3.783 \(\int \frac{4+x}{\left (6 x-x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{12-7 x}{9 \sqrt{6 x-x^2}} \]

[Out]

-(12 - 7*x)/(9*Sqrt[6*x - x^2])

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Rubi [A]  time = 0.0183545, antiderivative size = 22, 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 \[ -\frac{12-7 x}{9 \sqrt{6 x-x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(4 + x)/(6*x - x^2)^(3/2),x]

[Out]

-(12 - 7*x)/(9*Sqrt[6*x - x^2])

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Rubi in Sympy [A]  time = 1.87237, size = 17, normalized size = 0.77 \[ - \frac{- 28 x + 48}{36 \sqrt{- x^{2} + 6 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4+x)/(-x**2+6*x)**(3/2),x)

[Out]

-(-28*x + 48)/(36*sqrt(-x**2 + 6*x))

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Mathematica [A]  time = 0.0249244, size = 19, normalized size = 0.86 \[ \frac{7 x-12}{9 \sqrt{-(x-6) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(4 + x)/(6*x - x^2)^(3/2),x]

[Out]

(-12 + 7*x)/(9*Sqrt[-((-6 + x)*x)])

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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \[ -{\frac{x \left ( -6+x \right ) \left ( -12+7\,x \right ) }{9} \left ( -{x}^{2}+6\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4+x)/(-x^2+6*x)^(3/2),x)

[Out]

-1/9*x*(-6+x)*(-12+7*x)/(-x^2+6*x)^(3/2)

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Maxima [A]  time = 0.739876, size = 38, normalized size = 1.73 \[ \frac{7 \, x}{9 \, \sqrt{-x^{2} + 6 \, x}} - \frac{4}{3 \, \sqrt{-x^{2} + 6 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 4)/(-x^2 + 6*x)^(3/2),x, algorithm="maxima")

[Out]

7/9*x/sqrt(-x^2 + 6*x) - 4/3/sqrt(-x^2 + 6*x)

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Fricas [A]  time = 0.25948, size = 24, normalized size = 1.09 \[ \frac{7 \, x - 12}{9 \, \sqrt{-x^{2} + 6 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 4)/(-x^2 + 6*x)^(3/2),x, algorithm="fricas")

[Out]

1/9*(7*x - 12)/sqrt(-x^2 + 6*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 4}{\left (- x \left (x - 6\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4+x)/(-x**2+6*x)**(3/2),x)

[Out]

Integral((x + 4)/(-x*(x - 6))**(3/2), x)

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GIAC/XCAS [A]  time = 0.268446, size = 36, normalized size = 1.64 \[ -\frac{\sqrt{-x^{2} + 6 \, x}{\left (7 \, x - 12\right )}}{9 \,{\left (x^{2} - 6 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 4)/(-x^2 + 6*x)^(3/2),x, algorithm="giac")

[Out]

-1/9*sqrt(-x^2 + 6*x)*(7*x - 12)/(x^2 - 6*x)