3.1156 \(\int \frac{1}{\sqrt{3-6 x} \sqrt{2+4 x}} \, dx\)

Optimal. Leaf size=13 \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]

[Out]

ArcSin[2*x]/(2*Sqrt[6])

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Rubi [A]  time = 0.0177504, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

ArcSin[2*x]/(2*Sqrt[6])

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Rubi in Sympy [A]  time = 2.95253, size = 10, normalized size = 0.77 \[ \frac{\sqrt{6} \operatorname{asin}{\left (2 x \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(6)*asin(2*x)/12

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Mathematica [A]  time = 0.0252982, size = 13, normalized size = 1. \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

ArcSin[2*x]/(2*Sqrt[6])

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Maple [B]  time = 0.005, size = 37, normalized size = 2.9 \[{\frac{\arcsin \left ( 2\,x \right ) \sqrt{6}}{12}\sqrt{ \left ( 2+4\,x \right ) \left ( 3-6\,x \right ) }{\frac{1}{\sqrt{3-6\,x}}}{\frac{1}{\sqrt{2+4\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((2+4*x)*(3-6*x))^(1/2)/(2+4*x)^(1/2)/(3-6*x)^(1/2)*6^(1/2)*arcsin(2*x)

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Maxima [A]  time = 1.49726, size = 12, normalized size = 0.92 \[ \frac{1}{12} \, \sqrt{6} \arcsin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(6)*arcsin(2*x)

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Fricas [A]  time = 0.208616, size = 38, normalized size = 2.92 \[ -\frac{1}{12} \, \sqrt{6} \arctan \left (\frac{\sqrt{6} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{12 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(6)*arctan(1/12*sqrt(6)*sqrt(4*x + 2)*sqrt(-6*x + 3)/x)

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Sympy [A]  time = 19.3453, size = 41, normalized size = 3.15 \[ \begin{cases} - \frac{\sqrt{6} i \operatorname{acosh}{\left (\sqrt{x + \frac{1}{2}} \right )}}{6} & \text{for}\: \left |{x + \frac{1}{2}}\right | > 1 \\\frac{\sqrt{6} \operatorname{asin}{\left (\sqrt{x + \frac{1}{2}} \right )}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(6)*I*acosh(sqrt(x + 1/2))/6, Abs(x + 1/2) > 1), (sqrt(6)*asin(s
qrt(x + 1/2))/6, True))

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GIAC/XCAS [A]  time = 0.215817, size = 20, normalized size = 1.54 \[ \frac{1}{6} \, \sqrt{6} \arcsin \left (\frac{1}{2} \, \sqrt{4 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(6)*arcsin(1/2*sqrt(4*x + 2))