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

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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0524295, 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} \[ \frac{\sqrt{c} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

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]
*Sqrt[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 = 0.0483519, size = 90, normalized size = 1. \[ \frac{\sqrt{-a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2}} \]

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])

________________________________________________________________________________________

Maple [B]  time = 0.015, size = 171, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}-bc{x}^{2}-ac \right ) d} \left ( -a{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) d-bc{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) +bc{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) \right ) \sqrt{-b{x}^{2}-a}\sqrt{-d{x}^{2}+c}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \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]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

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}}{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)/(d*x^2 - c), x)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)