3.291 \(\int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{\frac{d x^2}{c}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

[Out]

(Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

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Rubi [A]  time = 0.0200224, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {421, 419} \[ \frac{\sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

Rule 421

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

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{4-x^2} \sqrt{c+d x^2}} \, dx &=\frac{\sqrt{1+\frac{d x^2}{c}} \int \frac{1}{\sqrt{4-x^2} \sqrt{1+\frac{d x^2}{c}}} \, dx}{\sqrt{c+d x^2}}\\ &=\frac{\sqrt{1+\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0379465, size = 40, normalized size = 1.03 \[ \frac{\sqrt{\frac{c+d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

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Maple [A]  time = 0.022, size = 38, normalized size = 1. \begin{align*}{\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(d*x^2 + c)*sqrt(-x^2 + 4)/(d*x^4 + (c - 4*d)*x^2 - 4*c), x)

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Sympy [A]  time = 2.21516, size = 20, normalized size = 0.51 \begin{align*} \begin{cases} \frac{F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{\sqrt{c}} & \text{for}\: x > -2 \wedge x < 2 \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Piecewise((elliptic_f(asin(x/2), -4*d/c)/sqrt(c), (x > -2) & (x < 2)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)