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 = 0.01856, 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 \[ \frac{x}{6 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}} \]

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])

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Rubi in Sympy [A]  time = 2.86188, size = 19, normalized size = 0.68 \[ \frac{x}{6 \sqrt{- 6 x + 3} \sqrt{4 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-6*x)**(3/2)/(2+4*x)**(3/2),x)

[Out]

x/(6*sqrt(-6*x + 3)*sqrt(4*x + 2))

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Mathematica [A]  time = 0.049143, 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 = 0.004, size = 28, normalized size = 1. \[ -{ \left ( -1+2\,x \right ) \left ( 1+2\,x \right ) x \left ( 3-6\,x \right ) ^{-{\frac{3}{2}}} \left ( 2+4\,x \right ) ^{-{\frac{3}{2}}}} \]

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]

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

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Maxima [A]  time = 1.33208, size = 16, normalized size = 0.57 \[ \frac{x}{6 \, \sqrt{-24 \, x^{2} + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x/sqrt(-24*x^2 + 6)

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Fricas [A]  time = 0.207193, size = 35, normalized size = 1.25 \[ -\frac{\sqrt{4 \, x + 2} x \sqrt{-6 \, x + 3}}{36 \,{\left (4 \, x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((4*x + 2)^(3/2)*(-6*x + 3)^(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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219771, size = 96, normalized size = 3.43 \[ -\frac{\sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}}{288 \, \sqrt{4 \, x + 2}} - \frac{\sqrt{6} \sqrt{4 \, x + 2} \sqrt{-4 \, x + 2}}{288 \,{\left (2 \, x - 1\right )}} + \frac{\sqrt{6} \sqrt{4 \, x + 2}}{288 \,{\left (\sqrt{-4 \, x + 2} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((4*x + 2)^(3/2)*(-6*x + 3)^(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) + 1/288*sqrt(6)*sqrt(4*x + 2)/(sqrt(-4*x + 2) - 2)