3.1131 \(\int \frac{1}{\sqrt{1-x} (1+x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0246339, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.32363, size = 31, normalized size = 0.76 \[ - \frac{\sqrt{- x + 1}}{3 \sqrt{x + 1}} - \frac{\sqrt{- x + 1}}{3 \left (x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0122937, size = 23, normalized size = 0.56 \[ -\frac{\sqrt{1-x} (x+2)}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 18, normalized size = 0.4 \[ -{\frac{2+x}{3}\sqrt{1-x} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.48436, size = 51, normalized size = 1.24 \[ -\frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205187, size = 89, normalized size = 2.17 \[ -\frac{x^{3} - 3 \, x^{2} + 3 \,{\left (x^{2} + 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x}{3 \,{\left (x^{3} +{\left (x^{2} + 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.8231, size = 66, normalized size = 1.61 \[ \begin{cases} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208558, size = 120, normalized size = 2.93 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{48 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{16 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{48 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 3/16*(sqrt(2) - sqrt(-x + 1))/sq
rt(x + 1) - 1/48*(x + 1)^(3/2)*(9*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(
2) - sqrt(-x + 1))^3