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

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

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Rubi [A]  time = 0.01, 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] /; F
reeQ[{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 = 0.02, size = 37, normalized size = 0.65 \begin {gather*} \frac {x \left (8 x^2-3\right )}{108 \sqrt {6-12 x} (2 x-1) (2 x+1)^{3/2}} \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))/(108*Sqrt[6 - 12*x]*(-1 + 2*x)*(1 + 2*x)^(3/2))

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IntegrateAlgebraic [B]  time = 0.85, size = 334, normalized size = 5.86 \begin {gather*} \frac {\left (\sqrt {2} \sqrt {2 x+1}-2\right )^9 \left (\frac {91 \left (4 x^2-4 x+1\right )}{294912 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^4}+\frac {35 \left (8 x^3-12 x^2+6 x-1\right )}{36864 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^6}+\frac {91 \left (16 x^4-32 x^3+24 x^2-8 x+1\right )}{73728 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^8}+\frac {5 \left (32 x^5-80 x^4+80 x^3-40 x^2+10 x-1\right )}{18432 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{10}}-\frac {64 x^6-192 x^5+240 x^4-160 x^3+60 x^2-12 x+1}{55296 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{12}}+\frac {5 (2 x-1)}{294912 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^2}-\frac {1}{3538944 \sqrt {3}}\right )}{(1-2 x)^{3/2} \left (-2 x+\sqrt {2} \sqrt {2 x+1}-1\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + Sqrt[2]*Sqrt[1 + 2*x])^9*(-1/3538944*1/Sqrt[3] - (1 - 12*x + 60*x^2 - 160*x^3 + 240*x^4 - 192*x^5 + 64*
x^6)/(55296*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^12) + (5*(-1 + 10*x - 40*x^2 + 80*x^3 - 80*x^4 + 32*x^5))/(18
432*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^10) + (91*(1 - 8*x + 24*x^2 - 32*x^3 + 16*x^4))/(73728*Sqrt[3]*(-2 +
Sqrt[2]*Sqrt[1 + 2*x])^8) + (35*(-1 + 6*x - 12*x^2 + 8*x^3))/(36864*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^6) +
(91*(1 - 4*x + 4*x^2))/(294912*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^4) + (5*(-1 + 2*x))/(294912*Sqrt[3]*(-2 +
Sqrt[2]*Sqrt[1 + 2*x])^2)))/((1 - 2*x)^(3/2)*(-1 - 2*x + Sqrt[2]*Sqrt[1 + 2*x])^3)

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fricas [A]  time = 1.31, 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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giac [B]  time = 1.02, size = 128, normalized size = 2.25 \begin {gather*} -\frac {1}{82944} \, \sqrt {6} {\left (\frac {{\left (\sqrt {-4 \, x + 2} - 2\right )}^{3}}{{\left (4 \, x + 2\right )}^{\frac {3}{2}}} + \frac {33 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}}{\sqrt {4 \, x + 2}}\right )} - \frac {{\left (4 \, \sqrt {6} {\left (2 \, x + 1\right )} - 9 \, \sqrt {6}\right )} \sqrt {4 \, x + 2} \sqrt {-4 \, x + 2}}{10368 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {6} {\left (4 \, x + 2\right )}^{\frac {3}{2}} {\left (\frac {33 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{2}}{2 \, x + 1} + 2\right )}}{165888 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{3}} \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="giac")

[Out]

-1/82944*sqrt(6)*((sqrt(-4*x + 2) - 2)^3/(4*x + 2)^(3/2) + 33*(sqrt(-4*x + 2) - 2)/sqrt(4*x + 2)) - 1/10368*(4
*sqrt(6)*(2*x + 1) - 9*sqrt(6))*sqrt(4*x + 2)*sqrt(-4*x + 2)/(2*x - 1)^2 + 1/165888*sqrt(6)*(4*x + 2)^(3/2)*(3
3*(sqrt(-4*x + 2) - 2)^2/(2*x + 1) + 2)/(sqrt(-4*x + 2) - 2)^3

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maple [A]  time = 0.00, size = 35, normalized size = 0.61 \begin {gather*} \frac {\left (2 x -1\right ) \left (2 x +1\right ) \left (8 x^{2}-3\right ) x}{3 \left (-6 x +3\right )^{\frac {5}{2}} \left (4 x +2\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*x-1)*(2*x+1)*x*(8*x^2-3)/(-6*x+3)^(5/2)/(4*x+2)^(5/2)

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maxima [A]  time = 1.28, 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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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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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