3.1158 \(\int \frac{1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0062288, antiderivative size = 57, normalized size of antiderivative = 1., 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} \[ \frac{x}{54 \sqrt{6} \sqrt{1-2 x} \sqrt{2 x+1}}+\frac{x}{108 \sqrt{6} (1-2 x)^{3/2} (2 x+1)^{3/2}} \]

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 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)^{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*}

Mathematica [A]  time = 0.0243268, size = 37, normalized size = 0.65 \[ \frac{x \left (8 x^2-3\right )}{108 \sqrt{6-12 x} (2 x-1) (2 x+1)^{3/2}} \]

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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Maple [A]  time = 0.004, size = 35, normalized size = 0.6 \begin{align*}{\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.968932, size = 34, normalized size = 0.6 \begin{align*} \frac{x}{54 \, \sqrt{-24 \, x^{2} + 6}} + \frac{x}{18 \,{\left (-24 \, x^{2} + 6\right )}^{\frac{3}{2}}} \end{align*}

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 = 1.55853, size = 97, normalized size = 1.7 \begin{align*} -\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{align*}

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(-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)**(5/2)/(4*x+2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.08276, size = 174, normalized size = 3.05 \begin{align*} -\frac{\sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}^{3}}{82944 \,{\left (4 \, x + 2\right )}^{\frac{3}{2}}} - \frac{11 \, \sqrt{6}{\left (\sqrt{-4 \, x + 2} - 2\right )}}{27648 \, \sqrt{4 \, x + 2}} - \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{align*}

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) - 11/27648*sqrt(6)*(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)*(33*(sqrt(-4*x + 2) - 2)^2/(2*x + 1) + 2)/(sqrt(-4*x + 2) - 2)^3