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

Optimal. Leaf size=90 \[ \frac{\sqrt{c} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c}} \]

[Out]

(Sqrt[c]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sq
rt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2])

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Rubi [A]  time = 0.0519834, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {427, 426, 424} \[ \frac{\sqrt{c} \sqrt{b x^2-a} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[-a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(Sqrt[d]*Sq
rt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2])

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

Mathematica [A]  time = 0.0463408, size = 90, normalized size = 1. \[ \frac{\sqrt{b x^2-a} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{d x^2-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 111, normalized size = 1.2 \begin{align*}{\frac{a}{-bd{x}^{4}+ad{x}^{2}+bc{x}^{2}-ac}\sqrt{b{x}^{2}-a}\sqrt{d{x}^{2}-c}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^2 - a)/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{b x^{2} - a}}{\sqrt{d x^{2} - c}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-a + b*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{b x^{2} - a}}{\sqrt{d x^{2} - c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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