\(\int \frac {e^{\frac {1}{16} ((1-2 x) \log (2)+(4-8 x) \log (\frac {-20+6 x-4 x^2+x^3}{-4+x}))} (-8+80 x-160 x^2+68 x^3-8 x^4+(-80+44 x-22 x^2+8 x^3-x^4) \log (2)+(-320+176 x-88 x^2+32 x^3-4 x^4) \log (\frac {-20+6 x-4 x^2+x^3}{-4+x}))}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx\) [1707]

Optimal result

Integrand size = 148, antiderivative size = 31 \[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=e^{\frac {1}{4} (1-2 x) \left (\frac {\log (2)}{4}+\log \left (5+\frac {x}{-4+x}+x^2\right )\right )} \]

[Out]

exp(1/4*(1-2*x)*(1/4*ln(2)+ln(5+x/(x-4)+x^2)))

Rubi [F]

\[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=\int \frac {\exp \left (\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx \]

[In]

Int[(E^(((1 - 2*x)*Log[2] + (4 - 8*x)*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)])/16)*(-8 + 80*x - 160*x^2 + 68*x
^3 - 8*x^4 + (-80 + 44*x - 22*x^2 + 8*x^3 - x^4)*Log[2] + (-320 + 176*x - 88*x^2 + 32*x^3 - 4*x^4)*Log[(-20 +
6*x - 4*x^2 + x^3)/(-4 + x)]))/(640 - 352*x + 176*x^2 - 64*x^3 + 8*x^4),x]

[Out]

-Defer[Int][E^(-1/16*((-1 + 2*x)*(Log[2] + 4*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))), x] - (Log[2]*Defer[In
t][E^(-1/16*((-1 + 2*x)*(Log[2] + 4*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))), x])/8 + (7*Defer[Int][1/(E^(((
-1 + 2*x)*(Log[2] + 4*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))/16)*(-4 + x)), x])/4 - (57*Defer[Int][1/(E^(((
-1 + 2*x)*(Log[2] + 4*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))/16)*(-20 + 6*x - 4*x^2 + x^3)), x])/2 + 4*Defe
r[Int][x/(E^(((-1 + 2*x)*(Log[2] + 4*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))/16)*(-20 + 6*x - 4*x^2 + x^3)),
x] - (5*Defer[Int][x^2/(E^(((-1 + 2*x)*(Log[2] + 4*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))/16)*(-20 + 6*x -
4*x^2 + x^3)), x])/4 - Defer[Int][Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]/E^(((-1 + 2*x)*(Log[2] + 4*Log[(-20
+ 6*x - 4*x^2 + x^3)/(-4 + x)]))/16), x]/2

