\(\int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx\) [1167]

Optimal result

Integrand size = 17, antiderivative size = 26 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {3+x}} \]

[Out]

1/36*x*2^(1/2)/(3-x)^(1/2)/(3+x)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, 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} \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {x+3}} \]

[In]

Int[1/((6 - 2*x)^(3/2)*(3 + x)^(3/2)),x]

[Out]

x/(18*Sqrt[2]*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}{18 \sqrt {2} \sqrt {3-x} \sqrt {3+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{18 \sqrt {2} \sqrt {9-x^2}} \]

[In]

Integrate[1/((6 - 2*x)^(3/2)*(3 + x)^(3/2)),x]

[Out]

x/(18*Sqrt[2]*Sqrt[9 - x^2])

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73

method	result	size
gosper	\(-\frac {\left (-3+x \right ) x}{9 \sqrt {3+x}\, \left (6-2 x \right )^{\frac {3}{2}}}\)	\(19\)
default	\(\frac {1}{6 \sqrt {6-2 x}\, \sqrt {3+x}}-\frac {\sqrt {6-2 x}}{36 \sqrt {3+x}}\)	\(30\)
risch	\(\frac {\sqrt {\left (3+x \right ) \left (6-2 x \right )}\, \sqrt {2}\, x}{36 \sqrt {3+x}\, \sqrt {6-2 x}\, \sqrt {-\left (-3+x \right ) \left (3+x \right )}}\)	\(40\)

[In]

int(1/(6-2*x)^(3/2)/(3+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=-\frac {\sqrt {x + 3} x \sqrt {-2 \, x + 6}}{36 \, {\left (x^{2} - 9\right )}} \]

[In]

integrate(1/(6-2*x)^(3/2)/(3+x)^(3/2),x, algorithm="fricas")

[Out]

-1/36*sqrt(x + 3)*x*sqrt(-2*x + 6)/(x^2 - 9)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=\begin {cases} - \frac {\sqrt {2} \sqrt {-1 + \frac {6}{x + 3}} \left (x + 3\right )}{36 x - 108} + \frac {3 \sqrt {2} \sqrt {-1 + \frac {6}{x + 3}}}{36 x - 108} & \text {for}\: \frac {1}{\left |{x + 3}\right |} > \frac {1}{6} \\- \frac {\sqrt {2} i}{36 \sqrt {1 - \frac {6}{x + 3}}} + \frac {\sqrt {2} i}{12 \sqrt {1 - \frac {6}{x + 3}} \left (x + 3\right )} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(6-2*x)**(3/2)/(3+x)**(3/2),x)

[Out]

Piecewise((-sqrt(2)*sqrt(-1 + 6/(x + 3))*(x + 3)/(36*x - 108) + 3*sqrt(2)*sqrt(-1 + 6/(x + 3))/(36*x - 108), 1
/Abs(x + 3) > 1/6), (-sqrt(2)*I/(36*sqrt(1 - 6/(x + 3))) + sqrt(2)*I/(12*sqrt(1 - 6/(x + 3))*(x + 3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=\frac {x}{18 \, \sqrt {-2 \, x^{2} + 18}} \]

[In]

integrate(1/(6-2*x)^(3/2)/(3+x)^(3/2),x, algorithm="maxima")

[Out]

1/18*x/sqrt(-2*x^2 + 18)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=\frac {\sqrt {2} {\left (\sqrt {6} - \sqrt {-x + 3}\right )}}{144 \, \sqrt {x + 3}} - \frac {\sqrt {2} \sqrt {x + 3} \sqrt {-x + 3}}{72 \, {\left (x - 3\right )}} - \frac {\sqrt {2} \sqrt {x + 3}}{144 \, {\left (\sqrt {6} - \sqrt {-x + 3}\right )}} \]

[In]

integrate(1/(6-2*x)^(3/2)/(3+x)^(3/2),x, algorithm="giac")

[Out]

1/144*sqrt(2)*(sqrt(6) - sqrt(-x + 3))/sqrt(x + 3) - 1/72*sqrt(2)*sqrt(x + 3)*sqrt(-x + 3)/(x - 3) - 1/144*sqr
t(2)*sqrt(x + 3)/(sqrt(6) - sqrt(-x + 3))

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx=-\frac {x\,\sqrt {6-2\,x}}{\left (36\,x-108\right )\,\sqrt {x+3}} \]

[In]

int(1/((6 - 2*x)^(3/2)*(x + 3)^(3/2)),x)

[Out]

-(x*(6 - 2*x)^(1/2))/((36*x - 108)*(x + 3)^(1/2))