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

Optimal. Leaf size=13 \[ 2 \log (x+2)-\log (x+1) \]

[Out]

-ln(1+x)+2*ln(2+x)

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {632, 31} \[ 2 \log (x+2)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[x/(2 + 3*x + x^2),x]

[Out]

-Log[1 + x] + 2*Log[2 + x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rubi steps

\begin {align*} \int \frac {x}{2+3 x+x^2} \, dx &=2 \int \frac {1}{2+x} \, dx-\int \frac {1}{1+x} \, dx\\ &=-\log (1+x)+2 \log (2+x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ 2 \log (x+2)-\log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(2 + 3*x + x^2),x]

[Out]

-Log[1 + x] + 2*Log[2 + x]

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fricas [A]  time = 0.41, size = 13, normalized size = 1.00 \[ 2 \, \log \left (x + 2\right ) - \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x + 2) - log(x + 1)

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giac [A]  time = 1.27, size = 15, normalized size = 1.15 \[ 2 \, \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(abs(x + 2)) - log(abs(x + 1))

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maple [A]  time = 0.00, size = 14, normalized size = 1.08 \[ -\ln \left (x +1\right )+2 \ln \left (x +2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+3*x+2),x)

[Out]

-ln(x+1)+2*ln(x+2)

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maxima [A]  time = 0.49, size = 13, normalized size = 1.00 \[ 2 \, \log \left (x + 2\right ) - \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x + 2) - log(x + 1)

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mupad [B]  time = 0.05, size = 13, normalized size = 1.00 \[ 2\,\ln \left (x+2\right )-\ln \left (x+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(3*x + x^2 + 2),x)

[Out]

2*log(x + 2) - log(x + 1)

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sympy [A]  time = 0.10, size = 10, normalized size = 0.77 \[ - \log {\left (x + 1 \right )} + 2 \log {\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1) + 2*log(x + 2)

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