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx \\ & = \int \left (\frac {17 \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x^3}{2 (-4+x) \left (-20+6 x-4 x^2+x^3\right )}+\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-80+44 x-22 x^2+8 x^3-x^4}+\frac {10 \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x}{80-44 x+22 x^2-8 x^3+x^4}-\frac {20 \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x^2}{80-44 x+22 x^2-8 x^3+x^4}-\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x^4}{80-44 x+22 x^2-8 x^3+x^4}-\frac {1}{8} \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \log (2)-\frac {1}{2} \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right ) \, dx \\ & = -\left (\frac {1}{2} \int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx\right )+\frac {17}{2} \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x^3}{(-4+x) \left (-20+6 x-4 x^2+x^3\right )} \, dx+10 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x}{80-44 x+22 x^2-8 x^3+x^4} \, dx-20 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x^2}{80-44 x+22 x^2-8 x^3+x^4} \, dx-\frac {1}{8} \log (2) \int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \, dx+\int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-80+44 x-22 x^2+8 x^3-x^4} \, dx-\int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) x^4}{80-44 x+22 x^2-8 x^3+x^4} \, dx \\ & = -\left (\frac {1}{2} \int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx\right )+\frac {17}{2} \int \left (\frac {16 \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x}+\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-80+4 x-15 x^2\right )}{-20+6 x-4 x^2+x^3}\right ) \, dx+10 \int \left (\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x}+\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-5-x^2\right )}{-20+6 x-4 x^2+x^3}\right ) \, dx-20 \int \left (\frac {4 \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x}+\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-20+x-4 x^2\right )}{-20+6 x-4 x^2+x^3}\right ) \, dx-\frac {1}{8} \log (2) \int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \, dx+\int \left (-\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{4 (-4+x)}+\frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (6+x^2\right )}{4 \left (-20+6 x-4 x^2+x^3\right )}\right ) \, dx-\int \left (\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )-\frac {2 \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (40-22 x+11 x^2-4 x^3\right )}{80-44 x+22 x^2-8 x^3+x^4}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x} \, dx\right )+\frac {1}{4} \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (6+x^2\right )}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{2} \int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx+2 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (40-22 x+11 x^2-4 x^3\right )}{80-44 x+22 x^2-8 x^3+x^4} \, dx+\frac {17}{2} \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-80+4 x-15 x^2\right )}{-20+6 x-4 x^2+x^3} \, dx+10 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x} \, dx+10 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-5-x^2\right )}{-20+6 x-4 x^2+x^3} \, dx-20 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \left (-20+x-4 x^2\right )}{-20+6 x-4 x^2+x^3} \, dx-80 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x} \, dx+136 \int \frac {\exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right )}{-4+x} \, dx-\frac {1}{8} \log (2) \int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \, dx-\int \exp \left (-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx\right )+\frac {1}{4} \int \left (\frac {6 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3}+\frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3}\right ) \, dx-\frac {1}{2} \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx+2 \int \left (-\frac {32 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x}+\frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (150-5 x+28 x^2\right )}{-20+6 x-4 x^2+x^3}\right ) \, dx+\frac {17}{2} \int \left (-\frac {80 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3}+\frac {4 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3}-\frac {15 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3}\right ) \, dx+10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+10 \int \left (-\frac {5 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3}-\frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3}\right ) \, dx-20 \int \left (-\frac {20 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3}+\frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3}-\frac {4 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3}\right ) \, dx-80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+136 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-\frac {1}{8} \log (2) \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx-\int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx\right )+\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{2} \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx+\frac {3}{2} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx+2 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (150-5 x+28 x^2\right )}{-20+6 x-4 x^2+x^3} \, dx+10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-20 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx+34 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx-50 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-64 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-\frac {255}{2} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx+136 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+400 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-680 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{8} \log (2) \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx-\int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx\right )+\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{2} \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx+\frac {3}{2} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx+2 \int \left (\frac {150 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3}-\frac {5 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3}+\frac {28 e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3}\right ) \, dx+10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-20 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx+34 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx-50 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-64 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-\frac {255}{2} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx+136 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+400 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-680 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{8} \log (2) \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx-\int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx\right )+\frac {1}{4} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{2} \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right ) \, dx+\frac {3}{2} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx+10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx-10 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-20 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx+34 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x}{-20+6 x-4 x^2+x^3} \, dx-50 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx+56 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-64 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx-80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+80 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx-\frac {255}{2} \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} x^2}{-20+6 x-4 x^2+x^3} \, dx+136 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-4+x} \, dx+300 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx+400 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-680 \int \frac {e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}}{-20+6 x-4 x^2+x^3} \, dx-\frac {1}{8} \log (2) \int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx-\int e^{-\frac {1}{16} (-1+2 x) \left (\log (2)+4 \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=\int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx \]

[In]

Integrate[(E^(((1 - 2*x)*Log[2] + (4 - 8*x)*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)])/16)*(-8 + 80*x - 160*x^2
+ 68*x^3 - 8*x^4 + (-80 + 44*x - 22*x^2 + 8*x^3 - x^4)*Log[2] + (-320 + 176*x - 88*x^2 + 32*x^3 - 4*x^4)*Log[(
-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))/(640 - 352*x + 176*x^2 - 64*x^3 + 8*x^4),x]

[Out]

Integrate[(E^(((1 - 2*x)*Log[2] + (4 - 8*x)*Log[(-20 + 6*x - 4*x^2 + x^3)/(-4 + x)])/16)*(-8 + 80*x - 160*x^2
+ 68*x^3 - 8*x^4 + (-80 + 44*x - 22*x^2 + 8*x^3 - x^4)*Log[2] + (-320 + 176*x - 88*x^2 + 32*x^3 - 4*x^4)*Log[(
-20 + 6*x - 4*x^2 + x^3)/(-4 + x)]))/(640 - 352*x + 176*x^2 - 64*x^3 + 8*x^4), x]

