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

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {74} \begin {gather*} \frac {1}{3} (x-1)^{3/2} (x+1)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin {align*} \int \sqrt {-1+x} x \sqrt {1+x} \, dx &=\frac {1}{3} (-1+x)^{3/2} (1+x)^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{3} (x-1)^{3/2} (x+1)^{3/2} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.04, size = 31, normalized size = 1.72 \begin {gather*} -\frac {8 (x-1)^{3/2}}{3 (x+1)^{3/2} \left (\frac {x-1}{x+1}-1\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

(-8*(-1 + x)^(3/2))/(3*(1 + x)^(3/2)*(-1 + (-1 + x)/(1 + x))^3)

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fricas [A]  time = 1.34, size = 17, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-1+x)^(1/2)*(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)

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giac [B]  time = 1.32, size = 39, normalized size = 2.17 \begin {gather*} \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} {\left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(x - 1) + 1/2*sqrt(x + 1)*sqrt(x - 1)*(x - 2)

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maple [A]  time = 0.00, size = 13, normalized size = 0.72 \begin {gather*} \frac {\left (x -1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 9, normalized size = 0.50 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-1+x)^(1/2)*(1+x)^(1/2),x, algorithm="maxima")

[Out]

1/3*(x^2 - 1)^(3/2)

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mupad [B]  time = 1.06, size = 17, normalized size = 0.94 \begin {gather*} \frac {\left (x^2-1\right )\,\sqrt {x-1}\,\sqrt {x+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x - 1)^(1/2)*(x + 1)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x - 1} \sqrt {x + 1}\, dx \end {gather*}

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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