3.9.42 \(\int \sqrt {-1+x} x \sqrt {1+x} \, dx\) [842]

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

[Out]

1/3*(-1+x)^(3/2)*(1+x)^(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 = {75} \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 75

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 = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{3} (-1+x)^{3/2} (1+x)^{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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Maple [A]
time = 0.10, size = 18, normalized size = 1.00

method	result	size
gosper	\(\frac {\left (-1+x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(13\)
default	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\)	\(18\)
risch	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, 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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Fricas [A]
time = 0.80, 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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 1.22, 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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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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