3.821 \(\int \frac{\sqrt{1+x}}{(1-x)^{5/2} x} \, dx\)

Optimal. Leaf size=59 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}+\frac{2 \sqrt{x+1}}{\sqrt{1-x}}-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0743007, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}}+\frac{2 \sqrt{x+1}}{\sqrt{1-x}}-\tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x]/((1 - x)^(5/2)*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.56273, size = 44, normalized size = 0.75 \[ - \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} + \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(1/2)/(1-x)**(5/2)/x,x)

[Out]

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

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Mathematica [A]  time = 0.0676802, size = 92, normalized size = 1.56 \[ \frac{\sqrt{1-x^2} (7-5 x)}{3 (x-1)^2}+\log \left (1-\sqrt{x+1}\right )-\log \left (\sqrt{1-x}-\sqrt{x+1}+2\right )-\log \left (\sqrt{x+1}+1\right )+\log \left (\sqrt{1-x}+\sqrt{x+1}+2\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[1 + x]/((1 - x)^(5/2)*x),x]

[Out]

((7 - 5*x)*Sqrt[1 - x^2])/(3*(-1 + x)^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 [B]  time = 0.017, size = 93, normalized size = 1.6 \[ -{\frac{1}{3\, \left ( -1+x \right ) ^{2}} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}-6\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x+5\,x\sqrt{-{x}^{2}+1}+3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) -7\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(1-x)^(5/2)/x,x)

[Out]

-1/3*(3*arctanh(1/(-x^2+1)^(1/2))*x^2-6*arctanh(1/(-x^2+1)^(1/2))*x+5*x*(-x^2+1)
^(1/2)+3*arctanh(1/(-x^2+1)^(1/2))-7*(-x^2+1)^(1/2))*(1-x)^(1/2)*(1+x)^(1/2)/(-1
+x)^2/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.34463, size = 95, normalized size = 1.61 \[ \frac{5 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{1}{\sqrt{-x^{2} + 1}} + \frac{4 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{4}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(x*(-x + 1)^(5/2)),x, algorithm="maxima")

[Out]

5/3*x/sqrt(-x^2 + 1) + 1/sqrt(-x^2 + 1) + 4/3*x/(-x^2 + 1)^(3/2) + 4/3/(-x^2 + 1
)^(3/2) - log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.233293, size = 166, normalized size = 2.81 \[ \frac{2 \, x^{3} + 12 \, x^{2} - 6 \,{\left (2 \, x^{2} - 3 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x + 2\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 18 \, x}{3 \,{\left (x^{3} -{\left (x^{2} - 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(x*(-x + 1)^(5/2)),x, algorithm="fricas")

[Out]

1/3*(2*x^3 + 12*x^2 - 6*(2*x^2 - 3*x)*sqrt(x + 1)*sqrt(-x + 1) + 3*(x^3 - (x^2 -
3*x + 2)*sqrt(x + 1)*sqrt(-x + 1) - 3*x + 2)*log((sqrt(x + 1)*sqrt(-x + 1) - 1)
/x) - 18*x)/(x^3 - (x^2 - 3*x + 2)*sqrt(x + 1)*sqrt(-x + 1) - 3*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1}}{x \left (- x + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**(1/2)/(1-x)**(5/2)/x,x)

[Out]

Integral(sqrt(x + 1)/(x*(-x + 1)**(5/2)), 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(sqrt(x + 1)/(x*(-x + 1)^(5/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError