3.1070 \(\int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx\)

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

[Out]

(1 + x)^(3/2)/(3*(1 - x)^(3/2))

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Rubi [A]  time = 0.011888, 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 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(1 + x)^(3/2)/(3*(1 - x)^(3/2))

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Rubi in Sympy [A]  time = 2.51115, size = 14, normalized size = 0.7 \[ \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

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))

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Mathematica [A]  time = 0.017617, size = 20, normalized size = 1. \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(1 + x)^(3/2)/(3*(1 - x)^(3/2))

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Maple [A]  time = 0.006, 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)

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Maxima [A]  time = 1.33976, 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)

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Fricas [A]  time = 0.205381, size = 76, normalized size = 3.8 \[ \frac{2 \,{\left (x^{3} + 3 \, \sqrt{x + 1} x \sqrt{-x + 1} - 3 \, x\right )}}{3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x + 2\right )}} \]

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)

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Sympy [A]  time = 10.0849, size = 61, normalized size = 3.05 \[ \begin{cases} \frac{i \left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{x - 1} \left (x + 1\right ) - 6 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{- x + 1} \left (x + 1\right ) - 6 \sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209565, size = 26, normalized size = 1.3 \[ \frac{{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} \]

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)*sqrt(-x + 1)/(x - 1)^2