3.719 \(\int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^4} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\sqrt{1-x} (x+1)^{5/2}}{3 x^3}-\frac{\sqrt{1-x} (x+1)^{3/2}}{3 x^2}-\frac{\sqrt{1-x} \sqrt{x+1}}{x}-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

-((Sqrt[1 - x]*Sqrt[1 + x])/x) - (Sqrt[1 - x]*(1 + x)^(3/2))/(3*x^2) - (Sqrt[1 -
x]*(1 + x)^(5/2))/(3*x^3) - ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]

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Rubi [A]  time = 0.10995, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt{1-x} (x+1)^{5/2}}{3 x^3}-\frac{\sqrt{1-x} (x+1)^{3/2}}{3 x^2}-\frac{\sqrt{1-x} \sqrt{x+1}}{x}-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^(3/2)/(Sqrt[1 - x]*x^4),x]

[Out]

-((Sqrt[1 - x]*Sqrt[1 + x])/x) - (Sqrt[1 - x]*(1 + x)^(3/2))/(3*x^2) - (Sqrt[1 -
x]*(1 + x)^(5/2))/(3*x^3) - ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]

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Rubi in Sympy [A]  time = 7.88908, size = 68, normalized size = 0.77 \[ - \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} - \frac{\sqrt{- x + 1} \sqrt{x + 1}}{x} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{3 x^{2}} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(3/2)/x**4/(1-x)**(1/2),x)

[Out]

-atanh(sqrt(-x + 1)*sqrt(x + 1)) - sqrt(-x + 1)*sqrt(x + 1)/x - sqrt(-x + 1)*(x
+ 1)**(3/2)/(3*x**2) - sqrt(-x + 1)*(x + 1)**(5/2)/(3*x**3)

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Mathematica [A]  time = 0.0596915, size = 47, normalized size = 0.53 \[ -\log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \left (5 x^2+3 x+1\right )}{3 x^3}+\log (x) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + x)^(3/2)/(Sqrt[1 - x]*x^4),x]

[Out]

-(Sqrt[1 - x^2]*(1 + 3*x + 5*x^2))/(3*x^3) + Log[x] - Log[1 + Sqrt[1 - x^2]]

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Maple [A]  time = 0.017, size = 78, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{1-x}\sqrt{1+x} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{3}+5\,{x}^{2}\sqrt{-{x}^{2}+1}+3\,x\sqrt{-{x}^{2}+1}+\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(3/2)/x^4/(1-x)^(1/2),x)

[Out]

-1/3*(1+x)^(1/2)*(1-x)^(1/2)*(3*arctanh(1/(-x^2+1)^(1/2))*x^3+5*x^2*(-x^2+1)^(1/
2)+3*x*(-x^2+1)^(1/2)+(-x^2+1)^(1/2))/x^3/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.48965, size = 92, normalized size = 1.05 \[ -\frac{5 \, \sqrt{-x^{2} + 1}}{3 \, x} - \frac{\sqrt{-x^{2} + 1}}{x^{2}} - \frac{\sqrt{-x^{2} + 1}}{3 \, x^{3}} - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(x^4*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

-5/3*sqrt(-x^2 + 1)/x - sqrt(-x^2 + 1)/x^2 - 1/3*sqrt(-x^2 + 1)/x^3 - log(2*sqrt
(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.231596, size = 212, normalized size = 2.41 \[ -\frac{5 \, x^{6} + 3 \, x^{5} - 24 \, x^{4} - 15 \, x^{3} + 15 \, x^{2} +{\left (15 \, x^{4} + 9 \, x^{3} - 17 \, x^{2} - 12 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \,{\left (3 \, x^{5} - 4 \, x^{3} -{\left (x^{5} - 4 \, x^{3}\right )} \sqrt{x + 1} \sqrt{-x + 1}\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 12 \, x + 4}{3 \,{\left (3 \, x^{5} - 4 \, x^{3} -{\left (x^{5} - 4 \, x^{3}\right )} \sqrt{x + 1} \sqrt{-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(x^4*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

-1/3*(5*x^6 + 3*x^5 - 24*x^4 - 15*x^3 + 15*x^2 + (15*x^4 + 9*x^3 - 17*x^2 - 12*x
- 4)*sqrt(x + 1)*sqrt(-x + 1) - 3*(3*x^5 - 4*x^3 - (x^5 - 4*x^3)*sqrt(x + 1)*sq
rt(-x + 1))*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 12*x + 4)/(3*x^5 - 4*x^3 - (
x^5 - 4*x^3)*sqrt(x + 1)*sqrt(-x + 1))

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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**4/(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^4*sqrt(-x + 1)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError