3.11.3 \(\int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx\) [1003]

Optimal. Leaf size=47 \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}}-\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}} \]

[Out]

1/12*EllipticE(1/2*x*6^(1/2),2/3*I*6^(1/2))*3^(1/2)-1/12*EllipticF(1/2*x*6^(1/2),2/3*I*6^(1/2))*3^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 47, 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}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}}-\frac {F\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.28, size = 40, normalized size = 0.85 \begin {gather*} \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )-F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(EllipticE[ArcSin[Sqrt[3/2]*x], -8/3] - EllipticF[ArcSin[Sqrt[3/2]*x], -8/3])/(4*Sqrt[3])

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Maple [A]
time = 0.13, size = 35, normalized size = 0.74

method	result	size
default	\(-\frac {\sqrt {3}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )\right )}{12}\)	\(35\)
elliptic	\(-\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (4 x^{2}+1\right )}\, \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \left (\EllipticF \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )-\EllipticE \left (\frac {x \sqrt {6}}{2}, \frac {2 i \sqrt {6}}{3}\right )\right )}{24 \sqrt {-3 x^{2}+2}\, \sqrt {-12 x^{4}+5 x^{2}+2}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*3^(1/2)*(EllipticF(1/2*x*6^(1/2),2/3*I*6^(1/2))-EllipticE(1/2*x*6^(1/2),2/3*I*6^(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/(-3*x^2+2)^(1/2)/(4*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.23, size = 23, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{12 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {2 - 3 x^{2}} \sqrt {4 x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{\sqrt {2-3\,x^2}\,\sqrt {4\,x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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