3.579 \(\int \frac{2+3 x}{\left (4+x^2\right )^{3/2}} \, dx\)

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 = 0.0181213, 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 \[ -\frac{6-x}{2 \sqrt{x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

-(6 - x)/(2*Sqrt[4 + x^2])

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Rubi in Sympy [A]  time = 2.76351, size = 15, normalized size = 0.83 \[ - \frac{- 2 x + 12}{4 \sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)/(x**2+4)**(3/2),x)

[Out]

-(-2*x + 12)/(4*sqrt(x**2 + 4))

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Mathematica [A]  time = 0.0161588, 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 = 0.005, size = 13, normalized size = 0.7 \[{\frac{x-6}{2}{\frac{1}{\sqrt{{x}^{2}+4}}}} \]

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 = 0.711882, size = 27, normalized size = 1.5 \[ \frac{x}{2 \, \sqrt{x^{2} + 4}} - \frac{3}{\sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)/(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 [A]  time = 0.215697, size = 43, normalized size = 2.39 \[ \frac{3 \, x - 3 \, \sqrt{x^{2} + 4} + 2}{x^{2} - \sqrt{x^{2} + 4} x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.0285, 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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GIAC/XCAS [A]  time = 0.212426, size = 16, normalized size = 0.89 \[ \frac{x - 6}{2 \, \sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x - 6)/sqrt(x^2 + 4)