### 3.139 $$\int \frac{x}{(4+x^2)^{5/2}} \, dx$$

Optimal. Leaf size=13 $-\frac{1}{3 \left (x^2+4\right )^{3/2}}$

[Out]

-1/(3*(4 + x^2)^(3/2))

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Rubi [A]  time = 0.0020326, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {261} $-\frac{1}{3 \left (x^2+4\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/(4 + x^2)^(5/2),x]

[Out]

-1/(3*(4 + x^2)^(3/2))

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 \frac{x}{\left (4+x^2\right )^{5/2}} \, dx &=-\frac{1}{3 \left (4+x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0021071, size = 13, normalized size = 1. $-\frac{1}{3 \left (x^2+4\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/(4 + x^2)^(5/2),x]

[Out]

-1/(3*(4 + x^2)^(3/2))

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \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]

int(x/(x^2+4)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.920904, size = 12, normalized size = 0.92 \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)^(5/2),x, algorithm="maxima")

[Out]

-1/3/(x^2 + 4)^(3/2)

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Fricas [B]  time = 2.36814, size = 53, normalized size = 4.08 \begin{align*} -\frac{\sqrt{x^{2} + 4}}{3 \,{\left (x^{4} + 8 \, x^{2} + 16\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+4)^(5/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(x^2 + 4)/(x^4 + 8*x^2 + 16)

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Sympy [B]  time = 2.32755, size = 26, normalized size = 2. \begin{align*} - \frac{1}{3 x^{2} \sqrt{x^{2} + 4} + 12 \sqrt{x^{2} + 4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+4)**(5/2),x)

[Out]

-1/(3*x**2*sqrt(x**2 + 4) + 12*sqrt(x**2 + 4))

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Giac [A]  time = 1.05664, size = 12, normalized size = 0.92 \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)^(5/2),x, algorithm="giac")

[Out]

-1/3/(x^2 + 4)^(3/2)