3.2.17 \(\int \frac {1}{\sqrt {2+4 x+3 x^2}} \, dx\) [117]

Optimal. Leaf size=18 \[ \frac {\sinh ^{-1}\left (\frac {2+3 x}{\sqrt {2}}\right )}{\sqrt {3}} \]

[Out]

1/3*arcsinh(1/2*(2+3*x)*2^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 221} \begin {gather*} \frac {\sinh ^{-1}\left (\frac {3 x+2}{\sqrt {2}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

ArcSinh[(2 + 3*x)/Sqrt[2]]/Sqrt[3]

Rule 221

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 633

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]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 27, normalized size = 1.50 \begin {gather*} -\frac {\log \left (-2-3 x+\sqrt {6+12 x+9 x^2}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

-(Log[-2 - 3*x + Sqrt[6 + 12*x + 9*x^2]]/Sqrt[3])

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Maple [A]
time = 0.62, size = 15, normalized size = 0.83

method	result	size
default	\(\frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{3}\)	\(15\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {3 x^{2}+4 x +2}-2 \RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*3^(1/2)*arcsinh(3/2*2^(1/2)*(x+2/3))

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Maxima [A]
time = 0.54, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x + 2))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).
time = 2.26, size = 38, normalized size = 2.11 \begin {gather*} \frac {1}{6} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 4*x + 2)*(3*x + 2) - 9*x^2 - 12*x - 5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {3 x^{2} + 4 x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(3*x**2 + 4*x + 2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (16) = 32\).
time = 1.73, size = 53, normalized size = 2.94 \begin {gather*} \frac {1}{6} \, \sqrt {3 \, x^{2} + 4 \, x + 2} {\left (3 \, x + 2\right )} - \frac {1}{9} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3*x^2 + 4*x + 2)*(3*x + 2) - 1/9*sqrt(3)*log(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x + 2)) - 2)

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Mupad [B]
time = 0.22, size = 26, normalized size = 1.44 \begin {gather*} \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x+\frac {2}{3}\right )+\sqrt {3\,x^2+4\,x+2}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x + 3*x^2 + 2)^(1/2),x)

[Out]

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

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