\(\int \frac {-75+140 x+20 x^3+e^{2 x} (-3 x^2+9 x^4+6 x^5)+e^x (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6)}{25+e^x (-10 x-10 x^2)+e^{2 x} (x^2+2 x^3+x^4)} \, dx\) [6871]

Optimal result

Integrand size = 99, antiderivative size = 32 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=3-(1-x) \left (3 x+\frac {x}{-e^x+\frac {5}{x+x^2}}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=\int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx \]

[In]

Int[(-75 + 140*x + 20*x^3 + E^(2*x)*(-3*x^2 + 9*x^4 + 6*x^5) + E^x*(30*x - 29*x^2 - 61*x^3 - 4*x^4 - x^5 + x^6
))/(25 + E^x*(-10*x - 10*x^2) + E^(2*x)*(x^2 + 2*x^3 + x^4)),x]

[Out]

(3*(1 - 2*x)^2)/4 - 5*Defer[Int][x/(-5 + E^x*x + E^x*x^2)^2, x] - 10*Defer[Int][x^2/(-5 + E^x*x + E^x*x^2)^2,
x] + 10*Defer[Int][x^3/(-5 + E^x*x + E^x*x^2)^2, x] + 5*Defer[Int][x^4/(-5 + E^x*x + E^x*x^2)^2, x] + Defer[In
t][x/(-5 + E^x*x + E^x*x^2), x] - 2*Defer[Int][x^2/(-5 + E^x*x + E^x*x^2), x] - 2*Defer[Int][x^3/(-5 + E^x*x +
E^x*x^2), x] + Defer[Int][x^4/(-5 + E^x*x + E^x*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{\left (5-e^x x-e^x x^2\right )^2} \, dx \\ & = \int \left (3 (-1+2 x)+\frac {x \left (1-2 x-2 x^2+x^3\right )}{-5+e^x x+e^x x^2}+\frac {5 x \left (-1-2 x+2 x^2+x^3\right )}{\left (-5+e^x x+e^x x^2\right )^2}\right ) \, dx \\ & = \frac {3}{4} (1-2 x)^2+5 \int \frac {x \left (-1-2 x+2 x^2+x^3\right )}{\left (-5+e^x x+e^x x^2\right )^2} \, dx+\int \frac {x \left (1-2 x-2 x^2+x^3\right )}{-5+e^x x+e^x x^2} \, dx \\ & = \frac {3}{4} (1-2 x)^2+5 \int \left (-\frac {x}{\left (-5+e^x x+e^x x^2\right )^2}-\frac {2 x^2}{\left (-5+e^x x+e^x x^2\right )^2}+\frac {2 x^3}{\left (-5+e^x x+e^x x^2\right )^2}+\frac {x^4}{\left (-5+e^x x+e^x x^2\right )^2}\right ) \, dx+\int \left (\frac {x}{-5+e^x x+e^x x^2}-\frac {2 x^2}{-5+e^x x+e^x x^2}-\frac {2 x^3}{-5+e^x x+e^x x^2}+\frac {x^4}{-5+e^x x+e^x x^2}\right ) \, dx \\ & = \frac {3}{4} (1-2 x)^2-2 \int \frac {x^2}{-5+e^x x+e^x x^2} \, dx-2 \int \frac {x^3}{-5+e^x x+e^x x^2} \, dx-5 \int \frac {x}{\left (-5+e^x x+e^x x^2\right )^2} \, dx+5 \int \frac {x^4}{\left (-5+e^x x+e^x x^2\right )^2} \, dx-10 \int \frac {x^2}{\left (-5+e^x x+e^x x^2\right )^2} \, dx+10 \int \frac {x^3}{\left (-5+e^x x+e^x x^2\right )^2} \, dx+\int \frac {x}{-5+e^x x+e^x x^2} \, dx+\int \frac {x^4}{-5+e^x x+e^x x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=x \left (-3+3 x+\frac {x-x^3}{-5+e^x x (1+x)}\right ) \]

[In]

