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]

1/6480*x/(1-2*x)^(5/2)/(1+2*x)^(5/2)*6^(1/2)+1/4860*x/(1-2*x)^(3/2)/(1+2*x)^(3/2)*6^(1/2)+1/2430*x*6^(1/2)/(1-
2*x)^(1/2)/(1+2*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 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)^{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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.42, size = 49, normalized size = 0.58 \[ -\frac {{\left (128 \, x^{5} - 80 \, x^{3} + 15 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{19440 \, {\left (64 \, x^{6} - 48 \, x^{4} + 12 \, x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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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giac [B]  time = 1.02, size = 181, normalized size = 2.13 \[ -\frac {1}{39813120} \, \sqrt {6} {\left (\frac {3 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{5}}{{\left (4 \, x + 2\right )}^{\frac {5}{2}}} + \frac {85 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{3}}{{\left (4 \, x + 2\right )}^{\frac {3}{2}}} + \frac {2130 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}}{\sqrt {4 \, x + 2}}\right )} - \frac {{\left ({\left (64 \, \sqrt {6} {\left (2 \, x + 1\right )} - 275 \, \sqrt {6}\right )} {\left (2 \, x + 1\right )} + 300 \, \sqrt {6}\right )} \sqrt {4 \, x + 2} \sqrt {-4 \, x + 2}}{1244160 \, {\left (2 \, x - 1\right )}^{3}} + \frac {\sqrt {6} {\left (\frac {1065 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{4}}{{\left (2 \, x + 1\right )}^{2}} + \frac {85 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{2}}{2 \, x + 1} + 6\right )} {\left (4 \, x + 2\right )}^{\frac {5}{2}}}{79626240 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/39813120*sqrt(6)*(3*(sqrt(-4*x + 2) - 2)^5/(4*x + 2)^(5/2) + 85*(sqrt(-4*x + 2) - 2)^3/(4*x + 2)^(3/2) + 21
30*(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 + 8
5*(sqrt(-4*x + 2) - 2)^2/(2*x + 1) + 6)*(4*x + 2)^(5/2)/(sqrt(-4*x + 2) - 2)^5

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maple [A]  time = 0.00, size = 40, normalized size = 0.47 \[ -\frac {\left (2 x -1\right ) \left (2 x +1\right ) \left (128 x^{4}-80 x^{2}+15\right ) x}{15 \left (-6 x +3\right )^{\frac {7}{2}} \left (4 x +2\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 37, normalized size = 0.44 \[ \frac {x}{405 \, \sqrt {-24 \, x^{2} + 6}} + \frac {x}{135 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}}} + \frac {x}{30 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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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mupad [B]  time = 0.45, size = 66, normalized size = 0.78 \[ -\frac {15\,x\,\sqrt {3-6\,x}-80\,x^3\,\sqrt {3-6\,x}+128\,x^5\,\sqrt {3-6\,x}}{\left (\left (6\,x-3\right )\,\left (240\,x+360\right )+1440\right )\,\sqrt {4\,x+2}\,{\left (6\,x-3\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(15*x*(3 - 6*x)^(1/2) - 80*x^3*(3 - 6*x)^(1/2) + 128*x^5*(3 - 6*x)^(1/2))/(((6*x - 3)*(240*x + 360) + 1440)*(
4*x + 2)^(1/2)*(6*x - 3)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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