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]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.98, size = 25, normalized size = 1.39 \[ \frac {x^{2} + \sqrt {x^{2} + 4} {\left (x - 6\right )} + 4}{2 \, {\left (x^{2} + 4\right )}} \]

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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giac [A]  time = 0.19, size = 12, normalized size = 0.67 \[ \frac {x - 6}{2 \, \sqrt {x^{2} + 4}} \]

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)

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maple [A]  time = 0.04, size = 13, normalized size = 0.72 \[ \frac {x -6}{2 \sqrt {x^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 20, normalized size = 1.11 \[ \frac {x}{2 \, \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \]

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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mupad [B]  time = 0.34, size = 12, normalized size = 0.67 \[ \frac {x-6}{2\,\sqrt {x^2+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)/(x^2 + 4)^(3/2),x)

[Out]

(x - 6)/(2*(x^2 + 4)^(1/2))

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sympy [A]  time = 4.20, size = 20, normalized size = 1.11 \[ \frac {x}{2 \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \]

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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