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

Optimal. Leaf size=92 \[ \frac{\left (\sqrt{10} x^2+2\right ) \sqrt{\frac{5 x^4+5 x^2+2}{\left (\sqrt{10} x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{5}{2}} x\right ),\frac{1}{8} \left (4-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{5 x^4+5 x^2+2}} \]

[Out]

((2 + Sqrt[10]*x^2)*Sqrt[(2 + 5*x^2 + 5*x^4)/(2 + Sqrt[10]*x^2)^2]*EllipticF[2*ArcTan[(5/2)^(1/4)*x], (4 - Sqr
t[10])/8])/(2*10^(1/4)*Sqrt[2 + 5*x^2 + 5*x^4])

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Rubi [A]  time = 0.0258335, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1103} \[ \frac{\left (\sqrt{10} x^2+2\right ) \sqrt{\frac{5 x^4+5 x^2+2}{\left (\sqrt{10} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{5}{2}} x\right )|\frac{1}{8} \left (4-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{5 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 + 5*x^4],x]

[Out]

((2 + Sqrt[10]*x^2)*Sqrt[(2 + 5*x^2 + 5*x^4)/(2 + Sqrt[10]*x^2)^2]*EllipticF[2*ArcTan[(5/2)^(1/4)*x], (4 - Sqr
t[10])/8])/(2*10^(1/4)*Sqrt[2 + 5*x^2 + 5*x^4])

Rule 1103

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+5 x^2+5 x^4}} \, dx &=\frac{\left (2+\sqrt{10} x^2\right ) \sqrt{\frac{2+5 x^2+5 x^4}{\left (2+\sqrt{10} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{5}{2}} x\right )|\frac{1}{8} \left (4-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{2+5 x^2+5 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.114411, size = 144, normalized size = 1.57 \[ -\frac{i \sqrt{1-\frac{10 x^2}{-5-i \sqrt{15}}} \sqrt{1-\frac{10 x^2}{-5+i \sqrt{15}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{10}{-5-i \sqrt{15}}} x\right ),\frac{-5-i \sqrt{15}}{-5+i \sqrt{15}}\right )}{\sqrt{10} \sqrt{-\frac{1}{-5-i \sqrt{15}}} \sqrt{5 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - (10*x^2)/(-5 - I*Sqrt[15])]*Sqrt[1 - (10*x^2)/(-5 + I*Sqrt[15])]*EllipticF[I*ArcSinh[Sqrt[-10/(
-5 - I*Sqrt[15])]*x], (-5 - I*Sqrt[15])/(-5 + I*Sqrt[15])])/(Sqrt[10]*Sqrt[-(-5 - I*Sqrt[15])^(-1)]*Sqrt[2 + 5
*x^2 + 5*x^4])

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Maple [C]  time = 0.749, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{1- \left ( -5/4+i/4\sqrt{15} \right ){x}^{2}}\sqrt{1- \left ( -5/4-i/4\sqrt{15} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+i\sqrt{15}},1/2\,\sqrt{1+i\sqrt{15}} \right ) }{\sqrt{-5+i\sqrt{15}}\sqrt{5\,{x}^{4}+5\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-5+I*15^(1/2))^(1/2)*(1-(-5/4+1/4*I*15^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*I*15^(1/2))*x^2)^(1/2)/(5*x^4+5*x^2+2
)^(1/2)*EllipticF(1/2*x*(-5+I*15^(1/2))^(1/2),1/2*(1+I*15^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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