3.658 \(\int x \left (1+\sqrt{1-x^2}\right ) \, dx\)

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

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Rubi [A]  time = 0.0159755, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*(1 + Sqrt[1 - x^2]),x]

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0188422, size = 23, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(1 + Sqrt[1 - x^2]),x]

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Maple [A]  time = 0.001, size = 18, normalized size = 0.8 \[{\frac{{x}^{2}}{2}}-{\frac{1}{3} \left ( -{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(1+(-x^2+1)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.718612, size = 23, normalized size = 1. \[ \frac{1}{2} \, x^{2} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 - 1/3*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 0.264869, size = 70, normalized size = 3.04 \[ \frac{2 \, x^{6} + 3 \, \sqrt{-x^{2} + 1} x^{4} - 3 \, x^{4}}{6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{-x^{2} + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.459458, size = 27, normalized size = 1.17 \[ \frac{x^{2} \sqrt{- x^{2} + 1}}{3} + \frac{x^{2}}{2} - \frac{\sqrt{- x^{2} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261545, size = 24, normalized size = 1.04 \[ \frac{1}{2} \, x^{2} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 - 1/3*(-x^2 + 1)^(3/2) - 1/2