3.12.57 \(\int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\) [1157]

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

[Out]

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

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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 [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(28)=56\).
time = 0.69, size = 59, normalized size = 2.11 \begin {gather*} \frac {x \left (3+2 x-2 \sqrt {2+4 x}\right )}{6 \sqrt {3-6 x} \left (-4+3 \sqrt {2+4 x}+2 x \left (-4+\sqrt {2+4 x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 37.42, size = 90, normalized size = 3.21 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-\frac {I}{36}\right ) \sqrt {6} x}{\sqrt {-1+2 x} \sqrt {1+2 x}},\text {Abs}\left [\frac {1}{2}+x\right ]>1\right \}\right \},\frac {-2 \sqrt {6} \left (\frac {1}{2}+x\right ) \sqrt {\frac {1}{2}-x}}{-144 \sqrt {\frac {1}{2}+x}+144 \left (\frac {1}{2}+x\right )^{\frac {3}{2}}}+\frac {\sqrt {6} \sqrt {\frac {1}{2}-x}}{-144 \sqrt {\frac {1}{2}+x}+144 \left (\frac {1}{2}+x\right )^{\frac {3}{2}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{(-I / 36) Sqrt[6] x / (Sqrt[-1 + 2 x] Sqrt[1 + 2 x]), Abs[1 / 2 + x] > 1}}, -2 Sqrt[6] (1 / 2 + x)
Sqrt[1 / 2 - x] / (-144 Sqrt[1 / 2 + x] + 144 (1 / 2 + x) ^ (3 / 2)) + Sqrt[6] Sqrt[1 / 2 - x] / (-144 Sqrt[1
/ 2 + x] + 144 (1 / 2 + x) ^ (3 / 2))]

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Maple [A]
time = 0.15, size = 34, normalized size = 1.21

method	result	size
gosper	\(-\frac {\left (2 x -1\right ) \left (1+2 x \right ) x}{\left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {3}{2}}}\)	\(28\)
default	\(\frac {1}{12 \sqrt {3-6 x}\, \sqrt {2+4 x}}-\frac {\sqrt {3-6 x}}{36 \sqrt {2+4 x}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12/(3-6*x)^(1/2)/(2+4*x)^(1/2)-1/36/(2+4*x)^(1/2)*(3-6*x)^(1/2)

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Maxima [A]
time = 0.27, 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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Fricas [A]
time = 0.29, 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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Sympy [C] Result contains complex when optimal does not.
time = 44.74, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
time = 0.00, size = 112, normalized size = 4.00 \begin {gather*} \frac {\frac {\sqrt {-2 x+1}}{24 \left (-2 \sqrt {2 x+1}+2 \sqrt {2}\right )}-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{96 \sqrt {-2 x+1}}-\frac {\sqrt {-2 x+1} \sqrt {2 x+1}}{24 \left (2 x+1\right )}}{\sqrt {3} \sqrt {2}} \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]

-1/288*sqrt(3)*sqrt(2)*((sqrt(2) - sqrt(2*x + 1))/sqrt(-2*x + 1) + 2*sqrt(-2*x + 1)/sqrt(2*x + 1) - sqrt(-2*x
+ 1)/(sqrt(2) - sqrt(2*x + 1)))

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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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