3.837 \(\int \sqrt{1-x} x \sqrt{1+x} \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2} \]

[Out]

-((1 - x)^(3/2)*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.00981708, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x]*x*Sqrt[1 + x],x]

[Out]

-((1 - x)^(3/2)*(1 + x)^(3/2))/3

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Rubi in Sympy [A]  time = 2.25037, size = 15, normalized size = 0.75 \[ - \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x + 1)**(3/2)*(x + 1)**(3/2)/3

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Mathematica [A]  time = 0.00502341, size = 15, normalized size = 0.75 \[ -\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x]*x*Sqrt[1 + x],x]

[Out]

-(1 - x^2)^(3/2)/3

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \[ -{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.48879, size = 15, normalized size = 0.75 \[ -\frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 0.222294, size = 90, normalized size = 4.5 \[ \frac{x^{6} - 6 \, x^{4} + 6 \, x^{2} + 3 \,{\left (x^{4} - 2 \, x^{2}\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 32.9118, size = 95, normalized size = 4.75 \[ - 2 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*Piecewise((x*sqrt(-x + 1)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >
= -1) & (x < 1))) + 2*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 = 0.224866, size = 23, normalized size = 1.15 \[ \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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