3.103.47 \(\int \frac {e^{2 x} (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} (27+6831 x+1215 x^2-19 x^3+2 x^4))}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 (2304+768 x+64 x^2)}{9-6 x+x^2}} (-27 x^2+27 x^3-9 x^4+x^5)+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} (162 x-216 x^2+108 x^3-24 x^4+2 x^5)} \, dx\)

Optimal. Leaf size=34 \[ \frac {e^{2 x}}{-3+x+e^{16 \left (3+\frac {3+5 x}{3-x}\right )^2} x} \]

________________________________________________________________________________________

Rubi [F]  time = 10.49, 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^{2 x} \left (189-243 x+117 x^2-25 x^3+2 x^4+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5+e^{\frac {2 \left (2304+768 x+64 x^2\right )}{9-6 x+x^2}} \left (-27 x^2+27 x^3-9 x^4+x^5\right )+e^{\frac {2304+768 x+64 x^2}{9-6 x+x^2}} \left (162 x-216 x^2+108 x^3-24 x^4+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(189 - 243*x + 117*x^2 - 25*x^3 + 2*x^4 + E^((2304 + 768*x + 64*x^2)/(9 - 6*x + x^2))*(27 + 6831*
x + 1215*x^2 - 19*x^3 + 2*x^4)))/(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 + E^((2*(2304 + 768*x + 64*x^
2))/(9 - 6*x + x^2))*(-27*x^2 + 27*x^3 - 9*x^4 + x^5) + E^((2304 + 768*x + 64*x^2)/(9 - 6*x + x^2))*(162*x - 2
16*x^2 + 108*x^3 - 24*x^4 + 2*x^5)),x]

[Out]

-10368*Defer[Int][E^(2*x)/((-3 + x)^2*(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x)^2), x] - 1152*Defer[Int][E^(2
*x)/((-3 + x)*(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x)^2), x] - 3*Defer[Int][E^(2*x)/(x*(-3 + x + E^((64*(6
+ x)^2)/(-3 + x)^2)*x)^2), x] + 2*Defer[Int][E^(2*x)/(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x), x] + 10368*De
fer[Int][E^(2*x)/((-3 + x)^3*(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x)), x] + 1152*Defer[Int][E^(2*x)/((-3 +
x)^2*(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x)), x] - Defer[Int][E^(2*x)/(x*(-3 + x + E^((64*(6 + x)^2)/(-3 +
x)^2)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (-(-3+x)^3 (-7+2 x)-e^{\frac {64 (6+x)^2}{(-3+x)^2}} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )\right )}{(3-x)^3 \left (3-x-e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2} \, dx\\ &=\int \left (-\frac {3 e^{2 x} \left (9+2298 x+385 x^2\right )}{(-3+x)^2 x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2}+\frac {e^{2 x} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )}{(-3+x)^3 x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )}\right ) \, dx\\ &=-\left (3 \int \frac {e^{2 x} \left (9+2298 x+385 x^2\right )}{(-3+x)^2 x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2} \, dx\right )+\int \frac {e^{2 x} \left (27+6831 x+1215 x^2-19 x^3+2 x^4\right )}{(-3+x)^3 x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )} \, dx\\ &=-\left (3 \int \left (\frac {3456 e^{2 x}}{(-3+x)^2 \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2}+\frac {384 e^{2 x}}{(-3+x) \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2}+\frac {e^{2 x}}{x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2}\right ) \, dx\right )+\int \left (\frac {2 e^{2 x}}{-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x}+\frac {10368 e^{2 x}}{(-3+x)^3 \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )}+\frac {1152 e^{2 x}}{(-3+x)^2 \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )}-\frac {e^{2 x}}{x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )}\right ) \, dx\\ &=2 \int \frac {e^{2 x}}{-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x} \, dx-3 \int \frac {e^{2 x}}{x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2} \, dx-1152 \int \frac {e^{2 x}}{(-3+x) \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2} \, dx+1152 \int \frac {e^{2 x}}{(-3+x)^2 \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )} \, dx-10368 \int \frac {e^{2 x}}{(-3+x)^2 \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )^2} \, dx+10368 \int \frac {e^{2 x}}{(-3+x)^3 \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )} \, dx-\int \frac {e^{2 x}}{x \left (-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.80, size = 27, normalized size = 0.79 \begin {gather*} \frac {e^{2 x}}{-3+x+e^{\frac {64 (6+x)^2}{(-3+x)^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(189 - 243*x + 117*x^2 - 25*x^3 + 2*x^4 + E^((2304 + 768*x + 64*x^2)/(9 - 6*x + x^2))*(27 +
6831*x + 1215*x^2 - 19*x^3 + 2*x^4)))/(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 + E^((2*(2304 + 768*x +
64*x^2))/(9 - 6*x + x^2))*(-27*x^2 + 27*x^3 - 9*x^4 + x^5) + E^((2304 + 768*x + 64*x^2)/(9 - 6*x + x^2))*(162
*x - 216*x^2 + 108*x^3 - 24*x^4 + 2*x^5)),x]

[Out]

E^(2*x)/(-3 + x + E^((64*(6 + x)^2)/(-3 + x)^2)*x)

