3.809 \(\int (1+x) \sqrt{1-x^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{3} \left (1-x^2\right )^{3/2}+\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

(x*Sqrt[1 - x^2])/2 - (1 - x^2)^(3/2)/3 + ArcSin[x]/2

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Rubi [A]  time = 0.027036, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{3} \left (1-x^2\right )^{3/2}+\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)*Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 - x^2])/2 - (1 - x^2)^(3/2)/3 + ArcSin[x]/2

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Rubi in Sympy [A]  time = 3.27272, size = 26, normalized size = 0.68 \[ \frac{x \sqrt{- x^{2} + 1}}{2} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0304317, size = 31, normalized size = 0.82 \[ \frac{1}{6} \left (\sqrt{1-x^2} \left (2 x^2+3 x-2\right )+3 \sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - x^2]*(-2 + 3*x + 2*x^2) + 3*ArcSin[x])/6

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Maple [A]  time = 0.007, size = 29, normalized size = 0.8 \[ -{\frac{1}{3} \left ( -{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{\arcsin \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.784371, size = 38, normalized size = 1. \[ -\frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21287, size = 171, normalized size = 4.5 \[ \frac{2 \, x^{6} + 3 \, x^{5} - 12 \, x^{4} - 15 \, x^{3} + 12 \, x^{2} - 6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{-x^{2} + 1} - 4\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + 3 \,{\left (2 \, x^{4} + 3 \, x^{3} - 4 \, x^{2} - 4 \, x\right )} \sqrt{-x^{2} + 1} + 12 \, x}{6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{-x^{2} + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*x^6 + 3*x^5 - 12*x^4 - 15*x^3 + 12*x^2 - 6*(3*x^2 - (x^2 - 4)*sqrt(-x^2 +
1) - 4)*arctan((sqrt(-x^2 + 1) - 1)/x) + 3*(2*x^4 + 3*x^3 - 4*x^2 - 4*x)*sqrt(-
x^2 + 1) + 12*x)/(3*x^2 - (x^2 - 4)*sqrt(-x^2 + 1) - 4)

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Sympy [A]  time = 0.486017, size = 39, normalized size = 1.03 \[ \frac{x^{2} \sqrt{- x^{2} + 1}}{3} + \frac{x \sqrt{- x^{2} + 1}}{2} - \frac{\sqrt{- x^{2} + 1}}{3} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223647, size = 34, normalized size = 0.89 \[ \frac{1}{6} \,{\left ({\left (2 \, x + 3\right )} x - 2\right )} \sqrt{-x^{2} + 1} + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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