Optimal. Leaf size=191 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{c+d x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.382336, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{d \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}}-\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{b x^2-a} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-a + b*x^2]/Sqrt[c + d*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 75.6895, size = 162, normalized size = 0.85 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c + d x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{d \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{- a + b x^{2}}} - \frac{\sqrt{a} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 + \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{- a + b x^{2}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2-a)**(1/2)/(d*x**2+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0756661, size = 90, normalized size = 0.47 \[ \frac{\sqrt{b x^2-a} \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-a + b*x^2]/Sqrt[c + d*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.022, size = 109, normalized size = 0.6 \[{\frac{a}{-bd{x}^{4}+ad{x}^{2}-c{x}^{2}b+ac}\sqrt{b{x}^{2}-a}\sqrt{d{x}^{2}+c}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{-{\frac{b{x}^{2}-a}{a}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]
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]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} - a}}{\sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 - a)/sqrt(d*x^2 + c),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{2} - a}}{\sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 - a)/sqrt(d*x^2 + c),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- a + b x^{2}}}{\sqrt{c + d x^{2}}}\, dx \]
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]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} - a}}{\sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 - a)/sqrt(d*x^2 + c),x, algorithm="giac")
[Out]