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*Sq
rt[6]*Sqrt[1 - 2*x]*Sqrt[1 + 2*x])

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Rubi [A]  time = 0.0110413, 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, Rules used = {40, 39} \[ \frac{x}{405 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}}+\frac{x}{810 \sqrt{6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac{x}{1080 \sqrt{6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \]

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*Sq
rt[6]*Sqrt[1 - 2*x]*Sqrt[1 + 2*x])

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]

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)^{7/2} (2+4 x)^{7/2}} \, dx &=\frac{x}{1080 \sqrt{6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac{2}{15} \int \frac{1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx\\ &=\frac{x}{1080 \sqrt{6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac{x}{810 \sqrt{6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac{2}{135} \int \frac{1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\\ &=\frac{x}{1080 \sqrt{6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac{x}{810 \sqrt{6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac{x}{405 \sqrt{6} \sqrt{1-2 x} \sqrt{1+2 x}}\\ \end{align*}

Mathematica [A]  time = 0.0304035, 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 = 0.002, size = 40, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,x-1 \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}}}} \end{align*}

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*(2*x-1)*(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 = 1.01568, size = 50, normalized size = 0.59 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)^(7/2)/(4*x+2)^(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 = 1.54768, size = 130, normalized size = 1.53 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)^(7/2)/(4*x+2)^(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. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.1112, size = 248, normalized size = 2.92 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-6*x)^(7/2)/(4*x+2)^(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*(sqrt(-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