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

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

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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] /; 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 \begin {gather*} \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}} \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))/(3240*Sqrt[6 - 12*x]*(1 - 2*x)^2*(1 + 2*x)^(5/2))

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IntegrateAlgebraic [B]  time = 1.25, size = 616, normalized size = 7.25 \begin {gather*} \frac {\left (\sqrt {2} \sqrt {2 x+1}-2\right )^{15} \left (-\frac {347 \left (4 x^2-4 x+1\right )}{2717908992 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^4}-\frac {539 \left (8 x^3-12 x^2+6 x-1\right )}{169869312 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^6}-\frac {2101 \left (16 x^4-32 x^3+24 x^2-8 x+1\right )}{113246208 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^8}-\frac {7469 \left (32 x^5-80 x^4+80 x^3-40 x^2+10 x-1\right )}{141557760 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{10}}-\frac {2101 \left (64 x^6-192 x^5+240 x^4-160 x^3+60 x^2-12 x+1\right )}{28311552 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{12}}-\frac {539 \left (128 x^7-448 x^6+672 x^5-560 x^4+280 x^3-84 x^2+14 x-1\right )}{10616832 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{14}}-\frac {347 \left (256 x^8-1024 x^7+1792 x^6-1792 x^5+1120 x^4-448 x^3+112 x^2-16 x+1\right )}{42467328 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{16}}+\frac {7 \left (512 x^9-2304 x^8+4608 x^7-5376 x^6+4032 x^5-2016 x^4+672 x^3-144 x^2+18 x-1\right )}{10616832 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{18}}-\frac {1024 x^{10}-5120 x^9+11520 x^8-15360 x^7+13440 x^6-8064 x^5+3360 x^4-960 x^3+180 x^2-20 x+1}{17694720 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{20}}+\frac {7 (2 x-1)}{2717908992 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^2}-\frac {1}{18119393280 \sqrt {3}}\right )}{(1-2 x)^{5/2} \left (-2 x+\sqrt {2} \sqrt {2 x+1}-1\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + Sqrt[2]*Sqrt[1 + 2*x])^15*(-1/18119393280*1/Sqrt[3] - (1 - 20*x + 180*x^2 - 960*x^3 + 3360*x^4 - 8064*x
^5 + 13440*x^6 - 15360*x^7 + 11520*x^8 - 5120*x^9 + 1024*x^10)/(17694720*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^
20) + (7*(-1 + 18*x - 144*x^2 + 672*x^3 - 2016*x^4 + 4032*x^5 - 5376*x^6 + 4608*x^7 - 2304*x^8 + 512*x^9))/(10
616832*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^18) - (347*(1 - 16*x + 112*x^2 - 448*x^3 + 1120*x^4 - 1792*x^5 + 1
792*x^6 - 1024*x^7 + 256*x^8))/(42467328*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^16) - (539*(-1 + 14*x - 84*x^2 +
280*x^3 - 560*x^4 + 672*x^5 - 448*x^6 + 128*x^7))/(10616832*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^14) - (2101*
(1 - 12*x + 60*x^2 - 160*x^3 + 240*x^4 - 192*x^5 + 64*x^6))/(28311552*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^12)
- (7469*(-1 + 10*x - 40*x^2 + 80*x^3 - 80*x^4 + 32*x^5))/(141557760*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^10)
- (2101*(1 - 8*x + 24*x^2 - 32*x^3 + 16*x^4))/(113246208*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^8) - (539*(-1 +
6*x - 12*x^2 + 8*x^3))/(169869312*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^6) - (347*(1 - 4*x + 4*x^2))/(271790899
2*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^4) + (7*(-1 + 2*x))/(2717908992*Sqrt[3]*(-2 + Sqrt[2]*Sqrt[1 + 2*x])^2)
))/((1 - 2*x)^(5/2)*(-1 - 2*x + Sqrt[2]*Sqrt[1 + 2*x])^5)

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fricas [A]  time = 1.26, 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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giac [B]  time = 1.02, size = 181, normalized size = 2.13 \begin {gather*} -\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}} \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="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 \begin {gather*} -\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}}} \end {gather*}

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 \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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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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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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