\(\int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x))}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 (-20 x^2+40 x^3)} \, dx\) [1289]

Optimal result

Integrand size = 108, antiderivative size = 27 \[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=1+e^{\frac {3}{2} \left (\frac {4}{x}+\frac {\log (x)}{-5+e^4+10 x}\right )} \]

[Out]

exp(3/2*ln(x)/(10*x+exp(4)-5)+6/x)+1

Rubi [F]

\[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=\int \frac {\exp \left (\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}\right ) \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx \]

[In]

Int[(E^((-60 + 12*E^4 + 120*x + 3*x*Log[x])/(-10*x + 2*E^4*x + 20*x^2))*(-300 - 12*E^8 + E^4*(120 - 237*x) + 1
185*x - 1170*x^2 - 30*x^2*Log[x]))/(50*x^2 + 2*E^8*x^2 - 200*x^3 + 200*x^4 + E^4*(-20*x^2 + 40*x^3)),x]

[Out]

(-117*Defer[Int][E^(6/x)*x^(-2 + 3/(2*(-5 + E^4 + 10*x))), x])/20 - 15*Log[x]*Defer[Int][(E^(6/x)*x^(3/(2*(-5
+ E^4 + 10*x))))/(-5 + E^4 + 10*x)^2, x] + (3*(5 - E^4)*Defer[Int][(E^(6/x)*x^(-2 + 3/(2*(-5 + E^4 + 10*x))))/
(-5 + E^4 + 10*x), x])/20 + 15*Defer[Int][Defer[Int][(E^(6/x)*x^(3/(2*(-5 + E^4 + 10*x))))/(-5 + E^4 + 10*x)^2
, x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}\right ) \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{\left (50+2 e^8\right ) x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx \\ & = \int \frac {3 e^{6/x} x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}} \left (-4 e^8-e^4 (-40+79 x)-5 \left (20-79 x+78 x^2\right )-10 x^2 \log (x)\right )}{2 \left (5-e^4-10 x\right )^2} \, dx \\ & = \frac {3}{2} \int \frac {e^{6/x} x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}} \left (-4 e^8-e^4 (-40+79 x)-5 \left (20-79 x+78 x^2\right )-10 x^2 \log (x)\right )}{\left (5-e^4-10 x\right )^2} \, dx \\ & = \frac {3}{2} \int \left (\frac {e^{6/x} \left (20-4 e^4-39 x\right ) x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}}}{-5+e^4+10 x}-\frac {10 e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}} \log (x)}{\left (-5+e^4+10 x\right )^2}\right ) \, dx \\ & = \frac {3}{2} \int \frac {e^{6/x} \left (20-4 e^4-39 x\right ) x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}}}{-5+e^4+10 x} \, dx-15 \int \frac {e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}} \log (x)}{\left (-5+e^4+10 x\right )^2} \, dx \\ & = \frac {3}{2} \int \left (-\frac {39}{10} e^{6/x} x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}}+\frac {e^{6/x} \left (5-e^4\right ) x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}}}{10 \left (-5+e^4+10 x\right )}\right ) \, dx+15 \int \frac {\int \frac {e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}}}{\left (-5+e^4+10 x\right )^2} \, dx}{x} \, dx-(15 \log (x)) \int \frac {e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}}}{\left (-5+e^4+10 x\right )^2} \, dx \\ & = -\left (\frac {117}{20} \int e^{6/x} x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}} \, dx\right )+15 \int \frac {\int \frac {e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}}}{\left (-5+e^4+10 x\right )^2} \, dx}{x} \, dx+\frac {1}{20} \left (3 \left (5-e^4\right )\right ) \int \frac {e^{6/x} x^{-2+\frac {3}{2 \left (-5+e^4+10 x\right )}}}{-5+e^4+10 x} \, dx-(15 \log (x)) \int \frac {e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}}}{\left (-5+e^4+10 x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=e^{6/x} x^{\frac {3}{2 \left (-5+e^4+10 x\right )}} \]

[In]

Integrate[(E^((-60 + 12*E^4 + 120*x + 3*x*Log[x])/(-10*x + 2*E^4*x + 20*x^2))*(-300 - 12*E^8 + E^4*(120 - 237*
x) + 1185*x - 1170*x^2 - 30*x^2*Log[x]))/(50*x^2 + 2*E^8*x^2 - 200*x^3 + 200*x^4 + E^4*(-20*x^2 + 40*x^3)),x]

[Out]

E^(6/x)*x^(3/(2*(-5 + E^4 + 10*x)))

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
risch	\(x^{\frac {3}{2 \left (10 x +{\mathrm e}^{4}-5\right )}} {\mathrm e}^{\frac {6}{x}}\)	\(21\)
parallelrisch	\({\mathrm e}^{\frac {\frac {3 x \ln \left (x \right )}{2}+6 \,{\mathrm e}^{4}+60 x -30}{x \left (10 x +{\mathrm e}^{4}-5\right )}}\)	\(29\)
norman	\(\frac {\left ({\mathrm e}^{4}-5\right ) x \,{\mathrm e}^{\frac {3 x \ln \left (x \right )+12 \,{\mathrm e}^{4}+120 x -60}{2 x \,{\mathrm e}^{4}+20 x^{2}-10 x}}+10 x^{2} {\mathrm e}^{\frac {3 x \ln \left (x \right )+12 \,{\mathrm e}^{4}+120 x -60}{2 x \,{\mathrm e}^{4}+20 x^{2}-10 x}}}{x \left (10 x +{\mathrm e}^{4}-5\right )}\)	\(90\)

[In]

int((-30*x^2*ln(x)-12*exp(4)^2+(-237*x+120)*exp(4)-1170*x^2+1185*x-300)*exp((3*x*ln(x)+12*exp(4)+120*x-60)/(2*
x*exp(4)+20*x^2-10*x))/(2*x^2*exp(4)^2+(40*x^3-20*x^2)*exp(4)+200*x^4-200*x^3+50*x^2),x,method=_RETURNVERBOSE)

[Out]

x^(3/2/(10*x+exp(4)-5))*exp(6/x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=e^{\left (\frac {3 \, {\left (x \log \left (x\right ) + 40 \, x + 4 \, e^{4} - 20\right )}}{2 \, {\left (10 \, x^{2} + x e^{4} - 5 \, x\right )}}\right )} \]

[In]

integrate((-30*x^2*log(x)-12*exp(4)^2+(-237*x+120)*exp(4)-1170*x^2+1185*x-300)*exp((3*x*log(x)+12*exp(4)+120*x
-60)/(2*x*exp(4)+20*x^2-10*x))/(2*x^2*exp(4)^2+(40*x^3-20*x^2)*exp(4)+200*x^4-200*x^3+50*x^2),x, algorithm="fr
icas")

[Out]

e^(3/2*(x*log(x) + 40*x + 4*e^4 - 20)/(10*x^2 + x*e^4 - 5*x))

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=e^{\frac {3 x \log {\left (x \right )} + 120 x - 60 + 12 e^{4}}{20 x^{2} - 10 x + 2 x e^{4}}} \]

[In]

integrate((-30*x**2*ln(x)-12*exp(4)**2+(-237*x+120)*exp(4)-1170*x**2+1185*x-300)*exp((3*x*ln(x)+12*exp(4)+120*
x-60)/(2*x*exp(4)+20*x**2-10*x))/(2*x**2*exp(4)**2+(40*x**3-20*x**2)*exp(4)+200*x**4-200*x**3+50*x**2),x)

[Out]

exp((3*x*log(x) + 120*x - 60 + 12*exp(4))/(20*x**2 - 10*x + 2*x*exp(4)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).

