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

Optimal. Leaf size=44 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0994702, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{x}+\sin ^{-1}(x)-2 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.60404, size = 34, normalized size = 0.77 \[ \operatorname{asin}{\left (x \right )} - 2 \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} - \frac{\sqrt{- x + 1} \sqrt{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0539123, size = 51, normalized size = 1.16 \[ -\frac{\sqrt{1-x^2}}{x}-2 \log \left (\sqrt{1-x^2}+1\right )+\tan ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )+2 \log (x) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.017, size = 55, normalized size = 1.3 \[{\frac{1}{x}\sqrt{1-x}\sqrt{1+x} \left ( \arcsin \left ( x \right ) x-2\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x-\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.47844, size = 57, normalized size = 1.3 \[ -\frac{\sqrt{-x^{2} + 1}}{x} + \arcsin \left (x\right ) - 2 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222333, size = 161, normalized size = 3.66 \[ \frac{x^{2} - 2 \,{\left (\sqrt{x + 1} x \sqrt{-x + 1} - x\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 2 \,{\left (\sqrt{x + 1} x \sqrt{-x + 1} - x\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + \sqrt{x + 1} \sqrt{-x + 1} - 1}{\sqrt{x + 1} x \sqrt{-x + 1} - x} \]

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]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + 1\right )^{\frac{3}{2}}}{x^{2} \sqrt{- x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 1)**(3/2)/(x**2*sqrt(-x + 1)), x)

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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^2*sqrt(-x + 1)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError