\(\int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\) [1157]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=\frac {x}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {39} \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=\frac {x}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}} \]

[In]

Int[1/((3 - 6*x)^(3/2)*(2 + 4*x)^(3/2)),x]

[Out]

x/(6*Sqrt[6]*Sqrt[1 - 2*x]*Sqrt[1 + 2*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}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=\frac {x}{6 \sqrt {6-24 x^2}} \]

[In]

Integrate[1/((3 - 6*x)^(3/2)*(2 + 4*x)^(3/2)),x]

[Out]

x/(6*Sqrt[6 - 24*x^2])

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {\left (-1+2 x \right ) \left (1+2 x \right ) x}{\left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {3}{2}}}\)	\(28\)
default	\(\frac {1}{12 \sqrt {3-6 x}\, \sqrt {2+4 x}}-\frac {\sqrt {3-6 x}}{36 \sqrt {2+4 x}}\)	\(34\)

[In]

int(1/(3-6*x)^(3/2)/(2+4*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-(-1+2*x)*(1+2*x)*x/(3-6*x)^(3/2)/(2+4*x)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=-\frac {\sqrt {4 \, x + 2} x \sqrt {-6 \, x + 3}}{36 \, {\left (4 \, x^{2} - 1\right )}} \]

[In]

integrate(1/(3-6*x)^(3/2)/(2+4*x)^(3/2),x, algorithm="fricas")

[Out]

-1/36*sqrt(4*x + 2)*x*sqrt(-6*x + 3)/(4*x^2 - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 52.68 (sec) , antiderivative size = 156, normalized size of antiderivative = 5.57 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=\begin {cases} - \frac {2 \sqrt {6} i \sqrt {x - \frac {1}{2}} \left (x + \frac {1}{2}\right )}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} + \frac {\sqrt {6} i \sqrt {x - \frac {1}{2}}}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} & \text {for}\: \left |{x + \frac {1}{2}}\right | > 1 \\- \frac {2 \sqrt {6} \sqrt {\frac {1}{2} - x} \left (x + \frac {1}{2}\right )}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} + \frac {\sqrt {6} \sqrt {\frac {1}{2} - x}}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(3-6*x)**(3/2)/(2+4*x)**(3/2),x)

[Out]

Piecewise((-2*sqrt(6)*I*sqrt(x - 1/2)*(x + 1/2)/(144*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)) + sqrt(6)*I*sqrt(x
- 1/2)/(144*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)), Abs(x + 1/2) > 1), (-2*sqrt(6)*sqrt(1/2 - x)*(x + 1/2)/(144
*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)) + sqrt(6)*sqrt(1/2 - x)/(144*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)), Tru
e))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=\frac {x}{6 \, \sqrt {-24 \, x^{2} + 6}} \]

[In]

integrate(1/(3-6*x)^(3/2)/(2+4*x)^(3/2),x, algorithm="maxima")

[Out]

1/6*x/sqrt(-24*x^2 + 6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (20) = 40\).

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.82 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=\frac {\sqrt {6} {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}}{288 \, \sqrt {2 \, x + 1}} - \frac {\sqrt {6} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1}}{144 \, {\left (2 \, x - 1\right )}} - \frac {\sqrt {6} \sqrt {2 \, x + 1}}{288 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}} \]

[In]

integrate(1/(3-6*x)^(3/2)/(2+4*x)^(3/2),x, algorithm="giac")

[Out]

1/288*sqrt(6)*(sqrt(2) - sqrt(-2*x + 1))/sqrt(2*x + 1) - 1/144*sqrt(6)*sqrt(2*x + 1)*sqrt(-2*x + 1)/(2*x - 1)
- 1/288*sqrt(6)*sqrt(2*x + 1)/(sqrt(2) - sqrt(-2*x + 1))

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx=-\frac {x\,\sqrt {3-6\,x}}{\sqrt {4\,x+2}\,\left (36\,x-18\right )} \]

[In]

int(1/((4*x + 2)^(3/2)*(3 - 6*x)^(3/2)),x)

[Out]

-(x*(3 - 6*x)^(1/2))/((4*x + 2)^(1/2)*(36*x - 18))