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

Optimal. Leaf size=35 \[ \frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-\frac{1}{3} \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-6\right ) \]

[Out]

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

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Rubi [A]  time = 0.0314943, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {493, 424, 419} \[ \frac{1}{3} \sqrt{2} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-\frac{1}{3} \sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[4 - x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

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

Rule 493

Int[(x_)^(n_)/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/b, Int[Sqrt[a +
b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c,
d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) &&  !(EqQ[n, 2] && SimplerSqrtQ[-(b/a), -(d/c)])

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]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

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

Mathematica [A]  time = 0.0300229, size = 28, normalized size = 0.8 \[ \frac{1}{3} \sqrt{2} \left (E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-6\right )-\text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-6\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.02, size = 29, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{3} \left ({\it EllipticF} \left ({\frac{x}{2}},i\sqrt{6} \right ) -{\it EllipticE} \left ({\frac{x}{2}},i\sqrt{6} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 4} x^{2}}{3 \, x^{4} - 10 \, x^{2} - 8}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(3*x^2 + 2)*sqrt(-x^2 + 4)*x^2/(3*x^4 - 10*x^2 - 8), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(sqrt(-(x - 2)*(x + 2))*sqrt(3*x**2 + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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