3.54.10 \(\int \frac {5+2 x}{4+5 x+x^2} \, dx\)

Optimal. Leaf size=13 \[ 2+e^5+\log ((1+x) (4+x)) \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.69, 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+5 x+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 2*x)/(4 + 5*x + x^2),x]

[Out]

Log[4 + 5*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+5 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.69 \begin {gather*} \log \left (4+5 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 2*x)/(4 + 5*x + x^2),x]

[Out]

Log[4 + 5*x + x^2]

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fricas [A]  time = 0.71, size = 9, normalized size = 0.69 \begin {gather*} \log \left (x^{2} + 5 \, x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x^2+5*x+4),x, algorithm="fricas")

[Out]

log(x^2 + 5*x + 4)

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giac [A]  time = 0.20, size = 10, normalized size = 0.77 \begin {gather*} \log \left ({\left | x^{2} + 5 \, x + 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x^2+5*x+4),x, algorithm="giac")

[Out]

log(abs(x^2 + 5*x + 4))

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maple [A]  time = 0.26, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\ln \left (x^{2}+5 x +4\right )\)	\(10\)
default	\(\ln \left (x^{2}+5 x +4\right )\)	\(10\)
norman	\(\ln \left (x +1\right )+\ln \left (4+x \right )\)	\(10\)
risch	\(\ln \left (x^{2}+5 x +4\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+2*x)/(x^2+5*x+4),x,method=_RETURNVERBOSE)

[Out]

ln(x^2+5*x+4)

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maxima [A]  time = 0.44, size = 9, normalized size = 0.69 \begin {gather*} \log \left (x^{2} + 5 \, x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x^2+5*x+4),x, algorithm="maxima")

[Out]

log(x^2 + 5*x + 4)

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mupad [B]  time = 3.43, size = 9, normalized size = 0.69 \begin {gather*} \ln \left (x^2+5\,x+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 5)/(5*x + x^2 + 4),x)

[Out]

log(5*x + x^2 + 4)

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sympy [A]  time = 0.07, size = 8, normalized size = 0.62 \begin {gather*} \log {\left (x^{2} + 5 x + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)/(x**2+5*x+4),x)

[Out]

log(x**2 + 5*x + 4)

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