\(\int \frac {1}{\sqrt {-1-x^2} \sqrt {2+5 x^2}} \, dx\) [248]

Optimal result

Integrand size = 23, antiderivative size = 53 \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {\sqrt {2+5 x^2} \operatorname {EllipticF}\left (\arctan (x),-\frac {3}{2}\right )}{\sqrt {2} \sqrt {-1-x^2} \sqrt {\frac {2+5 x^2}{1+x^2}}} \]

[Out]

1/2*(1/(x^2+1))^(1/2)*(x^2+1)^(1/2)*EllipticF(x/(x^2+1)^(1/2),1/2*I*6^(1/2))*(5*x^2+2)^(1/2)*2^(1/2)/(-x^2-1)^
(1/2)/((5*x^2+2)/(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 = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {429} \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {\sqrt {5 x^2+2} \operatorname {EllipticF}\left (\arctan (x),-\frac {3}{2}\right )}{\sqrt {2} \sqrt {-x^2-1} \sqrt {\frac {5 x^2+2}{x^2+1}}} \]

[In]

Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 5*x^2]),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2+5 x^2} F\left (\tan ^{-1}(x)|-\frac {3}{2}\right )}{\sqrt {2} \sqrt {-1-x^2} \sqrt {\frac {2+5 x^2}{1+x^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+5 x^2}} \, dx=-\frac {i \sqrt {1+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}(x),\frac {5}{2}\right )}{\sqrt {2} \sqrt {-1-x^2}} \]

[In]

Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 5*x^2]),x]

[Out]

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

Maple [A] (verified)

Time = 2.88 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {i F\left (\frac {i x \sqrt {10}}{2}, \frac {\sqrt {10}}{5}\right ) \sqrt {5}\, \sqrt {-x^{2}-1}}{5 \sqrt {x^{2}+1}}\)	\(36\)
elliptic	\(-\frac {i \sqrt {-\left (x^{2}+1\right ) \left (5 x^{2}+2\right )}\, \sqrt {10}\, \sqrt {10 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i x \sqrt {10}}{2}, \frac {\sqrt {10}}{5}\right )}{10 \sqrt {-x^{2}-1}\, \sqrt {5 x^{2}+2}\, \sqrt {-5 x^{4}-7 x^{2}-2}}\)	\(84\)

[In]

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

[Out]

1/5*I*EllipticF(1/2*I*x*10^(1/2),1/5*10^(1/2))/(x^2+1)^(1/2)*5^(1/2)*(-x^2-1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2+5 x^2}} \, dx=\frac {1}{2} i \, \sqrt {-2} F(\arcsin \left (i \, x\right )\,|\,\frac {5}{2}) \]

[In]

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

[Out]

1/2*I*sqrt(-2)*elliptic_f(arcsin(I*x), 5/2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(1/((- x^2 - 1)^(1/2)*(5*x^2 + 2)^(1/2)),x)

[Out]

int(1/((- x^2 - 1)^(1/2)*(5*x^2 + 2)^(1/2)), x)