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

Optimal. Leaf size=19 \[ x^2-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

[Out]

x^2 - (2*(1 - x^2)^(3/2))/3

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Rubi [A]  time = 0.0971286, antiderivative size = 19, 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 \[ x^2-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.022019, size = 24, normalized size = 1.26 \[ \frac{1}{3} \left (x^2-1\right ) \left (2 \sqrt{1-x^2}+3\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-1 + x^2)*(3 + 2*Sqrt[1 - x^2]))/3

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Maple [A]  time = 0.004, size = 24, normalized size = 1.3 \[{x}^{2}+{\frac{2\,{x}^{2}-2}{3}\sqrt{1-x}\sqrt{1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.78458, size = 20, normalized size = 1.05 \[ x^{2} - \frac{2}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2 - 2/3*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 0.268092, size = 78, normalized size = 4.11 \[ \frac{2 \, x^{6} + 3 \, \sqrt{x + 1} x^{4} \sqrt{-x + 1} - 3 \, x^{4}}{3 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(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 = 53.0237, size = 99, normalized size = 5.21 \[ x^{2} - 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 4 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2 - 4*Piecewise((x*sqrt(-x + 1)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2
, (x >= -1) & (x < 1))) + 4*Piecewise((x*sqrt(-x + 1)*sqrt(x + 1)/4 - (-x + 1)**
(3/2)*(x + 1)**(3/2)/6 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.286396, size = 36, normalized size = 1.89 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} +{\left (x + 1\right )}^{2} - 2 \, x - 2 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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