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

Optimal. Leaf size=87 \[ \frac{14 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{5 \sqrt{x+1}}{3 \sqrt{1-x} x}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.151866, antiderivative size = 87, 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 \[ \frac{14 \sqrt{x+1}}{3 \sqrt{1-x}}-\frac{5 \sqrt{x+1}}{3 \sqrt{1-x} x}+\frac{2 \sqrt{x+1}}{3 (1-x)^{3/2} x}-3 \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.5186, size = 65, normalized size = 0.75 \[ - 3 \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )} + \frac{14 \sqrt{x + 1}}{3 \sqrt{- x + 1}} + \frac{5 \sqrt{x + 1}}{3 \left (- x + 1\right )^{\frac{3}{2}}} - \frac{\sqrt{x + 1}}{x \left (- x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*atanh(sqrt(-x + 1)*sqrt(x + 1)) + 14*sqrt(x + 1)/(3*sqrt(-x + 1)) + 5*sqrt(x
+ 1)/(3*(-x + 1)**(3/2)) - sqrt(x + 1)/(x*(-x + 1)**(3/2))

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Mathematica [A]  time = 0.0731584, size = 54, normalized size = 0.62 \[ -\frac{\sqrt{1-x^2} \left (14 x^2-19 x+3\right )}{3 (x-1)^2 x}-3 \log \left (\sqrt{1-x^2}+1\right )+3 \log (x) \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[1 - x^2]*(3 - 19*x + 14*x^2))/(3*(-1 + x)^2*x) + 3*Log[x] - 3*Log[1 + Sqr
t[1 - x^2]]

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Maple [A]  time = 0.019, size = 113, normalized size = 1.3 \[ -{\frac{1}{3\,x \left ( -1+x \right ) ^{2}} \left ( 9\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{3}-18\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{2}+14\,{x}^{2}\sqrt{-{x}^{2}+1}+9\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) x-19\,x\sqrt{-{x}^{2}+1}+3\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(9*arctanh(1/(-x^2+1)^(1/2))*x^3-18*arctanh(1/(-x^2+1)^(1/2))*x^2+14*x^2*(-
x^2+1)^(1/2)+9*arctanh(1/(-x^2+1)^(1/2))*x-19*x*(-x^2+1)^(1/2)+3*(-x^2+1)^(1/2))
*(1-x)^(1/2)*(1+x)^(1/2)/x/(-1+x)^2/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.35192, size = 116, normalized size = 1.33 \[ \frac{14 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{3}{\sqrt{-x^{2} + 1}} + \frac{7 \, x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{4}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{1}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x} - 3 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14/3*x/sqrt(-x^2 + 1) + 3/sqrt(-x^2 + 1) + 7/3*x/(-x^2 + 1)^(3/2) + 4/3/(-x^2 +
1)^(3/2) - 1/((-x^2 + 1)^(3/2)*x) - 3*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.227914, size = 242, normalized size = 2.78 \[ \frac{27 \, x^{5} - 29 \, x^{4} - 69 \, x^{3} + 69 \, x^{2} -{\left (x^{4} - 60 \, x^{3} + 63 \, x^{2} + 18 \, x - 12\right )} \sqrt{x + 1} \sqrt{-x + 1} + 9 \,{\left (x^{5} - 4 \, x^{4} + x^{3} + 6 \, x^{2} +{\left (x^{4} + x^{3} - 6 \, x^{2} + 4 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4 \, x\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 18 \, x - 12}{3 \,{\left (x^{5} - 4 \, x^{4} + x^{3} + 6 \, x^{2} +{\left (x^{4} + x^{3} - 6 \, x^{2} + 4 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 4 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(27*x^5 - 29*x^4 - 69*x^3 + 69*x^2 - (x^4 - 60*x^3 + 63*x^2 + 18*x - 12)*sqr
t(x + 1)*sqrt(-x + 1) + 9*(x^5 - 4*x^4 + x^3 + 6*x^2 + (x^4 + x^3 - 6*x^2 + 4*x)
*sqrt(x + 1)*sqrt(-x + 1) - 4*x)*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 18*x -
12)/(x^5 - 4*x^4 + x^3 + 6*x^2 + (x^4 + x^3 - 6*x^2 + 4*x)*sqrt(x + 1)*sqrt(-x +
1) - 4*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError