3.1412 \(\int \frac{5-x}{\left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{15 x+2}{6 \sqrt{3 x^2+2}} \]

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

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Rubi [A]  time = 0.0173552, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{15 x+2}{6 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)/(2 + 3*x^2)^(3/2),x]

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

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Rubi in Sympy [A]  time = 3.03964, size = 15, normalized size = 0.75 \[ \frac{15 x + 2}{6 \sqrt{3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3*x**2+2)**(3/2),x)

[Out]

(15*x + 2)/(6*sqrt(3*x**2 + 2))

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Mathematica [A]  time = 0.0147064, size = 20, normalized size = 1. \[ \frac{15 x+2}{6 \sqrt{3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)/(2 + 3*x^2)^(3/2),x]

[Out]

(2 + 15*x)/(6*Sqrt[2 + 3*x^2])

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Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \[{\frac{2+15\,x}{6}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3*x^2+2)^(3/2),x)

[Out]

1/6*(2+15*x)/(3*x^2+2)^(1/2)

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Maxima [A]  time = 0.682963, size = 32, normalized size = 1.6 \[ \frac{5 \, x}{2 \, \sqrt{3 \, x^{2} + 2}} + \frac{1}{3 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(3*x^2 + 2)^(3/2),x, algorithm="maxima")

[Out]

5/2*x/sqrt(3*x^2 + 2) + 1/3/sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.268294, size = 22, normalized size = 1.1 \[ \frac{15 \, x + 2}{6 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(3*x^2 + 2)^(3/2),x, algorithm="fricas")

[Out]

1/6*(15*x + 2)/sqrt(3*x^2 + 2)

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Sympy [A]  time = 32.5824, size = 27, normalized size = 1.35 \[ \frac{5 x}{2 \sqrt{3 x^{2} + 2}} + \frac{1}{3 \sqrt{3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)/(3*x**2+2)**(3/2),x)

[Out]

5*x/(2*sqrt(3*x**2 + 2)) + 1/(3*sqrt(3*x**2 + 2))

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GIAC/XCAS [A]  time = 0.307519, size = 22, normalized size = 1.1 \[ \frac{15 \, x + 2}{6 \, \sqrt{3 \, x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/(3*x^2 + 2)^(3/2),x, algorithm="giac")

[Out]

1/6*(15*x + 2)/sqrt(3*x^2 + 2)