\(\int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} (8+e^{5-x} (-5000 x-625 x^2))}{64+16 x+x^2} \, dx\) [5138]

Optimal result

Integrand size = 52, antiderivative size = 26 \[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=x \left (-3+\frac {1+e^{625 e^{5-x}}-x}{8+x}\right ) \]

[Out]

x*((1-x+exp(625*exp(5-x)))/(x+8)-3)

Rubi [F]

\[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=\int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx \]

[In]

Int[(-184 - 64*x - 4*x^2 + E^(625*E^(5 - x))*(8 + E^(5 - x)*(-5000*x - 625*x^2)))/(64 + 16*x + x^2),x]

[Out]

E^(625*E^(5 - x)) - 4*x - 72/(8 + x) + 8*Defer[Int][E^(625*E^(5 - x))/(8 + x)^2, x] + 5000*Defer[Int][E^(5 + 6
25*E^(5 - x) - x)/(8 + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{(8+x)^2} \, dx \\ & = \int \left (-\frac {184}{(8+x)^2}+\frac {8 e^{625 e^{5-x}}}{(8+x)^2}-\frac {64 x}{(8+x)^2}-\frac {4 x^2}{(8+x)^2}-\frac {625 e^{5+625 e^{5-x}-x} x}{8+x}\right ) \, dx \\ & = \frac {184}{8+x}-4 \int \frac {x^2}{(8+x)^2} \, dx+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx-64 \int \frac {x}{(8+x)^2} \, dx-625 \int \frac {e^{5+625 e^{5-x}-x} x}{8+x} \, dx \\ & = \frac {184}{8+x}-4 \int \left (1+\frac {64}{(8+x)^2}-\frac {16}{8+x}\right ) \, dx+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx-64 \int \left (-\frac {8}{(8+x)^2}+\frac {1}{8+x}\right ) \, dx-625 \int \left (e^{5+625 e^{5-x}-x}-\frac {8 e^{5+625 e^{5-x}-x}}{8+x}\right ) \, dx \\ & = -4 x-\frac {72}{8+x}+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx-625 \int e^{5+625 e^{5-x}-x} \, dx+5000 \int \frac {e^{5+625 e^{5-x}-x}}{8+x} \, dx \\ & = -4 x-\frac {72}{8+x}+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx+625 \text {Subst}\left (\int e^{5+625 e^5 x} \, dx,x,e^{-x}\right )+5000 \int \frac {e^{5+625 e^{5-x}-x}}{8+x} \, dx \\ & = e^{625 e^{5-x}}-4 x-\frac {72}{8+x}+8 \int \frac {e^{625 e^{5-x}}}{(8+x)^2} \, dx+5000 \int \frac {e^{5+625 e^{5-x}-x}}{8+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=\frac {-72+\left (-32+e^{625 e^{5-x}}\right ) x-4 x^2}{8+x} \]

[In]

Integrate[(-184 - 64*x - 4*x^2 + E^(625*E^(5 - x))*(8 + E^(5 - x)*(-5000*x - 625*x^2)))/(64 + 16*x + x^2),x]

[Out]

(-72 + (-32 + E^(625*E^(5 - x)))*x - 4*x^2)/(8 + x)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
norman	\(\frac {x \,{\mathrm e}^{625 \,{\mathrm e}^{5-x}}-4 x^{2}+184}{x +8}\)	\(25\)
parallelrisch	\(-\frac {4 x^{2}-x \,{\mathrm e}^{625 \,{\mathrm e}^{5-x}}-184}{x +8}\)	\(27\)
risch	\(-\frac {72}{x +8}-4 x +\frac {x \,{\mathrm e}^{625 \,{\mathrm e}^{5-x}}}{x +8}\)	\(28\)
parts	\(\frac {x^{2} {\mathrm e}^{625 \,{\mathrm e}^{5-x}}+8 x \,{\mathrm e}^{625 \,{\mathrm e}^{5-x}}}{\left (x +8\right )^{2}}-\frac {72}{x +8}-4 x\)	\(44\)

[In]

int((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x,method=_RETURNVERBOSE)

[Out]

(x*exp(625*exp(5-x))-4*x^2+184)/(x+8)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=-\frac {4 \, x^{2} - x e^{\left (625 \, e^{\left (-x + 5\right )}\right )} + 32 \, x + 72}{x + 8} \]

[In]

integrate((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x, algorithm="fricas
")

[Out]

-(4*x^2 - x*e^(625*e^(-x + 5)) + 32*x + 72)/(x + 8)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=- 4 x + \frac {x e^{625 e^{5 - x}}}{x + 8} - \frac {72}{x + 8} \]

[In]

integrate((((-625*x**2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x**2-64*x-184)/(x**2+16*x+64),x)

[Out]

-4*x + x*exp(625*exp(5 - x))/(x + 8) - 72/(x + 8)

Maxima [F]

\[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=\int { -\frac {4 \, x^{2} + {\left (625 \, {\left (x^{2} + 8 \, x\right )} e^{\left (-x + 5\right )} - 8\right )} e^{\left (625 \, e^{\left (-x + 5\right )}\right )} + 64 \, x + 184}{x^{2} + 16 \, x + 64} \,d x } \]

[In]

integrate((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x, algorithm="maxima
")

[Out]

-4*x - 72/(x + 8) + integrate(-(625*x^2*e^5 + 5000*x*e^5 - 8*e^x)*e^(-x + 625*e^(-x + 5))/(x^2 + 16*x + 64), x
)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=-\frac {4 \, x^{2} - x e^{\left (625 \, e^{\left (-x + 5\right )}\right )} + 32 \, x + 72}{x + 8} \]

[In]

integrate((((-625*x^2-5000*x)*exp(5-x)+8)*exp(625*exp(5-x))-4*x^2-64*x-184)/(x^2+16*x+64),x, algorithm="giac")

[Out]

-(4*x^2 - x*e^(625*e^(-x + 5)) + 32*x + 72)/(x + 8)

Mupad [B] (verification not implemented)

Time = 11.64 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {-184-64 x-4 x^2+e^{625 e^{5-x}} \left (8+e^{5-x} \left (-5000 x-625 x^2\right )\right )}{64+16 x+x^2} \, dx=-\frac {23\,x-x\,{\mathrm {e}}^{625\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^5}+4\,x^2}{x+8} \]

[In]

int(-(64*x + exp(625*exp(5 - x))*(exp(5 - x)*(5000*x + 625*x^2) - 8) + 4*x^2 + 184)/(16*x + x^2 + 64),x)

[Out]

-(23*x - x*exp(625*exp(-x)*exp(5)) + 4*x^2)/(x + 8)