Optimal. Leaf size=80 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
[Out]
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Rubi [A] time = 0.0924041, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+1}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{3 \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 13.6039, size = 70, normalized size = 0.88 \[ \frac{x \sqrt{3 x^{2} + 2}}{3 \sqrt{x^{2} + 1}} - \frac{\sqrt{2} \sqrt{3 x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | - \frac{1}{2}\right )}{3 \sqrt{\frac{3 x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(x**2+1)**(1/2)/(3*x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0402996, size = 34, normalized size = 0.42 \[ -\frac{1}{3} i \sqrt{2} \left (E\left (i \sinh ^{-1}(x)|\frac{3}{2}\right )-F\left (i \sinh ^{-1}(x)|\frac{3}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]),x]
[Out]
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Maple [A] time = 0.019, size = 36, normalized size = 0.5 \[{\frac{i}{3}} \left ({\it EllipticF} \left ( ix,{\frac{\sqrt{3}\sqrt{2}}{2}} \right ) -{\it EllipticE} \left ( ix,{\frac{\sqrt{3}\sqrt{2}}{2}} \right ) \right ) \sqrt{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(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. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{x^{2} + 1} \sqrt{3 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(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. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="giac")
[Out]