3.1066 \(\int (1-x)^{3/2} \sqrt{1+x} \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac{1}{2} \sqrt{1-x} x \sqrt{x+1}+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0302339, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{3} (1-x)^{3/2} (x+1)^{3/2}+\frac{1}{2} \sqrt{1-x} x \sqrt{x+1}+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(3/2)*Sqrt[1 + x],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.50007, size = 36, normalized size = 0.75 \[ \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(3/2)*(1+x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0264536, size = 40, normalized size = 0.83 \[ \frac{1}{6} \sqrt{1-x^2} \left (-2 x^2+3 x+2\right )+\sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(3/2)*Sqrt[1 + x],x]

[Out]

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

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Maple [B]  time = 0.005, size = 71, normalized size = 1.5 \[{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(3/2)*(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49216, size = 38, normalized size = 0.79 \[ \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*(-x + 1)^(3/2),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 = 0.209754, size = 189, normalized size = 3.94 \[ -\frac{2 \, x^{6} - 3 \, x^{5} - 12 \, x^{4} + 15 \, x^{3} + 12 \, x^{2} + 3 \,{\left (2 \, x^{4} - 3 \, x^{3} - 4 \, x^{2} + 4 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 12 \, x}{6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*(-x + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(2*x^6 - 3*x^5 - 12*x^4 + 15*x^3 + 12*x^2 + 3*(2*x^4 - 3*x^3 - 4*x^2 + 4*x)
*sqrt(x + 1)*sqrt(-x + 1) + 6*(3*x^2 - (x^2 - 4)*sqrt(x + 1)*sqrt(-x + 1) - 4)*a
rctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 12*x)/(3*x^2 - (x^2 - 4)*sqrt(x + 1)*s
qrt(-x + 1) - 4)

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Sympy [A]  time = 17.8997, size = 168, normalized size = 3.5 \[ \begin{cases} - i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{x - 1}} + \frac{11 i \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{x - 1}} - \frac{17 i \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{x - 1}} + \frac{i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{- x + 1}} - \frac{11 \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{- x + 1}} + \frac{17 \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{- x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**(3/2)*(1+x)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(7/2)/(3*sqrt(x - 1)) +
11*I*(x + 1)**(5/2)/(6*sqrt(x - 1)) - 17*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) + I*sq
rt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)
**(7/2)/(3*sqrt(-x + 1)) - 11*(x + 1)**(5/2)/(6*sqrt(-x + 1)) + 17*(x + 1)**(3/2
)/(6*sqrt(-x + 1)) - sqrt(x + 1)/sqrt(-x + 1), True))

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GIAC/XCAS [A]  time = 0.214512, size = 59, normalized size = 1.23 \[ -\frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*(-x + 1)^(3/2),x, algorithm="giac")

[Out]

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