Time = 0.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.33 \[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=e^{\left (-\frac {60 \, e^{4}}{10 \, x {\left (e^{4} - 5\right )} + e^{8} - 10 \, e^{4} + 25} + \frac {3 \, \log \left (x\right )}{2 \, {\left (10 \, x + e^{4} - 5\right )}} + \frac {300}{10 \, x {\left (e^{4} - 5\right )} + e^{8} - 10 \, e^{4} + 25} + \frac {60}{10 \, x + e^{4} - 5} + \frac {6 \, e^{4}}{x {\left (e^{4} - 5\right )}} - \frac {30}{x {\left (e^{4} - 5\right )}}\right )} \]

[In]

integrate((-30*x^2*log(x)-12*exp(4)^2+(-237*x+120)*exp(4)-1170*x^2+1185*x-300)*exp((3*x*log(x)+12*exp(4)+120*x
-60)/(2*x*exp(4)+20*x^2-10*x))/(2*x^2*exp(4)^2+(40*x^3-20*x^2)*exp(4)+200*x^4-200*x^3+50*x^2),x, algorithm="ma
xima")

[Out]

e^(-60*e^4/(10*x*(e^4 - 5) + e^8 - 10*e^4 + 25) + 3/2*log(x)/(10*x + e^4 - 5) + 300/(10*x*(e^4 - 5) + e^8 - 10
*e^4 + 25) + 60/(10*x + e^4 - 5) + 6*e^4/(x*(e^4 - 5)) - 30/(x*(e^4 - 5)))

Giac [F]

\[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=\int { -\frac {3 \, {\left (10 \, x^{2} \log \left (x\right ) + 390 \, x^{2} + {\left (79 \, x - 40\right )} e^{4} - 395 \, x + 4 \, e^{8} + 100\right )} e^{\left (\frac {3 \, {\left (x \log \left (x\right ) + 40 \, x + 4 \, e^{4} - 20\right )}}{2 \, {\left (10 \, x^{2} + x e^{4} - 5 \, x\right )}}\right )}}{2 \, {\left (100 \, x^{4} - 100 \, x^{3} + x^{2} e^{8} + 25 \, x^{2} + 10 \, {\left (2 \, x^{3} - x^{2}\right )} e^{4}\right )}} \,d x } \]

[In]

integrate((-30*x^2*log(x)-12*exp(4)^2+(-237*x+120)*exp(4)-1170*x^2+1185*x-300)*exp((3*x*log(x)+12*exp(4)+120*x
-60)/(2*x*exp(4)+20*x^2-10*x))/(2*x^2*exp(4)^2+(40*x^3-20*x^2)*exp(4)+200*x^4-200*x^3+50*x^2),x, algorithm="gi
ac")

[Out]

integrate(-3/2*(10*x^2*log(x) + 390*x^2 + (79*x - 40)*e^4 - 395*x + 4*e^8 + 100)*e^(3/2*(x*log(x) + 40*x + 4*e
^4 - 20)/(10*x^2 + x*e^4 - 5*x))/(100*x^4 - 100*x^3 + x^2*e^8 + 25*x^2 + 10*(2*x^3 - x^2)*e^4), x)

Mupad [B] (verification not implemented)

Time = 9.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int \frac {e^{\frac {-60+12 e^4+120 x+3 x \log (x)}{-10 x+2 e^4 x+20 x^2}} \left (-300-12 e^8+e^4 (120-237 x)+1185 x-1170 x^2-30 x^2 \log (x)\right )}{50 x^2+2 e^8 x^2-200 x^3+200 x^4+e^4 \left (-20 x^2+40 x^3\right )} \, dx=x^{\frac {3}{20\,x+2\,{\mathrm {e}}^4-10}}\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^4}{x\,{\mathrm {e}}^4-5\,x+10\,x^2}}\,{\mathrm {e}}^{\frac {60}{10\,x+{\mathrm {e}}^4-5}}\,{\mathrm {e}}^{-\frac {30}{x\,{\mathrm {e}}^4-5\,x+10\,x^2}} \]

[In]

int(-(exp((120*x + 12*exp(4) + 3*x*log(x) - 60)/(2*x*exp(4) - 10*x + 20*x^2))*(12*exp(8) - 1185*x + 30*x^2*log
(x) + 1170*x^2 + exp(4)*(237*x - 120) + 300))/(2*x^2*exp(8) - exp(4)*(20*x^2 - 40*x^3) + 50*x^2 - 200*x^3 + 20
0*x^4),x)

[Out]

x^(3/(20*x + 2*exp(4) - 10))*exp((6*exp(4))/(x*exp(4) - 5*x + 10*x^2))*exp(60/(10*x + exp(4) - 5))*exp(-30/(x*
exp(4) - 5*x + 10*x^2))