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

Optimal. Leaf size=91 \[ \frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}} \]

[Out]

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

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Rubi [A]  time = 0.0577295, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {423, 426, 424, 421, 419} \[ \frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 - x^2]/Sqrt[c + d*x^2],x]

[Out]

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

Rule 423

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

Rule 426

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

Rule 424

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

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

Mathematica [A]  time = 0.0380096, size = 60, normalized size = 0.66 \[ \frac{2 \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c}{4 d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.022, size = 78, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( c{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) +4\,{\it EllipticF} \left ( x/2,2\,\sqrt{-{\frac{d}{c}}} \right ) d-c{\it EllipticE} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(x - 2)*(x + 2))/sqrt(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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