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

Optimal. Leaf size=20 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right ),-\frac{2}{5}\right )}{\sqrt{5}} \]

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

________________________________________________________________________________________

Rubi [A]  time = 0.0091487, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {419} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2-5 x^2} \sqrt{1+x^2}} \, dx &=\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0065552, size = 20, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right ),-\frac{2}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[Sqrt[5/2]*x], -2/5]/Sqrt[5]

________________________________________________________________________________________

Maple [A]  time = 0.035, size = 19, normalized size = 1. \begin{align*}{\frac{\sqrt{5}}{5}{\it EllipticF} \left ({\frac{x\sqrt{10}}{2}},{\frac{i}{5}}\sqrt{10} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}{5 \, x^{4} + 3 \, x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.56159, size = 36, normalized size = 1.8 \begin{align*} \begin{cases} \frac{\sqrt{5} F\left (\operatorname{asin}{\left (\frac{\sqrt{10} x}{2} \right )}\middle | - \frac{2}{5}\right )}{5} & \text{for}\: x > - \frac{\sqrt{10}}{5} \wedge x < \frac{\sqrt{10}}{5} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(5)*elliptic_f(asin(sqrt(10)*x/2), -2/5)/5, (x > -sqrt(10)/5) & (x < sqrt(10)/5)))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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