3.55 \(\int \frac {1}{\sqrt {5-4 x+3 x^2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}} \]

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Rubi [A]  time = 0.01, antiderivative size = 19, 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 = {619, 215} \[ -\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

-(ArcSinh[(2 - 3*x)/Sqrt[11]]/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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {5-4 x+3 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{44}}} \, dx,x,-4+6 x\right )}{2 \sqrt {33}}\\ &=-\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.95 \[ \frac {\sinh ^{-1}\left (\frac {3 x-2}{\sqrt {11}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

ArcSinh[(-2 + 3*x)/Sqrt[11]]/Sqrt[3]

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IntegrateAlgebraic [A]  time = 0.08, size = 33, normalized size = 1.74 \[ -\frac {\log \left (\sqrt {3} \sqrt {3 x^2-4 x+5}-3 x+2\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

-(Log[2 - 3*x + Sqrt[3]*Sqrt[5 - 4*x + 3*x^2]]/Sqrt[3])

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fricas [B]  time = 0.97, size = 38, normalized size = 2.00 \[ \frac {1}{6} \, \sqrt {3} \log \left (-2 \, \sqrt {3} \sqrt {3 \, x^{2} - 4 \, x + 5} {\left (3 \, x - 2\right )} - 18 \, x^{2} + 24 \, x - 19\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4*x+5)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-2*sqrt(3)*sqrt(3*x^2 - 4*x + 5)*(3*x - 2) - 18*x^2 + 24*x - 19)

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giac [B]  time = 1.08, size = 33, normalized size = 1.74 \[ -\frac {1}{3} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - 4 \, x + 5}\right )} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4*x+5)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.40, size = 15, normalized size = 0.79

method	result	size
default	\(\frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {11}\, \left (x -\frac {2}{3}\right )}{11}\right )}{3}\)	\(15\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {3 x^{2}-4 x +5}\right )}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2-4*x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*3^(1/2)*arcsinh(3/11*11^(1/2)*(x-2/3))

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maxima [A]  time = 1.00, size = 16, normalized size = 0.84 \[ \frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{11} \, \sqrt {11} {\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4*x+5)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arcsinh(1/11*sqrt(11)*(3*x - 2))

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mupad [B]  time = 0.29, size = 26, normalized size = 1.37 \[ \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x-\frac {2}{3}\right )+\sqrt {3\,x^2-4\,x+5}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2 - 4*x + 5)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 x^{2} - 4 x + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2-4*x+5)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**2 - 4*x + 5), x)

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