### 3.590 $$\int \frac{2+3 x}{(4+x^2)^{3/2}} \, dx$$

Optimal. Leaf size=18 $-\frac{6-x}{2 \sqrt{x^2+4}}$

[Out]

-(6 - x)/(2*Sqrt[4 + x^2])

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Rubi [A]  time = 0.0040183, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {637} $-\frac{6-x}{2 \sqrt{x^2+4}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + 3*x)/(4 + x^2)^(3/2),x]

[Out]

-(6 - x)/(2*Sqrt[4 + x^2])

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{2+3 x}{\left (4+x^2\right )^{3/2}} \, dx &=-\frac{6-x}{2 \sqrt{4+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0091034, size = 16, normalized size = 0.89 $\frac{x-6}{2 \sqrt{x^2+4}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + 3*x)/(4 + x^2)^(3/2),x]

[Out]

(-6 + x)/(2*Sqrt[4 + x^2])

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Maple [A]  time = 0.039, size = 13, normalized size = 0.7 \begin{align*}{\frac{x-6}{2}{\frac{1}{\sqrt{{x}^{2}+4}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)/(x^2+4)^(3/2),x)

[Out]

1/2*(x-6)/(x^2+4)^(1/2)

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Maxima [A]  time = 1.20106, size = 27, normalized size = 1.5 \begin{align*} \frac{x}{2 \, \sqrt{x^{2} + 4}} - \frac{3}{\sqrt{x^{2} + 4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x^2+4)^(3/2),x, algorithm="maxima")

[Out]

1/2*x/sqrt(x^2 + 4) - 3/sqrt(x^2 + 4)

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Fricas [B]  time = 1.80335, size = 66, normalized size = 3.67 \begin{align*} \frac{x^{2} + \sqrt{x^{2} + 4}{\left (x - 6\right )} + 4}{2 \,{\left (x^{2} + 4\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x^2+4)^(3/2),x, algorithm="fricas")

[Out]

1/2*(x^2 + sqrt(x^2 + 4)*(x - 6) + 4)/(x^2 + 4)

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Sympy [A]  time = 2.66411, size = 20, normalized size = 1.11 \begin{align*} \frac{x}{2 \sqrt{x^{2} + 4}} - \frac{3}{\sqrt{x^{2} + 4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x**2+4)**(3/2),x)

[Out]

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

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Giac [A]  time = 2.20806, size = 16, normalized size = 0.89 \begin{align*} \frac{x - 6}{2 \, \sqrt{x^{2} + 4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(x^2+4)^(3/2),x, algorithm="giac")

[Out]

1/2*(x - 6)/sqrt(x^2 + 4)