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

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

[Out]

1/2*(1/(4+x^2*(5+17^(1/2))))^(1/2)*(4+x^2*(5+17^(1/2)))^(3/2)*EllipticF(x*(5+17^(1/2))^(1/2)/(4+x^2*(5+17^(1/2
)))^(1/2),1/2*(-17+5*17^(1/2))^(1/2))*((4+x^2*(5-17^(1/2)))/(4+x^2*(5+17^(1/2))))^(1/2)/(x^4+5*x^2+2)^(1/2)/(5
+17^(1/2))^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.08, size = 103, normalized size = 0.95 \[ -\frac {i \sqrt {2 x^2-\sqrt {17}+5} \sqrt {2 x^2+\sqrt {17}+5} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {17}}} x\right )|\frac {21}{4}+\frac {5 \sqrt {17}}{4}\right )}{\sqrt {2 \left (5-\sqrt {17}\right )} \sqrt {x^4+5 x^2+2}} \]

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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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

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)

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maple [A]  time = 0.08, size = 76, normalized size = 0.70 \[ \frac {2 \sqrt {-\left (-\frac {5}{4}+\frac {\sqrt {17}}{4}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{4}-\frac {\sqrt {17}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-5+\sqrt {17}}\, x}{2}, \frac {5 \sqrt {2}}{4}+\frac {\sqrt {34}}{4}\right )}{\sqrt {-5+\sqrt {17}}\, \sqrt {x^{4}+5 x^{2}+2}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 5 \, x^{2} + 2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {x^4+5\,x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2 + x^4 + 2)^(1/2),x)

[Out]

int(1/(5*x^2 + x^4 + 2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{4} + 5 x^{2} + 2}}\, dx \]

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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