Maple [A] (verified)

Time = 3.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10

method	result	size
risch	\(\left (\frac {x^{3}-4 x^{2}+6 x -20}{x -4}\right )^{\frac {1}{4}-\frac {x}{2}} 2^{\frac {1}{16}-\frac {x}{8}}\)	\(34\)
norman	\({\mathrm e}^{\frac {\left (-8 x +4\right ) \ln \left (\frac {x^{3}-4 x^{2}+6 x -20}{x -4}\right )}{16}+\frac {\left (1-2 x \right ) \ln \left (2\right )}{16}}\)	\(39\)
parallelrisch	\({\mathrm e}^{\frac {\left (-8 x +4\right ) \ln \left (\frac {x^{3}-4 x^{2}+6 x -20}{x -4}\right )}{16}+\frac {\left (1-2 x \right ) \ln \left (2\right )}{16}}\)	\(39\)

[In]

int(((-4*x^4+32*x^3-88*x^2+176*x-320)*ln((x^3-4*x^2+6*x-20)/(x-4))+(-x^4+8*x^3-22*x^2+44*x-80)*ln(2)-8*x^4+68*
x^3-160*x^2+80*x-8)*exp(1/16*(-8*x+4)*ln((x^3-4*x^2+6*x-20)/(x-4))+1/16*(1-2*x)*ln(2))/(8*x^4-64*x^3+176*x^2-3
52*x+640),x,method=_RETURNVERBOSE)

[Out]

((x^3-4*x^2+6*x-20)/(x-4))^(1/4-1/2*x)*2^(1/16-1/8*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=e^{\left (-\frac {1}{16} \, {\left (2 \, x - 1\right )} \log \left (2\right ) - \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{3} - 4 \, x^{2} + 6 \, x - 20}{x - 4}\right )\right )} \]

[In]

integrate(((-4*x^4+32*x^3-88*x^2+176*x-320)*log((x^3-4*x^2+6*x-20)/(x-4))+(-x^4+8*x^3-22*x^2+44*x-80)*log(2)-8
*x^4+68*x^3-160*x^2+80*x-8)*exp(1/16*(-8*x+4)*log((x^3-4*x^2+6*x-20)/(x-4))+1/16*(1-2*x)*log(2))/(8*x^4-64*x^3
+176*x^2-352*x+640),x, algorithm="fricas")

[Out]

e^(-1/16*(2*x - 1)*log(2) - 1/4*(2*x - 1)*log((x^3 - 4*x^2 + 6*x - 20)/(x - 4)))

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=e^{\left (\frac {1}{16} - \frac {x}{8}\right ) \log {\left (2 \right )} + \left (\frac {1}{4} - \frac {x}{2}\right ) \log {\left (\frac {x^{3} - 4 x^{2} + 6 x - 20}{x - 4} \right )}} \]

[In]

integrate(((-4*x**4+32*x**3-88*x**2+176*x-320)*ln((x**3-4*x**2+6*x-20)/(x-4))+(-x**4+8*x**3-22*x**2+44*x-80)*l
n(2)-8*x**4+68*x**3-160*x**2+80*x-8)*exp(1/16*(-8*x+4)*ln((x**3-4*x**2+6*x-20)/(x-4))+1/16*(1-2*x)*ln(2))/(8*x
**4-64*x**3+176*x**2-352*x+640),x)

[Out]

exp((1/16 - x/8)*log(2) + (1/4 - x/2)*log((x**3 - 4*x**2 + 6*x - 20)/(x - 4)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (26) = 52\).

