3.10.97 \(\int \frac {x^2}{\sqrt {4-x^2} \sqrt {2+3 x^2}} \, dx\) [997]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {507, 435, 430} \begin {gather*} \frac {1}{3} \sqrt {2} E\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-6\right )-\frac {1}{3} \sqrt {2} F\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-6\right ) \end {gather*}

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 430

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

Rule 435

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

Rule 507

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])

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*}

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Mathematica [A]
time = 0.29, size = 28, normalized size = 0.80 \begin {gather*} \frac {1}{3} \sqrt {2} \left (E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right )-F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-6\right )\right ) \end {gather*}

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 = 0.12, size = 29, normalized size = 0.83

method	result	size
default	\(-\frac {\left (\EllipticF \left (\frac {x}{2}, i \sqrt {6}\right )-\EllipticE \left (\frac {x}{2}, i \sqrt {6}\right )\right ) \sqrt {2}}{3}\)	\(29\)
elliptic	\(-\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (x^{2}-4\right )}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (\frac {x}{2}, i \sqrt {6}\right )-\EllipticE \left (\frac {x}{2}, i \sqrt {6}\right )\right )}{3 \sqrt {3 x^{2}+2}\, \sqrt {-3 x^{4}+10 x^{2}+8}}\)	\(74\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [A]
time = 0.31, size = 23, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 4}}{3 \, x} \end {gather*}

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]

-1/3*sqrt(3*x^2 + 2)*sqrt(-x^2 + 4)/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right )} \sqrt {3 x^{2} + 2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {3\,x^2+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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