3.103.78 \(\int \frac {3+2 x}{1+3 x+x^2} \, dx\)

Optimal. Leaf size=19 \[ \log \left (\frac {1}{2} \left (1-x+x^2+\log \left (e^{4 x}\right )\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.47, 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+3 x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(1 + 3*x + x^2),x]

[Out]

Log[1 + 3*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 (1+3 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.47 \begin {gather*} \log \left (1+3 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(1 + 3*x + x^2),x]

[Out]

Log[1 + 3*x + x^2]

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fricas [A]  time = 1.24, size = 9, normalized size = 0.47 \begin {gather*} \log \left (x^{2} + 3 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+3)/(x^2+3*x+1),x, algorithm="fricas")

[Out]

log(x^2 + 3*x + 1)

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giac [A]  time = 0.12, size = 10, normalized size = 0.53 \begin {gather*} \log \left ({\left | x^{2} + 3 \, x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+3)/(x^2+3*x+1),x, algorithm="giac")

[Out]

log(abs(x^2 + 3*x + 1))

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maple [A]  time = 0.10, size = 10, normalized size = 0.53

method	result	size
derivativedivides	\(\ln \left (x^{2}+3 x +1\right )\)	\(10\)
default	\(\ln \left (x^{2}+3 x +1\right )\)	\(10\)
norman	\(\ln \left (x^{2}+3 x +1\right )\)	\(10\)
risch	\(\ln \left (x^{2}+3 x +1\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+3)/(x^2+3*x+1),x,method=_RETURNVERBOSE)

[Out]

ln(x^2+3*x+1)

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maxima [A]  time = 0.36, size = 9, normalized size = 0.47 \begin {gather*} \log \left (x^{2} + 3 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+3)/(x^2+3*x+1),x, algorithm="maxima")

[Out]

log(x^2 + 3*x + 1)

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mupad [B]  time = 0.05, size = 9, normalized size = 0.47 \begin {gather*} \ln \left (x^2+3\,x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3)/(3*x + x^2 + 1),x)

[Out]

log(3*x + x^2 + 1)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.42 \begin {gather*} \log {\left (x^{2} + 3 x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+3)/(x**2+3*x+1),x)

[Out]

log(x**2 + 3*x + 1)

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