Integrate[(-75 + 140*x + 20*x^3 + E^(2*x)*(-3*x^2 + 9*x^4 + 6*x^5) + E^x*(30*x - 29*x^2 - 61*x^3 - 4*x^4 - x^5
+ x^6))/(25 + E^x*(-10*x - 10*x^2) + E^(2*x)*(x^2 + 2*x^3 + x^4)),x]

[Out]

x*(-3 + 3*x + (x - x^3)/(-5 + E^x*x*(1 + x)))

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09

method	result	size
risch	\(3 x^{2}-3 x -\frac {\left (-1+x \right ) x^{2} \left (1+x \right )}{{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x -5}\)	\(35\)
norman	\(\frac {3 \,{\mathrm e}^{x} x +15 x -14 x^{2}-x^{4}+3 \,{\mathrm e}^{x} x^{4}-15}{{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x -5}\)	\(43\)
parallelrisch	\(\frac {3 \,{\mathrm e}^{x} x +15 x -14 x^{2}-x^{4}+3 \,{\mathrm e}^{x} x^{4}-15}{{\mathrm e}^{x} x^{2}+{\mathrm e}^{x} x -5}\)	\(43\)

[In]

int(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*
exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=-\frac {x^{4} + 14 \, x^{2} - 3 \, {\left (x^{4} - x^{2}\right )} e^{x} - 15 \, x}{{\left (x^{2} + x\right )} e^{x} - 5} \]

[In]

integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3
+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x, algorithm="fricas")

[Out]

-(x^4 + 14*x^2 - 3*(x^4 - x^2)*e^x - 15*x)/((x^2 + x)*e^x - 5)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=3 x^{2} - 3 x + \frac {- x^{4} + x^{2}}{\left (x^{2} + x\right ) e^{x} - 5} \]

[In]

integrate(((6*x**5+9*x**4-3*x**2)*exp(x)**2+(x**6-x**5-4*x**4-61*x**3-29*x**2+30*x)*exp(x)+20*x**3+140*x-75)/(
(x**4+2*x**3+x**2)*exp(x)**2+(-10*x**2-10*x)*exp(x)+25),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=-\frac {x^{4} + 14 \, x^{2} - 3 \, {\left (x^{4} - x^{2}\right )} e^{x} - 15 \, x}{{\left (x^{2} + x\right )} e^{x} - 5} \]

[In]

integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3
+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x, algorithm="maxima")

[Out]

-(x^4 + 14*x^2 - 3*(x^4 - x^2)*e^x - 15*x)/((x^2 + x)*e^x - 5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=\frac {3 \, x^{4} e^{x} - x^{4} - 3 \, x^{2} e^{x} - 14 \, x^{2} + 15 \, x}{x^{2} e^{x} + x e^{x} - 5} \]

[In]

integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3
+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x, algorithm="giac")

[Out]

(3*x^4*e^x - x^4 - 3*x^2*e^x - 14*x^2 + 15*x)/(x^2*e^x + x*e^x - 5)

Mupad [B] (verification not implemented)

Time = 12.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {-75+140 x+20 x^3+e^{2 x} \left (-3 x^2+9 x^4+6 x^5\right )+e^x \left (30 x-29 x^2-61 x^3-4 x^4-x^5+x^6\right )}{25+e^x \left (-10 x-10 x^2\right )+e^{2 x} \left (x^2+2 x^3+x^4\right )} \, dx=-\frac {x\,\left (x-1\right )\,\left (x-3\,x^2\,{\mathrm {e}}^x-3\,x\,{\mathrm {e}}^x+x^2+15\right )}{x^2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x-5} \]

[In]

int((140*x - exp(x)*(29*x^2 - 30*x + 61*x^3 + 4*x^4 + x^5 - x^6) + exp(2*x)*(9*x^4 - 3*x^2 + 6*x^5) + 20*x^3 -
75)/(exp(2*x)*(x^2 + 2*x^3 + x^4) - exp(x)*(10*x + 10*x^2) + 25),x)

[Out]

-(x*(x - 1)*(x - 3*x^2*exp(x) - 3*x*exp(x) + x^2 + 15))/(x^2*exp(x) + x*exp(x) - 5)