3.1108 \(\int \frac{(1-x)^{3/2}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{3}{2} \sqrt{x+1} \sqrt{1-x}+\frac{3}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0338033, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{3}{2} \sqrt{x+1} \sqrt{1-x}+\frac{3}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.11313, size = 37, normalized size = 0.79 \[ \frac{\left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{2} + \frac{3 \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{3 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0177763, size = 35, normalized size = 0.74 \[ 3 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{2} (x-4) \sqrt{1-x^2} \]

Antiderivative was successfully verified.

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

[Out]

-((-4 + x)*Sqrt[1 - x^2])/2 + 3*ArcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [A]  time = 0.005, size = 57, normalized size = 1.2 \[{\frac{1}{2} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{3}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{3\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(3/2)/(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.50338, size = 38, normalized size = 0.81 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 1} x + 2 \, \sqrt{-x^{2} + 1} + \frac{3}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207315, size = 140, normalized size = 2.98 \[ \frac{2 \, x^{3} - 4 \, x^{2} -{\left (x^{3} - 4 \, x^{2} - 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.8808, size = 139, normalized size = 2.96 \[ \begin{cases} - 3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} + \frac{7 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} - \frac{5 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x + 1}} - \frac{7 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x + 1}} + \frac{5 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21596, size = 59, normalized size = 1.26 \[ -\frac{1}{2} \, \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + 3 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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