3.21.80 \(\int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac {1}{3} \sqrt {3 x^2+4 x+2}-\frac {2 \sinh ^{-1}\left (\frac {3 x+2}{\sqrt {2}}\right )}{3 \sqrt {3}} \]

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {640, 619, 215} \begin {gather*} \frac {1}{3} \sqrt {3 x^2+4 x+2}-\frac {2 \sinh ^{-1}\left (\frac {3 x+2}{\sqrt {2}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

Sqrt[2 + 4*x + 3*x^2]/3 - (2*ArcSinh[(2 + 3*x)/Sqrt[2]])/(3*Sqrt[3])

Rule 215

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

Rule 619

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

Rule 640

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

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx &=\frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {2}{3} \int \frac {1}{\sqrt {2+4 x+3 x^2}} \, dx\\ &=\frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{8}}} \, dx,x,4+6 x\right )}{3 \sqrt {6}}\\ &=\frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {2 \sinh ^{-1}\left (\frac {2+3 x}{\sqrt {2}}\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {1}{9} \left (3 \sqrt {3 x^2+4 x+2}-2 \sqrt {3} \sinh ^{-1}\left (\frac {3 x+2}{\sqrt {2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

(3*Sqrt[2 + 4*x + 3*x^2] - 2*Sqrt[3]*ArcSinh[(2 + 3*x)/Sqrt[2]])/9

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IntegrateAlgebraic [A]  time = 0.11, size = 54, normalized size = 1.35 \begin {gather*} \frac {1}{3} \sqrt {3 x^2+4 x+2}+\frac {2 \log \left (\sqrt {3} \sqrt {3 x^2+4 x+2}-3 x-2\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

Sqrt[2 + 4*x + 3*x^2]/3 + (2*Log[-2 - 3*x + Sqrt[3]*Sqrt[2 + 4*x + 3*x^2]])/(3*Sqrt[3])

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fricas [A]  time = 0.41, size = 52, normalized size = 1.30 \begin {gather*} \frac {1}{9} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 4 \, x + 2} {\left (3 \, x + 2\right )} - 9 \, x^{2} - 12 \, x - 5\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+4*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 4*x + 2)*(3*x + 2) - 9*x^2 - 12*x - 5) + 1/3*sqrt(3*x^2 + 4*x + 2)

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giac [A]  time = 0.18, size = 48, normalized size = 1.20 \begin {gather*} \frac {2}{9} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+4*x+2)^(1/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*log(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x + 2)) - 2) + 1/3*sqrt(3*x^2 + 4*x + 2)

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maple [A]  time = 0.06, size = 30, normalized size = 0.75 \begin {gather*} -\frac {2 \sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}+\frac {\sqrt {3 x^{2}+4 x +2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(3*x^2+4*x+2)^(1/2),x)

[Out]

1/3*(3*x^2+4*x+2)^(1/2)-2/9*3^(1/2)*arcsinh(3/2*2^(1/2)*(x+2/3))

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maxima [A]  time = 1.77, size = 31, normalized size = 0.78 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x^2+4*x+2)^(1/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x + 2)) + 1/3*sqrt(3*x^2 + 4*x + 2)

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mupad [B]  time = 0.11, size = 44, normalized size = 1.10 \begin {gather*} \frac {\sqrt {3\,x^2+4\,x+2}}{3}-\frac {2\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+4\,x+2}+\frac {\sqrt {3}\,\left (3\,x+2\right )}{3}\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x + 3*x^2 + 2)^(1/2),x)

[Out]

(4*x + 3*x^2 + 2)^(1/2)/3 - (2*3^(1/2)*log((4*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 2))/3))/9

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {3 x^{2} + 4 x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3*x**2+4*x+2)**(1/2),x)

[Out]

Integral(x/sqrt(3*x**2 + 4*x + 2), x)

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