3.12.56 \(\int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx\) [1156]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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, 222} \begin {gather*} \frac {\text {ArcSin}(2 x)}{2 \sqrt {6}} \end {gather*}

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 222

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(13)=26\).
time = 0.04, size = 27, normalized size = 2.08 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {1-4 x^2}}{1+2 x}\right )}{\sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTan[Sqrt[1 - 4*x^2]/(1 + 2*x)]/Sqrt[6])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(36\) vs. \(2(9)=18\).
time = 0.16, size = 37, normalized size = 2.85

method	result	size
default	\(\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{12 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{12} \, \sqrt {6} \arcsin \left (2 \, x\right ) \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (9) = 18\).
time = 1.20, size = 28, normalized size = 2.15 \begin {gather*} -\frac {1}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \end {gather*}

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)

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Sympy [C] Result contains complex when optimal does not.
time = 1.00, size = 41, normalized size = 3.15 \begin {gather*} \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 {gather*}

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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (9) = 18\).
time = 1.05, size = 21, normalized size = 1.62 \begin {gather*} \frac {1}{6} \, \sqrt {3} \sqrt {2} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right ) \end {gather*}

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(3)*sqrt(2)*arcsin(1/2*sqrt(2)*sqrt(2*x + 1))

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Mupad [B]
time = 0.05, size = 40, normalized size = 3.08 \begin {gather*} -\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {24}\,\left (\sqrt {3}-\sqrt {3-6\,x}\right )}{6\,\left (\sqrt {2}-\sqrt {4\,x+2}\right )}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6^(1/2)*atan((24^(1/2)*(3^(1/2) - (3 - 6*x)^(1/2)))/(6*(2^(1/2) - (4*x + 2)^(1/2)))))/3

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