\(\int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x (14-7 x^2)}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x (6-2 x-2 x^2)} \, dx\) [1292]

Optimal result

Integrand size = 78, antiderivative size = 27 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=e^{10}+3 x+\frac {(4-x) x}{-3-e^x+x+x^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=\int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx \]

[In]

Int[(15 + 3*E^(2*x) - 12*x - 20*x^2 + 6*x^3 + 3*x^4 + E^x*(14 - 7*x^2))/(9 + E^(2*x) - 6*x - 5*x^2 + 2*x^3 + x
^4 + E^x*(6 - 2*x - 2*x^2)),x]

[Out]

3*x - 4*Defer[Int][(3 + E^x - x - x^2)^(-1), x] - 16*Defer[Int][x/(-3 - E^x + x + x^2)^2, x] + 5*Defer[Int][x^
3/(-3 - E^x + x + x^2)^2, x] - Defer[Int][x^4/(-3 - E^x + x + x^2)^2, x] - 6*Defer[Int][x/(-3 - E^x + x + x^2)
, x] + Defer[Int][x^2/(-3 - E^x + x + x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4-7 e^x \left (-2+x^2\right )}{\left (3+e^x-x-x^2\right )^2} \, dx \\ & = \int \left (3-\frac {4-6 x+x^2}{3+e^x-x-x^2}-\frac {x \left (16-5 x^2+x^3\right )}{\left (-3-e^x+x+x^2\right )^2}\right ) \, dx \\ & = 3 x-\int \frac {4-6 x+x^2}{3+e^x-x-x^2} \, dx-\int \frac {x \left (16-5 x^2+x^3\right )}{\left (-3-e^x+x+x^2\right )^2} \, dx \\ & = 3 x-\int \left (\frac {16 x}{\left (-3-e^x+x+x^2\right )^2}-\frac {5 x^3}{\left (-3-e^x+x+x^2\right )^2}+\frac {x^4}{\left (-3-e^x+x+x^2\right )^2}\right ) \, dx-\int \left (\frac {4}{3+e^x-x-x^2}+\frac {6 x}{-3-e^x+x+x^2}-\frac {x^2}{-3-e^x+x+x^2}\right ) \, dx \\ & = 3 x-4 \int \frac {1}{3+e^x-x-x^2} \, dx+5 \int \frac {x^3}{\left (-3-e^x+x+x^2\right )^2} \, dx-6 \int \frac {x}{-3-e^x+x+x^2} \, dx-16 \int \frac {x}{\left (-3-e^x+x+x^2\right )^2} \, dx-\int \frac {x^4}{\left (-3-e^x+x+x^2\right )^2} \, dx+\int \frac {x^2}{-3-e^x+x+x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=x \left (3-\frac {-4+x}{-3-e^x+x+x^2}\right ) \]

[In]

Integrate[(15 + 3*E^(2*x) - 12*x - 20*x^2 + 6*x^3 + 3*x^4 + E^x*(14 - 7*x^2))/(9 + E^(2*x) - 6*x - 5*x^2 + 2*x
^3 + x^4 + E^x*(6 - 2*x - 2*x^2)),x]

[Out]

x*(3 - (-4 + x)/(-3 - E^x + x + x^2))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
risch	\(3 x -\frac {\left (x -4\right ) x}{x^{2}-3-{\mathrm e}^{x}+x}\)	\(23\)
norman	\(\frac {-7 x +2 \,{\mathrm e}^{x}+3 x^{3}-3 \,{\mathrm e}^{x} x +6}{x^{2}-3-{\mathrm e}^{x}+x}\)	\(33\)
parallelrisch	\(\frac {3 x^{3}+2 x^{2}-3 \,{\mathrm e}^{x} x -5 x}{x^{2}-3-{\mathrm e}^{x}+x}\)	\(33\)

[In]

int((3*exp(x)^2+(-7*x^2+14)*exp(x)+3*x^4+6*x^3-20*x^2-12*x+15)/(exp(x)^2+(-2*x^2-2*x+6)*exp(x)+x^4+2*x^3-5*x^2
-6*x+9),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=\frac {3 \, x^{3} + 2 \, x^{2} - 3 \, x e^{x} - 5 \, x}{x^{2} + x - e^{x} - 3} \]

[In]

integrate((3*exp(x)^2+(-7*x^2+14)*exp(x)+3*x^4+6*x^3-20*x^2-12*x+15)/(exp(x)^2+(-2*x^2-2*x+6)*exp(x)+x^4+2*x^3
-5*x^2-6*x+9),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=3 x + \frac {x^{2} - 4 x}{- x^{2} - x + e^{x} + 3} \]

[In]

integrate((3*exp(x)**2+(-7*x**2+14)*exp(x)+3*x**4+6*x**3-20*x**2-12*x+15)/(exp(x)**2+(-2*x**2-2*x+6)*exp(x)+x*
*4+2*x**3-5*x**2-6*x+9),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=\frac {3 \, x^{3} + 2 \, x^{2} - 3 \, x e^{x} - 5 \, x}{x^{2} + x - e^{x} - 3} \]

[In]

integrate((3*exp(x)^2+(-7*x^2+14)*exp(x)+3*x^4+6*x^3-20*x^2-12*x+15)/(exp(x)^2+(-2*x^2-2*x+6)*exp(x)+x^4+2*x^3
-5*x^2-6*x+9),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=\frac {3 \, x^{3} + 2 \, x^{2} - 3 \, x e^{x} - 5 \, x}{x^{2} + x - e^{x} - 3} \]

[In]

integrate((3*exp(x)^2+(-7*x^2+14)*exp(x)+3*x^4+6*x^3-20*x^2-12*x+15)/(exp(x)^2+(-2*x^2-2*x+6)*exp(x)+x^4+2*x^3
-5*x^2-6*x+9),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {15+3 e^{2 x}-12 x-20 x^2+6 x^3+3 x^4+e^x \left (14-7 x^2\right )}{9+e^{2 x}-6 x-5 x^2+2 x^3+x^4+e^x \left (6-2 x-2 x^2\right )} \, dx=3\,x-\frac {{\mathrm {e}}^x-5\,x+3}{x-{\mathrm {e}}^x+x^2-3} \]

[In]

int((3*exp(2*x) - 12*x - exp(x)*(7*x^2 - 14) - 20*x^2 + 6*x^3 + 3*x^4 + 15)/(exp(2*x) - 6*x - exp(x)*(2*x + 2*
x^2 - 6) - 5*x^2 + 2*x^3 + x^4 + 9),x)

[Out]

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