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

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

[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]
*Sqrt[-a + b*x^2]*Sqrt[1 + (d*x^2)/c])

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Rubi [A]  time = 0.0500475, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {427, 426, 424} \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}} \]

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]
*Sqrt[-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 = 0.0418449, size = 91, normalized size = 1. \[ \frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{\frac{b}{a}} \sqrt{b x^2-a} \sqrt{\frac{c+d x^2}{c}}} \]

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]*Sqrt[(c + d*x^2)/c])

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Maple [A]  time = 0.01, size = 110, normalized size = 1.2 \begin{align*}{\frac{c}{bd{x}^{4}-ad{x}^{2}+bc{x}^{2}-ac}\sqrt{-d{x}^{2}-c}\sqrt{b{x}^{2}-a}\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ){\frac{1}{\sqrt{{\frac{b}{a}}}}}} \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]

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

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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{-d x^{2} - c}}{\sqrt{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(-d*x^2 - c)/sqrt(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)