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

Optimal. Leaf size=90 \[ \frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{4 x^4+5 x^2+2}{\left (\sqrt{2} x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right ),\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}} \]

[Out]

((1 + Sqrt[2]*x^2)*Sqrt[(2 + 5*x^2 + 4*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticF[2*ArcTan[2^(1/4)*x], (8 - 5*Sqrt[2]
)/16])/(2*2^(3/4)*Sqrt[2 + 5*x^2 + 4*x^4])

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Rubi [A]  time = 0.0189061, antiderivative size = 90, 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} \[ \frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{4 x^4+5 x^2+2}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x^2 + 4*x^4],x]

[Out]

((1 + Sqrt[2]*x^2)*Sqrt[(2 + 5*x^2 + 4*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticF[2*ArcTan[2^(1/4)*x], (8 - 5*Sqrt[2]
)/16])/(2*2^(3/4)*Sqrt[2 + 5*x^2 + 4*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+4 x^4}} \, dx &=\frac{\left (1+\sqrt{2} x^2\right ) \sqrt{\frac{2+5 x^2+4 x^4}{\left (1+\sqrt{2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{2+5 x^2+4 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0882328, size = 147, normalized size = 1.63 \[ -\frac{i \sqrt{1-\frac{8 x^2}{-5-i \sqrt{7}}} \sqrt{1-\frac{8 x^2}{-5+i \sqrt{7}}} \text{EllipticF}\left (i \sinh ^{-1}\left (2 \sqrt{-\frac{2}{-5-i \sqrt{7}}} x\right ),\frac{-5-i \sqrt{7}}{-5+i \sqrt{7}}\right )}{2 \sqrt{2} \sqrt{-\frac{1}{-5-i \sqrt{7}}} \sqrt{4 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*Sqrt[1 - (8*x^2)/(-5 - I*Sqrt[7])]*Sqrt[1 - (8*x^2)/(-5 + I*Sqrt[7])]*EllipticF[I*ArcSinh[2*Sqrt[-2/(-
5 - I*Sqrt[7])]*x], (-5 - I*Sqrt[7])/(-5 + I*Sqrt[7])])/(Sqrt[2]*Sqrt[-(-5 - I*Sqrt[7])^(-1)]*Sqrt[2 + 5*x^2 +
4*x^4])

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Maple [C]  time = 0.743, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{1- \left ( -5/4+i/4\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -5/4-i/4\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+i\sqrt{7}},1/4\,\sqrt{9+5\,i\sqrt{7}} \right ) }{\sqrt{-5+i\sqrt{7}}\sqrt{4\,{x}^{4}+5\,{x}^{2}+2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-5+I*7^(1/2))^(1/2)*(1-(-5/4+1/4*I*7^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*I*7^(1/2))*x^2)^(1/2)/(4*x^4+5*x^2+2)^(
1/2)*EllipticF(1/2*x*(-5+I*7^(1/2))^(1/2),1/4*(9+5*I*7^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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