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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.162079, 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 \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.4818, size = 75, normalized size = 0.86 \[ \frac{\sqrt{c} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a + b x^{2}} \sqrt{c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(1/2)/(-d*x**2+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.10233, size = 89, normalized size = 1.02 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} F\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]

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 = 0.044, size = 106, normalized size = 1.2 \[ -{\frac{1}{bd{x}^{4}+ad{x}^{2}-c{x}^{2}b-ac}{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \]

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. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*sqrt(-d*x^2 + c)),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. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{2} + a} \sqrt{-d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x^{2}} \sqrt{c - d x^{2}}}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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