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

Optimal. Leaf size=18 \[ -\frac{(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

[Out]

-(1 + x)^(3/2)/(3*(-1 + x)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.011009, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

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 = 2.05364, size = 15, normalized size = 0.83 \[ - \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \left (x - 1\right )^{\frac{3}{2}}} \]

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 = 0.0161908, size = 17, normalized size = 0.94 \[ -\frac{1}{3 \left (\frac{x-1}{x+1}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*((-1 + x)/(1 + x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 13, normalized size = 0.7 \[ -{\frac{1}{3} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( -1+x \right ) ^{-{\frac{3}{2}}}} \]

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 = 1.34322, size = 46, normalized size = 2.56 \[ -\frac{2 \, \sqrt{x^{2} - 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{\sqrt{x^{2} - 1}}{3 \,{\left (x - 1\right )}} \]

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 = 0.232634, size = 78, normalized size = 4.33 \[ -\frac{2 \,{\left (3 \, \sqrt{x + 1} \sqrt{x - 1} x - 3 \, x^{2} + 1\right )}}{3 \,{\left (2 \, x^{3} -{\left (2 \, x^{2} - 3 \, x + 1\right )} \sqrt{x + 1} \sqrt{x - 1} - 3 \, x^{2} + 1\right )}} \]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.0555, size = 61, normalized size = 3.39 \[ \begin{cases} - \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{x - 1} \left (x + 1\right ) - 6 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{i \left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{- x + 1} \left (x + 1\right ) - 6 \sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217879, size = 16, normalized size = 0.89 \[ -\frac{{\left (x + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x - 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(x + 1)^(3/2)/(x - 1)^(3/2)