3.1122 \(\int \frac{1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{3 (1-x)^{3/2} \sqrt{x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0246828, antiderivative size = 42, 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 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{1}{3 (1-x)^{3/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.96472, size = 34, normalized size = 0.81 \[ \frac{2 x}{3 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{1}{3 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0239578, size = 30, normalized size = 0.71 \[ \frac{-2 x^2+2 x+1}{3 (1-x)^{3/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.6 \[ -{\frac{2\,{x}^{2}-2\,x-1}{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34685, size = 54, normalized size = 1.29 \[ \frac{2 \, x}{3 \, \sqrt{-x^{2} + 1}} - \frac{1}{3 \,{\left (\sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204938, size = 122, normalized size = 2.9 \[ -\frac{2 \, x^{4} - 4 \, x^{3} - 3 \, x^{2} +{\left (x^{3} + 3 \, x^{2} - 6 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \, x}{3 \,{\left (2 \, x^{3} - 2 \, x^{2} -{\left (x^{3} - x^{2} - 2 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(2*x^4 - 4*x^3 - 3*x^2 + (x^3 + 3*x^2 - 6*x)*sqrt(x + 1)*sqrt(-x + 1) + 6*x
)/(2*x^3 - 2*x^2 - (x^3 - x^2 - 2*x + 2)*sqrt(x + 1)*sqrt(-x + 1) - 2*x + 2)

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Sympy [A]  time = 88.3379, size = 160, normalized size = 3.81 \[ \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac{6 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac{3 \sqrt{-1 + \frac{2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac{6 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac{3 i \sqrt{1 - \frac{2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-12*x + 3*(x + 1)**2) + 6*sqrt(-1
+ 2/(x + 1))*(x + 1)/(-12*x + 3*(x + 1)**2) - 3*sqrt(-1 + 2/(x + 1))/(-12*x + 3
*(x + 1)**2), 2*Abs(1/(x + 1)) > 1), (-2*I*sqrt(1 - 2/(x + 1))*(x + 1)**2/(-12*x
+ 3*(x + 1)**2) + 6*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-12*x + 3*(x + 1)**2) - 3*I*
sqrt(1 - 2/(x + 1))/(-12*x + 3*(x + 1)**2), True))

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GIAC/XCAS [A]  time = 0.207192, size = 90, normalized size = 2.14 \[ \frac{\sqrt{2} - \sqrt{-x + 1}}{8 \, \sqrt{x + 1}} - \frac{{\left (5 \, x - 7\right )} \sqrt{x + 1} \sqrt{-x + 1}}{12 \,{\left (x - 1\right )}^{2}} - \frac{\sqrt{x + 1}}{8 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/12*(5*x - 7)*sqrt(x + 1)*sqrt(-x +
1)/(x - 1)^2 - 1/8*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1))