### 3.120 $$\int \frac{x}{\sqrt{4+x^2}} \, dx$$

Optimal. Leaf size=9 $\sqrt{x^2+4}$

[Out]

Sqrt[4 + x^2]

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Rubi [A]  time = 0.0016157, antiderivative size = 9, 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} $\sqrt{x^2+4}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[4 + x^2],x]

[Out]

Sqrt[4 + x^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}{\sqrt{4+x^2}} \, dx &=\sqrt{4+x^2}\\ \end{align*}

Mathematica [A]  time = 0.0009114, size = 9, normalized size = 1. $\sqrt{x^2+4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[4 + x^2],x]

[Out]

Sqrt[4 + x^2]

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Maple [A]  time = 0.003, size = 8, normalized size = 0.9 \begin{align*} \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]

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

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Maxima [A]  time = 0.924973, size = 9, normalized size = 1. \begin{align*} \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="maxima")

[Out]

sqrt(x^2 + 4)

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Fricas [A]  time = 1.79617, size = 20, normalized size = 2.22 \begin{align*} \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]

sqrt(x^2 + 4)

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Sympy [A]  time = 0.125309, size = 7, normalized size = 0.78 \begin{align*} \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)

[Out]

sqrt(x**2 + 4)

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Giac [A]  time = 1.05176, size = 9, normalized size = 1. \begin{align*} \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="giac")

[Out]

sqrt(x^2 + 4)