3.111 \(\int \frac{-x+x^3}{\sqrt{-2+x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]

[Out]

Sqrt[-2 + x^2] + (-2 + x^2)^(3/2)/3

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Rubi [A]  time = 0.0526903, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-2 + x^2] + (-2 + x^2)^(3/2)/3

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Rubi in Sympy [A]  time = 7.20262, size = 17, normalized size = 0.74 \[ \frac{\left (x^{2} - 2\right )^{\frac{3}{2}}}{3} + \sqrt{x^{2} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**2 - 2)**(3/2)/3 + sqrt(x**2 - 2)

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Mathematica [A]  time = 0.0121626, size = 18, normalized size = 0.78 \[ \frac{1}{3} \sqrt{x^2-2} \left (x^2+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-2 + x^2]*(1 + x^2))/3

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Maple [A]  time = 0.008, size = 15, normalized size = 0.7 \[{\frac{{x}^{2}+1}{3}\sqrt{{x}^{2}-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-x)/(x^2-2)^(1/2),x)

[Out]

1/3*(x^2+1)*(x^2-2)^(1/2)

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Maxima [A]  time = 1.32909, size = 30, normalized size = 1.3 \[ \frac{1}{3} \, \sqrt{x^{2} - 2} x^{2} + \frac{1}{3} \, \sqrt{x^{2} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(x^2 - 2)*x^2 + 1/3*sqrt(x^2 - 2)

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Fricas [A]  time = 0.23735, size = 93, normalized size = 4.04 \[ -\frac{2 \, x^{6} - 3 \, x^{4} - 3 \, x^{2} -{\left (2 \, x^{5} - x^{3} - 3 \, x\right )} \sqrt{x^{2} - 2} + 2}{3 \,{\left (2 \, x^{3} -{\left (2 \, x^{2} - 1\right )} \sqrt{x^{2} - 2} - 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(2*x^6 - 3*x^4 - 3*x^2 - (2*x^5 - x^3 - 3*x)*sqrt(x^2 - 2) + 2)/(2*x^3 - (2
*x^2 - 1)*sqrt(x^2 - 2) - 3*x)

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Sympy [A]  time = 0.480152, size = 22, normalized size = 0.96 \[ \frac{x^{2} \sqrt{x^{2} - 2}}{3} + \frac{\sqrt{x^{2} - 2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(x**2 - 2)/3 + sqrt(x**2 - 2)/3

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GIAC/XCAS [A]  time = 0.214884, size = 23, normalized size = 1. \[ \frac{1}{3} \,{\left (x^{2} - 2\right )}^{\frac{3}{2}} + \sqrt{x^{2} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^2 - 2)^(3/2) + sqrt(x^2 - 2)