3.278 \(\int \frac{\sqrt{c-d x^2}}{\sqrt{a-b x^2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}} \]

[Out]

(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqr
t[a - b*x^2]*Sqrt[1 - (d*x^2)/c])

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Rubi [A]  time = 0.0517505, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {427, 426, 424} \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - d*x^2]/Sqrt[a - b*x^2],x]

[Out]

(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sqr
t[a - b*x^2]*Sqrt[1 - (d*x^2)/c])

Rule 427

Int[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[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

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]

Rubi steps

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

Mathematica [A]  time = 0.0555552, size = 88, normalized size = 1. \[ \frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{c-d x^2} E\left (\sin ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{\frac{c-d x^2}{c}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - d*x^2]/Sqrt[a - b*x^2],x]

[Out]

(Sqrt[(a - b*x^2)/a]*Sqrt[c - d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[a - b*x^2]*S
qrt[(c - d*x^2)/c])

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Maple [B]  time = 0.012, size = 164, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( bd{x}^{4}-ad{x}^{2}-bc{x}^{2}+ac \right ) b} \left ( -ad{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) +c{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) b+ad{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) \right ) \sqrt{-d{x}^{2}+c}\sqrt{-b{x}^{2}+a}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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