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

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 = 0.0014717, 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, Rules used = {74} \[ \frac{1}{3} (x-1)^{3/2} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

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*}

Mathematica [A]  time = 0.0037917, 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 = 0.002, size = 13, normalized size = 0.7 \begin{align*}{\frac{1}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}} \end{align*}

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 = 1.17424, size = 12, normalized size = 0.67 \begin{align*} \frac{1}{3} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} \end{align*}

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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Fricas [A]  time = 1.55008, size = 53, normalized size = 2.94 \begin{align*} \frac{1}{3} \,{\left (x^{2} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} \end{align*}

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

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 [A]  time = 1.85908, size = 16, normalized size = 0.89 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )}^{\frac{3}{2}} \end{align*}

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]

1/3*(x + 1)^(3/2)*(x - 1)^(3/2)