\(\int \frac {e^{179 x/3} (-6-349 x)+e^x (-12-6 x+3 x^2)}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} (18 x^3+9 x^4)+e^{176 x/3} (36 x^3+36 x^4+9 x^5)} \, dx\) [6702]

Optimal result

Integrand size = 105, antiderivative size = 19 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \]

[Out]

exp(x)/x^2/(exp(88/3*x)^2+2+x)^2

Rubi [F]

\[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx \]

[In]

Int[(E^((179*x)/3)*(-6 - 349*x) + E^x*(-12 - 6*x + 3*x^2))/(24*x^3 + 3*E^(176*x)*x^3 + 36*x^4 + 18*x^5 + 3*x^6
+ E^((352*x)/3)*(18*x^3 + 9*x^4) + E^((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5)),x]

[Out]

(698*Defer[Int][E^x/(x^2*(2 + E^((176*x)/3) + x)^3), x])/3 + (352*Defer[Int][E^x/(x*(2 + E^((176*x)/3) + x)^3)
, x])/3 - 2*Defer[Int][E^x/(x^3*(2 + E^((176*x)/3) + x)^2), x] - (349*Defer[Int][E^x/(x^2*(2 + E^((176*x)/3) +
x)^2), x])/3

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (-12-6 x+3 x^2-e^{176 x/3} (6+349 x)\right )}{3 x^3 \left (2+e^{176 x/3}+x\right )^3} \, dx \\ & = \frac {1}{3} \int \frac {e^x \left (-12-6 x+3 x^2-e^{176 x/3} (6+349 x)\right )}{x^3 \left (2+e^{176 x/3}+x\right )^3} \, dx \\ & = \frac {1}{3} \int \left (\frac {2 e^x (349+176 x)}{x^2 \left (2+e^{176 x/3}+x\right )^3}-\frac {e^x (6+349 x)}{x^3 \left (2+e^{176 x/3}+x\right )^2}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {e^x (6+349 x)}{x^3 \left (2+e^{176 x/3}+x\right )^2} \, dx\right )+\frac {2}{3} \int \frac {e^x (349+176 x)}{x^2 \left (2+e^{176 x/3}+x\right )^3} \, dx \\ & = -\left (\frac {1}{3} \int \left (\frac {6 e^x}{x^3 \left (2+e^{176 x/3}+x\right )^2}+\frac {349 e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2}\right ) \, dx\right )+\frac {2}{3} \int \left (\frac {349 e^x}{x^2 \left (2+e^{176 x/3}+x\right )^3}+\frac {176 e^x}{x \left (2+e^{176 x/3}+x\right )^3}\right ) \, dx \\ & = -\left (2 \int \frac {e^x}{x^3 \left (2+e^{176 x/3}+x\right )^2} \, dx\right )-\frac {349}{3} \int \frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \, dx+\frac {352}{3} \int \frac {e^x}{x \left (2+e^{176 x/3}+x\right )^3} \, dx+\frac {698}{3} \int \frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \]

[In]

Integrate[(E^((179*x)/3)*(-6 - 349*x) + E^x*(-12 - 6*x + 3*x^2))/(24*x^3 + 3*E^(176*x)*x^3 + 36*x^4 + 18*x^5 +
3*x^6 + E^((352*x)/3)*(18*x^3 + 9*x^4) + E^((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5)),x]

[Out]

E^x/(x^2*(2 + E^((176*x)/3) + x)^2)

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {{\mathrm e}^{x}}{x^{2} \left ({\mathrm e}^{\frac {176 x}{3}}+x +2\right )^{2}}\)	\(16\)
parallelrisch	\(\frac {{\mathrm e}^{x}}{x^{2} \left ({\mathrm e}^{\frac {352 x}{3}}+2 \,{\mathrm e}^{\frac {176 x}{3}} x +x^{2}+4 \,{\mathrm e}^{\frac {176 x}{3}}+4 x +4\right )}\)	\(40\)

[In]

int(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*exp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3*x)^4+
(9*x^5+36*x^4+36*x^3)*exp(88/3*x)^2+3*x^6+18*x^5+36*x^4+24*x^3),x,method=_RETURNVERBOSE)

[Out]

1/x^2*exp(x)/(exp(176/3*x)+x+2)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).

Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^{x}}{x^{4} + 4 \, x^{3} + x^{2} e^{\left (\frac {352}{3} \, x\right )} + 4 \, x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (\frac {176}{3} \, x\right )}} \]

[In]

integrate(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*exp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3
*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x)^2+3*x^6+18*x^5+36*x^4+24*x^3),x, algorithm="fricas")

[Out]

e^x/(x^4 + 4*x^3 + x^2*e^(352/3*x) + 4*x^2 + 2*(x^3 + 2*x^2)*e^(176/3*x))

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\text {Timed out} \]

[In]

integrate(((-349*x-6)*exp(x)*exp(88/3*x)**2+(3*x**2-6*x-12)*exp(x))/(3*x**3*exp(88/3*x)**6+(9*x**4+18*x**3)*ex
p(88/3*x)**4+(9*x**5+36*x**4+36*x**3)*exp(88/3*x)**2+3*x**6+18*x**5+36*x**4+24*x**3),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).

Time = 1.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\frac {e^{x}}{x^{4} + 4 \, x^{3} + x^{2} e^{\left (\frac {352}{3} \, x\right )} + 4 \, x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left (\frac {176}{3} \, x\right )}} \]

[In]

integrate(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*exp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3
*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x)^2+3*x^6+18*x^5+36*x^4+24*x^3),x, algorithm="maxima")

[Out]

e^x/(x^4 + 4*x^3 + x^2*e^(352/3*x) + 4*x^2 + 2*(x^3 + 2*x^2)*e^(176/3*x))

Giac [F]

\[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\int { -\frac {{\left (349 \, x + 6\right )} e^{\left (\frac {179}{3} \, x\right )} - 3 \, {\left (x^{2} - 2 \, x - 4\right )} e^{x}}{3 \, {\left (x^{6} + 6 \, x^{5} + 12 \, x^{4} + x^{3} e^{\left (176 \, x\right )} + 8 \, x^{3} + 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{\left (\frac {352}{3} \, x\right )} + 3 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} e^{\left (\frac {176}{3} \, x\right )}\right )}} \,d x } \]

[In]

integrate(((-349*x-6)*exp(x)*exp(88/3*x)^2+(3*x^2-6*x-12)*exp(x))/(3*x^3*exp(88/3*x)^6+(9*x^4+18*x^3)*exp(88/3
*x)^4+(9*x^5+36*x^4+36*x^3)*exp(88/3*x)^2+3*x^6+18*x^5+36*x^4+24*x^3),x, algorithm="giac")

[Out]

integrate(-1/3*((349*x + 6)*e^(179/3*x) - 3*(x^2 - 2*x - 4)*e^x)/(x^6 + 6*x^5 + 12*x^4 + x^3*e^(176*x) + 8*x^3
+ 3*(x^4 + 2*x^3)*e^(352/3*x) + 3*(x^5 + 4*x^4 + 4*x^3)*e^(176/3*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{179 x/3} (-6-349 x)+e^x \left (-12-6 x+3 x^2\right )}{24 x^3+3 e^{176 x} x^3+36 x^4+18 x^5+3 x^6+e^{352 x/3} \left (18 x^3+9 x^4\right )+e^{176 x/3} \left (36 x^3+36 x^4+9 x^5\right )} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (-3\,x^2+6\,x+12\right )+{\mathrm {e}}^{\frac {176\,x}{3}}\,{\mathrm {e}}^x\,\left (349\,x+6\right )}{{\mathrm {e}}^{\frac {352\,x}{3}}\,\left (9\,x^4+18\,x^3\right )+3\,x^3\,{\mathrm {e}}^{176\,x}+{\mathrm {e}}^{\frac {176\,x}{3}}\,\left (9\,x^5+36\,x^4+36\,x^3\right )+24\,x^3+36\,x^4+18\,x^5+3\,x^6} \,d x \]

[In]

int(-(exp(x)*(6*x - 3*x^2 + 12) + exp((176*x)/3)*exp(x)*(349*x + 6))/(exp((352*x)/3)*(18*x^3 + 9*x^4) + 3*x^3*
exp(176*x) + exp((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5) + 24*x^3 + 36*x^4 + 18*x^5 + 3*x^6),x)

[Out]

int(-(exp(x)*(6*x - 3*x^2 + 12) + exp((176*x)/3)*exp(x)*(349*x + 6))/(exp((352*x)/3)*(18*x^3 + 9*x^4) + 3*x^3*
exp(176*x) + exp((176*x)/3)*(36*x^3 + 36*x^4 + 9*x^5) + 24*x^3 + 36*x^4 + 18*x^5 + 3*x^6), x)