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

Optimal. Leaf size=108 \[ \frac{\sqrt{\frac{\left (5-\sqrt{17}\right ) x^2+4}{\left (5+\sqrt{17}\right ) x^2+4}} \left (\left (5+\sqrt{17}\right ) x^2+4\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{1}{2} \sqrt{5+\sqrt{17}} x\right ),\frac{1}{4} \left (5 \sqrt{17}-17\right )\right )}{2 \sqrt{5+\sqrt{17}} \sqrt{x^4+5 x^2+2}} \]

[Out]

(Sqrt[(4 + (5 - Sqrt[17])*x^2)/(4 + (5 + Sqrt[17])*x^2)]*(4 + (5 + Sqrt[17])*x^2)*EllipticF[ArcTan[(Sqrt[5 + S
qrt[17]]*x)/2], (-17 + 5*Sqrt[17])/4])/(2*Sqrt[5 + Sqrt[17]]*Sqrt[2 + 5*x^2 + x^4])

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Rubi [A]  time = 0.0420829, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1099} \[ \frac{\sqrt{\frac{\left (5-\sqrt{17}\right ) x^2+4}{\left (5+\sqrt{17}\right ) x^2+4}} \left (\left (5+\sqrt{17}\right ) x^2+4\right ) F\left (\tan ^{-1}\left (\frac{1}{2} \sqrt{5+\sqrt{17}} x\right )|\frac{1}{4} \left (-17+5 \sqrt{17}\right )\right )}{2 \sqrt{5+\sqrt{17}} \sqrt{x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 + x^4],x]

[Out]

(Sqrt[(4 + (5 - Sqrt[17])*x^2)/(4 + (5 + Sqrt[17])*x^2)]*(4 + (5 + Sqrt[17])*x^2)*EllipticF[ArcTan[(Sqrt[5 + S
qrt[17]]*x)/2], (-17 + 5*Sqrt[17])/4])/(2*Sqrt[5 + Sqrt[17]]*Sqrt[2 + 5*x^2 + x^4])

Rule 1099

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[((2*a + (b +
q)*x^2)*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)]
)/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSq
rtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+5 x^2+x^4}} \, dx &=\frac{\sqrt{\frac{4+\left (5-\sqrt{17}\right ) x^2}{4+\left (5+\sqrt{17}\right ) x^2}} \left (4+\left (5+\sqrt{17}\right ) x^2\right ) F\left (\tan ^{-1}\left (\frac{1}{2} \sqrt{5+\sqrt{17}} x\right )|\frac{1}{4} \left (-17+5 \sqrt{17}\right )\right )}{2 \sqrt{5+\sqrt{17}} \sqrt{2+5 x^2+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0818699, size = 103, normalized size = 0.95 \[ -\frac{i \sqrt{2 x^2-\sqrt{17}+5} \sqrt{2 x^2+\sqrt{17}+5} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{17}}} x\right ),\frac{21}{4}+\frac{5 \sqrt{17}}{4}\right )}{\sqrt{2 \left (5-\sqrt{17}\right )} \sqrt{x^4+5 x^2+2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[2 + 5*x^2 + x^4],x]

[Out]

((-I)*Sqrt[5 - Sqrt[17] + 2*x^2]*Sqrt[5 + Sqrt[17] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[17])]*x], 21/
4 + (5*Sqrt[17])/4])/(Sqrt[2*(5 - Sqrt[17])]*Sqrt[2 + 5*x^2 + x^4])

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Maple [A]  time = 0.226, size = 76, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -5/4+1/4\,\sqrt{17} \right ){x}^{2}}\sqrt{1- \left ( -5/4-1/4\,\sqrt{17} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+\sqrt{17}},5/4\,\sqrt{2}+1/4\,\sqrt{34} \right ) }{\sqrt{-5+\sqrt{17}}\sqrt{{x}^{4}+5\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-5+17^(1/2))^(1/2)*(1-(-5/4+1/4*17^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*17^(1/2))*x^2)^(1/2)/(x^4+5*x^2+2)^(1/2)*
EllipticF(1/2*x*(-5+17^(1/2))^(1/2),5/4*2^(1/2)+1/4*34^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(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{1}{\sqrt{x^{4} + 5 \, x^{2} + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(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{x^{4} + 5 x^{2} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(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{x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(x^4 + 5*x^2 + 2), x)