\(\int \frac {x^2}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx\) [995]

Optimal result

Integrand size = 26, antiderivative size = 31 \[ \int \frac {x^2}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx=\frac {1}{3} \sqrt {2} E\left (\arcsin (x)\left |-\frac {3}{2}\right .\right )-\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),-\frac {3}{2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, 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} \[ \int \frac {x^2}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx=\frac {1}{3} \sqrt {2} E\left (\arcsin (x)\left |-\frac {3}{2}\right .\right )-\frac {1}{3} \sqrt {2} \operatorname {EllipticF}\left (\arcsin (x),-\frac {3}{2}\right ) \]

[In]

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

[Out]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx=\frac {i \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )\right )}{\sqrt {3}} \]

[In]

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

[Out]

(I*(EllipticE[I*ArcSinh[Sqrt[3/2]*x], -2/3] - EllipticF[I*ArcSinh[Sqrt[3/2]*x], -2/3]))/Sqrt[3]

Maple [A] (verified)

Time = 3.43 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\left (F\left (x , \frac {i \sqrt {6}}{2}\right )-E\left (x , \frac {i \sqrt {6}}{2}\right )\right ) \sqrt {2}}{3}\)	\(25\)
elliptic	\(-\frac {\sqrt {-\left (3 x^{2}+2\right ) \left (x^{2}-1\right )}\, \sqrt {6 x^{2}+4}\, \left (F\left (x , \frac {i \sqrt {6}}{2}\right )-E\left (x , \frac {i \sqrt {6}}{2}\right )\right )}{3 \sqrt {3 x^{2}+2}\, \sqrt {-3 x^{4}+x^{2}+2}}\)	\(68\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {x^2}{\sqrt {1-x^2} \sqrt {2+3 x^2}} \, dx=-\frac {\sqrt {-3} x E(\arcsin \left (\frac {1}{x}\right )\,|\,-\frac {2}{3}) - \sqrt {-3} x F(\arcsin \left (\frac {1}{x}\right )\,|\,-\frac {2}{3}) + \sqrt {3 \, x^{2} + 2} \sqrt {-x^{2} + 1}}{3 \, x} \]

[In]

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

[Out]

-1/3*(sqrt(-3)*x*elliptic_e(arcsin(1/x), -2/3) - sqrt(-3)*x*elliptic_f(arcsin(1/x), -2/3) + sqrt(3*x^2 + 2)*sq
rt(-x^2 + 1))/x

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/(sqrt(-(x - 1)*(x + 1))*sqrt(3*x**2 + 2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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