3.830 \(\int \sqrt{-1+x} x \sqrt{1+x} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{3} (x-1)^{3/2} (x+1)^{3/2} \]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.00877585, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{3} (x-1)^{3/2} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

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Rubi in Sympy [A]  time = 1.91793, size = 14, normalized size = 0.78 \[ \frac{\left (x - 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)**(3/2)*(x + 1)**(3/2)/3

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Mathematica [A]  time = 0.0113245, size = 18, normalized size = 1. \[ \frac{1}{3} (x-1)^{3/2} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

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Maple [A]  time = 0.004, size = 13, normalized size = 0.7 \[{\frac{1}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35172, size = 12, normalized size = 0.67 \[ \frac{1}{3} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^2 - 1)^(3/2)

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Fricas [A]  time = 0.229816, size = 101, normalized size = 5.61 \[ -\frac{4 \, x^{6} - 9 \, x^{4} -{\left (4 \, x^{5} - 7 \, x^{3} + 3 \, x\right )} \sqrt{x + 1} \sqrt{x - 1} + 6 \, x^{2} - 1}{3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} - 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(4*x^6 - 9*x^4 - (4*x^5 - 7*x^3 + 3*x)*sqrt(x + 1)*sqrt(x - 1) + 6*x^2 - 1)
/(4*x^3 - (4*x^2 - 1)*sqrt(x + 1)*sqrt(x - 1) - 3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232017, size = 16, normalized size = 0.89 \[ \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x + 1)^(3/2)*(x - 1)^(3/2)