3.1002 \(\int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4+x^2}} \, dx\)

Optimal. Leaf size=43 \[ \frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}}-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {493, 424, 419} \[ \frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}}-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticE[ArcSin[Sqrt[3/2]*x], -1/6])/Sqrt[3] - (2*EllipticF[ArcSin[Sqrt[3/2]*x], -1/6])/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 {4+x^2}} \, dx &=-\left (4 \int \frac {1}{\sqrt {2-3 x^2} \sqrt {4+x^2}} \, dx\right )+\int \frac {\sqrt {4+x^2}}{\sqrt {2-3 x^2}} \, dx\\ &=\frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}}-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 38, normalized size = 0.88 \[ \frac {2 \left (E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )-F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(EllipticE[ArcSin[Sqrt[3/2]*x], -1/6] - EllipticF[ArcSin[Sqrt[3/2]*x], -1/6]))/Sqrt[3]

________________________________________________________________________________________

fricas [F]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} + 4} \sqrt {-3 \, x^{2} + 2} x^{2}}{3 \, x^{4} + 10 \, x^{2} - 8}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 35, normalized size = 0.81 \[ -\frac {2 \sqrt {3}\, \left (-\EllipticE \left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{6}\right )+\EllipticF \left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{6}\right )\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________