3.1006 \(\int \frac{x^2}{\sqrt{2+3 x^2} \sqrt{1+4 x^2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + 4*x^2]) - (Sqrt[2 + 3*x^2]*EllipticE[ArcTan[2*x]
, 5/8])/(3*Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + 4*x^2)]*Sqrt[1 + 4*x^2])

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Rubi [A]  time = 0.107021, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x \sqrt{3 x^2+2}}{3 \sqrt{4 x^2+1}}-\frac{\sqrt{3 x^2+2} E\left (\tan ^{-1}(2 x)|\frac{5}{8}\right )}{3 \sqrt{2} \sqrt{\frac{3 x^2+2}{4 x^2+1}} \sqrt{4 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[2 + 3*x^2])/(3*Sqrt[1 + 4*x^2]) - (Sqrt[2 + 3*x^2]*EllipticE[ArcTan[2*x]
, 5/8])/(3*Sqrt[2]*Sqrt[(2 + 3*x^2)/(1 + 4*x^2)]*Sqrt[1 + 4*x^2])

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Rubi in Sympy [A]  time = 13.7355, size = 70, normalized size = 0.8 \[ \frac{x \sqrt{3 x^{2} + 2}}{3 \sqrt{4 x^{2} + 1}} - \frac{\sqrt{3 x^{2} + 2} E\left (\operatorname{atan}{\left (2 x \right )}\middle | \frac{5}{8}\right )}{6 \sqrt{\frac{3 x^{2} + 2}{8 x^{2} + 2}} \sqrt{4 x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(3*x**2 + 2)/(3*sqrt(4*x**2 + 1)) - sqrt(3*x**2 + 2)*elliptic_e(atan(2*x),
5/8)/(6*sqrt((3*x**2 + 2)/(8*x**2 + 2))*sqrt(4*x**2 + 1))

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Mathematica [C]  time = 0.0430924, size = 50, normalized size = 0.57 \[ -\frac{i \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{8}{3}\right )\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((-I/4)*(EllipticE[I*ArcSinh[Sqrt[3/2]*x], 8/3] - EllipticF[I*ArcSinh[Sqrt[3/2]*
x], 8/3]))/Sqrt[3]

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Maple [C]  time = 0.025, size = 48, normalized size = 0.6 \[{\frac{i}{12}} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{3}\sqrt{2}x,{\frac{2\,\sqrt{3}\sqrt{2}}{3}} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{3}\sqrt{2}x,{\frac{2\,\sqrt{3}\sqrt{2}}{3}} \right ) \right ) \sqrt{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*I*(EllipticF(1/2*I*3^(1/2)*2^(1/2)*x,2/3*3^(1/2)*2^(1/2))-EllipticE(1/2*I*3
^(1/2)*2^(1/2)*x,2/3*3^(1/2)*2^(1/2)))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{3 x^{2} + 2} \sqrt{4 x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(sqrt(3*x**2 + 2)*sqrt(4*x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{4 \, x^{2} + 1} \sqrt{3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(4*x^2 + 1)*sqrt(3*x^2 + 2)), x)