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

Optimal. Leaf size=19 \[ \frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \]

________________________________________________________________________________________

Rubi [F]  time = 1.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.38, size = 19, normalized size = 1.00 \begin {gather*} \frac {e^x}{x^2 \left (2+e^{176 x/3}+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [B]  time = 0.71, size = 42, normalized size = 2.21 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.08, size = 16, normalized size = 0.84

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

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maxima [B]  time = 1.21, size = 42, normalized size = 2.21 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________