3.1080 \(\int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{1}{2} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{2} \sqrt{1-x} \sqrt{x+1}+\frac{3}{2} \sin ^{-1}(x) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0328027, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{1}{2} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{2} \sqrt{1-x} \sqrt{x+1}+\frac{3}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.91353, size = 37, normalized size = 0.79 \[ - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{2} - \frac{3 \sqrt{- x + 1} \sqrt{x + 1}}{2} + \frac{3 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.018742, size = 35, normalized size = 0.74 \[ 3 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{2} (x+4) \sqrt{1-x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 57, normalized size = 1.2 \[ -{\frac{1}{2}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{3}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{3\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(3/2)/(1-x)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.50134, size = 38, normalized size = 0.81 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 1} x - 2 \, \sqrt{-x^{2} + 1} + \frac{3}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.213906, size = 140, normalized size = 2.98 \[ \frac{2 \, x^{3} + 4 \, x^{2} -{\left (x^{3} + 4 \, x^{2} - 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*x^3 + 4*x^2 - (x^3 + 4*x^2 - 2*x)*sqrt(x + 1)*sqrt(-x + 1) - 6*(x^2 + 2*s
qrt(x + 1)*sqrt(-x + 1) - 2)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 2*x)/(x^
2 + 2*sqrt(x + 1)*sqrt(-x + 1) - 2)

_______________________________________________________________________________________

Sympy [A]  time = 12.5616, size = 136, normalized size = 2.89 \[ \begin{cases} - 3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} - \frac{i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} + \frac{3 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x + 1}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x + 1}} - \frac{3 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214303, size = 42, normalized size = 0.89 \[ -\frac{1}{2} \,{\left (x + 4\right )} \sqrt{x + 1} \sqrt{-x + 1} + 3 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(x + 4)*sqrt(x + 1)*sqrt(-x + 1) + 3*arcsin(1/2*sqrt(2)*sqrt(x + 1))