Optimal. Leaf size=159 \[ \frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
[Out]
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Rubi [A] time = 0.571441, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^2]/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)/(f*x**2+e)**(1/2),x)
[Out]
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Mathematica [A] time = 0.121682, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[Sqrt[a + b*x^2]/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
[Out]
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Maple [F] time = 0.098, size = 0, normalized size = 0. \[ \int{1\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2}}}{\sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="giac")
[Out]