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]

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

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.43, size = 22, normalized size = 1.05 \[ -\frac {\sqrt {x + 3} x \sqrt {-x + 3}}{9 \, {\left (x^{2} - 9\right )}} \]

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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giac [B]  time = 0.90, size = 62, normalized size = 2.95 \[ \frac {\sqrt {6} - \sqrt {-x + 3}}{36 \, \sqrt {x + 3}} - \frac {\sqrt {x + 3} \sqrt {-x + 3}}{18 \, {\left (x - 3\right )}} - \frac {\sqrt {x + 3}}{36 \, {\left (\sqrt {6} - \sqrt {-x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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))

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {x}{9 \sqrt {-x +3}\, \sqrt {x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 12, normalized size = 0.57 \[ \frac {x}{9 \, \sqrt {-x^{2} + 9}} \]

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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mupad [B]  time = 0.36, size = 22, normalized size = 1.05 \[ -\frac {x\,\sqrt {3-x}}{\left (9\,x-27\right )\,\sqrt {x+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((3 - x)^(3/2)*(x + 3)^(3/2)),x)

[Out]

-(x*(3 - x)^(1/2))/((9*x - 27)*(x + 3)^(1/2))

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sympy [A]  time = 1.80, size = 73, normalized size = 3.48 \[ \begin {cases} \frac {1}{9 \sqrt {-1 + \frac {6}{x + 3}}} - \frac {1}{3 \sqrt {-1 + \frac {6}{x + 3}} \left (x + 3\right )} & \text {for}\: \frac {6}{\left |{x + 3}\right |} > 1 \\\frac {i \sqrt {1 - \frac {6}{x + 3}} \left (x + 3\right )}{27 - 9 x} - \frac {3 i \sqrt {1 - \frac {6}{x + 3}}}{27 - 9 x} & \text {otherwise} \end {cases} \]

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

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