\(\int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx\) [16]

Optimal result

Integrand size = 16, antiderivative size = 10 \[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=\operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-6\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430} \[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=\operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-6\right ) \]

[In]

Int[1/Sqrt[2 + 5*x^2 - 3*x^4],x]

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -6]

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 1109

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {12-6 x^2} \sqrt {2+6 x^2}} \, dx \\ & = F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 6.50 \[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=-\frac {i \sqrt {1-\frac {x^2}{2}} \sqrt {1+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {3} x\right ),-\frac {1}{6}\right )}{\sqrt {3} \sqrt {2+5 x^2-3 x^4}} \]

[In]

Integrate[1/Sqrt[2 + 5*x^2 - 3*x^4],x]

[Out]

((-I)*Sqrt[1 - x^2/2]*Sqrt[1 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3]*x], -1/6])/(Sqrt[3]*Sqrt[2 + 5*x^2 - 3*x^4])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (13 ) = 26\).

Time = 0.54 (sec) , antiderivative size = 51, normalized size of antiderivative = 5.10

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {3 x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {6}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}+2}}\)	\(51\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {3 x^{2}+1}\, F\left (\frac {x \sqrt {2}}{2}, i \sqrt {6}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}+2}}\)	\(51\)

[In]

int(1/(-3*x^4+5*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*2^(1/2)*(-2*x^2+4)^(1/2)*(3*x^2+1)^(1/2)/(-3*x^4+5*x^2+2)^(1/2)*EllipticF(1/2*x*2^(1/2),I*6^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x\right )\,|\,-6) \]

[In]

integrate(1/(-3*x^4+5*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

elliptic_f(arcsin(1/2*sqrt(2)*x), -6)

Sympy [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {- 3 x^{4} + 5 x^{2} + 2}}\, dx \]

[In]

integrate(1/(-3*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**4 + 5*x**2 + 2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-3*x^4+5*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-3*x^4+5*x^2+2)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {-3\,x^4+5\,x^2+2}} \,d x \]

[In]

int(1/(5*x^2 - 3*x^4 + 2)^(1/2),x)

[Out]

int(1/(5*x^2 - 3*x^4 + 2)^(1/2), x)