3.1.39 \(\int \frac {1}{\sqrt {3-5 x^2-2 x^4}} \, dx\) [39]

Optimal. Leaf size=18 \[ \frac {F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {6}} \]

[Out]

1/6*EllipticF(x*2^(1/2),1/6*I*6^(1/2))*6^(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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430} \begin {gather*} \frac {F\left (\text {ArcSin}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 5*x^2 - 2*x^4],x]

[Out]

EllipticF[ArcSin[Sqrt[2]*x], -1/6]/Sqrt[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 {3-5 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {2-4 x^2} \sqrt {12+4 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {6}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(18)=36\).
time = 10.02, size = 54, normalized size = 3.00 \begin {gather*} \frac {\sqrt {1-2 x^2} \sqrt {3+x^2} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {6} \sqrt {3-5 x^2-2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x^2]*Sqrt[3 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/6])/(Sqrt[6]*Sqrt[3 - 5*x^2 - 2*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. 49 vs. \(2 (17 ) = 34\).
time = 0.04, size = 50, normalized size = 2.78

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (\sqrt {2}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {-2 x^{4}-5 x^{2}+3}}\)	\(50\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {3 x^{2}+9}\, \EllipticF \left (\sqrt {2}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {-2 x^{4}-5 x^{2}+3}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*2^(1/2)*(-2*x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-2*x^4-5*x^2+3)^(1/2)*EllipticF(2^(1/2)*x,1/6*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/(-2*x^4-5*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.09, size = 15, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {3} \sqrt {2} {\rm ellipticF}\left (\sqrt {2} x, -\frac {1}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*sqrt(2)*ellipticF(sqrt(2)*x, -1/6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4-5\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3 - 2*x^4 - 5*x^2)^(1/2),x)

[Out]

int(1/(3 - 2*x^4 - 5*x^2)^(1/2), x)

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