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

Optimal. Leaf size=23 \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Rubi [A]  time = 0.008017, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {14, 261} \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*(1 + Sqrt[1 - x]*Sqrt[1 + x]),x]

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0040071, size = 23, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Maple [A]  time = 0.003, size = 26, normalized size = 1.1 \begin{align*}{\frac{{x}^{2}-1}{3}\sqrt{1-x}\sqrt{1+x}}+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.54099, size = 23, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} - \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+(1-x)^(1/2)*(1+x)^(1/2)),x, algorithm="maxima")

[Out]

1/2*x^2 - 1/3*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 1.63926, size = 68, normalized size = 2.96 \begin{align*} \frac{1}{2} \, x^{2} + \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+(1-x)^(1/2)*(1+x)^(1/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 71.2532, size = 105, normalized size = 4.57 \begin{align*} - x + \frac{\left (x + 1\right )^{2}}{2} - 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 ) - 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+(1-x)**(1/2)*(1+x)**(1/2)),x)

[Out]

-x + (x + 1)**2/2 - 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))) - 1

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Giac [A]  time = 1.156, size = 39, normalized size = 1.7 \begin{align*} \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{2} \,{\left (x + 1\right )}^{2} - x - 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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