### 3.127 $$\int x \sqrt{4-x^2} \, dx$$

Optimal. Leaf size=15 $-\frac{1}{3} \left (4-x^2\right )^{3/2}$

[Out]

-(4 - x^2)^(3/2)/3

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Rubi [A]  time = 0.0022245, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {261} $-\frac{1}{3} \left (4-x^2\right )^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Sqrt[4 - x^2],x]

[Out]

-(4 - x^2)^(3/2)/3

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 \sqrt{4-x^2} \, dx &=-\frac{1}{3} \left (4-x^2\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0015562, size = 15, normalized size = 1. $-\frac{1}{3} \left (4-x^2\right )^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Sqrt[4 - x^2],x]

[Out]

-(4 - x^2)^(3/2)/3

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Maple [A]  time = 0.003, size = 18, normalized size = 1.2 \begin{align*}{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) }{3}\sqrt{-{x}^{2}+4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-x^2+4)^(1/2),x)

[Out]

1/3*(-2+x)*(2+x)*(-x^2+4)^(1/2)

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Maxima [A]  time = 0.919963, size = 15, normalized size = 1. \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+4)^(1/2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 4)^(3/2)

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Fricas [A]  time = 2.00834, size = 41, normalized size = 2.73 \begin{align*} \frac{1}{3} \,{\left (x^{2} - 4\right )} \sqrt{-x^{2} + 4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+4)^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^2 - 4)*sqrt(-x^2 + 4)

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Sympy [B]  time = 0.180193, size = 24, normalized size = 1.6 \begin{align*} \frac{x^{2} \sqrt{4 - x^{2}}}{3} - \frac{4 \sqrt{4 - x^{2}}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x**2+4)**(1/2),x)

[Out]

x**2*sqrt(4 - x**2)/3 - 4*sqrt(4 - x**2)/3

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Giac [A]  time = 1.07207, size = 15, normalized size = 1. \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+4)^(1/2),x, algorithm="giac")

[Out]

-1/3*(-x^2 + 4)^(3/2)