\(\int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx\) [1000]

Optimal result

Integrand size = 26, antiderivative size = 35 \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=-\frac {E\left (\arcsin (2 x)\left |\frac {3}{8}\right .\right )}{3 \sqrt {2}}+\frac {\operatorname {EllipticF}\left (\arcsin (2 x),\frac {3}{8}\right )}{3 \sqrt {2}} \]

[Out]

-1/6*EllipticE(2*x,1/4*6^(1/2))*2^(1/2)+1/6*EllipticF(2*x,1/4*6^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.02 (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} \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (2 x),\frac {3}{8}\right )}{3 \sqrt {2}}-\frac {E\left (\arcsin (2 x)\left |\frac {3}{8}\right .\right )}{3 \sqrt {2}} \]

[In]

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

[Out]

-1/3*EllipticE[ArcSin[2*x], 3/8]/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*} \text {integral}& = -\left (\frac {1}{3} \int \frac {\sqrt {2-3 x^2}}{\sqrt {1-4 x^2}} \, dx\right )+\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*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=\frac {-E\left (\arcsin (2 x)\left |\frac {3}{8}\right .\right )+\operatorname {EllipticF}\left (\arcsin (2 x),\frac {3}{8}\right )}{3 \sqrt {2}} \]

[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])

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {\sqrt {2}\, \left (F\left (2 x , \frac {\sqrt {6}}{4}\right )-E\left (2 x , \frac {\sqrt {6}}{4}\right )\right )}{6}\)	\(27\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}-2\right ) \left (4 x^{2}-1\right )}\, \sqrt {-6 x^{2}+4}\, \left (F\left (2 x , \frac {\sqrt {6}}{4}\right )-E\left (2 x , \frac {\sqrt {6}}{4}\right )\right )}{6 \sqrt {-3 x^{2}+2}\, \sqrt {12 x^{4}-11 x^{2}+2}}\)	\(73\)

[In]

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

[Out]

1/6*2^(1/2)*(EllipticF(2*x,1/4*6^(1/2))-EllipticE(2*x,1/4*6^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (29) = 58\).

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=\frac {4 \, \sqrt {2} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {3}{8}) - 4 \, \sqrt {2} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {3}{8}) + 3 \, \sqrt {-3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}}{36 \, x} \]

[In]

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

[Out]

1/36*(4*sqrt(2)*x*elliptic_e(arcsin(1/3*sqrt(3)*sqrt(2)/x), 3/8) - 4*sqrt(2)*x*elliptic_f(arcsin(1/3*sqrt(3)*s
qrt(2)/x), 3/8) + 3*sqrt(-3*x^2 + 2)*sqrt(-4*x^2 + 1))/x

Sympy [F]

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

[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(2 - 3*x**2)), x)

Maxima [F]

\[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {-3 \, x^{2} + 2} \sqrt {-4 \, x^{2} + 1}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {1-4 x^2} \sqrt {2-3 x^2}} \, dx=\int \frac {x^2}{\sqrt {2-3\,x^2}\,\sqrt {1-4\,x^2}} \,d x \]

[In]

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

[Out]

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