right )+2 \ln \relax (x )\)	\(10\)
risch	\(\ln \left (4+x \right )+2 \ln \relax (x )\)	\(10\)
meijerg	\(\ln \left (1+\frac {x}{4}\right )+2 \ln \relax (x )-4 \ln \relax (2)\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+8)/(x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

ln(4+x)+2*ln(x)

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maxima [A]  time = 0.39, size = 9, normalized size = 0.69 \begin {gather*} \log \left (x + 4\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+8)/(x^2+4*x),x, algorithm="maxima")

[Out]

log(x + 4) + 2*log(x)

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mupad [B]  time = 3.51, size = 9, normalized size = 0.69 \begin {gather*} \ln \left (x+4\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 8)/(4*x + x^2),x)

[Out]

log(x + 4) + 2*log(x)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.62 \begin {gather*} 2 \log {\relax (x )} + \log {\left (x + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x+8)/(x**2+4*x),x)

[Out]

2*log(x) + log(x + 4)

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