\(\int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} (8+48 x+4 x^2-16 x^3+4 x^4)}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx\) [7612]

Optimal result

Integrand size = 74, antiderivative size = 28 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2 e^{4+\frac {x+\frac {2}{(3-x) (1+x)}}{2+x}} \]

[Out]

2*exp((x+2/(1+x)/(-x+3))/(2+x)+4)

Rubi [F]

\[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=\int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx \]

[In]

Int[(E^((-26 - 31*x - 2*x^2 + 5*x^3)/(-6 - 7*x + x^3))*(8 + 48*x + 4*x^2 - 16*x^3 + 4*x^4))/(36 + 84*x + 49*x^
2 - 12*x^3 - 14*x^4 + x^6),x]

[Out]

Defer[Int][E^((-26 - 31*x - 2*x^2 + 5*x^3)/(-6 - 7*x + x^3))/(-3 + x)^2, x]/5 - Defer[Int][E^((-26 - 31*x - 2*
x^2 + 5*x^3)/(-6 - 7*x + x^3))/(1 + x)^2, x] + (24*Defer[Int][E^((-26 - 31*x - 2*x^2 + 5*x^3)/(-6 - 7*x + x^3)
)/(2 + x)^2, x])/5

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{\left (6+7 x-x^3\right )^2} \, dx \\ & = \int \left (\frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}}}{5 (-3+x)^2}-\frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}}}{(1+x)^2}+\frac {24 e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}}}{5 (2+x)^2}\right ) \, dx \\ & = \frac {1}{5} \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}}}{(-3+x)^2} \, dx+\frac {24}{5} \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}}}{(2+x)^2} \, dx-\int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}}}{(1+x)^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2 e^{5-\frac {2 \left (-2-2 x+x^2\right )}{-6-7 x+x^3}} \]

[In]

Integrate[(E^((-26 - 31*x - 2*x^2 + 5*x^3)/(-6 - 7*x + x^3))*(8 + 48*x + 4*x^2 - 16*x^3 + 4*x^4))/(36 + 84*x +
49*x^2 - 12*x^3 - 14*x^4 + x^6),x]

[Out]

2*E^(5 - (2*(-2 - 2*x + x^2))/(-6 - 7*x + x^3))

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
gosper	\(2 \,{\mathrm e}^{\frac {5 x^{3}-2 x^{2}-31 x -26}{x^{3}-7 x -6}}\)	\(30\)
parallelrisch	\(2 \,{\mathrm e}^{\frac {5 x^{3}-2 x^{2}-31 x -26}{x^{3}-7 x -6}}\)	\(30\)
risch	\(2 \,{\mathrm e}^{\frac {5 x^{3}-2 x^{2}-31 x -26}{\left (-3+x \right ) \left (2+x \right ) \left (1+x \right )}}\)	\(35\)
norman	\(\frac {-14 x \,{\mathrm e}^{\frac {5 x^{3}-2 x^{2}-31 x -26}{x^{3}-7 x -6}}+2 x^{3} {\mathrm e}^{\frac {5 x^{3}-2 x^{2}-31 x -26}{x^{3}-7 x -6}}-12 \,{\mathrm e}^{\frac {5 x^{3}-2 x^{2}-31 x -26}{x^{3}-7 x -6}}}{x^{3}-7 x -6}\)	\(104\)

[In]

int((4*x^4-16*x^3+4*x^2+48*x+8)*exp((5*x^3-2*x^2-31*x-26)/(x^3-7*x-6))/(x^6-14*x^4-12*x^3+49*x^2+84*x+36),x,me
thod=_RETURNVERBOSE)

[Out]

2*exp((5*x^3-2*x^2-31*x-26)/(x^3-7*x-6))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2 \, e^{\left (\frac {5 \, x^{3} - 2 \, x^{2} - 31 \, x - 26}{x^{3} - 7 \, x - 6}\right )} \]

[In]

integrate((4*x^4-16*x^3+4*x^2+48*x+8)*exp((5*x^3-2*x^2-31*x-26)/(x^3-7*x-6))/(x^6-14*x^4-12*x^3+49*x^2+84*x+36
),x, algorithm="fricas")

[Out]

2*e^((5*x^3 - 2*x^2 - 31*x - 26)/(x^3 - 7*x - 6))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2 e^{\frac {5 x^{3} - 2 x^{2} - 31 x - 26}{x^{3} - 7 x - 6}} \]

[In]

integrate((4*x**4-16*x**3+4*x**2+48*x+8)*exp((5*x**3-2*x**2-31*x-26)/(x**3-7*x-6))/(x**6-14*x**4-12*x**3+49*x*
*2+84*x+36),x)

[Out]

2*exp((5*x**3 - 2*x**2 - 31*x - 26)/(x**3 - 7*x - 6))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2 \, e^{\left (-\frac {12}{5 \, {\left (x + 2\right )}} + \frac {1}{2 \, {\left (x + 1\right )}} - \frac {1}{10 \, {\left (x - 3\right )}} + 5\right )} \]

[In]

integrate((4*x^4-16*x^3+4*x^2+48*x+8)*exp((5*x^3-2*x^2-31*x-26)/(x^3-7*x-6))/(x^6-14*x^4-12*x^3+49*x^2+84*x+36
),x, algorithm="maxima")

[Out]

2*e^(-12/5/(x + 2) + 1/2/(x + 1) - 1/10/(x - 3) + 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2 \, e^{\left (\frac {5 \, x^{3}}{x^{3} - 7 \, x - 6} - \frac {2 \, x^{2}}{x^{3} - 7 \, x - 6} - \frac {31 \, x}{x^{3} - 7 \, x - 6} - \frac {26}{x^{3} - 7 \, x - 6}\right )} \]

[In]

integrate((4*x^4-16*x^3+4*x^2+48*x+8)*exp((5*x^3-2*x^2-31*x-26)/(x^3-7*x-6))/(x^6-14*x^4-12*x^3+49*x^2+84*x+36
),x, algorithm="giac")

[Out]

2*e^(5*x^3/(x^3 - 7*x - 6) - 2*x^2/(x^3 - 7*x - 6) - 31*x/(x^3 - 7*x - 6) - 26/(x^3 - 7*x - 6))

Mupad [B] (verification not implemented)

Time = 12.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\frac {-26-31 x-2 x^2+5 x^3}{-6-7 x+x^3}} \left (8+48 x+4 x^2-16 x^3+4 x^4\right )}{36+84 x+49 x^2-12 x^3-14 x^4+x^6} \, dx=2\,{\mathrm {e}}^{\frac {-5\,x^3+2\,x^2+31\,x+26}{-x^3+7\,x+6}} \]

[In]

int((exp((31*x + 2*x^2 - 5*x^3 + 26)/(7*x - x^3 + 6))*(48*x + 4*x^2 - 16*x^3 + 4*x^4 + 8))/(84*x + 49*x^2 - 12
*x^3 - 14*x^4 + x^6 + 36),x)

[Out]

2*exp((31*x + 2*x^2 - 5*x^3 + 26)/(7*x - x^3 + 6))