3.1095 \(\int \frac{(1+x)^{5/2}}{(1-x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0484262, 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 \[ \frac{2 (x+1)^{5/2}}{\sqrt{1-x}}+\frac{5}{2} \sqrt{1-x} (x+1)^{3/2}+\frac{15}{2} \sqrt{1-x} \sqrt{x+1}-\frac{15}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.59423, size = 54, normalized size = 0.83 \[ \frac{5 \sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{2} + \frac{15 \sqrt{- x + 1} \sqrt{x + 1}}{2} - \frac{15 \operatorname{asin}{\left (x \right )}}{2} + \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{\sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.045349, size = 45, normalized size = 0.69 \[ \frac{\sqrt{1-x^2} \left (x^2+7 x-24\right )}{2 (x-1)}-15 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, 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 = 1.50447, 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 = 0.206803, size = 198, normalized size = 3.05 \[ -\frac{x^{5} + 10 \, x^{4} - 29 \, x^{3} - 18 \, x^{2} -{\left (x^{4} + 5 \, x^{3} - 18 \, x^{2} + 68 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 30 \,{\left (x^{3} - 3 \, x^{2} +{\left (x^{2} + 2 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x + 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 68 \, x}{2 \,{\left (x^{3} - 3 \, x^{2} +{\left (x^{2} + 2 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x + 4\right )}} \]

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 = 66.609, size = 139, normalized size = 2.14 \[ \begin{cases} 15 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} + \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} - \frac{15 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 15 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x + 1}} - \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x + 1}} + \frac{15 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

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))
+ 5*I*(x + 1)**(3/2)/(2*sqrt(x - 1)) - 15*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)) - 5*(
x + 1)**(3/2)/(2*sqrt(-x + 1)) + 15*sqrt(x + 1)/sqrt(-x + 1), True))

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GIAC/XCAS [A]  time = 0.210807, size = 57, normalized size = 0.88 \[ \frac{{\left ({\left (x + 6\right )}{\left (x + 1\right )} - 30\right )} \sqrt{x + 1} \sqrt{-x + 1}}{2 \,{\left (x - 1\right )}} - 15 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

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*((x + 6)*(x + 1) - 30)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 15*arcsin(1/2*sqrt
(2)*sqrt(x + 1))