3.2.27 \(\int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx\) [127]

Optimal. Leaf size=10 \[ F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right ) \]

[Out]

EllipticF(1/2*x*2^(1/2),I*6^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430} \begin {gather*} F\left (\left .\text {ArcSin}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 - 3*x^4],x]

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -6]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 1109

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx &=\left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {12-6 x^2} \sqrt {2+6 x^2}} \, dx\\ &=F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.02, size = 65, normalized size = 6.50 \begin {gather*} -\frac {i \sqrt {1-\frac {x^2}{2}} \sqrt {1+3 x^2} F\left (i \sinh ^{-1}\left (\sqrt {3} x\right )|-\frac {1}{6}\right )}{\sqrt {3} \sqrt {2+5 x^2-3 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - x^2/2]*Sqrt[1 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3]*x], -1/6])/(Sqrt[3]*Sqrt[2 + 5*x^2 - 3*x^4])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (13 ) = 26\).
time = 0.02, size = 51, normalized size = 5.10

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {3 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {6}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}+2}}\)	\(51\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {3 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {6}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}+2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*2^(1/2)*(-2*x^2+4)^(1/2)*(3*x^2+1)^(1/2)/(-3*x^4+5*x^2+2)^(1/2)*EllipticF(1/2*2^(1/2)*x,I*6^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.07, size = 8, normalized size = 0.80 \begin {gather*} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x, -6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ellipticF(1/2*sqrt(2)*x, -6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \int \frac {1}{\sqrt {-3\,x^4+5\,x^2+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2 - 3*x^4 + 2)^(1/2),x)

[Out]

int(1/(5*x^2 - 3*x^4 + 2)^(1/2), x)

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