\(\int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx\) [1165]

Optimal result

Integrand size = 17, antiderivative size = 21 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{9 \sqrt {3-x} \sqrt {3+x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {39} \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{9 \sqrt {3-x} \sqrt {x+3}} \]

[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*} \text {integral}& = \frac {x}{9 \sqrt {3-x} \sqrt {3+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{9 \sqrt {9-x^2}} \]

[In]

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

[Out]

x/(9*Sqrt[9 - x^2])

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {x}{9 \sqrt {3-x}\, \sqrt {3+x}}\)	\(16\)
default	\(\frac {1}{3 \sqrt {3-x}\, \sqrt {3+x}}-\frac {\sqrt {3-x}}{9 \sqrt {3+x}}\)	\(30\)
risch	\(\frac {\sqrt {\left (3+x \right ) \left (3-x \right )}\, x}{9 \sqrt {3-x}\, \sqrt {3+x}\, \sqrt {-\left (-3+x \right ) \left (3+x \right )}}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=-\frac {\sqrt {x + 3} x \sqrt {-x + 3}}{9 \, {\left (x^{2} - 9\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.38 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=\begin {cases} - \frac {\sqrt {-1 + \frac {6}{x + 3}} \left (x + 3\right )}{9 x - 27} + \frac {3 \sqrt {-1 + \frac {6}{x + 3}}}{9 x - 27} & \text {for}\: \frac {1}{\left |{x + 3}\right |} > \frac {1}{6} \\- \frac {i}{9 \sqrt {1 - \frac {6}{x + 3}}} + \frac {i}{3 \sqrt {1 - \frac {6}{x + 3}} \left (x + 3\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{9 \, \sqrt {-x^{2} + 9}} \]

[In]

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

[Out]

1/9*x/sqrt(-x^2 + 9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (15) = 30\).

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=\frac {\sqrt {6} - \sqrt {-x + 3}}{36 \, \sqrt {x + 3}} - \frac {\sqrt {x + 3} \sqrt {-x + 3}}{18 \, {\left (x - 3\right )}} - \frac {\sqrt {x + 3}}{36 \, {\left (\sqrt {6} - \sqrt {-x + 3}\right )}} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(3-x)^{3/2} (3+x)^{3/2}} \, dx=-\frac {x\,\sqrt {3-x}}{\left (9\,x-27\right )\,\sqrt {x+3}} \]

[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))