3.720 \(\int \frac{(1+x)^{3/2}}{\sqrt{1-x} x^5} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{x+1}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{x+1}}{8 x^2}-\frac{4 \sqrt{1-x} \sqrt{x+1}}{3 x}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x])/(4*x^4) - (2*Sqrt[1 - x]*Sqrt[1 + x])/(3*x^3) - (7*Sq
rt[1 - x]*Sqrt[1 + x])/(8*x^2) - (4*Sqrt[1 - x]*Sqrt[1 + x])/(3*x) - (7*ArcTanh[
Sqrt[1 - x]*Sqrt[1 + x]])/8

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Rubi [A]  time = 0.196955, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{1-x} \sqrt{x+1}}{4 x^4}-\frac{2 \sqrt{1-x} \sqrt{x+1}}{3 x^3}-\frac{7 \sqrt{1-x} \sqrt{x+1}}{8 x^2}-\frac{4 \sqrt{1-x} \sqrt{x+1}}{3 x}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^(3/2)/(Sqrt[1 - x]*x^5),x]

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x])/(4*x^4) - (2*Sqrt[1 - x]*Sqrt[1 + x])/(3*x^3) - (7*Sq
rt[1 - x]*Sqrt[1 + x])/(8*x^2) - (4*Sqrt[1 - x]*Sqrt[1 + x])/(3*x) - (7*ArcTanh[
Sqrt[1 - x]*Sqrt[1 + x]])/8

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Rubi in Sympy [A]  time = 15.2837, size = 97, normalized size = 0.84 \[ - \frac{7 \operatorname{atanh}{\left (\sqrt{- x + 1} \sqrt{x + 1} \right )}}{8} - \frac{4 \sqrt{- x + 1} \sqrt{x + 1}}{3 x} - \frac{7 \sqrt{- x + 1} \sqrt{x + 1}}{8 x^{2}} - \frac{2 \sqrt{- x + 1} \sqrt{x + 1}}{3 x^{3}} - \frac{\sqrt{- x + 1} \sqrt{x + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(3/2)/x**5/(1-x)**(1/2),x)

[Out]

-7*atanh(sqrt(-x + 1)*sqrt(x + 1))/8 - 4*sqrt(-x + 1)*sqrt(x + 1)/(3*x) - 7*sqrt
(-x + 1)*sqrt(x + 1)/(8*x**2) - 2*sqrt(-x + 1)*sqrt(x + 1)/(3*x**3) - sqrt(-x +
1)*sqrt(x + 1)/(4*x**4)

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Mathematica [A]  time = 0.0712087, size = 58, normalized size = 0.5 \[ -\frac{7}{8} \log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \left (32 x^3+21 x^2+16 x+6\right )}{24 x^4}+\frac{7 \log (x)}{8} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + x)^(3/2)/(Sqrt[1 - x]*x^5),x]

[Out]

-(Sqrt[1 - x^2]*(6 + 16*x + 21*x^2 + 32*x^3))/(24*x^4) + (7*Log[x])/8 - (7*Log[1
+ Sqrt[1 - x^2]])/8

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Maple [A]  time = 0.02, size = 94, normalized size = 0.8 \[ -{\frac{1}{24\,{x}^{4}}\sqrt{1-x}\sqrt{1+x} \left ( 21\,{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ){x}^{4}+32\,{x}^{3}\sqrt{-{x}^{2}+1}+21\,{x}^{2}\sqrt{-{x}^{2}+1}+16\,x\sqrt{-{x}^{2}+1}+6\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(3/2)/x^5/(1-x)^(1/2),x)

[Out]

-1/24*(1+x)^(1/2)*(1-x)^(1/2)*(21*arctanh(1/(-x^2+1)^(1/2))*x^4+32*x^3*(-x^2+1)^
(1/2)+21*x^2*(-x^2+1)^(1/2)+16*x*(-x^2+1)^(1/2)+6*(-x^2+1)^(1/2))/x^4/(-x^2+1)^(
1/2)

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Maxima [A]  time = 1.52057, size = 111, normalized size = 0.97 \[ -\frac{4 \, \sqrt{-x^{2} + 1}}{3 \, x} - \frac{7 \, \sqrt{-x^{2} + 1}}{8 \, x^{2}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3 \, x^{3}} - \frac{\sqrt{-x^{2} + 1}}{4 \, x^{4}} - \frac{7}{8} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(x^5*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

-4/3*sqrt(-x^2 + 1)/x - 7/8*sqrt(-x^2 + 1)/x^2 - 2/3*sqrt(-x^2 + 1)/x^3 - 1/4*sq
rt(-x^2 + 1)/x^4 - 7/8*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.228576, size = 248, normalized size = 2.16 \[ \frac{128 \, x^{7} + 84 \, x^{6} - 320 \, x^{5} - 228 \, x^{4} + 64 \, x^{3} + 96 \, x^{2} -{\left (32 \, x^{7} + 21 \, x^{6} - 240 \, x^{5} - 162 \, x^{4} + 128 \, x^{3} + 120 \, x^{2} + 128 \, x + 48\right )} \sqrt{x + 1} \sqrt{-x + 1} + 21 \,{\left (x^{8} - 8 \, x^{6} + 8 \, x^{4} + 4 \,{\left (x^{6} - 2 \, x^{4}\right )} \sqrt{x + 1} \sqrt{-x + 1}\right )} \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 128 \, x + 48}{24 \,{\left (x^{8} - 8 \, x^{6} + 8 \, x^{4} + 4 \,{\left (x^{6} - 2 \, x^{4}\right )} \sqrt{x + 1} \sqrt{-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(x^5*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

1/24*(128*x^7 + 84*x^6 - 320*x^5 - 228*x^4 + 64*x^3 + 96*x^2 - (32*x^7 + 21*x^6
- 240*x^5 - 162*x^4 + 128*x^3 + 120*x^2 + 128*x + 48)*sqrt(x + 1)*sqrt(-x + 1) +
21*(x^8 - 8*x^6 + 8*x^4 + 4*(x^6 - 2*x^4)*sqrt(x + 1)*sqrt(-x + 1))*log((sqrt(x
+ 1)*sqrt(-x + 1) - 1)/x) + 128*x + 48)/(x^8 - 8*x^6 + 8*x^4 + 4*(x^6 - 2*x^4)*
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**5/(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^5*sqrt(-x + 1)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError