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

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

[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)

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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} \begin {gather*} \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}} \end {gather*}

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] /;
FreeQ[{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 = 1.05, size = 149, normalized size = 1.75 \begin {gather*} \frac {x \left (15-80 x^2+128 x^4\right ) \left (3363+32 x^5-2378 \sqrt {2+4 x}+x \left (13930-7472 \sqrt {2+4 x}\right )-80 x^4 \left (-19+2 \sqrt {2+4 x}\right )-80 x^3 \left (-121+28 \sqrt {2+4 x}\right )-8 x^2 \left (-2375+894 \sqrt {2+4 x}\right )\right )}{810 \sqrt {3-6 x} (1-2 x)^2 \left (-4+3 \sqrt {2+4 x}+2 x \left (-4+\sqrt {2+4 x}\right )\right )^5} \end {gather*}

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)*(3363 + 32*x^5 - 2378*Sqrt[2 + 4*x] + x*(13930 - 7472*Sqrt[2 + 4*x]) - 80*x^4*(-19
+ 2*Sqrt[2 + 4*x]) - 80*x^3*(-121 + 28*Sqrt[2 + 4*x]) - 8*x^2*(-2375 + 894*Sqrt[2 + 4*x])))/(810*Sqrt[3 - 6*x]
*(1 - 2*x)^2*(-4 + 3*Sqrt[2 + 4*x] + 2*x*(-4 + Sqrt[2 + 4*x]))^5)

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [A]
time = 0.14, size = 98, normalized size = 1.15

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60/(3-6*x)^(5/2)/(2+4*x)^(5/2)+1/108/(3-6*x)^(3/2)/(2+4*x)^(5/2)+1/81/(3-6*x)^(1/2)/(2+4*x)^(5/2)-1/405/(2+4
*x)^(5/2)*(3-6*x)^(1/2)-1/1215/(2+4*x)^(3/2)*(3-6*x)^(1/2)-1/2430/(2+4*x)^(1/2)*(3-6*x)^(1/2)

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Maxima [A]
time = 0.27, size = 37, normalized size = 0.44 \begin {gather*} \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 {gather*}

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 = 0.29, size = 49, normalized size = 0.58 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (61) = 122\).
time = 0.02, size = 339, normalized size = 3.99 \begin {gather*} \frac {\frac {\frac {1}{5}\cdot 38294359833110460235776 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{5}+217001372387625941336064 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{3}-\frac {2718899548150842676740096 \left (-2 \sqrt {2 x+1}+2 \sqrt {2}\right )}{\sqrt {-2 x+1}}}{16940199370653408073579757568}+\frac {-2130 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{4}-85 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{2}-3}{6635520 \left (-\frac {-2 \sqrt {2 x+1}+2 \sqrt {2}}{2 \sqrt {-2 x+1}}\right )^{5}}+\frac {2 \left (\left (\frac {55}{41472}-\frac {1}{3240} \sqrt {-2 x+1} \sqrt {-2 x+1}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {5}{3456}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}}{\left (2 x+1\right )^{3}}}{\sqrt {3} \sqrt {2}} \end {gather*}

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]

-1/39813120*sqrt(3)*sqrt(2)*(3*(sqrt(2) - sqrt(2*x + 1))^5/(-2*x + 1)^(5/2) + 85*(sqrt(2) - sqrt(2*x + 1))^3/(
-2*x + 1)^(3/2) + 2130*(sqrt(2) - sqrt(2*x + 1))/sqrt(-2*x + 1) + 64*((128*x + 211)*(2*x - 1) + 300)*sqrt(-2*x
+ 1)/(2*x + 1)^(5/2) - (2130*(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)

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Mupad [B]
time = 0.45, size = 66, normalized size = 0.78 \begin {gather*} -\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} \end {gather*}

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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