3.12.58 \(\int \frac {1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx\) [1158]

Optimal. Leaf size=57 \[ \frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{54 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}} \]

[Out]

1/648*x/(1-2*x)^(3/2)/(1+2*x)^(3/2)*6^(1/2)+1/324*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 = 57, 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 = {40, 39} \begin {gather*} \frac {x}{54 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((3 - 6*x)^(5/2)*(2 + 4*x)^(5/2)),x]

[Out]

x/(108*Sqrt[6]*(1 - 2*x)^(3/2)*(1 + 2*x)^(3/2)) + x/(54*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]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx &=\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {1}{9} \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\\ &=\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{54 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}}\\ \end {align*}

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Mathematica [A]
time = 0.80, size = 107, normalized size = 1.88 \begin {gather*} \frac {x \left (-3+8 x^2\right ) \left (99+8 x^3-70 \sqrt {2+4 x}+x \left (246-104 \sqrt {2+4 x}\right )+x^2 \left (132-24 \sqrt {2+4 x}\right )\right )}{54 \sqrt {3-6 x} (-1+2 x) \left (-4+3 \sqrt {2+4 x}+2 x \left (-4+\sqrt {2+4 x}\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((3 - 6*x)^(5/2)*(2 + 4*x)^(5/2)),x]

[Out]

(x*(-3 + 8*x^2)*(99 + 8*x^3 - 70*Sqrt[2 + 4*x] + x*(246 - 104*Sqrt[2 + 4*x]) + x^2*(132 - 24*Sqrt[2 + 4*x])))/
(54*Sqrt[3 - 6*x]*(-1 + 2*x)*(-4 + 3*Sqrt[2 + 4*x] + 2*x*(-4 + Sqrt[2 + 4*x]))^3)

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((3 - 6*x)^(5/2)*(2 + 4*x)^(5/2)),x]')

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Maple [A]
time = 0.15, size = 66, normalized size = 1.16

method	result	size
gosper	\(\frac {\left (2 x -1\right ) \left (1+2 x \right ) x \left (8 x^{2}-3\right )}{3 \left (3-6 x \right )^{\frac {5}{2}} \left (2+4 x \right )^{\frac {5}{2}}}\)	\(35\)
default	\(\frac {1}{36 \left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {3}{2}}}+\frac {1}{36 \sqrt {3-6 x}\, \left (2+4 x \right )^{\frac {3}{2}}}-\frac {\sqrt {3-6 x}}{162 \left (2+4 x \right )^{\frac {3}{2}}}-\frac {\sqrt {3-6 x}}{324 \sqrt {2+4 x}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3-6*x)^(5/2)/(2+4*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/36/(3-6*x)^(3/2)/(2+4*x)^(3/2)+1/36/(3-6*x)^(1/2)/(2+4*x)^(3/2)-1/162/(2+4*x)^(3/2)*(3-6*x)^(1/2)-1/324/(2+4
*x)^(1/2)*(3-6*x)^(1/2)

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Maxima [A]
time = 0.26, size = 25, normalized size = 0.44 \begin {gather*} \frac {x}{54 \, \sqrt {-24 \, x^{2} + 6}} + \frac {x}{18 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)^(5/2)/(4*x+2)^(5/2),x, algorithm="maxima")

[Out]

1/54*x/sqrt(-24*x^2 + 6) + 1/18*x/(-24*x^2 + 6)^(3/2)

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Fricas [A]
time = 0.29, size = 39, normalized size = 0.68 \begin {gather*} -\frac {{\left (8 \, x^{3} - 3 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{648 \, {\left (16 \, x^{4} - 8 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)^(5/2)/(4*x+2)^(5/2),x, algorithm="fricas")

[Out]

-1/648*(8*x^3 - 3*x)*sqrt(4*x + 2)*sqrt(-6*x + 3)/(16*x^4 - 8*x^2 + 1)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)**(5/2)/(4*x+2)**(5/2),x)

[Out]

Exception raised: SystemError

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (41) = 82\).
time = 0.01, size = 232, normalized size = 4.07 \begin {gather*} \frac {-\frac {-7077888 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{3}+\frac {116785152 \left (-2 \sqrt {2 x+1}+2 \sqrt {2}\right )}{\sqrt {-2 x+1}}}{97844723712}-\frac {33 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{2}+1}{13824 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{3}}-\frac {2 \left (\frac 1{192}-\frac {1}{432} \sqrt {-2 x+1} \sqrt {-2 x+1}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}}{\left (2 x+1\right )^{2}}}{\sqrt {3} \sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)^(5/2)/(4*x+2)^(5/2),x)

[Out]

-1/82944*sqrt(3)*sqrt(2)*((sqrt(2) - sqrt(2*x + 1))^3/(-2*x + 1)^(3/2) + 33*(sqrt(2) - sqrt(2*x + 1))/sqrt(-2*
x + 1) + 16*(8*x + 5)*sqrt(-2*x + 1)/(2*x + 1)^(3/2) + (-2*x + 1)^(3/2)*(33*(sqrt(2) - sqrt(2*x + 1))^2/(2*x -
1) - 1)/(sqrt(2) - sqrt(2*x + 1))^3)

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Mupad [B]
time = 0.31, size = 49, normalized size = 0.86 \begin {gather*} -\frac {3\,x\,\sqrt {3-6\,x}-8\,x^3\,\sqrt {3-6\,x}}{\sqrt {4\,x+2}\,\left (-2592\,x^3+1296\,x^2+648\,x-324\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((4*x + 2)^(5/2)*(3 - 6*x)^(5/2)),x)

[Out]

-(3*x*(3 - 6*x)^(1/2) - 8*x^3*(3 - 6*x)^(1/2))/((4*x + 2)^(1/2)*(648*x + 1296*x^2 - 2592*x^3 - 324))

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