Optimal. Leaf size=88 \[ \frac{\sqrt{c} \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.171007, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{\sqrt{c} \sqrt{1-\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 54.7996, size = 73, normalized size = 0.83 \[ \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | \frac{a d}{b c}\right )}{\sqrt{b} \sqrt{a - b x^{2}} \sqrt{c - d x^{2}}} \]
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]
_______________________________________________________________________________________
Mathematica [A] time = 0.0925442, size = 88, normalized size = 1. \[ \frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} F\left (\sin ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{\sqrt{\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a - b*x^2]*Sqrt[c - d*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.041, size = 108, 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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]