3.1117 \(\int \frac{(1-x)^{5/2}}{(1+x)^{3/2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 (1-x)^{5/2}}{\sqrt{x+1}}-\frac{5}{2} \sqrt{x+1} (1-x)^{3/2}-\frac{15}{2} \sqrt{x+1} \sqrt{1-x}-\frac{15}{2} \sin ^{-1}(x) \]

[Out]

(-2*(1 - x)^(5/2))/Sqrt[1 + x] - (15*Sqrt[1 - x]*Sqrt[1 + x])/2 - (5*(1 - x)^(3/
2)*Sqrt[1 + x])/2 - (15*ArcSin[x])/2

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Rubi [A]  time = 0.0458641, antiderivative size = 65, 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 \[ -\frac{2 (1-x)^{5/2}}{\sqrt{x+1}}-\frac{5}{2} \sqrt{x+1} (1-x)^{3/2}-\frac{15}{2} \sqrt{x+1} \sqrt{1-x}-\frac{15}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - x)^(5/2))/Sqrt[1 + x] - (15*Sqrt[1 - x]*Sqrt[1 + x])/2 - (5*(1 - x)^(3/
2)*Sqrt[1 + x])/2 - (15*ArcSin[x])/2

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Rubi in Sympy [A]  time = 6.73245, size = 56, normalized size = 0.86 \[ - \frac{2 \left (- x + 1\right )^{\frac{5}{2}}}{\sqrt{x + 1}} - \frac{5 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{2} - \frac{15 \sqrt{- x + 1} \sqrt{x + 1}}{2} - \frac{15 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-x + 1)**(5/2)/sqrt(x + 1) - 5*(-x + 1)**(3/2)*sqrt(x + 1)/2 - 15*sqrt(-x +
1)*sqrt(x + 1)/2 - 15*asin(x)/2

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Mathematica [A]  time = 0.040746, size = 45, normalized size = 0.69 \[ \frac{\sqrt{1-x} \left (x^2-7 x-24\right )}{2 \sqrt{x+1}}-15 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - x]*(-24 - 7*x + x^2))/(2*Sqrt[1 + x]) - 15*ArcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [A]  time = 0.026, size = 77, normalized size = 1.2 \[ -{\frac{{x}^{3}-8\,{x}^{2}-17\,x+24}{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}-{\frac{15\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(5/2)/(1+x)^(3/2),x)

[Out]

-1/2*(x^3-8*x^2-17*x+24)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(
1+x)^(1/2)-15/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.48817, size = 76, normalized size = 1.17 \[ -\frac{x^{3}}{2 \, \sqrt{-x^{2} + 1}} + \frac{4 \, x^{2}}{\sqrt{-x^{2} + 1}} + \frac{17 \, x}{2 \, \sqrt{-x^{2} + 1}} - \frac{12}{\sqrt{-x^{2} + 1}} - \frac{15}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x^3/sqrt(-x^2 + 1) + 4*x^2/sqrt(-x^2 + 1) + 17/2*x/sqrt(-x^2 + 1) - 12/sqrt
(-x^2 + 1) - 15/2*arcsin(x)

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Fricas [A]  time = 0.212006, size = 200, normalized size = 3.08 \[ \frac{x^{5} - 10 \, x^{4} - 29 \, x^{3} + 18 \, x^{2} +{\left (x^{4} - 5 \, x^{3} - 18 \, x^{2} - 68 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 30 \,{\left (x^{3} + 3 \, x^{2} -{\left (x^{2} - 2 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 68 \, x}{2 \,{\left (x^{3} + 3 \, x^{2} -{\left (x^{2} - 2 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^5 - 10*x^4 - 29*x^3 + 18*x^2 + (x^4 - 5*x^3 - 18*x^2 - 68*x)*sqrt(x + 1)*
sqrt(-x + 1) + 30*(x^3 + 3*x^2 - (x^2 - 2*x - 4)*sqrt(x + 1)*sqrt(-x + 1) - 2*x
- 4)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 68*x)/(x^3 + 3*x^2 - (x^2 - 2*x
- 4)*sqrt(x + 1)*sqrt(-x + 1) - 2*x - 4)

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Sympy [A]  time = 63.9608, size = 168, normalized size = 2.58 \[ \begin{cases} 15 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} - \frac{11 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} + \frac{i \sqrt{x + 1}}{\sqrt{x - 1}} + \frac{16 i}{\sqrt{x - 1} \sqrt{x + 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x + 1}} + \frac{11 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{- x + 1}} - \frac{16}{\sqrt{- x + 1} \sqrt{x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((15*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(5/2)/(2*sqrt(x - 1))
- 11*I*(x + 1)**(3/2)/(2*sqrt(x - 1)) + I*sqrt(x + 1)/sqrt(x - 1) + 16*I/(sqrt(x
- 1)*sqrt(x + 1)), Abs(x + 1)/2 > 1), (-15*asin(sqrt(2)*sqrt(x + 1)/2) - (x + 1
)**(5/2)/(2*sqrt(-x + 1)) + 11*(x + 1)**(3/2)/(2*sqrt(-x + 1)) - sqrt(x + 1)/sqr
t(-x + 1) - 16/(sqrt(-x + 1)*sqrt(x + 1)), True))

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GIAC/XCAS [A]  time = 0.229625, size = 99, normalized size = 1.52 \[ \frac{1}{2} \, \sqrt{x + 1}{\left (x - 8\right )} \sqrt{-x + 1} + \frac{4 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{\sqrt{x + 1}} - \frac{4 \, \sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}} - 15 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x + 1)*(x - 8)*sqrt(-x + 1) + 4*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) -
4*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 15*arcsin(1/2*sqrt(2)*sqrt(x + 1))