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 33, normalized size = 0.97 \begin {gather*} \frac {e^{\left (2 \, x\right )}}{x e^{\left (\frac {64 \, {\left (x^{2} + 12 \, x + 36\right )}}{x^{2} - 6 \, x + 9}\right )} + x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+2*x^4-25*x^3+117*x^2-243*x+1
89)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+1
62*x)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, algorithm="fricas")

[Out]

e^(2*x)/(x*e^(64*(x^2 + 12*x + 36)/(x^2 - 6*x + 9)) + x - 3)

________________________________________________________________________________________

giac [A]  time = 0.60, size = 34, normalized size = 1.00 \begin {gather*} \frac {e^{\left (2 \, x\right )}}{x e^{\left (-\frac {192 \, {\left (x^{2} - 12 \, x\right )}}{x^{2} - 6 \, x + 9} + 256\right )} + x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+2*x^4-25*x^3+117*x^2-243*x+1
89)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+1
62*x)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, algorithm="giac")

[Out]

e^(2*x)/(x*e^(-192*(x^2 - 12*x)/(x^2 - 6*x + 9) + 256) + x - 3)

________________________________________________________________________________________

maple [A]  time = 0.44, size = 26, normalized size = 0.76

method	result	size
risch	\(\frac {{\mathrm e}^{2 x}}{x \,{\mathrm e}^{\frac {64 \left (x +6\right )^{2}}{\left (x -3\right )^{2}}}+x -3}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+2*x^4-25*x^3+117*x^2-243*x+189)*ex
p(x)^2/((x^5-9*x^4+27*x^3-27*x^2)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+162*x)*
exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-243),x,method=_RETURNVERBOSE)

[Out]

exp(2*x)/(x*exp(64*(x+6)^2/(x-3)^2)+x-3)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 34, normalized size = 1.00 \begin {gather*} \frac {e^{\left (2 \, x\right )}}{x e^{\left (\frac {5184}{x^{2} - 6 \, x + 9} + \frac {1152}{x - 3} + 64\right )} + x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-19*x^3+1215*x^2+6831*x+27)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+2*x^4-25*x^3+117*x^2-243*x+1
89)*exp(x)^2/((x^5-9*x^4+27*x^3-27*x^2)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))^2+(2*x^5-24*x^4+108*x^3-216*x^2+1
62*x)*exp((64*x^2+768*x+2304)/(x^2-6*x+9))+x^5-15*x^4+90*x^3-270*x^2+405*x-243),x, algorithm="maxima")

[Out]

e^(2*x)/(x*e^(5184/(x^2 - 6*x + 9) + 1152/(x - 3) + 64) + x - 3)

________________________________________________________________________________________

mupad [B]  time = 6.22, size = 116, normalized size = 3.41 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (385\,x^2+2298\,x+9\right )\,{\left (x^3-9\,x^2+27\,x-27\right )}^2}{{\left (x-3\right )}^2\,\left (x+x\,{\mathrm {e}}^{\frac {768\,x}{x^2-6\,x+9}+\frac {2304}{x^2-6\,x+9}+\frac {64\,x^2}{x^2-6\,x+9}}-3\right )\,\left (385\,x^6-2322\,x^5-6777\,x^4+82404\,x^3-216513\,x^2+185166\,x+729\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(117*x^2 - 243*x - 25*x^3 + 2*x^4 + exp((768*x + 64*x^2 + 2304)/(x^2 - 6*x + 9))*(6831*x + 1215*
x^2 - 19*x^3 + 2*x^4 + 27) + 189))/(405*x + exp((768*x + 64*x^2 + 2304)/(x^2 - 6*x + 9))*(162*x - 216*x^2 + 10
8*x^3 - 24*x^4 + 2*x^5) - exp((2*(768*x + 64*x^2 + 2304))/(x^2 - 6*x + 9))*(27*x^2 - 27*x^3 + 9*x^4 - x^5) - 2
70*x^2 + 90*x^3 - 15*x^4 + x^5 - 243),x)

[Out]

(exp(2*x)*(2298*x + 385*x^2 + 9)*(27*x - 9*x^2 + x^3 - 27)^2)/((x - 3)^2*(x + x*exp((768*x)/(x^2 - 6*x + 9) +
2304/(x^2 - 6*x + 9) + (64*x^2)/(x^2 - 6*x + 9)) - 3)*(185166*x - 216513*x^2 + 82404*x^3 - 6777*x^4 - 2322*x^5
+ 385*x^6 + 729))

________________________________________________________________________________________

sympy [A]  time = 0.45, size = 29, normalized size = 0.85 \begin {gather*} \frac {e^{2 x}}{x e^{\frac {64 x^{2} + 768 x + 2304}{x^{2} - 6 x + 9}} + x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4-19*x**3+1215*x**2+6831*x+27)*exp((64*x**2+768*x+2304)/(x**2-6*x+9))+2*x**4-25*x**3+117*x**2
-243*x+189)*exp(x)**2/((x**5-9*x**4+27*x**3-27*x**2)*exp((64*x**2+768*x+2304)/(x**2-6*x+9))**2+(2*x**5-24*x**4
+108*x**3-216*x**2+162*x)*exp((64*x**2+768*x+2304)/(x**2-6*x+9))+x**5-15*x**4+90*x**3-270*x**2+405*x-243),x)

[Out]

exp(2*x)/(x*exp((64*x**2 + 768*x + 2304)/(x**2 - 6*x + 9)) + x - 3)

________________________________________________________________________________________