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

Optimal. Leaf size=69 \[ -\frac{\sqrt{1-x} (x+1)^{3/2}}{2 x^2}-\frac{3 \sqrt{1-x} \sqrt{x+1}}{2 x}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0795779, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\sqrt{1-x} (x+1)^{3/2}}{2 x^2}-\frac{3 \sqrt{1-x} \sqrt{x+1}}{2 x}-\frac{3}{2} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.29285, size = 56, normalized size = 0.81 \[ - \frac{3 \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )}}{2} - \frac{3 \sqrt{- x + 1} \sqrt{x + 1}}{2 x} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0513493, size = 46, normalized size = 0.67 \[ \frac{1}{2} \left (-\frac{\sqrt{1-x^2} (4 x+1)}{x^2}-3 \log \left (\sqrt{1-x^2}+1\right )+3 \log (x)\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.017, size = 64, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{1-x}\sqrt{1+x} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}+4\,x\sqrt{-{x}^{2}+1}+\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(1+x)^(1/2)*(1-x)^(1/2)*(3*arctanh(1/(-x^2+1)^(1/2))*x^2+4*x*(-x^2+1)^(1/2)
+(-x^2+1)^(1/2))/x^2/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.50172, size = 73, normalized size = 1.06 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{x} - \frac{\sqrt{-x^{2} + 1}}{2 \, x^{2}} - \frac{3}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

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]

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

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Fricas [A]  time = 0.232311, size = 162, normalized size = 2.35 \[ \frac{8 \, x^{3} + 2 \, x^{2} -{\left (4 \, x^{3} + x^{2} - 8 \, x - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 3 \,{\left (x^{4} + 2 \, \sqrt{x + 1} x^{2} \sqrt{-x + 1} - 2 \, x^{2}\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 8 \, x - 2}{2 \,{\left (x^{4} + 2 \, \sqrt{x + 1} x^{2} \sqrt{-x + 1} - 2 \, x^{2}\right )}} \]

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]

1/2*(8*x^3 + 2*x^2 - (4*x^3 + x^2 - 8*x - 2)*sqrt(x + 1)*sqrt(-x + 1) + 3*(x^4 +
2*sqrt(x + 1)*x^2*sqrt(-x + 1) - 2*x^2)*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) -
8*x - 2)/(x^4 + 2*sqrt(x + 1)*x^2*sqrt(-x + 1) - 2*x^2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError