3.1165 \(\int \frac{1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{x}{9 \sqrt{3-x} \sqrt{x+3}} \]

[Out]

x/(9*Sqrt[3 - x]*Sqrt[3 + x])

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Rubi [A]  time = 0.0124701, antiderivative size = 21, 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 \[ \frac{x}{9 \sqrt{3-x} \sqrt{x+3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((3 - x)^(3/2)*(3 + x)^(3/2)),x]

[Out]

x/(9*Sqrt[3 - x]*Sqrt[3 + x])

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Rubi in Sympy [A]  time = 2.68114, size = 15, normalized size = 0.71 \[ \frac{x}{9 \sqrt{- x + 3} \sqrt{x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-x)**(3/2)/(3+x)**(3/2),x)

[Out]

x/(9*sqrt(-x + 3)*sqrt(x + 3))

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Mathematica [A]  time = 0.011753, size = 16, normalized size = 0.76 \[ \frac{x}{9 \sqrt{9-x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((3 - x)^(3/2)*(3 + x)^(3/2)),x]

[Out]

x/(9*Sqrt[9 - x^2])

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \[{\frac{x}{9}{\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{3+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-x)^(3/2)/(3+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.34142, size = 16, normalized size = 0.76 \[ \frac{x}{9 \, \sqrt{-x^{2} + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 3)^(3/2)*(-x + 3)^(3/2)),x, algorithm="maxima")

[Out]

1/9*x/sqrt(-x^2 + 9)

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Fricas [A]  time = 0.204027, size = 55, normalized size = 2.62 \[ -\frac{\sqrt{x + 3} x \sqrt{-x + 3} - 3 \, x}{9 \,{\left (x^{2} + 3 \, \sqrt{x + 3} \sqrt{-x + 3} - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 3)^(3/2)*(-x + 3)^(3/2)),x, algorithm="fricas")

[Out]

-1/9*(sqrt(x + 3)*x*sqrt(-x + 3) - 3*x)/(x^2 + 3*sqrt(x + 3)*sqrt(-x + 3) - 9)

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Sympy [A]  time = 11.6488, size = 75, normalized size = 3.57 \[ \begin{cases} \frac{1}{9 \sqrt{-1 + \frac{6}{x + 3}}} - \frac{1}{3 \sqrt{-1 + \frac{6}{x + 3}} \left (x + 3\right )} & \text{for}\: 6 \left |{\frac{1}{x + 3}}\right | > 1 \\- \frac{i \sqrt{1 - \frac{6}{x + 3}} \left (x + 3\right )}{9 x - 27} + \frac{3 i \sqrt{1 - \frac{6}{x + 3}}}{9 x - 27} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-x)**(3/2)/(3+x)**(3/2),x)

[Out]

Piecewise((1/(9*sqrt(-1 + 6/(x + 3))) - 1/(3*sqrt(-1 + 6/(x + 3))*(x + 3)), 6*Ab
s(1/(x + 3)) > 1), (-I*sqrt(1 - 6/(x + 3))*(x + 3)/(9*x - 27) + 3*I*sqrt(1 - 6/(
x + 3))/(9*x - 27), True))

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GIAC/XCAS [A]  time = 0.225636, size = 84, normalized size = 4. \[ \frac{\sqrt{6} - \sqrt{-x + 3}}{36 \, \sqrt{x + 3}} - \frac{\sqrt{x + 3} \sqrt{-x + 3}}{18 \,{\left (x - 3\right )}} - \frac{\sqrt{x + 3}}{36 \,{\left (\sqrt{6} - \sqrt{-x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 3)^(3/2)*(-x + 3)^(3/2)),x, algorithm="giac")

[Out]

1/36*(sqrt(6) - sqrt(-x + 3))/sqrt(x + 3) - 1/18*sqrt(x + 3)*sqrt(-x + 3)/(x - 3
) - 1/36*sqrt(x + 3)/(sqrt(6) - sqrt(-x + 3))