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 = 0.0016875, 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, Rules used = {39} \[ \frac{x}{9 \sqrt{3-x} \sqrt{x+3}} \]

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])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx &=\frac{x}{9 \sqrt{3-x} \sqrt{3+x}}\\ \end{align*}

Mathematica [A]  time = 0.0043093, 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 = 0.002, size = 16, normalized size = 0.8 \begin{align*}{\frac{x}{9}{\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{3+x}}}} \end{align*}

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 = 0.984483, size = 16, normalized size = 0.76 \begin{align*} \frac{x}{9 \, \sqrt{-x^{2} + 9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x/sqrt(-x^2 + 9)

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Fricas [A]  time = 1.48049, size = 58, normalized size = 2.76 \begin{align*} -\frac{\sqrt{x + 3} x \sqrt{-x + 3}}{9 \,{\left (x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.84558, size = 73, normalized size = 3.48 \begin{align*} \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}\: \frac{6}{\left |{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} \end{align*}

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/Abs(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 [B]  time = 1.07454, size = 84, normalized size = 4. \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(3/2)/(3+x)^(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))