3.1132 \(\int \frac{1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0247833, 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 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}-\frac{1}{3 \sqrt{1-x} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.41403, size = 34, normalized size = 0.81 \[ \frac{2 x}{3 \sqrt{- x + 1} \sqrt{x + 1}} - \frac{1}{3 \sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0224186, size = 30, normalized size = 0.71 \[ \frac{2 x^2+2 x-1}{3 \sqrt{1-x} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 25, normalized size = 0.6 \[{\frac{2\,{x}^{2}+2\,x-1}{3}{\frac{1}{\sqrt{1-x}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33862, size = 51, normalized size = 1.21 \[ \frac{2 \, x}{3 \, \sqrt{-x^{2} + 1}} - \frac{1}{3 \,{\left (\sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(5/2)*(-x + 1)^(3/2)),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 = 0.206081, size = 120, normalized size = 2.86 \[ -\frac{2 \, x^{4} + 4 \, x^{3} - 3 \, x^{2} -{\left (x^{3} - 3 \, x^{2} - 6 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x}{3 \,{\left (2 \, x^{3} + 2 \, x^{2} -{\left (x^{3} + x^{2} - 2 \, x - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(5/2)*(-x + 1)^(3/2)),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 = 86.1169, size = 167, normalized size = 3.98 \[ \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{\sqrt{-1 + \frac{2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{i \sqrt{1 - \frac{2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209701, size = 146, normalized size = 3.48 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{96 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{7 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{32 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1} \sqrt{-x + 1}}{4 \,{\left (x - 1\right )}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{21 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{96 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 7/32*(sqrt(2) - sqrt(-x + 1))/sq
rt(x + 1) - 1/4*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 1/96*(x + 1)^(3/2)*(21*(sqrt(
2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3