3.196 \(\int \frac{x \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.0639883, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 10.8309, size = 20, normalized size = 0.8 \[ \frac{22 x^{2} + 16}{26 \sqrt{x^{4} + 5 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(3*x**2+2)/(x**4+5*x**2+3)**(3/2),x)

[Out]

(22*x**2 + 16)/(26*sqrt(x**4 + 5*x**2 + 3))

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Mathematica [A]  time = 0.0216536, size = 25, normalized size = 1. \[ \frac{11 x^2+8}{13 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(2 + 3*x^2))/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(8 + 11*x^2)/(13*Sqrt[3 + 5*x^2 + x^4])

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Maple [A]  time = 0.007, size = 22, normalized size = 0.9 \[{\frac{11\,{x}^{2}+8}{13}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(3*x^2+2)/(x^4+5*x^2+3)^(3/2),x)

[Out]

1/13*(11*x^2+8)/(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.706359, size = 43, normalized size = 1.72 \[ \frac{11 \, x^{2}}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} + \frac{8}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x/(x^4 + 5*x^2 + 3)^(3/2),x, algorithm="maxima")

[Out]

11/13*x^2/sqrt(x^4 + 5*x^2 + 3) + 8/13/sqrt(x^4 + 5*x^2 + 3)

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Fricas [A]  time = 0.282032, size = 77, normalized size = 3.08 \[ \frac{3 \, x^{2} - 3 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 2}{2 \, x^{4} + 10 \, x^{2} - \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x/(x^4 + 5*x^2 + 3)^(3/2),x, algorithm="fricas")

[Out]

(3*x^2 - 3*sqrt(x^4 + 5*x^2 + 3) + 2)/(2*x^4 + 10*x^2 - sqrt(x^4 + 5*x^2 + 3)*(2
*x^2 + 5) + 6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(3*x**2+2)/(x**4+5*x**2+3)**(3/2),x)

[Out]

Integral(x*(3*x**2 + 2)/(x**4 + 5*x**2 + 3)**(3/2), x)

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GIAC/XCAS [A]  time = 0.281653, size = 28, normalized size = 1.12 \[ \frac{11 \, x^{2} + 8}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x/(x^4 + 5*x^2 + 3)^(3/2),x, algorithm="giac")

[Out]

1/13*(11*x^2 + 8)/sqrt(x^4 + 5*x^2 + 3)