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])

________________________________________________________________________________________

Rubi [A]  time = 0.0025095, 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, Rules used = {41, 216} \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-6 x} \sqrt{2+4 x}} \, dx &=\int \frac{1}{\sqrt{6-24 x^2}} \, dx\\ &=\frac{\sin ^{-1}(2 x)}{2 \sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.0151583, 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])

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 37, normalized size = 2.9 \begin{align*}{\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}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.43274, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{12} \, \sqrt{6} \arcsin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.46493, size = 90, normalized size = 6.92 \begin{align*} -\frac{1}{12} \, \sqrt{6} \arctan \left (\frac{\sqrt{6} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{12 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 5.28419, size = 41, normalized size = 3.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.07293, size = 20, normalized size = 1.54 \begin{align*} \frac{1}{6} \, \sqrt{6} \arcsin \left (\frac{1}{2} \, \sqrt{4 \, x + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(6)*arcsin(1/2*sqrt(4*x + 2))