\(\int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx\) [1159]

Optimal result

Integrand size = 19, antiderivative size = 85 \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {40, 39} \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=\frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \]

[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] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=\frac {x \left (15-80 x^2+128 x^4\right )}{3240 \sqrt {6} \left (1-4 x^2\right )^{5/2}} \]

[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]*(1 - 4*x^2)^(5/2))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.47

method	result	size
gosper	\(-\frac {\left (-1+2 x \right ) \left (1+2 x \right ) x \left (128 x^{4}-80 x^{2}+15\right )}{15 \left (3-6 x \right )^{\frac {7}{2}} \left (2+4 x \right )^{\frac {7}{2}}}\)	\(40\)
default	\(\frac {1}{60 \left (3-6 x \right )^{\frac {5}{2}} \left (2+4 x \right )^{\frac {5}{2}}}+\frac {1}{108 \left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {5}{2}}}+\frac {1}{81 \sqrt {3-6 x}\, \left (2+4 x \right )^{\frac {5}{2}}}-\frac {\sqrt {3-6 x}}{405 \left (2+4 x \right )^{\frac {5}{2}}}-\frac {\sqrt {3-6 x}}{1215 \left (2+4 x \right )^{\frac {3}{2}}}-\frac {\sqrt {3-6 x}}{2430 \sqrt {2+4 x}}\)	\(98\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=-\frac {{\left (128 \, x^{5} - 80 \, x^{3} + 15 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{19440 \, {\left (64 \, x^{6} - 48 \, x^{4} + 12 \, x^{2} - 1\right )}} \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=\frac {x}{405 \, \sqrt {-24 \, x^{2} + 6}} + \frac {x}{135 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}}} + \frac {x}{30 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}}} \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (61) = 122\).

Time = 0.34 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.45 \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=\frac {\sqrt {6} {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{5}}{13271040 \, {\left (2 \, x + 1\right )}^{\frac {5}{2}}} + \frac {17 \, \sqrt {6} {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{3}}{7962624 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} + \frac {71 \, \sqrt {6} {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}}{1327104 \, \sqrt {2 \, x + 1}} - \frac {{\left ({\left (64 \, \sqrt {6} {\left (2 \, x + 1\right )} - 275 \, \sqrt {6}\right )} {\left (2 \, x + 1\right )} + 300 \, \sqrt {6}\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1}}{622080 \, {\left (2 \, x - 1\right )}^{3}} - \frac {\sqrt {6} {\left (\frac {2130 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{4}}{{\left (2 \, x + 1\right )}^{2}} + \frac {85 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{2}}{2 \, x + 1} + 3\right )} {\left (2 \, x + 1\right )}^{\frac {5}{2}}}{39813120 \, {\left (\sqrt {2} - \sqrt {-2 \, x + 1}\right )}^{5}} \]

[In]

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

[Out]

1/13271040*sqrt(6)*(sqrt(2) - sqrt(-2*x + 1))^5/(2*x + 1)^(5/2) + 17/7962624*sqrt(6)*(sqrt(2) - sqrt(-2*x + 1)
)^3/(2*x + 1)^(3/2) + 71/1327104*sqrt(6)*(sqrt(2) - sqrt(-2*x + 1))/sqrt(2*x + 1) - 1/622080*((64*sqrt(6)*(2*x
+ 1) - 275*sqrt(6))*(2*x + 1) + 300*sqrt(6))*sqrt(2*x + 1)*sqrt(-2*x + 1)/(2*x - 1)^3 - 1/39813120*sqrt(6)*(2
130*(sqrt(2) - sqrt(-2*x + 1))^4/(2*x + 1)^2 + 85*(sqrt(2) - sqrt(-2*x + 1))^2/(2*x + 1) + 3)*(2*x + 1)^(5/2)/
(sqrt(2) - sqrt(-2*x + 1))^5

Mupad [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx=-\frac {15\,x\,\sqrt {3-6\,x}-80\,x^3\,\sqrt {3-6\,x}+128\,x^5\,\sqrt {3-6\,x}}{\left (\left (6\,x-3\right )\,\left (240\,x+360\right )+1440\right )\,\sqrt {4\,x+2}\,{\left (6\,x-3\right )}^3} \]

[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)