Optimal. Leaf size=58 \[ \frac{2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}} \]
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Rubi [A] time = 0.0756475, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2 \sqrt{x^2+1} \Pi \left (1-\frac{2 b}{a};\left .\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-1\right )}{a \sqrt{\frac{x^2+1}{x^2+2}} \sqrt{x^2+2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]
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Rubi in Sympy [A] time = 13.3641, size = 58, normalized size = 1. \[ \frac{2 \sqrt{2} \sqrt{x^{2} + 1} \Pi \left (1 - \frac{2 b}{a}; \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{a \sqrt{\frac{2 x^{2} + 2}{x^{2} + 2}} \sqrt{x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+2)**(1/2)/(x**2+1)**(1/2)/(b*x**2+a),x)
[Out]
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Mathematica [C] time = 0.101301, size = 50, normalized size = 0.86 \[ -\frac{i \left (a F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )-(a-2 b) \Pi \left (\frac{b}{a};i \sinh ^{-1}(x)|\frac{1}{2}\right )\right )}{\sqrt{2} a b} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.021, size = 64, normalized size = 1.1 \[{\frac{i}{ab} \left ( a{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) -2\,b{\it EllipticPi} \left ( i/2x\sqrt{2},2\,{\frac{b}{a}},\sqrt{2} \right ) -a{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+2)^(1/2)/(x^2+1)^(1/2)/(b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2)/((b*x^2 + a)*sqrt(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt{x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2)/((b*x^2 + a)*sqrt(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{\left (a + b x^{2}\right ) \sqrt{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+2)**(1/2)/(x**2+1)**(1/2)/(b*x**2+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 2)/((b*x^2 + a)*sqrt(x^2 + 1)),x, algorithm="giac")
[Out]