Optimal. Leaf size=198 \[ -\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{d} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.397058, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{d} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-c + d*x^2]/Sqrt[-a - b*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 94.7698, size = 167, normalized size = 0.84 \[ - \frac{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{- a - b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{- c + d x^{2}}} - \frac{\sqrt{c} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{- a - b x^{2}} \sqrt{- c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2-c)**(1/2)/(-b*x**2-a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0637675, size = 93, normalized size = 0.47 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{-a-b x^2} \sqrt{\frac{c-d x^2}{c}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-c + d*x^2]/Sqrt[-a - b*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.018, size = 167, normalized size = 0.8 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}-c{x}^{2}b-ac \right ) b}\sqrt{d{x}^{2}-c}\sqrt{-b{x}^{2}-a}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}} \left ( ad{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) +c{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) b-ad{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \right ){\frac{1}{\sqrt{{\frac{d}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2-c)^(1/2)/(-b*x^2-a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} - c}}{\sqrt{-b x^{2} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 - c)/sqrt(-b*x^2 - a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{2} - c}}{\sqrt{-b x^{2} - a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 - c)/sqrt(-b*x^2 - a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c + d x^{2}}}{\sqrt{- a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2-c)**(1/2)/(-b*x**2-a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} - c}}{\sqrt{-b x^{2} - a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^2 - c)/sqrt(-b*x^2 - a),x, algorithm="giac")
[Out]