3.10.99 \(\int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx\) [999]

Optimal. Leaf size=35 \[ \frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}-\frac {F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}} \]

[Out]

1/6*EllipticE(2*x,1/4*I*6^(1/2))*2^(1/2)-1/6*EllipticF(2*x,1/4*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 {E\left (\text {ArcSin}(2 x)\left |-\frac {3}{8}\right .\right )}{3 \sqrt {2}}-\frac {F\left (\text {ArcSin}(2 x)\left |-\frac {3}{8}\right .\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

EllipticE[ArcSin[2*x], -3/8]/(3*Sqrt[2]) - EllipticF[ArcSin[2*x], -3/8]/(3*Sqrt[2])

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 {1-4 x^2} \sqrt {2+3 x^2}} \, dx &=\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\sqrt {1-4 x^2}} \, dx-\frac {2}{3} \int \frac {1}{\sqrt {1-4 x^2} \sqrt {2+3 x^2}} \, dx\\ &=\frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}-\frac {F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 28, normalized size = 0.80 \begin {gather*} \frac {E\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )-F\left (\sin ^{-1}(2 x)|-\frac {3}{8}\right )}{3 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[1 - 4*x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

(EllipticE[ArcSin[2*x], -3/8] - EllipticF[ArcSin[2*x], -3/8])/(3*Sqrt[2])

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Maple [A]
time = 0.13, size = 29, normalized size = 0.83

method	result	size
default	\(-\frac {\left (\EllipticF \left (2 x , \frac {i \sqrt {6}}{4}\right )-\EllipticE \left (2 x , \frac {i \sqrt {6}}{4}\right )\right ) \sqrt {2}}{6}\)	\(29\)
elliptic	\(-\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (4 x^{2}-1\right )}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (2 x , \frac {i \sqrt {6}}{4}\right )-\EllipticE \left (2 x , \frac {i \sqrt {6}}{4}\right )\right )}{6 \sqrt {3 x^{2}+2}\, \sqrt {-12 x^{4}-5 x^{2}+2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(EllipticF(2*x,1/4*I*6^(1/2))-EllipticE(2*x,1/4*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/(-4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.27, size = 23, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}}{12 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3*x^2 + 2)*sqrt(-4*x^2 + 1)/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {- \left (2 x - 1\right ) \left (2 x + 1\right )} \sqrt {3 x^{2} + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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