3.1130 \(\int \frac{\sqrt{1-x}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

[Out]

-(1 - x)^(3/2)/(3*(1 + x)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0117837, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - x)^(3/2)/(3*(1 + x)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.38087, size = 15, normalized size = 0.75 \[ - \frac{\left (- x + 1\right )^{\frac{3}{2}}}{3 \left (x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x + 1)**(3/2)/(3*(x + 1)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0123056, size = 20, normalized size = 1. \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - x)^(3/2)/(3*(1 + x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[ -{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(1/2)/(1+x)^(5/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34178, size = 51, normalized size = 2.55 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.205144, size = 74, normalized size = 3.7 \[ -\frac{2 \,{\left (x^{3} + 3 \, \sqrt{x + 1} x \sqrt{-x + 1} - 3 \, x\right )}}{3 \,{\left (x^{3} +{\left (x^{2} + 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.7706, size = 66, normalized size = 3.3 \[ \begin{cases} \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\\frac{i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{2 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((sqrt(-1 + 2/(x + 1))/3 - 2*sqrt(-1 + 2/(x + 1))/(3*(x + 1)), 2*Abs(1/
(x + 1)) > 1), (I*sqrt(1 - 2/(x + 1))/3 - 2*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)), T
rue))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214232, size = 120, normalized size = 6. \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{24 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{\sqrt{2} - \sqrt{-x + 1}}{8 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{24 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 1/8*(sqrt(2) - sqrt(-x + 1))/sqr
t(x + 1) + 1/24*(x + 1)^(3/2)*(3*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2
) - sqrt(-x + 1))^3