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

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

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.57 \begin {gather*} \frac {x}{6 \sqrt {6-24 x^2}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.73, size = 80, normalized size = 2.86 \begin {gather*} \frac {x (2 x+3)-2 \sqrt {2} x \sqrt {2 x+1}}{6 \sqrt {3} \sqrt {1-2 x} (-8 x-4)+6 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1} (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*x*Sqrt[1 + 2*x] + x*(3 + 2*x))/(6*Sqrt[3]*(-4 - 8*x)*Sqrt[1 - 2*x] + 6*Sqrt[6]*Sqrt[1 - 2*x]*Sqrt[
1 + 2*x]*(3 + 2*x))

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fricas [A]  time = 1.39, size = 26, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {4 \, x + 2} x \sqrt {-6 \, x + 3}}{36 \, {\left (4 \, x^{2} - 1\right )}} \end {gather*}

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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giac [B]  time = 1.06, size = 71, normalized size = 2.54 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\left (2 x -1\right ) \left (2 x +1\right ) x}{\left (-6 x +3\right )^{\frac {3}{2}} \left (4 x +2\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 12, normalized size = 0.43 \begin {gather*} \frac {x}{6 \, \sqrt {-24 \, x^{2} + 6}} \end {gather*}

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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mupad [B]  time = 0.46, size = 24, normalized size = 0.86 \begin {gather*} -\frac {x\,\sqrt {3-6\,x}}{\sqrt {4\,x+2}\,\left (36\,x-18\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(3 - 6*x)^(1/2))/((4*x + 2)^(1/2)*(36*x - 18))

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sympy [B]  time = 85.28, size = 156, normalized size = 5.57 \begin {gather*} \begin {cases} - \frac {2 \sqrt {6} i \sqrt {x - \frac {1}{2}} \left (x + \frac {1}{2}\right )}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} + \frac {\sqrt {6} i \sqrt {x - \frac {1}{2}}}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} & \text {for}\: \left |{x + \frac {1}{2}}\right | > 1 \\- \frac {2 \sqrt {6} \sqrt {\frac {1}{2} - x} \left (x + \frac {1}{2}\right )}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} + \frac {\sqrt {6} \sqrt {\frac {1}{2} - x}}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} & \text {otherwise} \end {cases} \end {gather*}

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]

Piecewise((-2*sqrt(6)*I*sqrt(x - 1/2)*(x + 1/2)/(144*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)) + sqrt(6)*I*sqrt(x
- 1/2)/(144*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)), Abs(x + 1/2) > 1), (-2*sqrt(6)*sqrt(1/2 - x)*(x + 1/2)/(144
*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)) + sqrt(6)*sqrt(1/2 - x)/(144*(x + 1/2)**(3/2) - 144*sqrt(x + 1/2)), Tru
e))

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