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

Optimal. Leaf size=69 \[ \frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x) \]

[Out]

(-3*Sqrt[-1 + x]*Sqrt[1 + x])/8 + ((7 - 2*x)*(-1 + x)^(3/2)*Sqrt[1 + x])/24 + ((
-1 + x)^(3/2)*x^2*Sqrt[1 + x])/4 + (3*ArcCosh[x])/8

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Rubi [A]  time = 0.0860815, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[-1 + x]*Sqrt[1 + x])/8 + ((7 - 2*x)*(-1 + x)^(3/2)*Sqrt[1 + x])/24 + ((
-1 + x)^(3/2)*x^2*Sqrt[1 + x])/4 + (3*ArcCosh[x])/8

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Rubi in Sympy [A]  time = 4.4054, size = 61, normalized size = 0.88 \[ \frac{x^{2} \left (x - 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{4} + \frac{\left (- 2 x + 7\right ) \left (x - 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{24} - \frac{3 \sqrt{x - 1} \sqrt{x + 1}}{8} + \frac{3 \operatorname{acosh}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(x - 1)**(3/2)*sqrt(x + 1)/4 + (-2*x + 7)*(x - 1)**(3/2)*sqrt(x + 1)/24 - 3
*sqrt(x - 1)*sqrt(x + 1)/8 + 3*acosh(x)/8

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Mathematica [A]  time = 0.0214597, size = 74, normalized size = 1.07 \[ \frac{\sqrt{\frac{x-1}{x+1}} \left (\sqrt{x-1} \left (6 x^4-2 x^3+x^2-7 x-16\right )+18 \sqrt{x+1} \sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )\right )}{24 \sqrt{x-1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(-1 + x)/(1 + x)]*(Sqrt[-1 + x]*(-16 - 7*x + x^2 - 2*x^3 + 6*x^4) + 18*Sqr
t[1 + x]*ArcSinh[Sqrt[-1 + x]/Sqrt[2]]))/(24*Sqrt[-1 + x])

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Maple [A]  time = 0.014, size = 79, normalized size = 1.1 \[{\frac{1+x}{24}\sqrt{{\frac{-1+x}{1+x}}} \left ( 6\,x \left ({x}^{2}-1 \right ) ^{3/2}-8\, \left ( \left ( -1+x \right ) \left ( 1+x \right ) \right ) ^{3/2}+15\,x\sqrt{{x}^{2}-1}-24\,\sqrt{{x}^{2}-1}+9\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \right ){\frac{1}{\sqrt{ \left ( -1+x \right ) \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*((-1+x)/(1+x))^(1/2),x)

[Out]

1/24*((-1+x)/(1+x))^(1/2)*(1+x)*(6*x*(x^2-1)^(3/2)-8*((-1+x)*(1+x))^(3/2)+15*x*(
x^2-1)^(1/2)-24*(x^2-1)^(1/2)+9*ln(x+(x^2-1)^(1/2)))/((-1+x)*(1+x))^(1/2)

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Maxima [A]  time = 0.716802, size = 186, normalized size = 2.7 \[ -\frac{39 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{7}{2}} - 31 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{2}} + 49 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{3}{2}} - 9 \, \sqrt{\frac{x - 1}{x + 1}}}{12 \,{\left (\frac{4 \,{\left (x - 1\right )}}{x + 1} - \frac{6 \,{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{4 \,{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac{{\left (x - 1\right )}^{4}}{{\left (x + 1\right )}^{4}} - 1\right )}} + \frac{3}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{3}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(39*((x - 1)/(x + 1))^(7/2) - 31*((x - 1)/(x + 1))^(5/2) + 49*((x - 1)/(x
+ 1))^(3/2) - 9*sqrt((x - 1)/(x + 1)))/(4*(x - 1)/(x + 1) - 6*(x - 1)^2/(x + 1)^
2 + 4*(x - 1)^3/(x + 1)^3 - (x - 1)^4/(x + 1)^4 - 1) + 3/8*log(sqrt((x - 1)/(x +
1)) + 1) - 3/8*log(sqrt((x - 1)/(x + 1)) - 1)

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Fricas [A]  time = 0.276881, size = 86, normalized size = 1.25 \[ \frac{1}{24} \,{\left (6 \, x^{4} - 2 \, x^{3} + x^{2} - 7 \, x - 16\right )} \sqrt{\frac{x - 1}{x + 1}} + \frac{3}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \frac{3}{8} \, \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(6*x^4 - 2*x^3 + x^2 - 7*x - 16)*sqrt((x - 1)/(x + 1)) + 3/8*log(sqrt((x -
1)/(x + 1)) + 1) - 3/8*log(sqrt((x - 1)/(x + 1)) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3*sqrt((x - 1)/(x + 1)), x)

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GIAC/XCAS [A]  time = 0.270971, size = 84, normalized size = 1.22 \[ -\frac{3}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ){\rm sign}\left (x + 1\right ) + \frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x{\rm sign}\left (x + 1\right ) - 4 \,{\rm sign}\left (x + 1\right )\right )} x + 9 \,{\rm sign}\left (x + 1\right )\right )} x - 16 \,{\rm sign}\left (x + 1\right )\right )} \sqrt{x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*ln(abs(-x + sqrt(x^2 - 1)))*sign(x + 1) + 1/24*((2*(3*x*sign(x + 1) - 4*sig
n(x + 1))*x + 9*sign(x + 1))*x - 16*sign(x + 1))*sqrt(x^2 - 1)