3.814 \(\int x (1+\sqrt{1-x^2}) \, dx\)

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 = 0.0071066, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, 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^2]),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^2}\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 = 0.0115721, 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^2]),x]

[Out]

x^2/2 - (1 - x^2)^(3/2)/3

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Maple [A]  time = 0.002, size = 18, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{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]

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

[Out]

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

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Maxima [A]  time = 1.12028, 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+(-x^2+1)^(1/2)),x, algorithm="maxima")

[Out]

1/2*x^2 - 1/3*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 1.69514, size = 54, normalized size = 2.35 \begin{align*} \frac{1}{2} \, x^{2} + \frac{1}{3} \,{\left (x^{2} - 1\right )} \sqrt{-x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.181158, size = 27, normalized size = 1.17 \begin{align*} \frac{x^{2} \sqrt{1 - x^{2}}}{3} + \frac{x^{2}}{2} - \frac{\sqrt{1 - x^{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10405, size = 24, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/3*(-x^2 + 1)^(3/2) - 1/2