3.1157 \(\int \frac{1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{x}{6 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}} \]

[Out]

x/(6*Sqrt[6]*Sqrt[1 - 2*x]*Sqrt[1 + 2*x])

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Rubi [A]  time = 0.0019949, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {39} \[ \frac{x}{6 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}} \]

Antiderivative was successfully verified.

[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*} \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}}\\ \end{align*}

Mathematica [A]  time = 0.0156117, size = 16, normalized size = 0.57 \[ \frac{x}{6 \sqrt{6-24 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(6*Sqrt[6 - 24*x^2])

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Maple [A]  time = 0.003, size = 28, normalized size = 1. \begin{align*} -{ \left ( 2\,x-1 \right ) \left ( 1+2\,x \right ) x \left ( 3-6\,x \right ) ^{-{\frac{3}{2}}} \left ( 2+4\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.963899, size = 16, normalized size = 0.57 \begin{align*} \frac{x}{6 \, \sqrt{-24 \, x^{2} + 6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x/sqrt(-24*x^2 + 6)

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Fricas [A]  time = 1.55657, size = 68, normalized size = 2.43 \begin{align*} -\frac{\sqrt{4 \, x + 2} x \sqrt{-6 \, x + 3}}{36 \,{\left (4 \, x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.05424, size = 96, normalized size = 3.43 \begin{align*} -\frac{\sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}}{288 \, \sqrt{4 \, x + 2}} - \frac{\sqrt{6} \sqrt{4 \, x + 2} \sqrt{-4 \, x + 2}}{288 \,{\left (2 \, x - 1\right )}} + \frac{\sqrt{6} \sqrt{4 \, x + 2}}{288 \,{\left (\sqrt{-4 \, x + 2} - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*sqrt(6)*(sqrt(-4*x + 2) - 2)/sqrt(4*x + 2) - 1/288*sqrt(6)*sqrt(4*x + 2)*sqrt(-4*x + 2)/(2*x - 1) + 1/2
88*sqrt(6)*sqrt(4*x + 2)/(sqrt(-4*x + 2) - 2)