3.1159 \(\int \frac{1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{x}{405 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}}+\frac{x}{810 \sqrt{6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac{x}{1080 \sqrt{6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \]

[Out]

x/(1080*Sqrt[6]*(1 - 2*x)^(5/2)*(1 + 2*x)^(5/2)) + x/(810*Sqrt[6]*(1 - 2*x)^(3/2
)*(1 + 2*x)^(3/2)) + x/(405*Sqrt[6]*Sqrt[1 - 2*x]*Sqrt[1 + 2*x])

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Rubi [A]  time = 0.0585249, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x}{405 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}}+\frac{x}{810 \sqrt{6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac{x}{1080 \sqrt{6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((3 - 6*x)^(7/2)*(2 + 4*x)^(7/2)),x]

[Out]

x/(1080*Sqrt[6]*(1 - 2*x)^(5/2)*(1 + 2*x)^(5/2)) + x/(810*Sqrt[6]*(1 - 2*x)^(3/2
)*(1 + 2*x)^(3/2)) + x/(405*Sqrt[6]*Sqrt[1 - 2*x]*Sqrt[1 + 2*x])

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Rubi in Sympy [A]  time = 6.82881, size = 60, normalized size = 0.71 \[ \frac{x}{405 \sqrt{- 6 x + 3} \sqrt{4 x + 2}} + \frac{x}{135 \left (- 6 x + 3\right )^{\frac{3}{2}} \left (4 x + 2\right )^{\frac{3}{2}}} + \frac{x}{30 \left (- 6 x + 3\right )^{\frac{5}{2}} \left (4 x + 2\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-6*x)**(7/2)/(2+4*x)**(7/2),x)

[Out]

x/(405*sqrt(-6*x + 3)*sqrt(4*x + 2)) + x/(135*(-6*x + 3)**(3/2)*(4*x + 2)**(3/2)
) + x/(30*(-6*x + 3)**(5/2)*(4*x + 2)**(5/2))

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Mathematica [A]  time = 0.097901, size = 42, normalized size = 0.49 \[ \frac{x \left (128 x^4-80 x^2+15\right )}{3240 \sqrt{6-12 x} (1-2 x)^2 (2 x+1)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((3 - 6*x)^(7/2)*(2 + 4*x)^(7/2)),x]

[Out]

(x*(15 - 80*x^2 + 128*x^4))/(3240*Sqrt[6 - 12*x]*(1 - 2*x)^2*(1 + 2*x)^(5/2))

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Maple [A]  time = 0.006, size = 40, normalized size = 0.5 \[ -{\frac{ \left ( -1+2\,x \right ) \left ( 1+2\,x \right ) x \left ( 128\,{x}^{4}-80\,{x}^{2}+15 \right ) }{15} \left ( 3-6\,x \right ) ^{-{\frac{7}{2}}} \left ( 2+4\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-6*x)^(7/2)/(2+4*x)^(7/2),x)

[Out]

-1/15*(-1+2*x)*(1+2*x)*x*(128*x^4-80*x^2+15)/(3-6*x)^(7/2)/(2+4*x)^(7/2)

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Maxima [A]  time = 1.31932, size = 50, normalized size = 0.59 \[ \frac{x}{405 \, \sqrt{-24 \, x^{2} + 6}} + \frac{x}{135 \,{\left (-24 \, x^{2} + 6\right )}^{\frac{3}{2}}} + \frac{x}{30 \,{\left (-24 \, x^{2} + 6\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((4*x + 2)^(7/2)*(-6*x + 3)^(7/2)),x, algorithm="maxima")

[Out]

1/405*x/sqrt(-24*x^2 + 6) + 1/135*x/(-24*x^2 + 6)^(3/2) + 1/30*x/(-24*x^2 + 6)^(
5/2)

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Fricas [A]  time = 0.206792, size = 66, normalized size = 0.78 \[ -\frac{{\left (128 \, x^{5} - 80 \, x^{3} + 15 \, x\right )} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{19440 \,{\left (64 \, x^{6} - 48 \, x^{4} + 12 \, x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((4*x + 2)^(7/2)*(-6*x + 3)^(7/2)),x, algorithm="fricas")

[Out]

-1/19440*(128*x^5 - 80*x^3 + 15*x)*sqrt(4*x + 2)*sqrt(-6*x + 3)/(64*x^6 - 48*x^4
+ 12*x^2 - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-6*x)**(7/2)/(2+4*x)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230325, size = 248, normalized size = 2.92 \[ -\frac{\sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}^{5}}{13271040 \,{\left (4 \, x + 2\right )}^{\frac{5}{2}}} - \frac{17 \, \sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}^{3}}{7962624 \,{\left (4 \, x + 2\right )}^{\frac{3}{2}}} - \frac{71 \, \sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}}{1327104 \, \sqrt{4 \, x + 2}} - \frac{{\left ({\left (64 \, \sqrt{6}{\left (2 \, x + 1\right )} - 275 \, \sqrt{6}\right )}{\left (2 \, x + 1\right )} + 300 \, \sqrt{6}\right )} \sqrt{4 \, x + 2} \sqrt{-4 \, x + 2}}{1244160 \,{\left (2 \, x - 1\right )}^{3}} + \frac{\sqrt{6}{\left (\frac{1065 \,{\left (\sqrt{-4 \, x + 2} - 2\right )}^{4}}{{\left (2 \, x + 1\right )}^{2}} + \frac{85 \,{\left (\sqrt{-4 \, x + 2} - 2\right )}^{2}}{2 \, x + 1} + 6\right )}{\left (4 \, x + 2\right )}^{\frac{5}{2}}}{79626240 \,{\left (\sqrt{-4 \, x + 2} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((4*x + 2)^(7/2)*(-6*x + 3)^(7/2)),x, algorithm="giac")

[Out]

-1/13271040*sqrt(6)*(sqrt(-4*x + 2) - 2)^5/(4*x + 2)^(5/2) - 17/7962624*sqrt(6)*
(sqrt(-4*x + 2) - 2)^3/(4*x + 2)^(3/2) - 71/1327104*sqrt(6)*(sqrt(-4*x + 2) - 2)
/sqrt(4*x + 2) - 1/1244160*((64*sqrt(6)*(2*x + 1) - 275*sqrt(6))*(2*x + 1) + 300
*sqrt(6))*sqrt(4*x + 2)*sqrt(-4*x + 2)/(2*x - 1)^3 + 1/79626240*sqrt(6)*(1065*(s
qrt(-4*x + 2) - 2)^4/(2*x + 1)^2 + 85*(sqrt(-4*x + 2) - 2)^2/(2*x + 1) + 6)*(4*x
+ 2)^(5/2)/(sqrt(-4*x + 2) - 2)^5