3.27 \(\int \frac{1}{\sqrt{3+7 x^2-2 x^4}} \, dx\)

Optimal. Leaf size=45 \[ \sqrt{\frac{2}{\sqrt{73}-7}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right ),\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right ) \]

[Out]

Sqrt[2/(-7 + Sqrt[73])]*EllipticF[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12]

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Rubi [A]  time = 0.064989, 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} \[ \sqrt{\frac{2}{\sqrt{73}-7}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 7*x^2 - 2*x^4],x]

[Out]

Sqrt[2/(-7 + Sqrt[73])]*EllipticF[ArcSin[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12]

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{3+7 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{7+\sqrt{73}-4 x^2} \sqrt{-7+\sqrt{73}+4 x^2}} \, dx\\ &=\sqrt{\frac{2}{-7+\sqrt{73}}} F\left (\sin ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right )|\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right )\\ \end{align*}

Mathematica [C]  time = 0.0442848, size = 52, normalized size = 1.16 \[ -i \sqrt{\frac{2}{7+\sqrt{73}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{\sqrt{73}-7}}\right ),\frac{1}{12} \left (7 \sqrt{73}-61\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[3 + 7*x^2 - 2*x^4],x]

[Out]

(-I)*Sqrt[2/(7 + Sqrt[73])]*EllipticF[I*ArcSinh[(2*x)/Sqrt[-7 + Sqrt[73]]], (-61 + 7*Sqrt[73])/12]

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Maple [B]  time = 0.255, size = 84, normalized size = 1.9 \begin{align*} 6\,{\frac{\sqrt{1- \left ( -7/6+1/6\,\sqrt{73} \right ){x}^{2}}\sqrt{1- \left ( -1/6\,\sqrt{73}-7/6 \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-42+6\,\sqrt{73}},{\frac{7\,i}{12}}\sqrt{6}+i/12\sqrt{438} \right ) }{\sqrt{-42+6\,\sqrt{73}}\sqrt{-2\,{x}^{4}+7\,{x}^{2}+3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^4+7*x^2+3)^(1/2),x)

[Out]

6/(-42+6*73^(1/2))^(1/2)*(1-(-7/6+1/6*73^(1/2))*x^2)^(1/2)*(1-(-1/6*73^(1/2)-7/6)*x^2)^(1/2)/(-2*x^4+7*x^2+3)^
(1/2)*EllipticF(1/6*x*(-42+6*73^(1/2))^(1/2),7/12*I*6^(1/2)+1/12*I*438^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 7 \, x^{2} + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^4+7*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-2*x^4 + 7*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} + 7 \, x^{2} + 3}}{2 \, x^{4} - 7 \, x^{2} - 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^4+7*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-2*x^4 + 7*x^2 + 3)/(2*x^4 - 7*x^2 - 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 x^{4} + 7 x^{2} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x**4+7*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**4 + 7*x**2 + 3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-2 \, x^{4} + 7 \, x^{2} + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^4+7*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-2*x^4 + 7*x^2 + 3), x)