3.1004 \(\int \frac{x^2}{\sqrt{1+x^2} \sqrt{2+3 x^2}} \, dx\)

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]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTa
n[x], -1/2])/(3*Sqrt[1 + x^2]*Sqrt[(2 + 3*x^2)/(1 + x^2)])

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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]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + x^2]) - (Sqrt[2]*Sqrt[2 + 3*x^2]*EllipticE[ArcTa
n[x], -1/2])/(3*Sqrt[1 + x^2]*Sqrt[(2 + 3*x^2)/(1 + x^2)])

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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]

x*sqrt(3*x**2 + 2)/(3*sqrt(x**2 + 1)) - sqrt(2)*sqrt(3*x**2 + 2)*elliptic_e(atan
(x), -1/2)/(3*sqrt((3*x**2 + 2)/(x**2 + 1))*sqrt(x**2 + 1))

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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]

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

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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]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]

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]

integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 1}}, x\right ) \]

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]

integral(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 1)), x)

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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]

Integral(x**2/(sqrt(x**2 + 1)*sqrt(3*x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]

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]

integrate(x^2/(sqrt(3*x^2 + 2)*sqrt(x^2 + 1)), x)