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

Optimal. Leaf size=14 \[ \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {634, 618, 204, 628} \begin {gather*} \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 2*x)/(5 + 4*x + x^2),x]

[Out]

ArcTan[2 + x] + Log[5 + 4*x + x^2]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 {5+2 x}{5+4 x+x^2} \, dx &=\int \frac {1}{5+4 x+x^2} \, dx+\int \frac {4+2 x}{5+4 x+x^2} \, dx\\ &=\log \left (5+4 x+x^2\right )-2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,4+2 x\right )\\ &=\tan ^{-1}(2+x)+\log \left (5+4 x+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[2 + x] + Log[5 + 4*x + x^2]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5+2 x}{5+4 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 14, normalized size = 1.00 \begin {gather*} \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x + 2) + log(x^2 + 4*x + 5)

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giac [A]  time = 0.15, size = 14, normalized size = 1.00 \begin {gather*} \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x + 2) + log(x^2 + 4*x + 5)

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maple [A]  time = 0.05, size = 15, normalized size = 1.07 \begin {gather*} \arctan \left (x +2\right )+\ln \left (x^{2}+4 x +5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.42, size = 14, normalized size = 1.00 \begin {gather*} \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x + 2) + log(x^2 + 4*x + 5)

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mupad [B]  time = 1.18, size = 14, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (x+2\right )+\ln \left (x^2+4\,x+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 5)/(4*x + x^2 + 5),x)

[Out]

atan(x + 2) + log(4*x + x^2 + 5)

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sympy [A]  time = 0.12, size = 14, normalized size = 1.00 \begin {gather*} \log {\left (x^{2} + 4 x + 5 \right )} + \operatorname {atan}{\left (x + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 + 4*x + 5) + atan(x + 2)

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