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

Optimal. Leaf size=21 \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

[Out]

(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3]

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Rubi [A]  time = 0.0064591, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {424} \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 + x^2]/Sqrt[2 - 3*x^2],x]

[Out]

(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{4+x^2}}{\sqrt{2-3 x^2}} \, dx &=\frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0041913, size = 21, normalized size = 1. \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3]

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Maple [A]  time = 0.018, size = 19, normalized size = 0.9 \begin{align*}{\frac{2\,\sqrt{3}}{3}{\it EllipticE} \left ({\frac{x\sqrt{6}}{2}},{\frac{i}{6}}\sqrt{6} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*EllipticE(1/2*x*6^(1/2),1/6*I*6^(1/2))*3^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 4.51391, size = 37, normalized size = 1.76 \begin{align*} \begin{cases} \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | - \frac{1}{6}\right )}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*sqrt(3)*elliptic_e(asin(sqrt(6)*x/2), -1/6)/3, (x > -sqrt(6)/3) & (x < sqrt(6)/3)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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