3.1093 \(\int \sqrt{1-x} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{1}{4} (1-x)^{3/2} (x+1)^{5/2}-\frac{5}{12} (1-x)^{3/2} (x+1)^{3/2}+\frac{5}{8} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{8} \sin ^{-1}(x) \]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/8 - (5*(1 - x)^(3/2)*(1 + x)^(3/2))/12 - ((1 - x)^
(3/2)*(1 + x)^(5/2))/4 + (5*ArcSin[x])/8

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Rubi [A]  time = 0.0424691, antiderivative size = 68, 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 \[ -\frac{1}{4} (1-x)^{3/2} (x+1)^{5/2}-\frac{5}{12} (1-x)^{3/2} (x+1)^{3/2}+\frac{5}{8} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/8 - (5*(1 - x)^(3/2)*(1 + x)^(3/2))/12 - ((1 - x)^
(3/2)*(1 + x)^(5/2))/4 + (5*ArcSin[x])/8

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Rubi in Sympy [A]  time = 6.0378, size = 56, normalized size = 0.82 \[ \frac{5 x \sqrt{- x + 1} \sqrt{x + 1}}{8} - \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{5}{2}}}{4} - \frac{5 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{12} + \frac{5 \operatorname{asin}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x*sqrt(-x + 1)*sqrt(x + 1)/8 - (-x + 1)**(3/2)*(x + 1)**(5/2)/4 - 5*(-x + 1)**
(3/2)*(x + 1)**(3/2)/12 + 5*asin(x)/8

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Mathematica [A]  time = 0.0276654, size = 49, normalized size = 0.72 \[ \frac{1}{24} \sqrt{1-x^2} \left (6 x^3+16 x^2+9 x-16\right )+\frac{5}{4} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - x^2]*(-16 + 9*x + 16*x^2 + 6*x^3))/24 + (5*ArcSin[Sqrt[1 + x]/Sqrt[2]]
)/4

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Maple [A]  time = 0.006, size = 85, normalized size = 1.3 \[{\frac{1}{4}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{12}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{8}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(1/2)*(1+x)^(5/2),x)

[Out]

1/4*(1-x)^(1/2)*(1+x)^(7/2)-1/12*(1-x)^(1/2)*(1+x)^(5/2)-5/24*(1-x)^(1/2)*(1+x)^
(3/2)-5/8*(1-x)^(1/2)*(1+x)^(1/2)+5/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2
)*arcsin(x)

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Maxima [A]  time = 1.49014, size = 54, normalized size = 0.79 \[ -\frac{1}{4} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x - \frac{2}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{-x^{2} + 1} x + \frac{5}{8} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(-x^2 + 1)^(3/2)*x - 2/3*(-x^2 + 1)^(3/2) + 5/8*sqrt(-x^2 + 1)*x + 5/8*arcs
in(x)

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Fricas [A]  time = 0.208503, size = 224, normalized size = 3.29 \[ -\frac{24 \, x^{7} + 64 \, x^{6} - 36 \, x^{5} - 240 \, x^{4} - 60 \, x^{3} + 192 \, x^{2} -{\left (6 \, x^{7} + 16 \, x^{6} - 39 \, x^{5} - 144 \, x^{4} - 24 \, x^{3} + 192 \, x^{2} + 72 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 30 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 72 \, x}{24 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(24*x^7 + 64*x^6 - 36*x^5 - 240*x^4 - 60*x^3 + 192*x^2 - (6*x^7 + 16*x^6 -
39*x^5 - 144*x^4 - 24*x^3 + 192*x^2 + 72*x)*sqrt(x + 1)*sqrt(-x + 1) + 30*(x^4
- 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1) + 8)*arctan((sqrt(x + 1)*sqrt(-x
+ 1) - 1)/x) + 72*x)/(x^4 - 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1) + 8)

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Sympy [A]  time = 74.9852, size = 214, normalized size = 3.15 \[ \begin{cases} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} + \frac{i \left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{x - 1}} - \frac{7 i \left (x + 1\right )^{\frac{7}{2}}}{12 \sqrt{x - 1}} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{24 \sqrt{x - 1}} - \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{x - 1}} + \frac{5 i \sqrt{x + 1}}{4 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} - \frac{\left (x + 1\right )^{\frac{9}{2}}}{4 \sqrt{- x + 1}} + \frac{7 \left (x + 1\right )^{\frac{7}{2}}}{12 \sqrt{- x + 1}} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{24 \sqrt{- x + 1}} + \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{24 \sqrt{- x + 1}} - \frac{5 \sqrt{x + 1}}{4 \sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-5*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 + I*(x + 1)**(9/2)/(4*sqrt(x - 1)
) - 7*I*(x + 1)**(7/2)/(12*sqrt(x - 1)) - I*(x + 1)**(5/2)/(24*sqrt(x - 1)) - 5*
I*(x + 1)**(3/2)/(24*sqrt(x - 1)) + 5*I*sqrt(x + 1)/(4*sqrt(x - 1)), Abs(x + 1)/
2 > 1), (5*asin(sqrt(2)*sqrt(x + 1)/2)/4 - (x + 1)**(9/2)/(4*sqrt(-x + 1)) + 7*(
x + 1)**(7/2)/(12*sqrt(-x + 1)) + (x + 1)**(5/2)/(24*sqrt(-x + 1)) + 5*(x + 1)**
(3/2)/(24*sqrt(-x + 1)) - 5*sqrt(x + 1)/(4*sqrt(-x + 1)), True))

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GIAC/XCAS [A]  time = 0.221973, size = 103, normalized size = 1.51 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{8} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{5}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x + 1)^(3/2)*(x - 1)*sqrt(-x + 1) + 1/8*((2*(x + 1)*(x - 2) + 5)*(x + 1) -
1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 5/4*arcsin(1/2*sq
rt(2)*sqrt(x + 1))