\(\int \frac {e^{\frac {e^2 (20 x-21 x^2+x^3)+e^2 (-20+x) \log (x)}{-97+4 x}} (e^2 (1940-2117 x+4078 x^2-375 x^3+8 x^4)-17 e^2 x \log (x))}{9409 x-776 x^2+16 x^3} \, dx\) [6763]

Optimal result

Integrand size = 86, antiderivative size = 32 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {e^2 ((-1+x) x+\log (x))}{5-\frac {3-x}{20-x}}} \]

[Out]

exp((ln(x)+x*(-1+x))/(5-(-x+3)/(-x+20))*exp(2))

Rubi [F]

\[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=\int \frac {\exp \left (\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}\right ) \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx \]

[In]

Int[(E^((E^2*(20*x - 21*x^2 + x^3) + E^2*(-20 + x)*Log[x])/(-97 + 4*x))*(E^2*(1940 - 2117*x + 4078*x^2 - 375*x
^3 + 8*x^4) - 17*E^2*x*Log[x]))/(9409*x - 776*x^2 + 16*x^3),x]

[Out]

(13*Defer[Int][E^(2 + (E^2*(-20 + x)*(-x + x^2 + Log[x]))/(-97 + 4*x)), x])/16 + (20*Defer[Int][E^(2 + (E^2*(-
20 + x)*(-x + x^2 + Log[x]))/(-97 + 4*x))/x, x])/97 + Defer[Int][E^(2 + (E^2*(-20 + x)*(-x + x^2 + Log[x]))/(-
97 + 4*x))*x, x]/2 - (153357*Defer[Int][E^(2 + (E^2*(-20 + x)*(-x + x^2 + Log[x]))/(-97 + 4*x))/(-97 + 4*x)^2,
x])/16 + (17*Defer[Int][E^(2 + (E^2*(-20 + x)*(-x + x^2 + Log[x]))/(-97 + 4*x))/(-97 + 4*x), x])/97 - 17*Defe
r[Int][(E^(2 + (E^2*(-20 + x)*(-x + x^2 + Log[x]))/(-97 + 4*x))*Log[x])/(-97 + 4*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}\right ) \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{x \left (9409-776 x+16 x^2\right )} \, dx \\ & = \int \frac {\exp \left (\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}\right ) \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{x (-97+4 x)^2} \, dx \\ & = \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \left (1940-2117 x+4078 x^2-375 x^3+8 x^4-17 x \log (x)\right )}{(97-4 x)^2 x} \, dx \\ & = \int \left (-\frac {2117 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2}+\frac {1940 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{x (-97+4 x)^2}+\frac {4078 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x}{(-97+4 x)^2}-\frac {375 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x^2}{(-97+4 x)^2}+\frac {8 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x^3}{(-97+4 x)^2}-\frac {17 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \log (x)}{(-97+4 x)^2}\right ) \, dx \\ & = 8 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x^3}{(-97+4 x)^2} \, dx-17 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \log (x)}{(-97+4 x)^2} \, dx-375 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x^2}{(-97+4 x)^2} \, dx+1940 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{x (-97+4 x)^2} \, dx-2117 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx+4078 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x}{(-97+4 x)^2} \, dx \\ & = 8 \int \left (\frac {97}{32} \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )+\frac {1}{16} \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x+\frac {912673 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{64 (-97+4 x)^2}+\frac {28227 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{64 (-97+4 x)}\right ) \, dx-17 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \log (x)}{(-97+4 x)^2} \, dx-375 \int \left (\frac {1}{16} \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )+\frac {9409 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{16 (-97+4 x)^2}+\frac {97 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{8 (-97+4 x)}\right ) \, dx+1940 \int \left (\frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{9409 x}+\frac {4 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{97 (-97+4 x)^2}-\frac {4 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{9409 (-97+4 x)}\right ) \, dx-2117 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx+4078 \int \left (\frac {97 \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{4 (-97+4 x)^2}+\frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{4 (-97+4 x)}\right ) \, dx \\ & = \frac {20}{97} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{x} \, dx+\frac {1}{2} \int \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) x \, dx-\frac {80}{97} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{-97+4 x} \, dx-17 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \log (x)}{(-97+4 x)^2} \, dx-\frac {375}{16} \int \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \, dx+\frac {97}{4} \int \exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right ) \, dx+80 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx+\frac {2039}{2} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{-97+4 x} \, dx-2117 \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx+\frac {28227}{8} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{-97+4 x} \, dx-\frac {36375}{8} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{-97+4 x} \, dx+\frac {197783}{2} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx+\frac {912673}{8} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx-\frac {3528375}{16} \int \frac {\exp \left (2+\frac {e^2 (-20+x) \left (-x+x^2+\log (x)\right )}{-97+4 x}\right )}{(-97+4 x)^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {e^2 x \left (20-21 x+x^2\right )}{-97+4 x}} x^{\frac {e^2 (-20+x)}{-97+4 x}} \]

[In]

Integrate[(E^((E^2*(20*x - 21*x^2 + x^3) + E^2*(-20 + x)*Log[x])/(-97 + 4*x))*(E^2*(1940 - 2117*x + 4078*x^2 -
375*x^3 + 8*x^4) - 17*E^2*x*Log[x]))/(9409*x - 776*x^2 + 16*x^3),x]

[Out]

