3.1005 \(\int \frac{x^2}{\sqrt{4+x^2} \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+4}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\left .\tan ^{-1}\left (\frac{x}{2}\right )\right |-5\right )}{3 \sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{x^2+4}}} \]

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[4 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTa
n[x/2], -5])/(3*Sqrt[4 + x^2]*Sqrt[(2 + 3*x^2)/(4 + x^2)])

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Rubi [A]  time = 0.0952093, antiderivative size = 82, 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 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{x^2+4}}-\frac{\sqrt{2} \sqrt{3 x^2+2} E\left (\left .\tan ^{-1}\left (\frac{x}{2}\right )\right |-5\right )}{3 \sqrt{x^2+4} \sqrt{\frac{3 x^2+2}{x^2+4}}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[4 + x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[4 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTa
n[x/2], -5])/(3*Sqrt[4 + x^2]*Sqrt[(2 + 3*x^2)/(4 + x^2)])

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Rubi in Sympy [A]  time = 13.8343, size = 68, normalized size = 0.83 \[ \frac{x \sqrt{3 x^{2} + 2}}{3 \sqrt{x^{2} + 4}} - \frac{2 \sqrt{3 x^{2} + 2} E\left (\operatorname{atan}{\left (\frac{x}{2} \right )}\middle | -5\right )}{3 \sqrt{\frac{12 x^{2} + 8}{2 x^{2} + 8}} \sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**2+4)**(1/2)/(3*x**2+2)**(1/2),x)

[Out]

x*sqrt(3*x**2 + 2)/(3*sqrt(x**2 + 4)) - 2*sqrt(3*x**2 + 2)*elliptic_e(atan(x/2),
 -5)/(3*sqrt((12*x**2 + 8)/(2*x**2 + 8))*sqrt(x**2 + 4))

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Mathematica [C]  time = 0.039859, size = 38, normalized size = 0.46 \[ -\frac{1}{3} i \sqrt{2} \left (E\left (\left .i \sinh ^{-1}\left (\frac{x}{2}\right )\right |6\right )-F\left (\left .i \sinh ^{-1}\left (\frac{x}{2}\right )\right |6\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[4 + x^2]*Sqrt[2 + 3*x^2]),x]

[Out]

(-I/3)*Sqrt[2]*(EllipticE[I*ArcSinh[x/2], 6] - EllipticF[I*ArcSinh[x/2], 6])

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Maple [A]  time = 0.023, size = 34, normalized size = 0.4 \[{\frac{i}{3}} \left ({\it EllipticF} \left ({\frac{i}{2}}x,\sqrt{3}\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x,\sqrt{3}\sqrt{2} \right ) \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^2+4)^(1/2)/(3*x^2+2)^(1/2),x)

[Out]

1/3*I*(EllipticF(1/2*I*x,3^(1/2)*2^(1/2))-EllipticE(1/2*I*x,3^(1/2)*2^(1/2)))*2^
(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{x^{2} + 4} \sqrt{3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**2+4)**(1/2)/(3*x**2+2)**(1/2),x)

[Out]

Integral(x**2/(sqrt(x**2 + 4)*sqrt(3*x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 4)), x)