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]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \]

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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fricas [A]  time = 0.41, size = 22, normalized size = 0.96 \[ \frac {1}{2} \, x^{2} + \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {-x^{2} + 1} \]

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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giac [A]  time = 0.33, size = 18, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.78 \[ \frac {x^{2}}{2}-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3} \]

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 = 0.43, size = 17, normalized size = 0.74 \[ \frac {1}{2} \, x^{2} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]

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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mupad [B]  time = 0.03, size = 23, normalized size = 1.00 \[ \frac {x^2}{2}+\sqrt {1-x^2}\,\left (\frac {x^2}{3}-\frac {1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*((1 - x^2)^(1/2) + 1),x)

[Out]

x^2/2 + (1 - x^2)^(1/2)*(x^2/3 - 1/3)

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sympy [A]  time = 0.19, size = 27, normalized size = 1.17 \[ \frac {x^{2} \sqrt {1 - x^{2}}}{3} + \frac {x^{2}}{2} - \frac {\sqrt {1 - x^{2}}}{3} \]

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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