\(\int \frac {1}{\sqrt {-3-5 x^2-2 x^4}} \, dx\) [107]

Optimal result

Integrand size = 16, antiderivative size = 53 \[ \int \frac {1}{\sqrt {-3-5 x^2-2 x^4}} \, dx=\frac {\sqrt {3+2 x^2} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{3}\right )}{\sqrt {3} \sqrt {-1-x^2} \sqrt {\frac {3+2 x^2}{1+x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, 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, 429} \[ \int \frac {1}{\sqrt {-3-5 x^2-2 x^4}} \, dx=\frac {\sqrt {2 x^2+3} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{3}\right )}{\sqrt {3} \sqrt {-x^2-1} \sqrt {\frac {2 x^2+3}{x^2+1}}} \]

[In]

Int[1/Sqrt[-3 - 5*x^2 - 2*x^4],x]

[Out]

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

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[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 {2}\right ) \int \frac {1}{\sqrt {-4-4 x^2} \sqrt {6+4 x^2}} \, dx \\ & = \frac {\sqrt {3+2 x^2} F\left (\tan ^{-1}(x)|\frac {1}{3}\right )}{\sqrt {3} \sqrt {-1-x^2} \sqrt {\frac {3+2 x^2}{1+x^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\sqrt {-3-5 x^2-2 x^4}} \, dx=-\frac {i \sqrt {1+x^2} \sqrt {3+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{3}} x\right ),\frac {3}{2}\right )}{\sqrt {2} \sqrt {-3-5 x^2-2 x^4}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+9}\, F\left (i x , \frac {\sqrt {6}}{3}\right )}{3 \sqrt {-2 x^{4}-5 x^{2}-3}}\)	\(44\)
elliptic	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+9}\, F\left (i x , \frac {\sqrt {6}}{3}\right )}{3 \sqrt {-2 x^{4}-5 x^{2}-3}}\)	\(44\)

[In]

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

[Out]

-1/3*I*(x^2+1)^(1/2)*(6*x^2+9)^(1/2)/(-2*x^4-5*x^2-3)^(1/2)*EllipticF(I*x,1/3*6^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt {-3-5 x^2-2 x^4}} \, dx=\frac {1}{6} \, \sqrt {3} \sqrt {-2} \sqrt {-3} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} \sqrt {-2} x\right )\,|\,\frac {3}{2}) \]

[In]

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

[Out]

1/6*sqrt(3)*sqrt(-2)*sqrt(-3)*elliptic_f(arcsin(1/3*sqrt(3)*sqrt(-2)*x), 3/2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/(- 5*x^2 - 2*x^4 - 3)^(1/2),x)

[Out]

int(1/(- 5*x^2 - 2*x^4 - 3)^(1/2), x)