\(\int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx\) [998]

Optimal result

Integrand size = 26, antiderivative size = 35 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=-\frac {1}{3} \sqrt {2} E\left (\left .\arcsin \left (\frac {x}{2}\right )\right |6\right )+\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),6\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {507, 435, 430} \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),6\right )-\frac {1}{3} \sqrt {2} E\left (\left .\arcsin \left (\frac {x}{2}\right )\right |6\right ) \]

[In]

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

[Out]

-1/3*(Sqrt[2]*EllipticE[ArcSin[x/2], 6]) + (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*} \text {integral}& = -\left (\frac {1}{3} \int \frac {\sqrt {2-3 x^2}}{\sqrt {4-x^2}} \, dx\right )+\frac {2}{3} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {4-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*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=-\frac {2 \left (E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|\frac {1}{6}\right )-\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{6}\right )\right )}{\sqrt {3}} \]

[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]

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \sqrt {3}\, \left (F\left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )-E\left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )\right )}{3}\)	\(33\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (x^{2}-4\right )}\, \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \left (F\left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )-E\left (\frac {x \sqrt {6}}{2}, \frac {\sqrt {6}}{6}\right )\right )}{3 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{4}-14 x^{2}+8}}\)	\(80\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=\frac {8 \, \sqrt {3} x E(\arcsin \left (\frac {2}{x}\right )\,|\,\frac {1}{6}) - 8 \, \sqrt {3} x F(\arcsin \left (\frac {2}{x}\right )\,|\,\frac {1}{6}) + \sqrt {-x^{2} + 4} \sqrt {-3 \, x^{2} + 2}}{3 \, x} \]

[In]

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

[Out]

1/3*(8*sqrt(3)*x*elliptic_e(arcsin(2/x), 1/6) - 8*sqrt(3)*x*elliptic_f(arcsin(2/x), 1/6) + sqrt(-x^2 + 4)*sqrt
(-3*x^2 + 2))/x

Sympy [F]

\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=\int \frac {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right )} \sqrt {2 - 3 x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{2} + 4} \sqrt {-3 \, x^{2} + 2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-x^{2} + 4} \sqrt {-3 \, x^{2} + 2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {4-x^2}} \, dx=\int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {2-3\,x^2}} \,d x \]

[In]

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

[Out]

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