Time = 0.42 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=\frac {2^{\frac {1}{16}} {\left (x^{3} - 4 \, x^{2} + 6 \, x - 20\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{8} \, x \log \left (2\right ) - \frac {1}{2} \, x \log \left (x^{3} - 4 \, x^{2} + 6 \, x - 20\right ) + \frac {1}{2} \, x \log \left (x - 4\right )\right )}}{{\left (x - 4\right )}^{\frac {1}{4}}} \]

[In]

integrate(((-4*x^4+32*x^3-88*x^2+176*x-320)*log((x^3-4*x^2+6*x-20)/(x-4))+(-x^4+8*x^3-22*x^2+44*x-80)*log(2)-8
*x^4+68*x^3-160*x^2+80*x-8)*exp(1/16*(-8*x+4)*log((x^3-4*x^2+6*x-20)/(x-4))+1/16*(1-2*x)*log(2))/(8*x^4-64*x^3
+176*x^2-352*x+640),x, algorithm="maxima")

[Out]

2^(1/16)*(x^3 - 4*x^2 + 6*x - 20)^(1/4)*e^(-1/8*x*log(2) - 1/2*x*log(x^3 - 4*x^2 + 6*x - 20) + 1/2*x*log(x - 4
))/(x - 4)^(1/4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (26) = 52\).

Time = 0.64 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=e^{\left (-\frac {1}{8} \, x \log \left (2\right ) - \frac {1}{2} \, x \log \left (\frac {x^{3}}{x - 4} - \frac {4 \, x^{2}}{x - 4} + \frac {6 \, x}{x - 4} - \frac {20}{x - 4}\right ) + \frac {1}{16} \, \log \left (2\right ) + \frac {1}{4} \, \log \left (\frac {x^{3}}{x - 4} - \frac {4 \, x^{2}}{x - 4} + \frac {6 \, x}{x - 4} - \frac {20}{x - 4}\right )\right )} \]

[In]

integrate(((-4*x^4+32*x^3-88*x^2+176*x-320)*log((x^3-4*x^2+6*x-20)/(x-4))+(-x^4+8*x^3-22*x^2+44*x-80)*log(2)-8
*x^4+68*x^3-160*x^2+80*x-8)*exp(1/16*(-8*x+4)*log((x^3-4*x^2+6*x-20)/(x-4))+1/16*(1-2*x)*log(2))/(8*x^4-64*x^3
+176*x^2-352*x+640),x, algorithm="giac")

[Out]

e^(-1/8*x*log(2) - 1/2*x*log(x^3/(x - 4) - 4*x^2/(x - 4) + 6*x/(x - 4) - 20/(x - 4)) + 1/16*log(2) + 1/4*log(x
^3/(x - 4) - 4*x^2/(x - 4) + 6*x/(x - 4) - 20/(x - 4)))

Mupad [B] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.26 \[ \int \frac {e^{\frac {1}{16} \left ((1-2 x) \log (2)+(4-8 x) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )} \left (-8+80 x-160 x^2+68 x^3-8 x^4+\left (-80+44 x-22 x^2+8 x^3-x^4\right ) \log (2)+\left (-320+176 x-88 x^2+32 x^3-4 x^4\right ) \log \left (\frac {-20+6 x-4 x^2+x^3}{-4+x}\right )\right )}{640-352 x+176 x^2-64 x^3+8 x^4} \, dx=\frac {2^{\frac {1}{16}-\frac {x}{8}}\,{\left (\frac {6\,x}{x-4}-\frac {20}{x-4}-\frac {4\,x^2}{x-4}+\frac {x^3}{x-4}\right )}^{1/4}}{{\left (\frac {x^3-4\,x^2+6\,x-20}{x-4}\right )}^{x/2}} \]

[In]

int(-(exp(- (log(2)*(2*x - 1))/16 - (log((6*x - 4*x^2 + x^3 - 20)/(x - 4))*(8*x - 4))/16)*(log(2)*(22*x^2 - 44
*x - 8*x^3 + x^4 + 80) - 80*x + log((6*x - 4*x^2 + x^3 - 20)/(x - 4))*(88*x^2 - 176*x - 32*x^3 + 4*x^4 + 320)
+ 160*x^2 - 68*x^3 + 8*x^4 + 8))/(176*x^2 - 352*x - 64*x^3 + 8*x^4 + 640),x)

[Out]

(2^(1/16 - x/8)*((6*x)/(x - 4) - 20/(x - 4) - (4*x^2)/(x - 4) + x^3/(x - 4))^(1/4))/((6*x - 4*x^2 + x^3 - 20)/
(x - 4))^(x/2)