E^((E^2*x*(20 - 21*x + x^2))/(-97 + 4*x))*x^((E^2*(-20 + x))/(-97 + 4*x))

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{2} \left (x -20\right ) \left (x^{2}+\ln \left (x \right )-x \right )}{4 x -97}}\)	\(24\)
parallelrisch	\({\mathrm e}^{\frac {{\mathrm e}^{2} \left (x^{3}+x \ln \left (x \right )-21 x^{2}-20 \ln \left (x \right )+20 x \right )}{4 x -97}}\)	\(32\)
norman	\(\frac {4 x \,{\mathrm e}^{\frac {\left (x -20\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{3}-21 x^{2}+20 x \right ) {\mathrm e}^{2}}{4 x -97}}-97 \,{\mathrm e}^{\frac {\left (x -20\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{3}-21 x^{2}+20 x \right ) {\mathrm e}^{2}}{4 x -97}}}{4 x -97}\)	\(81\)

[In]

int((-17*x*exp(2)*ln(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp(((x-20)*exp(2)*ln(x)+(x^3-21*x^2+20*x
)*exp(2))/(4*x-97))/(16*x^3-776*x^2+9409*x),x,method=_RETURNVERBOSE)

[Out]

exp(exp(2)*(x-20)*(x^2+ln(x)-x)/(4*x-97))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {{\left (x - 20\right )} e^{2} \log \left (x\right ) + {\left (x^{3} - 21 \, x^{2} + 20 \, x\right )} e^{2}}{4 \, x - 97}\right )} \]

[In]

integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp(((x-20)*exp(2)*log(x)+(x^3-21*
x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-776*x^2+9409*x),x, algorithm="fricas")

[Out]

e^(((x - 20)*e^2*log(x) + (x^3 - 21*x^2 + 20*x)*e^2)/(4*x - 97))

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {\left (x - 20\right ) e^{2} \log {\left (x \right )} + \left (x^{3} - 21 x^{2} + 20 x\right ) e^{2}}{4 x - 97}} \]

[In]

integrate((-17*x*exp(2)*ln(x)+(8*x**4-375*x**3+4078*x**2-2117*x+1940)*exp(2))*exp(((x-20)*exp(2)*ln(x)+(x**3-2
1*x**2+20*x)*exp(2))/(4*x-97))/(16*x**3-776*x**2+9409*x),x)

[Out]

exp(((x - 20)*exp(2)*log(x) + (x**3 - 21*x**2 + 20*x)*exp(2))/(4*x - 97))

Maxima [A] (verification not implemented)

none

Time = 19.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {1}{4} \, x^{2} e^{2} + \frac {13}{16} \, x e^{2} + \frac {1}{4} \, e^{2} \log \left (x\right ) + \frac {17 \, e^{2} \log \left (x\right )}{4 \, {\left (4 \, x - 97\right )}} + \frac {153357 \, e^{2}}{64 \, {\left (4 \, x - 97\right )}} + \frac {1581}{64} \, e^{2}\right )} \]

[In]

integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp(((x-20)*exp(2)*log(x)+(x^3-21*
x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-776*x^2+9409*x),x, algorithm="maxima")

[Out]

e^(1/4*x^2*e^2 + 13/16*x*e^2 + 1/4*e^2*log(x) + 17/4*e^2*log(x)/(4*x - 97) + 153357/64*e^2/(4*x - 97) + 1581/6
4*e^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (26) = 52\).

Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {x^{3} e^{2}}{4 \, x - 97} - \frac {21 \, x^{2} e^{2}}{4 \, x - 97} + \frac {x e^{2} \log \left (x\right )}{4 \, x - 97} + \frac {20 \, x e^{2}}{4 \, x - 97} - \frac {20 \, e^{2} \log \left (x\right )}{4 \, x - 97}\right )} \]

[In]

integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp(((x-20)*exp(2)*log(x)+(x^3-21*
x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-776*x^2+9409*x),x, algorithm="giac")

[Out]

e^(x^3*e^2/(4*x - 97) - 21*x^2*e^2/(4*x - 97) + x*e^2*log(x)/(4*x - 97) + 20*x*e^2/(4*x - 97) - 20*e^2*log(x)/
(4*x - 97))

Mupad [B] (verification not implemented)

Time = 12.88 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=\frac {{\mathrm {e}}^{\frac {20\,x\,{\mathrm {e}}^2}{4\,x-97}}\,{\mathrm {e}}^{\frac {x^3\,{\mathrm {e}}^2}{4\,x-97}}\,{\mathrm {e}}^{-\frac {21\,x^2\,{\mathrm {e}}^2}{4\,x-97}}}{x^{\frac {20\,{\mathrm {e}}^2-x\,{\mathrm {e}}^2}{4\,x-97}}} \]

[In]

int((exp((exp(2)*(20*x - 21*x^2 + x^3) + exp(2)*log(x)*(x - 20))/(4*x - 97))*(exp(2)*(4078*x^2 - 2117*x - 375*
x^3 + 8*x^4 + 1940) - 17*x*exp(2)*log(x)))/(9409*x - 776*x^2 + 16*x^3),x)

[Out]

(exp((20*x*exp(2))/(4*x - 97))*exp((x^3*exp(2))/(4*x - 97))*exp(-(21*x^2*exp(2))/(4*x - 97)))/x^((20*exp(2) -
x*exp(2))/(4*x - 97))