3.1107 \(\int \frac{(1-x)^{5/2}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=67 \[ \frac{1}{3} \sqrt{x+1} (1-x)^{5/2}+\frac{5}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{5}{2} \sqrt{x+1} \sqrt{1-x}+\frac{5}{2} \sin ^{-1}(x) \]

[Out]

(5*Sqrt[1 - x]*Sqrt[1 + x])/2 + (5*(1 - x)^(3/2)*Sqrt[1 + x])/6 + ((1 - x)^(5/2)
*Sqrt[1 + x])/3 + (5*ArcSin[x])/2

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Rubi [A]  time = 0.0485606, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{3} \sqrt{x+1} (1-x)^{5/2}+\frac{5}{6} \sqrt{x+1} (1-x)^{3/2}+\frac{5}{2} \sqrt{x+1} \sqrt{1-x}+\frac{5}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - x]*Sqrt[1 + x])/2 + (5*(1 - x)^(3/2)*Sqrt[1 + x])/6 + ((1 - x)^(5/2)
*Sqrt[1 + x])/3 + (5*ArcSin[x])/2

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Rubi in Sympy [A]  time = 6.94129, size = 54, normalized size = 0.81 \[ \frac{\left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}}{3} + \frac{5 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{6} + \frac{5 \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{5 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x + 1)**(5/2)*sqrt(x + 1)/3 + 5*(-x + 1)**(3/2)*sqrt(x + 1)/6 + 5*sqrt(-x + 1)
*sqrt(x + 1)/2 + 5*asin(x)/2

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Mathematica [A]  time = 0.0253878, size = 42, normalized size = 0.63 \[ \frac{1}{6} \sqrt{1-x^2} \left (2 x^2-9 x+22\right )+5 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - x^2]*(22 - 9*x + 2*x^2))/6 + 5*ArcSin[Sqrt[1 + x]/Sqrt[2]]

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Maple [A]  time = 0.006, size = 71, normalized size = 1.1 \[{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{5}{2}}}\sqrt{1+x}}+{\frac{5}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{5}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(5/2)/(1+x)^(1/2),x)

[Out]

1/3*(1-x)^(5/2)*(1+x)^(1/2)+5/6*(1-x)^(3/2)*(1+x)^(1/2)+5/2*(1-x)^(1/2)*(1+x)^(1
/2)+5/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.51131, size = 57, normalized size = 0.85 \[ \frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{3}{2} \, \sqrt{-x^{2} + 1} x + \frac{11}{3} \, \sqrt{-x^{2} + 1} + \frac{5}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.209976, size = 189, normalized size = 2.82 \[ \frac{2 \, x^{6} - 9 \, x^{5} + 12 \, x^{4} + 45 \, x^{3} - 36 \, x^{2} + 3 \,{\left (2 \, x^{4} - 9 \, x^{3} + 12 \, x^{2} + 12 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 30 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 36 \, x}{6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*x^6 - 9*x^5 + 12*x^4 + 45*x^3 - 36*x^2 + 3*(2*x^4 - 9*x^3 + 12*x^2 + 12*x
)*sqrt(x + 1)*sqrt(-x + 1) - 30*(3*x^2 - (x^2 - 4)*sqrt(x + 1)*sqrt(-x + 1) - 4)
*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 36*x)/(3*x^2 - (x^2 - 4)*sqrt(x + 1)
*sqrt(-x + 1) - 4)

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Sympy [A]  time = 44.9529, size = 175, normalized size = 2.61 \[ \begin{cases} - 5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{x - 1}} - \frac{17 i \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{x - 1}} + \frac{59 i \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{x - 1}} - \frac{11 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{7}{2}}}{3 \sqrt{- x + 1}} + \frac{17 \left (x + 1\right )^{\frac{5}{2}}}{6 \sqrt{- x + 1}} - \frac{59 \left (x + 1\right )^{\frac{3}{2}}}{6 \sqrt{- x + 1}} + \frac{11 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-5*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(7/2)/(3*sqrt(x - 1))
- 17*I*(x + 1)**(5/2)/(6*sqrt(x - 1)) + 59*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) - 11
*I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (5*asin(sqrt(2)*sqrt(x + 1)/2) -
(x + 1)**(7/2)/(3*sqrt(-x + 1)) + 17*(x + 1)**(5/2)/(6*sqrt(-x + 1)) - 59*(x + 1
)**(3/2)/(6*sqrt(-x + 1)) + 11*sqrt(x + 1)/sqrt(-x + 1), True))

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GIAC/XCAS [A]  time = 0.219522, size = 93, normalized size = 1.39 \[ \frac{1}{6} \,{\left ({\left (2 \, x - 5\right )}{\left (x + 1\right )} + 9\right )} \sqrt{x + 1} \sqrt{-x + 1} - \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + 5 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*(x - 2)*sqrt(
-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 5*arcsin(1/2*sqrt(2)*sqrt(x + 1))