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

Optimal. Leaf size=88 \[ -\frac{1}{4} \sqrt{1-x} x (x+1)^{5/2}-\frac{1}{6} \sqrt{1-x} (x+1)^{5/2}-\frac{7}{24} \sqrt{1-x} (x+1)^{3/2}-\frac{7}{8} \sqrt{1-x} \sqrt{x+1}+\frac{7}{8} \sin ^{-1}(x) \]

[Out]

(-7*Sqrt[1 - x]*Sqrt[1 + x])/8 - (7*Sqrt[1 - x]*(1 + x)^(3/2))/24 - (Sqrt[1 - x]
*(1 + x)^(5/2))/6 - (Sqrt[1 - x]*x*(1 + x)^(5/2))/4 + (7*ArcSin[x])/8

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Rubi [A]  time = 0.0931823, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1}{4} \sqrt{1-x} x (x+1)^{5/2}-\frac{1}{6} \sqrt{1-x} (x+1)^{5/2}-\frac{7}{24} \sqrt{1-x} (x+1)^{3/2}-\frac{7}{8} \sqrt{1-x} \sqrt{x+1}+\frac{7}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-7*Sqrt[1 - x]*Sqrt[1 + x])/8 - (7*Sqrt[1 - x]*(1 + x)^(3/2))/24 - (Sqrt[1 - x]
*(1 + x)^(5/2))/6 - (Sqrt[1 - x]*x*(1 + x)^(5/2))/4 + (7*ArcSin[x])/8

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Rubi in Sympy [A]  time = 8.29439, size = 71, normalized size = 0.81 \[ - \frac{x \sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{4} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{6} - \frac{7 \sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{24} - \frac{7 \sqrt{- x + 1} \sqrt{x + 1}}{8} + \frac{7 \operatorname{asin}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(-x + 1)*(x + 1)**(5/2)/4 - sqrt(-x + 1)*(x + 1)**(5/2)/6 - 7*sqrt(-x + 1
)*(x + 1)**(3/2)/24 - 7*sqrt(-x + 1)*sqrt(x + 1)/8 + 7*asin(x)/8

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Mathematica [A]  time = 0.0403089, size = 49, normalized size = 0.56 \[ \frac{7}{4} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{24} \sqrt{1-x^2} \left (6 x^3+16 x^2+21 x+32\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - x^2]*(32 + 21*x + 16*x^2 + 6*x^3))/24 + (7*ArcSin[Sqrt[1 + x]/Sqrt[2]
])/4

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Maple [A]  time = 0.013, size = 80, normalized size = 0.9 \[{\frac{1}{24}\sqrt{1-x}\sqrt{1+x} \left ( -6\,{x}^{3}\sqrt{-{x}^{2}+1}-16\,{x}^{2}\sqrt{-{x}^{2}+1}-21\,x\sqrt{-{x}^{2}+1}+21\,\arcsin \left ( x \right ) -32\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(1+x)^(1/2)*(1-x)^(1/2)*(-6*x^3*(-x^2+1)^(1/2)-16*x^2*(-x^2+1)^(1/2)-21*x*(
-x^2+1)^(1/2)+21*arcsin(x)-32*(-x^2+1)^(1/2))/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.5267, size = 76, normalized size = 0.86 \[ -\frac{1}{4} \, \sqrt{-x^{2} + 1} x^{3} - \frac{2}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{7}{8} \, \sqrt{-x^{2} + 1} x - \frac{4}{3} \, \sqrt{-x^{2} + 1} + \frac{7}{8} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(-x^2 + 1)*x^3 - 2/3*sqrt(-x^2 + 1)*x^2 - 7/8*sqrt(-x^2 + 1)*x - 4/3*sq
rt(-x^2 + 1) + 7/8*arcsin(x)

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Fricas [A]  time = 0.213923, size = 211, normalized size = 2.4 \[ \frac{24 \, x^{7} + 64 \, x^{6} + 12 \, x^{5} - 96 \, x^{4} - 204 \, x^{3} -{\left (6 \, x^{7} + 16 \, x^{6} - 27 \, x^{5} - 96 \, x^{4} - 120 \, x^{3} + 168 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 42 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 168 \, x}{24 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(24*x^7 + 64*x^6 + 12*x^5 - 96*x^4 - 204*x^3 - (6*x^7 + 16*x^6 - 27*x^5 - 9
6*x^4 - 120*x^3 + 168*x)*sqrt(x + 1)*sqrt(-x + 1) - 42*(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) + 168*x
)/(x^4 - 8*x^2 + 4*(x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1) + 8)

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Sympy [A]  time = 84.4209, size = 240, normalized size = 2.73 \[ 2 \left (\begin{cases} - \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \sqrt{- x + 1} \sqrt{x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) - 4 \left (\begin{cases} - \frac{3 x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} - 2 \sqrt{- x + 1} \sqrt{x + 1} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} - \frac{7 x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{2 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} + \frac{\sqrt{- x + 1} \sqrt{x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 4 \sqrt{- x + 1} \sqrt{x + 1} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(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(s
qrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))) - 4*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))) + 2*Piecewise((-7*x*sqrt(
-x + 1)*sqrt(x + 1)/4 + 2*(-x + 1)**(3/2)*(x + 1)**(3/2)/3 + sqrt(-x + 1)*sqrt(x
+ 1)*(-5*x - 2*(x + 1)**3 + 6*(x + 1)**2 - 4)/16 - 4*sqrt(-x + 1)*sqrt(x + 1) +
35*asin(sqrt(2)*sqrt(x + 1)/2)/8, (x >= -1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.231855, size = 62, normalized size = 0.7 \[ -\frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )} + 7\right )}{\left (x + 1\right )} + 21\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{7}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*((2*(3*x + 2)*(x + 1) + 7)*(x + 1) + 21)*sqrt(x + 1)*sqrt(-x + 1) + 7/4*ar
csin(1/2*sqrt(2)*sqrt(x + 1))