\(\int \frac {e^{2+2 x} (216+180 x+36 x^2)+e^{3+3 x} (-24-36 x-18 x^2-3 x^3)}{1728+e^{1+x} (2592+2160 x+432 x^2)+e^{2+2 x} (1296+2160 x+1332 x^2+360 x^3+36 x^4)+e^{3+3 x} (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6)} \, dx\) [10097]

Optimal result

Integrand size = 127, antiderivative size = 23 \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\frac {3}{2 \left (3+x+\frac {12 e^{-1-x}}{2+x}\right )^2} \]

[Out]

3/2/(12/exp(1+x)/(2+x)+3+x)^2

Rubi [F]

\[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx \]

[In]

Int[(E^(2 + 2*x)*(216 + 180*x + 36*x^2) + E^(3 + 3*x)*(-24 - 36*x - 18*x^2 - 3*x^3))/(1728 + E^(1 + x)*(2592 +
2160*x + 432*x^2) + E^(2 + 2*x)*(1296 + 2160*x + 1332*x^2 + 360*x^3 + 36*x^4) + E^(3 + 3*x)*(216 + 540*x + 55
8*x^2 + 305*x^3 + 93*x^4 + 15*x^5 + x^6)),x]

[Out]

252*Defer[Int][E^(2 + 2*x)/(12 + E^(1 + x)*(6 + 5*x + x^2))^3, x] + 216*Defer[Int][(E^(2 + 2*x)*x)/(12 + E^(1
+ x)*(6 + 5*x + x^2))^3, x] + 36*Defer[Int][(E^(2 + 2*x)*x^2)/(12 + E^(1 + x)*(6 + 5*x + x^2))^3, x] + 36*Defe
r[Int][E^(2 + 2*x)/((3 + x)*(12 + E^(1 + x)*(6 + 5*x + x^2))^3), x] - 3*Defer[Int][E^(2 + 2*x)/(12 + E^(1 + x)
*(6 + 5*x + x^2))^2, x] - 3*Defer[Int][(E^(2 + 2*x)*x)/(12 + E^(1 + x)*(6 + 5*x + x^2))^2, x] - 3*Defer[Int][E
^(2 + 2*x)/((3 + x)*(12 + E^(1 + x)*(6 + 5*x + x^2))^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^{2+2 x} (2+x) \left (-e^{1+x} (2+x)^2+12 (3+x)\right )}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx \\ & = 3 \int \frac {e^{2+2 x} (2+x) \left (-e^{1+x} (2+x)^2+12 (3+x)\right )}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx \\ & = 3 \int \left (-\frac {e^{2+2 x} (2+x)^2}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}+\frac {12 e^{2+2 x} \left (22+25 x+9 x^2+x^3\right )}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}\right ) \, dx \\ & = -\left (3 \int \frac {e^{2+2 x} (2+x)^2}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx\right )+36 \int \frac {e^{2+2 x} \left (22+25 x+9 x^2+x^3\right )}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx \\ & = -\left (3 \int \frac {e^{2+2 x} (2+x)^2}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx\right )+36 \int \frac {e^{2+2 x} \left (22+25 x+9 x^2+x^3\right )}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx \\ & = -\left (3 \int \left (\frac {e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}+\frac {e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}+\frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}\right ) \, dx\right )+36 \int \left (\frac {7 e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}+\frac {6 e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}+\frac {e^{2+2 x} x^2}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}+\frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}\right ) \, dx \\ & = -\left (3 \int \frac {e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx\right )-3 \int \frac {e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx-3 \int \frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx+36 \int \frac {e^{2+2 x} x^2}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx+36 \int \frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx+216 \int \frac {e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx+252 \int \frac {e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx \\ & = -\left (3 \int \frac {e^{2+2 x}}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx\right )-3 \int \frac {e^{2+2 x} x}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx-3 \int \frac {e^{2+2 x}}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx+36 \int \frac {e^{2+2 x} x^2}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx+36 \int \frac {e^{2+2 x}}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx+216 \int \frac {e^{2+2 x} x}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx+252 \int \frac {e^{2+2 x}}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 11.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\frac {3 e^{2+2 x} (2+x)^2}{2 \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \]

[In]

Integrate[(E^(2 + 2*x)*(216 + 180*x + 36*x^2) + E^(3 + 3*x)*(-24 - 36*x - 18*x^2 - 3*x^3))/(1728 + E^(1 + x)*(
2592 + 2160*x + 432*x^2) + E^(2 + 2*x)*(1296 + 2160*x + 1332*x^2 + 360*x^3 + 36*x^4) + E^(3 + 3*x)*(216 + 540*
x + 558*x^2 + 305*x^3 + 93*x^4 + 15*x^5 + x^6)),x]

[Out]

(3*E^(2 + 2*x)*(2 + x)^2)/(2*(12 + E^(1 + x)*(6 + 5*x + x^2))^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).

Time = 1.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83

method	result	size
risch	\(\frac {3 \left (x^{2}+4 x +4\right ) {\mathrm e}^{2+2 x}}{2 \left (x^{2} {\mathrm e}^{1+x}+5 x \,{\mathrm e}^{1+x}+6 \,{\mathrm e}^{1+x}+12\right )^{2}}\)	\(42\)
norman	\(\frac {6 \,{\mathrm e}^{2+2 x}+6 x \,{\mathrm e}^{2+2 x}+\frac {3 x^{2} {\mathrm e}^{2+2 x}}{2}}{\left (x^{2} {\mathrm e}^{1+x}+5 x \,{\mathrm e}^{1+x}+6 \,{\mathrm e}^{1+x}+12\right )^{2}}\)	\(56\)
parallelrisch	\(\frac {3 x^{2} {\mathrm e}^{2+2 x}+12 x \,{\mathrm e}^{2+2 x}+12 \,{\mathrm e}^{2+2 x}}{2 x^{4} {\mathrm e}^{2+2 x}+20 x^{3} {\mathrm e}^{2+2 x}+74 x^{2} {\mathrm e}^{2+2 x}+48 x^{2} {\mathrm e}^{1+x}+120 x \,{\mathrm e}^{2+2 x}+240 x \,{\mathrm e}^{1+x}+72 \,{\mathrm e}^{2+2 x}+288 \,{\mathrm e}^{1+x}+288}\)	\(107\)

[In]

int(((-3*x^3-18*x^2-36*x-24)*exp(1+x)^3+(36*x^2+180*x+216)*exp(1+x)^2)/((x^6+15*x^5+93*x^4+305*x^3+558*x^2+540
*x+216)*exp(1+x)^3+(36*x^4+360*x^3+1332*x^2+2160*x+1296)*exp(1+x)^2+(432*x^2+2160*x+2592)*exp(1+x)+1728),x,met
hod=_RETURNVERBOSE)

[Out]

3/2*(x^2+4*x+4)*exp(2+2*x)/(x^2*exp(1+x)+5*x*exp(1+x)+6*exp(1+x)+12)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\frac {3 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x + 2\right )}}{2 \, {\left ({\left (x^{4} + 10 \, x^{3} + 37 \, x^{2} + 60 \, x + 36\right )} e^{\left (2 \, x + 2\right )} + 24 \, {\left (x^{2} + 5 \, x + 6\right )} e^{\left (x + 1\right )} + 144\right )}} \]

[In]

integrate(((-3*x^3-18*x^2-36*x-24)*exp(1+x)^3+(36*x^2+180*x+216)*exp(1+x)^2)/((x^6+15*x^5+93*x^4+305*x^3+558*x
^2+540*x+216)*exp(1+x)^3+(36*x^4+360*x^3+1332*x^2+2160*x+1296)*exp(1+x)^2+(432*x^2+2160*x+2592)*exp(1+x)+1728)
,x, algorithm="fricas")

[Out]

3/2*(x^2 + 4*x + 4)*e^(2*x + 2)/((x^4 + 10*x^3 + 37*x^2 + 60*x + 36)*e^(2*x + 2) + 24*(x^2 + 5*x + 6)*e^(x + 1
) + 144)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (19) = 38\).

Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.35 \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\frac {\left (- 36 x^{2} - 180 x - 216\right ) e^{x + 1} - 216}{144 x^{2} + 864 x + \left (24 x^{4} + 264 x^{3} + 1080 x^{2} + 1944 x + 1296\right ) e^{x + 1} + \left (x^{6} + 16 x^{5} + 106 x^{4} + 372 x^{3} + 729 x^{2} + 756 x + 324\right ) e^{2 x + 2} + 1296} + \frac {3}{2 x^{2} + 12 x + 18} \]

[In]

integrate(((-3*x**3-18*x**2-36*x-24)*exp(1+x)**3+(36*x**2+180*x+216)*exp(1+x)**2)/((x**6+15*x**5+93*x**4+305*x
**3+558*x**2+540*x+216)*exp(1+x)**3+(36*x**4+360*x**3+1332*x**2+2160*x+1296)*exp(1+x)**2+(432*x**2+2160*x+2592
)*exp(1+x)+1728),x)

[Out]

((-36*x**2 - 180*x - 216)*exp(x + 1) - 216)/(144*x**2 + 864*x + (24*x**4 + 264*x**3 + 1080*x**2 + 1944*x + 129
6)*exp(x + 1) + (x**6 + 16*x**5 + 106*x**4 + 372*x**3 + 729*x**2 + 756*x + 324)*exp(2*x + 2) + 1296) + 3/(2*x*
*2 + 12*x + 18)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.52 \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\frac {3 \, {\left (x^{2} e^{2} + 4 \, x e^{2} + 4 \, e^{2}\right )} e^{\left (2 \, x\right )}}{2 \, {\left ({\left (x^{4} e^{2} + 10 \, x^{3} e^{2} + 37 \, x^{2} e^{2} + 60 \, x e^{2} + 36 \, e^{2}\right )} e^{\left (2 \, x\right )} + 24 \, {\left (x^{2} e + 5 \, x e + 6 \, e\right )} e^{x} + 144\right )}} \]

[In]

integrate(((-3*x^3-18*x^2-36*x-24)*exp(1+x)^3+(36*x^2+180*x+216)*exp(1+x)^2)/((x^6+15*x^5+93*x^4+305*x^3+558*x
^2+540*x+216)*exp(1+x)^3+(36*x^4+360*x^3+1332*x^2+2160*x+1296)*exp(1+x)^2+(432*x^2+2160*x+2592)*exp(1+x)+1728)
,x, algorithm="maxima")

[Out]

3/2*(x^2*e^2 + 4*x*e^2 + 4*e^2)*e^(2*x)/((x^4*e^2 + 10*x^3*e^2 + 37*x^2*e^2 + 60*x*e^2 + 36*e^2)*e^(2*x) + 24*
(x^2*e + 5*x*e + 6*e)*e^x + 144)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (20) = 40\).

Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.57 \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\frac {3 \, {\left (x^{2} e^{\left (2 \, x + 2\right )} + 4 \, x e^{\left (2 \, x + 2\right )} + 4 \, e^{\left (2 \, x + 2\right )}\right )}}{2 \, {\left (x^{4} e^{\left (2 \, x + 2\right )} + 10 \, x^{3} e^{\left (2 \, x + 2\right )} + 37 \, x^{2} e^{\left (2 \, x + 2\right )} + 24 \, x^{2} e^{\left (x + 1\right )} + 60 \, x e^{\left (2 \, x + 2\right )} + 120 \, x e^{\left (x + 1\right )} + 36 \, e^{\left (2 \, x + 2\right )} + 144 \, e^{\left (x + 1\right )} + 144\right )}} \]

[In]

integrate(((-3*x^3-18*x^2-36*x-24)*exp(1+x)^3+(36*x^2+180*x+216)*exp(1+x)^2)/((x^6+15*x^5+93*x^4+305*x^3+558*x
^2+540*x+216)*exp(1+x)^3+(36*x^4+360*x^3+1332*x^2+2160*x+1296)*exp(1+x)^2+(432*x^2+2160*x+2592)*exp(1+x)+1728)
,x, algorithm="giac")

[Out]

3/2*(x^2*e^(2*x + 2) + 4*x*e^(2*x + 2) + 4*e^(2*x + 2))/(x^4*e^(2*x + 2) + 10*x^3*e^(2*x + 2) + 37*x^2*e^(2*x
+ 2) + 24*x^2*e^(x + 1) + 60*x*e^(2*x + 2) + 120*x*e^(x + 1) + 36*e^(2*x + 2) + 144*e^(x + 1) + 144)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x+2}\,\left (36\,x^2+180\,x+216\right )-{\mathrm {e}}^{3\,x+3}\,\left (3\,x^3+18\,x^2+36\,x+24\right )}{{\mathrm {e}}^{x+1}\,\left (432\,x^2+2160\,x+2592\right )+{\mathrm {e}}^{2\,x+2}\,\left (36\,x^4+360\,x^3+1332\,x^2+2160\,x+1296\right )+{\mathrm {e}}^{3\,x+3}\,\left (x^6+15\,x^5+93\,x^4+305\,x^3+558\,x^2+540\,x+216\right )+1728} \,d x \]

[In]

int((exp(2*x + 2)*(180*x + 36*x^2 + 216) - exp(3*x + 3)*(36*x + 18*x^2 + 3*x^3 + 24))/(exp(x + 1)*(2160*x + 43
2*x^2 + 2592) + exp(2*x + 2)*(2160*x + 1332*x^2 + 360*x^3 + 36*x^4 + 1296) + exp(3*x + 3)*(540*x + 558*x^2 + 3
05*x^3 + 93*x^4 + 15*x^5 + x^6 + 216) + 1728),x)

[Out]

int((exp(2*x + 2)*(180*x + 36*x^2 + 216) - exp(3*x + 3)*(36*x + 18*x^2 + 3*x^3 + 24))/(exp(x + 1)*(2160*x + 43
2*x^2 + 2592) + exp(2*x + 2)*(2160*x + 1332*x^2 + 360*x^3 + 36*x^4 + 1296) + exp(3*x + 3)*(540*x + 558*x^2 + 3
05*x^3 + 93*x^4 + 15*x^5 + x^6 + 216) + 1728), x)