3.1081 \(\int \frac{(1+x)^{3/2}}{(1-x)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 (x+1)^{3/2}}{\sqrt{1-x}}+3 \sqrt{1-x} \sqrt{x+1}-3 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.032995, antiderivative size = 41, 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 \[ \frac{2 (x+1)^{3/2}}{\sqrt{1-x}}+3 \sqrt{1-x} \sqrt{x+1}-3 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.33719, size = 34, normalized size = 0.83 \[ 3 \sqrt{- x + 1} \sqrt{x + 1} - 3 \operatorname{asin}{\left (x \right )} + \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0373503, size = 37, normalized size = 0.9 \[ \frac{(x-5) \sqrt{1-x^2}}{x-1}-6 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-5 + x)*Sqrt[1 - x^2])/(-1 + x) - 6*ArcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [B]  time = 0.028, size = 72, normalized size = 1.8 \[ -{({x}^{2}-4\,x-5)\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-3\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(3/2)/(1-x)^(3/2),x)

[Out]

-(x^2-4*x-5)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-3
*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49155, size = 57, normalized size = 1.39 \[ -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{2} - 2 \, x + 1} - \frac{6 \, \sqrt{-x^{2} + 1}}{x - 1} - 3 \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216187, size = 139, normalized size = 3.39 \[ \frac{x^{3} - x^{2} +{\left (x^{2} - 8 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (x^{2} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + x - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 8 \, x}{x^{2} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^3 - x^2 + (x^2 - 8*x)*sqrt(x + 1)*sqrt(-x + 1) + 6*(x^2 - sqrt(x + 1)*(x - 2)
*sqrt(-x + 1) + x - 2)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 8*x)/(x^2 - sq
rt(x + 1)*(x - 2)*sqrt(-x + 1) + x - 2)

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Sympy [A]  time = 12.1097, size = 100, normalized size = 2.44 \[ \begin{cases} 6 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} - \frac{6 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 6 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} + \frac{6 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((6*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(3/2)/sqrt(x - 1) - 6*I
*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (-6*asin(sqrt(2)*sqrt(x + 1)/2) - (
x + 1)**(3/2)/sqrt(-x + 1) + 6*sqrt(x + 1)/sqrt(-x + 1), True))

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GIAC/XCAS [A]  time = 0.212893, size = 47, normalized size = 1.15 \[ \frac{\sqrt{x + 1}{\left (x - 5\right )} \sqrt{-x + 1}}{x - 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 1)*(x - 5)*sqrt(-x + 1)/(x - 1) - 6*arcsin(1/2*sqrt(2)*sqrt(x + 1))