3.659 \(\int x \left (1+\sqrt{1-x} \sqrt{1+x}\right ) \, dx\)

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 = 0.0164698, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*(1 + Sqrt[1 - x]*Sqrt[1 + x]),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. \[ 2 \int ^{\sqrt{x + 1}} x \left (x^{2} - 1\right ) \left (x \sqrt{- x^{2} + 2} + 1\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*Integral(x*(x**2 - 1)*(x*sqrt(-x**2 + 2) + 1), (x, sqrt(x + 1)))

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Mathematica [A]  time = 0.00634142, 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]*Sqrt[1 + x]),x]

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Maple [A]  time = 0.002, size = 26, normalized size = 1.1 \[{\frac{{x}^{2}-1}{3}\sqrt{1-x}\sqrt{1+x}}+{\frac{{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.799495, 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 + 1)*sqrt(-x + 1) + 1),x, algorithm="maxima")

[Out]

1/2*x^2 - 1/3*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 0.264758, size = 78, normalized size = 3.39 \[ \frac{2 \, x^{6} + 3 \, \sqrt{x + 1} x^{4} \sqrt{-x + 1} - 3 \, x^{4}}{6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.76043, size = 63, normalized size = 2.74 \[ \begin{cases} \frac{i x^{2} \sqrt{x^{2} - 1}}{3} + \frac{x^{2}}{2} - \frac{i \sqrt{x^{2} - 1}}{3} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{x^{2} \sqrt{- x^{2} + 1}}{3} + \frac{x^{2}}{2} + \frac{\sqrt{- x^{2} + 1}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*x**2*sqrt(x**2 - 1)/3 + x**2/2 - I*sqrt(x**2 - 1)/3, Abs(x**2) > 1)
, (-x**2*sqrt(-x**2 + 1)/3 + x**2/2 + sqrt(-x**2 + 1)/3, True))

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GIAC/XCAS [A]  time = 0.270317, size = 39, normalized size = 1.7 \[ \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{2} \,{\left (x + 1\right )}^{2} - x - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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