Optimal. Leaf size=148 \[ \frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}+\frac{\sqrt{3 x^2+2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]
[Out]
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Rubi [A] time = 0.16047, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{4 \sqrt{3 x^2+2} x}{3 \sqrt{4 x^2+1}}+\frac{\sqrt{3 x^2+2} F\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{2 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}}-\frac{2 \sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 20.2569, size = 121, normalized size = 0.82 \[ \frac{4 x \sqrt{3 x^{2} + 2}}{3 \sqrt{4 x^{2} + 1}} - \frac{2 \sqrt{3 x^{2} + 2} E\left (\operatorname{atan}{\left (2 x \right )}\middle | \frac{5}{8}\right )}{3 \sqrt{\frac{3 x^{2} + 2}{8 x^{2} + 2}} \sqrt{4 x^{2} + 1}} + \frac{\sqrt{3 x^{2} + 2} F\left (\operatorname{atan}{\left (2 x \right )}\middle | \frac{5}{8}\right )}{4 \sqrt{\frac{3 x^{2} + 2}{8 x^{2} + 2}} \sqrt{4 x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0252582, size = 27, normalized size = 0.18 \[ -\frac{i E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + 4*x^2]/Sqrt[2 + 3*x^2],x]
[Out]
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Maple [C] time = 0.113, size = 26, normalized size = 0.2 \[ -{\frac{i}{3}}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{3}\sqrt{2}x,{\frac{2\,\sqrt{3}\sqrt{2}}{3}} \right ) \sqrt{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((4*x^2+1)^(1/2)/(3*x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 1)/sqrt(3*x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 1)/sqrt(3*x^2 + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 x^{2} + 1}}{\sqrt{3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{4 \, x^{2} + 1}}{\sqrt{3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(4*x^2 + 1)/sqrt(3*x^2 + 2),x, algorithm="giac")
[Out]