3.1129 \(\int \frac{(1-x)^{3/2}}{(1+x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0298746, antiderivative size = 41, 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 \[ -\frac{2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{x+1}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(3/2)/(1 + x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.4185, size = 34, normalized size = 0.83 \[ - \frac{2 \left (- x + 1\right )^{\frac{3}{2}}}{3 \left (x + 1\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- x + 1}}{\sqrt{x + 1}} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(3/2)/(1+x)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0409002, size = 42, normalized size = 1.02 \[ \frac{4 \sqrt{1-x} (2 x+1)}{3 (x+1)^{3/2}}+2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(3/2)/(1 + x)^(5/2),x]

[Out]

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

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Maple [B]  time = 0.029, size = 73, normalized size = 1.8 \[ -{\frac{8\,{x}^{2}-4\,x-4}{3}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(3/2)/(1+x)^(5/2),x)

[Out]

-4/3*(2*x^2-x-1)/(1+x)^(3/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(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 = 1.49791, size = 89, normalized size = 2.17 \[ -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{7 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(3/2)/(x + 1)^(5/2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 1)^(3/2)/(x^3 + 3*x^2 + 3*x + 1) - 2/3*sqrt(-x^2 + 1)/(x^2 + 2*x +
1) + 7/3*sqrt(-x^2 + 1)/(x + 1) + arcsin(x)

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Fricas [A]  time = 0.209612, size = 151, normalized size = 3.68 \[ -\frac{2 \,{\left (2 \, x^{3} - 6 \, \sqrt{x + 1} x^{2} \sqrt{-x + 1} + 6 \, x^{2} + 3 \,{\left (x^{3} +{\left (x^{2} + 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right )\right )}}{3 \,{\left (x^{3} +{\left (x^{2} + 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(3/2)/(x + 1)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 22.4582, size = 128, normalized size = 3.12 \[ \begin{cases} \frac{8 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{4 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + i \log{\left (\frac{1}{x + 1} \right )} + i \log{\left (x + 1 \right )} + 2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\\frac{8 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{4 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + i \log{\left (\frac{1}{x + 1} \right )} - 2 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**(3/2)/(1+x)**(5/2),x)

[Out]

Piecewise((8*sqrt(-1 + 2/(x + 1))/3 - 4*sqrt(-1 + 2/(x + 1))/(3*(x + 1)) + I*log
(1/(x + 1)) + I*log(x + 1) + 2*asin(sqrt(2)*sqrt(x + 1)/2), 2*Abs(1/(x + 1)) > 1
), (8*I*sqrt(1 - 2/(x + 1))/3 - 4*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)) + I*log(1/(x
+ 1)) - 2*I*log(sqrt(1 - 2/(x + 1)) + 1), True))

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GIAC/XCAS [A]  time = 0.224667, size = 138, normalized size = 3.37 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{12 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{4 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{15 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{12 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x + 1)^(3/2)/(x + 1)^(5/2),x, algorithm="giac")

[Out]

1/12*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 5/4*(sqrt(2) - sqrt(-x + 1))/sqr
t(x + 1) + 1/12*(x + 1)^(3/2)*(15*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(
2) - sqrt(-x + 1))^3 + 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))