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

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{3} (1-x)^{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 = 15, normalized size = 0.75 \begin {gather*} -\frac {1}{3} \left (1-x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(1 - x^2)^(3/2)

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IntegrateAlgebraic [A]  time = 0.07, size = 35, normalized size = 1.75 \begin {gather*} -\frac {8 (1-x)^{3/2}}{3 (x+1)^{3/2} \left (\frac {1-x}{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 = 1.45, size = 19, normalized size = 0.95 \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 = 1.40, size = 43, normalized size = 2.15 \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} {\left (x - 2\right )} \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="giac")

[Out]

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.75 \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 = 1.34, size = 11, normalized size = 0.55 \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 = 1.09, size = 19, normalized size = 0.95 \begin {gather*} \frac {\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1 - x)^(1/2)*(x + 1)^(1/2),x)

[Out]

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

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sympy [B]  time = 35.69, size = 95, normalized size = 4.75 \begin {gather*} - 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 {gather*}

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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