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

Optimal. Leaf size=45 \[ \sqrt{\frac{2}{\sqrt{41}-5}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{5+\sqrt{41}}}\right ),\frac{1}{8} \left (-33-5 \sqrt{41}\right )\right ) \]

[Out]

Sqrt[2/(-5 + Sqrt[41])]*EllipticF[ArcSin[(2*x)/Sqrt[5 + Sqrt[41]]], (-33 - 5*Sqrt[41])/8]

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Rubi [A]  time = 0.062061, antiderivative size = 45, normalized size of antiderivative = 1., 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 = {1095, 419} \[ \sqrt{\frac{2}{\sqrt{41}-5}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{5+\sqrt{41}}}\right )|\frac{1}{8} \left (-33-5 \sqrt{41}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 - 2*x^4],x]

[Out]

Sqrt[2/(-5 + Sqrt[41])]*EllipticF[ArcSin[(2*x)/Sqrt[5 + Sqrt[41]]], (-33 - 5*Sqrt[41])/8]

Rule 1095

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]

Rule 419

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+5 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{5+\sqrt{41}-4 x^2} \sqrt{-5+\sqrt{41}+4 x^2}} \, dx\\ &=\sqrt{\frac{2}{-5+\sqrt{41}}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{5+\sqrt{41}}}\right )|\frac{1}{8} \left (-33-5 \sqrt{41}\right )\right )\\ \end{align*}

Mathematica [C]  time = 0.0464176, size = 52, normalized size = 1.16 \[ -i \sqrt{\frac{2}{5+\sqrt{41}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{\sqrt{41}-5}}\right ),\frac{5 \sqrt{41}}{8}-\frac{33}{8}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-I)*Sqrt[2/(5 + Sqrt[41])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-5 + Sqrt[41]]], -33/8 + (5*Sqrt[41])/8]

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Maple [B]  time = 0.228, size = 76, normalized size = 1.7 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -5/4+1/4\,\sqrt{41} \right ){x}^{2}}\sqrt{1- \left ( -5/4-1/4\,\sqrt{41} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+\sqrt{41}},5/4\,i+i/4\sqrt{41} \right ) }{\sqrt{-5+\sqrt{41}}\sqrt{-2\,{x}^{4}+5\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^4+5*x^2+2)^(1/2),x)

[Out]

2/(-5+41^(1/2))^(1/2)*(1-(-5/4+1/4*41^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*41^(1/2))*x^2)^(1/2)/(-2*x^4+5*x^2+2)^(1/
2)*EllipticF(1/2*x*(-5+41^(1/2))^(1/2),5/4*I+1/4*I*41^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} + 5 \, x^{2} + 2}}{2 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-2*x^4 + 5*x^2 + 2)/(2*x^4 - 5*x^2 - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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