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

Optimal. Leaf size=61 \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0578356, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.03356, size = 46, normalized size = 0.75 \[ - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{3} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{3} - \sqrt{- x + 1} \sqrt{x + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0270792, size = 40, normalized size = 0.66 \[ 2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{3} \sqrt{1-x^2} \left (x^2+3 x+5\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 66, normalized size = 1.1 \[{\frac{1}{3}\sqrt{1-x}\sqrt{1+x} \left ( -{x}^{2}\sqrt{-{x}^{2}+1}-3\,x\sqrt{-{x}^{2}+1}+3\,\arcsin \left ( x \right ) -5\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50466, size = 54, normalized size = 0.89 \[ -\frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \sqrt{-x^{2} + 1} x - \frac{5}{3} \, \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217686, size = 177, normalized size = 2.9 \[ -\frac{x^{6} + 3 \, x^{5} - 15 \, x^{3} - 6 \, x^{2} + 3 \,{\left (x^{4} + 3 \, x^{3} + 2 \, x^{2} - 4 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 12 \, x}{3 \,{\left (3 \, x^{2} -{\left (x^{2} - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(x^6 + 3*x^5 - 15*x^3 - 6*x^2 + 3*(x^4 + 3*x^3 + 2*x^2 - 4*x)*sqrt(x + 1)*s
qrt(-x + 1) + 6*(3*x^2 - (x^2 - 4)*sqrt(x + 1)*sqrt(-x + 1) - 4)*arctan((sqrt(x
+ 1)*sqrt(-x + 1) - 1)/x) + 12*x)/(3*x^2 - (x^2 - 4)*sqrt(x + 1)*sqrt(-x + 1) -
4)

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Sympy [A]  time = 50.3717, size = 129, normalized size = 2.11 \[ - 2 \left (\begin{cases} - \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \sqrt{- x + 1} \sqrt{x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} - \frac{3 x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} - 2 \sqrt{- x + 1} \sqrt{x + 1} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*Piecewise((-x*sqrt(-x + 1)*sqrt(x + 1)/4 - sqrt(-x + 1)*sqrt(x + 1) + 3*asin(
sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))) + 2*Piecewise((-3*x*sqrt(-x + 1)
*sqrt(x + 1)/4 + (-x + 1)**(3/2)*(x + 1)**(3/2)/6 - 2*sqrt(-x + 1)*sqrt(x + 1) +
5*asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.224089, size = 50, normalized size = 0.82 \[ -\frac{1}{3} \,{\left ({\left (x + 2\right )}{\left (x + 1\right )} + 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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