3.216 \(\int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=87 \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0524197, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {421, 419} \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0649762, size = 89, normalized size = 1.02 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (x \sqrt{-\frac{b}{a}}\right ),-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 106, normalized size = 1.2 \begin{align*} -{\frac{1}{bd{x}^{4}+ad{x}^{2}-bc{x}^{2}-ac}{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(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}}{b d x^{4} -{\left (b c - a d\right )} x^{2} - a c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b*x**2+a)**(1/2)/(-d*x**2+c)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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