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

Optimal. Leaf size=43 \[ -\sqrt{1-x} \sqrt{x+1}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.101894, antiderivative size = 43, 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 \[ -\sqrt{1-x} \sqrt{x+1}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.82229, size = 32, normalized size = 0.74 \[ - \sqrt{- x + 1} \sqrt{x + 1} + 2 \operatorname{asin}{\left (x \right )} - \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0408519, size = 96, normalized size = 2.23 \[ -\sqrt{1-x^2}+\log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{1-x}-\sqrt{x+1}+2\right )-\log \left (\sqrt{x+1}+1\right )+\log \left (\sqrt{1-x}+\sqrt{x+1}+2\right )+4 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

-Sqrt[1 - x^2] + 4*ArcSin[Sqrt[1 + x]/Sqrt[2]] + Log[1 - Sqrt[1 + x]] - Log[2 +
Sqrt[1 - x] - Sqrt[1 + x]] - Log[1 + Sqrt[1 + x]] + Log[2 + Sqrt[1 - x] + Sqrt[1
+ x]]

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Maple [A]  time = 0.016, size = 51, normalized size = 1.2 \[{1\sqrt{1-x}\sqrt{1+x} \left ( -\sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) +2\,\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50378, size = 55, normalized size = 1.28 \[ -\sqrt{-x^{2} + 1} + 2 \, \arcsin \left (x\right ) - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22242, size = 128, normalized size = 2.98 \[ \frac{x^{2} - 4 \,{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) +{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right )}{\sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 1)**(3/2)/(x*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*sqrt(-x + 1)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError