Integrand size = 127, antiderivative size = 23 \begin {dmath*} \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} \end {dmath*}
Time = 11.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \begin {dmath*} \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} \end {dmath*}
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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{2 x+2} \left (36 x^2+180 x+216\right )+e^{3 x+3} \left (-3 x^3-18 x^2-36 x-24\right )}{e^{x+1} \left (432 x^2+2160 x+2592\right )+e^{2 x+2} \left (36 x^4+360 x^3+1332 x^2+2160 x+1296\right )+e^{3 x+3} \left (x^6+15 x^5+93 x^4+305 x^3+558 x^2+540 x+216\right )+1728} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 e^{2 x+2} (x+2) \left (12 (x+3)-e^{x+1} (x+2)^2\right )}{\left (e^{x+1} \left (x^2+5 x+6\right )+12\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int -\frac {e^{2 x+2} (x+2) \left (e^{x+1} (x+2)^2-12 (x+3)\right )}{\left (e^{x+1} \left (x^2+5 x+6\right )+12\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 \int \frac {e^{2 x+2} (x+2) \left (e^{x+1} (x+2)^2-12 (x+3)\right )}{\left (e^{x+1} \left (x^2+5 x+6\right )+12\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -3 \int \left (\frac {e^{2 x+2} (x+2)^2}{(x+3) \left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^2}-\frac {12 e^{2 x+2} \left (x^3+9 x^2+25 x+22\right )}{(x+3) \left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \left (-84 \int \frac {e^{2 x+2}}{\left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^3}dx-72 \int \frac {e^{2 x+2} x}{\left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^3}dx-12 \int \frac {e^{2 x+2} x^2}{\left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^3}dx-12 \int \frac {e^{2 x+2}}{(x+3) \left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^3}dx+\int \frac {e^{2 x+2}}{\left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^2}dx+\int \frac {e^{2 x+2} x}{\left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^2}dx+\int \frac {e^{2 x+2}}{(x+3) \left (e^{x+1} x^2+5 e^{x+1} x+6 e^{x+1}+12\right )^2}dx\right )\) |
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 + 558* x^2 + 305*x^3 + 93*x^4 + 15*x^5 + x^6)),x]
3.11.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
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 | \(\frac {3 x^{2} {\mathrm e}^{2+2 x}+12 x \,{\mathrm e}^{2+2 x}+12 \,{\mathrm e}^{2+2 x}}{2 x^{4} {\mathrm e}^{2+2 x}+20 x^{3} {\mathrm e}^{2+2 x}+74 x^{2} {\mathrm e}^{2+2 x}+48 x^{2} {\mathrm e}^{1+x}+120 x \,{\mathrm e}^{2+2 x}+240 x \,{\mathrm e}^{1+x}+72 \,{\mathrm e}^{2+2 x}+288 \,{\mathrm e}^{1+x}+288}\) | \(107\) |
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+133 2*x^2+2160*x+1296)*exp(1+x)^2+(432*x^2+2160*x+2592)*exp(1+x)+1728),x,metho d=_RETURNVERBOSE)
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 \begin {dmath*} \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 )}} \end {dmath*}
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=\
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)
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 \begin {dmath*} \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} \end {dmath*}
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+(3 6*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)
((-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 + 1296)*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)
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 \begin {dmath*} \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 )}} \end {dmath*}
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=\
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)
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 \begin {dmath*} \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 )}} \end {dmath*}
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=\
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)
Timed out. \begin {dmath*} \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 \end {dmath*}
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 + 432*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 + 305 *x^3 + 93*x^4 + 15*x^5 + x^6 + 216) + 1728),x)