Optimal. Leaf size=51 \[ \frac{\sqrt{5 x^2+2} F\left (\tan ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{5 x^2+2}{x^2+1}}} \]
[Out]
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Rubi [A] time = 0.0310118, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\sqrt{5 x^2+2} F\left (\tan ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{5 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 + x^2]*Sqrt[2 + 5*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 5.53648, size = 48, normalized size = 0.94 \[ \frac{\sqrt{2} \sqrt{5 x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | - \frac{3}{2}\right )}{2 \sqrt{\frac{5 x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**2+1)**(1/2)/(5*x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.029018, size = 19, normalized size = 0.37 \[ -\frac{i F\left (i \sinh ^{-1}(x)|\frac{5}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + 5*x^2]),x]
[Out]
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Maple [A] time = 0.102, size = 20, normalized size = 0.4 \[ -{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{\sqrt{2}\sqrt{5}}{2}} \right ) \sqrt{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^2+1)^(1/2)/(5*x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{5 \, x^{2} + 2} \sqrt{x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**2+1)**(1/2)/(5*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="giac")
[Out]