3.1133 \(\int \frac{1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{x}{3 (1-x)^{3/2} (x+1)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0237181, antiderivative size = 43, 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 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{x}{3 (1-x)^{3/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.87681, size = 34, normalized size = 0.79 \[ \frac{2 x}{3 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{x}{3 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0173428, size = 23, normalized size = 0.53 \[ -\frac{x \left (2 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(-3 + 2*x^2))/(3*(1 - x^2)^(3/2))

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Maple [A]  time = 0.004, size = 23, normalized size = 0.5 \[ -{\frac{x \left ( 2\,{x}^{2}-3 \right ) }{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.342, size = 34, normalized size = 0.79 \[ \frac{2 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206227, size = 116, normalized size = 2.7 \[ \frac{6 \, x^{5} - 17 \, x^{3} -{\left (2 \, x^{5} - 11 \, x^{3} + 12 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12 \, x}{3 \,{\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} +{\left (3 \, x^{4} - 7 \, x^{2} + 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(6*x^5 - 17*x^3 - (2*x^5 - 11*x^3 + 12*x)*sqrt(x + 1)*sqrt(-x + 1) + 12*x)/(
x^6 - 6*x^4 + 9*x^2 + (3*x^4 - 7*x^2 + 4)*sqrt(x + 1)*sqrt(-x + 1) - 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220602, size = 153, normalized size = 3.56 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{192 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{11 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{64 \, \sqrt{x + 1}} - \frac{{\left (4 \, x - 5\right )} \sqrt{x + 1} \sqrt{-x + 1}}{12 \,{\left (x - 1\right )}^{2}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{33 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{192 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 11/64*(sqrt(2) - sqrt(-x + 1))/
sqrt(x + 1) - 1/12*(4*x - 5)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 - 1/192*(x + 1)^
(3/2)*(33*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3