3.1003 \(\int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+4 x^2}} \, dx\)

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 = 0.03, 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 = {493, 424, 419} \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}}-\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}} \]

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 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)])

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 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)])

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 = 0.03, size = 40, normalized size = 0.85 \[ \frac {E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )-F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {8}{3}\right )}{4 \sqrt {3}} \]

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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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2} x^{2}}{12 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \]

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]

integral(-sqrt(4*x^2 + 1)*sqrt(-3*x^2 + 2)*x^2/(12*x^4 - 5*x^2 - 2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}\,{d x} \]

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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maple [A]  time = 0.04, size = 35, normalized size = 0.74 \[ -\frac {\sqrt {3}\, \left (-\EllipticE \left (\frac {\sqrt {6}\, x}{2}, \frac {2 i \sqrt {6}}{3}\right )+\EllipticF \left (\frac {\sqrt {6}\, x}{2}, \frac {2 i \sqrt {6}}{3}\right )\right )}{12} \]

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)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {4 \, x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}\,{d x} \]

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

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

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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