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

Optimal. Leaf size=110 \[ -\frac{1}{5} \sqrt{1-x} x^2 (x+1)^{5/2}-\frac{1}{10} \sqrt{1-x} (x+1)^{7/2}-\frac{1}{10} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{4} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{4} \sqrt{1-x} \sqrt{x+1}+\frac{3}{4} \sin ^{-1}(x) \]

[Out]

(-3*Sqrt[1 - x]*Sqrt[1 + x])/4 - (Sqrt[1 - x]*(1 + x)^(3/2))/4 - (Sqrt[1 - x]*(1
+ x)^(5/2))/10 - (Sqrt[1 - x]*x^2*(1 + x)^(5/2))/5 - (Sqrt[1 - x]*(1 + x)^(7/2)
)/10 + (3*ArcSin[x])/4

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Rubi [A]  time = 0.120356, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{1}{5} \sqrt{1-x} x^2 (x+1)^{5/2}-\frac{1}{10} \sqrt{1-x} (x+1)^{7/2}-\frac{1}{10} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{4} \sqrt{1-x} (x+1)^{3/2}-\frac{3}{4} \sqrt{1-x} \sqrt{x+1}+\frac{3}{4} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - x]*Sqrt[1 + x])/4 - (Sqrt[1 - x]*(1 + x)^(3/2))/4 - (Sqrt[1 - x]*(1
+ x)^(5/2))/10 - (Sqrt[1 - x]*x^2*(1 + x)^(5/2))/5 - (Sqrt[1 - x]*(1 + x)^(7/2)
)/10 + (3*ArcSin[x])/4

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Rubi in Sympy [A]  time = 10.587, size = 87, normalized size = 0.79 \[ - \frac{x^{2} \sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{5} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{7}{2}}}{10} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{10} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{4} - \frac{3 \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{3 \operatorname{asin}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*sqrt(-x + 1)*(x + 1)**(5/2)/5 - sqrt(-x + 1)*(x + 1)**(7/2)/10 - sqrt(-x +
1)*(x + 1)**(5/2)/10 - sqrt(-x + 1)*(x + 1)**(3/2)/4 - 3*sqrt(-x + 1)*sqrt(x +
1)/4 + 3*asin(x)/4

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Mathematica [A]  time = 0.0498716, size = 54, normalized size = 0.49 \[ \frac{3}{2} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{20} \sqrt{1-x^2} \left (4 x^4+10 x^3+12 x^2+15 x+24\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - x^2]*(24 + 15*x + 12*x^2 + 10*x^3 + 4*x^4))/20 + (3*ArcSin[Sqrt[1 + x
]/Sqrt[2]])/2

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Maple [A]  time = 0.012, size = 94, normalized size = 0.9 \[{\frac{1}{20}\sqrt{1-x}\sqrt{1+x} \left ( -4\,{x}^{4}\sqrt{-{x}^{2}+1}-10\,{x}^{3}\sqrt{-{x}^{2}+1}-12\,{x}^{2}\sqrt{-{x}^{2}+1}-15\,x\sqrt{-{x}^{2}+1}+15\,\arcsin \left ( x \right ) -24\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(1+x)^(1/2)*(1-x)^(1/2)*(-4*x^4*(-x^2+1)^(1/2)-10*x^3*(-x^2+1)^(1/2)-12*x^2
*(-x^2+1)^(1/2)-15*x*(-x^2+1)^(1/2)+15*arcsin(x)-24*(-x^2+1)^(1/2))/(-x^2+1)^(1/
2)

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Maxima [A]  time = 1.5149, size = 95, normalized size = 0.86 \[ -\frac{1}{5} \, \sqrt{-x^{2} + 1} x^{4} - \frac{1}{2} \, \sqrt{-x^{2} + 1} x^{3} - \frac{3}{5} \, \sqrt{-x^{2} + 1} x^{2} - \frac{3}{4} \, \sqrt{-x^{2} + 1} x - \frac{6}{5} \, \sqrt{-x^{2} + 1} + \frac{3}{4} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*sqrt(-x^2 + 1)*x^4 - 1/2*sqrt(-x^2 + 1)*x^3 - 3/5*sqrt(-x^2 + 1)*x^2 - 3/4*
sqrt(-x^2 + 1)*x - 6/5*sqrt(-x^2 + 1) + 3/4*arcsin(x)

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Fricas [A]  time = 0.213275, size = 257, normalized size = 2.34 \[ -\frac{4 \, x^{10} + 10 \, x^{9} - 40 \, x^{8} - 115 \, x^{7} - 20 \, x^{6} + 85 \, x^{5} + 80 \, x^{4} + 260 \, x^{3} + 5 \,{\left (4 \, x^{8} + 10 \, x^{7} - 4 \, x^{6} - 25 \, x^{5} - 16 \, x^{4} - 28 \, x^{3} + 48 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 30 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 16\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 240 \, x}{20 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(4*x^10 + 10*x^9 - 40*x^8 - 115*x^7 - 20*x^6 + 85*x^5 + 80*x^4 + 260*x^3 +
5*(4*x^8 + 10*x^7 - 4*x^6 - 25*x^5 - 16*x^4 - 28*x^3 + 48*x)*sqrt(x + 1)*sqrt(-
x + 1) + 30*(5*x^4 - 20*x^2 - (x^4 - 12*x^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) + 16)
*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 240*x)/(5*x^4 - 20*x^2 - (x^4 - 12*x
^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) + 16)

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Sympy [A]  time = 138.577, size = 366, normalized size = 3.33 \[ - 2 \left (\begin{cases} - \frac{x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \sqrt{- x + 1} \sqrt{x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 6 \left (\begin{cases} - \frac{3 x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{\left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} - 2 \sqrt{- x + 1} \sqrt{x + 1} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) - 6 \left (\begin{cases} - \frac{7 x \sqrt{- x + 1} \sqrt{x + 1}}{4} + \frac{2 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} + \frac{\sqrt{- x + 1} \sqrt{x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 4 \sqrt{- x + 1} \sqrt{x + 1} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} - \frac{15 x \sqrt{- x + 1} \sqrt{x + 1}}{4} - \frac{\left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{10} + 2 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}} + \frac{5 \sqrt{- x + 1} \sqrt{x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 8 \sqrt{- x + 1} \sqrt{x + 1} + \frac{63 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(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))) + 6*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))) - 6*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))) + 2*Piecewise((-15*x*s
qrt(-x + 1)*sqrt(x + 1)/4 - (-x + 1)**(5/2)*(x + 1)**(5/2)/10 + 2*(-x + 1)**(3/2
)*(x + 1)**(3/2) + 5*sqrt(-x + 1)*sqrt(x + 1)*(-5*x - 2*(x + 1)**3 + 6*(x + 1)**
2 - 4)/16 - 8*sqrt(-x + 1)*sqrt(x + 1) + 63*asin(sqrt(2)*sqrt(x + 1)/2)/8, (x >=
-1) & (x < 1)))

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GIAC/XCAS [A]  time = 0.239496, size = 70, normalized size = 0.64 \[ -\frac{1}{20} \,{\left ({\left (2 \,{\left ({\left (2 \, x - 1\right )}{\left (x + 1\right )} + 3\right )}{\left (x + 1\right )} + 5\right )}{\left (x + 1\right )} + 15\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{3}{2} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*((2*((2*x - 1)*(x + 1) + 3)*(x + 1) + 5)*(x + 1) + 15)*sqrt(x + 1)*sqrt(-x
+ 1) + 3/2*arcsin(1/2*sqrt(2)*sqrt(x + 1))