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

Optimal. Leaf size=20 \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2} \]

[Out]

-((1 - x)^(3/2)*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.0016797, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {74} \[ -\frac{1}{3} (1-x)^{3/2} (x+1)^{3/2} \]

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 = 0.0025343, size = 15, normalized size = 0.75 \[ -\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - x^2)^(3/2)/3

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \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 = 1.65474, size = 15, normalized size = 0.75 \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 = 1.53953, size = 54, normalized size = 2.7 \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 [B]  time = 48.2723, size = 95, normalized size = 4.75 \begin{align*} - 2 \left (\begin{cases} \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} - \frac{\left (1 - x\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} + \frac{\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \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]

-2*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))) + 2*Piecewise
((x*sqrt(1 - x)*sqrt(x + 1)/4 - (1 - x)**(3/2)*(x + 1)**(3/2)/6 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (
x < 1)))

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Giac [A]  time = 1.37071, size = 23, normalized size = 1.15 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \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="giac")

[Out]

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