\(\int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx\) [131]

Optimal result

Integrand size = 16, antiderivative size = 12 \[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {7}{2}\right )}{\sqrt {2}} \]

[Out]

1/2*EllipticF(x,1/2*I*14^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, 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} \[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin (x),-\frac {7}{2}\right )}{\sqrt {2}} \]

[In]

Int[1/Sqrt[2 + 5*x^2 - 7*x^4],x]

[Out]

EllipticF[ArcSin[x], -7/2]/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 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 {7}\right ) \int \frac {1}{\sqrt {14-14 x^2} \sqrt {4+14 x^2}} \, dx \\ & = \frac {F\left (\sin ^{-1}(x)|-\frac {7}{2}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 5.42 \[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=-\frac {i \sqrt {1-x^2} \sqrt {2+7 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {7}{2}} x\right ),-\frac {2}{7}\right )}{\sqrt {7} \sqrt {2+5 x^2-7 x^4}} \]

[In]

Integrate[1/Sqrt[2 + 5*x^2 - 7*x^4],x]

[Out]

((-I)*Sqrt[1 - x^2]*Sqrt[2 + 7*x^2]*EllipticF[I*ArcSinh[Sqrt[7/2]*x], -2/7])/(Sqrt[7]*Sqrt[2 + 5*x^2 - 7*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. 42 vs. \(2 (13 ) = 26\).

Time = 0.63 (sec) , antiderivative size = 43, normalized size of antiderivative = 3.58

method	result	size
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, F\left (x , \frac {i \sqrt {14}}{2}\right )}{2 \sqrt {-7 x^{4}+5 x^{2}+2}}\)	\(43\)
elliptic	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {14 x^{2}+4}\, F\left (x , \frac {i \sqrt {14}}{2}\right )}{2 \sqrt {-7 x^{4}+5 x^{2}+2}}\)	\(43\)

[In]

int(1/(-7*x^4+5*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(-x^2+1)^(1/2)*(14*x^2+4)^(1/2)/(-7*x^4+5*x^2+2)^(1/2)*EllipticF(x,1/2*I*14^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\frac {1}{2} \, \sqrt {2} F(\arcsin \left (x\right )\,|\,-\frac {7}{2}) \]

[In]

integrate(1/(-7*x^4+5*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*elliptic_f(arcsin(x), -7/2)

Sympy [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\int \frac {1}{\sqrt {- 7 x^{4} + 5 x^{2} + 2}}\, dx \]

[In]

integrate(1/(-7*x**4+5*x**2+2)**(1/2),x)

[Out]

Integral(1/sqrt(-7*x**4 + 5*x**2 + 2), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\int { \frac {1}{\sqrt {-7 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-7*x^4+5*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-7*x^4 + 5*x^2 + 2), x)

Giac [F]

\[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\int { \frac {1}{\sqrt {-7 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]

[In]

integrate(1/(-7*x^4+5*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-7*x^4 + 5*x^2 + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+5 x^2-7 x^4}} \, dx=\int \frac {1}{\sqrt {-7\,x^4+5\,x^2+2}} \,d x \]

[In]

int(1/(5*x^2 - 7*x^4 + 2)^(1/2),x)

[Out]

int(1/(5*x^2 - 7*x^4 + 2)^(1/2), x)