3.1096 \(\int \frac{(1+x)^{5/2}}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac{10 (x+1)^{3/2}}{3 \sqrt{1-x}}-5 \sqrt{1-x} \sqrt{x+1}+5 \sin ^{-1}(x) \]

[Out]

-5*Sqrt[1 - x]*Sqrt[1 + x] - (10*(1 + x)^(3/2))/(3*Sqrt[1 - x]) + (2*(1 + x)^(5/
2))/(3*(1 - x)^(3/2)) + 5*ArcSin[x]

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Rubi [A]  time = 0.0466106, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac{10 (x+1)^{3/2}}{3 \sqrt{1-x}}-5 \sqrt{1-x} \sqrt{x+1}+5 \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-5*Sqrt[1 - x]*Sqrt[1 + x] - (10*(1 + x)^(3/2))/(3*Sqrt[1 - x]) + (2*(1 + x)^(5/
2))/(3*(1 - x)^(3/2)) + 5*ArcSin[x]

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Rubi in Sympy [A]  time = 6.89366, size = 53, normalized size = 0.84 \[ - 5 \sqrt{- x + 1} \sqrt{x + 1} + 5 \operatorname{asin}{\left (x \right )} - \frac{10 \left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{- x + 1}} + \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-x + 1)*sqrt(x + 1) + 5*asin(x) - 10*(x + 1)**(3/2)/(3*sqrt(-x + 1)) + 2
*(x + 1)**(5/2)/(3*(-x + 1)**(3/2))

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Mathematica [A]  time = 0.0579416, size = 47, normalized size = 0.75 \[ 10 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{\sqrt{1-x^2} \left (3 x^2-34 x+23\right )}{3 (x-1)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - x^2]*(23 - 34*x + 3*x^2))/(3*(-1 + x)^2) + 10*ArcSin[Sqrt[1 + x]/Sqrt
[2]]

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Maple [A]  time = 0.03, size = 84, normalized size = 1.3 \[{\frac{3\,{x}^{3}-31\,{x}^{2}-11\,x+23}{-3+3\,x}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}+5\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(5/2)/(1-x)^(5/2),x)

[Out]

1/3*(3*x^3-31*x^2-11*x+23)/(-1+x)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x
)^(1/2)/(1+x)^(1/2)+5*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.51149, size = 134, normalized size = 2.13 \[ -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{10 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{35 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x - 1\right )}} + 5 \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x^2 + 1)^(5/2)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 5/3*(-x^2 + 1)^(3/2)/(x^3 -
3*x^2 + 3*x - 1) + 10/3*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 35/3*sqrt(-x^2 + 1)/(x
- 1) + 5*arcsin(x)

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Fricas [A]  time = 0.20986, size = 220, normalized size = 3.49 \[ \frac{3 \, x^{5} - 48 \, x^{4} + 7 \, x^{3} + 102 \, x^{2} -{\left (3 \, x^{4} - 17 \, x^{3} + 102 \, x^{2} - 48 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 30 \,{\left (x^{4} - 4 \, x^{3} + x^{2} +{\left (x^{3} + x^{2} - 6 \, x + 4\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \, x - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 48 \, x}{3 \,{\left (x^{4} - 4 \, x^{3} + x^{2} +{\left (x^{3} + x^{2} - 6 \, x + 4\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*x^5 - 48*x^4 + 7*x^3 + 102*x^2 - (3*x^4 - 17*x^3 + 102*x^2 - 48*x)*sqrt(x
+ 1)*sqrt(-x + 1) - 30*(x^4 - 4*x^3 + x^2 + (x^3 + x^2 - 6*x + 4)*sqrt(x + 1)*s
qrt(-x + 1) + 6*x - 4)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 48*x)/(x^4 - 4
*x^3 + x^2 + (x^3 + x^2 - 6*x + 4)*sqrt(x + 1)*sqrt(-x + 1) + 6*x - 4)

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Sympy [A]  time = 65.1956, size = 576, normalized size = 9.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-30*I*sqrt(x - 1)*(x + 1)**(27/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(3*sqr
t(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) + 15*pi*sqrt(x - 1)*(x
+ 1)**(27/2)/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) +
60*I*sqrt(x - 1)*(x + 1)**(25/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(x - 1)*(x
+ 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) - 30*pi*sqrt(x - 1)*(x + 1)**(25/2
)/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) - 3*I*(x + 1)*
*15/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) + 40*I*(x +
1)**14/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) - 60*I*(x
+ 1)**13/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)), Abs(x
+ 1)/2 > 1), (30*sqrt(-x + 1)*(x + 1)**(27/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sq
rt(-x + 1)*(x + 1)**(27/2) - 6*sqrt(-x + 1)*(x + 1)**(25/2)) - 60*sqrt(-x + 1)*(
x + 1)**(25/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(-x + 1)*(x + 1)**(27/2) - 6*s
qrt(-x + 1)*(x + 1)**(25/2)) + 3*(x + 1)**15/(3*sqrt(-x + 1)*(x + 1)**(27/2) - 6
*sqrt(-x + 1)*(x + 1)**(25/2)) - 40*(x + 1)**14/(3*sqrt(-x + 1)*(x + 1)**(27/2)
- 6*sqrt(-x + 1)*(x + 1)**(25/2)) + 60*(x + 1)**13/(3*sqrt(-x + 1)*(x + 1)**(27/
2) - 6*sqrt(-x + 1)*(x + 1)**(25/2)), True))

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GIAC/XCAS [A]  time = 0.212762, size = 59, normalized size = 0.94 \[ -\frac{{\left ({\left (3 \, x - 37\right )}{\left (x + 1\right )} + 60\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} + 10 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*((3*x - 37)*(x + 1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 10*arcsin(1/
2*sqrt(2)*sqrt(x + 1))