Integrand size = 146, antiderivative size = 29 \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\frac {1+x}{-x \left (4 e^x+x\right )+\frac {3}{x^2 (4+4 x)}} \end {dmath*}
Time = 5.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {8 x^2 (1+x)^2}{-6+8 x^4+8 x^5+32 e^x x^3 (1+x)} \end {dmath*}
Integrate[(24*x + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8 + E^ x*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8))/(9 - 24*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + E^(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + E^x*(- 96*x^3 - 96*x^4 + 128*x^7 + 256*x^8 + 128*x^9)),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.
\(\displaystyle \int \frac {16 x^8+64 x^7+80 x^6+32 x^5+48 x^3+72 x^2+e^x \left (64 x^8+192 x^7+256 x^6+192 x^5+64 x^4\right )+24 x}{16 x^{10}+32 x^9+16 x^8-24 x^5-24 x^4+e^{2 x} \left (256 x^8+512 x^7+256 x^6\right )+e^x \left (128 x^9+256 x^8+128 x^7-96 x^4-96 x^3\right )+9} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 x (x+1) \left (\left (8 e^x+2\right ) x^6+2 \left (8 e^x+3\right ) x^5+4 \left (4 e^x+1\right ) x^4+8 e^x x^3+6 x+3\right )}{\left (-4 x^5-4 x^4-16 e^x (x+1) x^3+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 \int \frac {x (x+1) \left (2 \left (1+4 e^x\right ) x^6+2 \left (3+8 e^x\right ) x^5+4 \left (1+4 e^x\right ) x^4+8 e^x x^3+6 x+3\right )}{\left (-4 x^5-4 x^4-16 e^x (x+1) x^3+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 8 \int \left (\frac {x \left (x^3+2 x^2+2 x+1\right )}{2 \left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )}-\frac {x \left (4 x^8+8 x^7-8 x^5-4 x^4-3 x^3-18 x^2-24 x-9\right )}{2 \left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \left (\frac {9}{2} \int \frac {x}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+9 \int \frac {x^3}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+\frac {3}{2} \int \frac {x^4}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+2 \int \frac {x^5}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+\frac {1}{2} \int \frac {x}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx+\int \frac {x^3}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx+\frac {1}{2} \int \frac {x^4}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx-2 \int \frac {x^9}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx-4 \int \frac {x^8}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+4 \int \frac {x^6}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+12 \int \frac {x^2}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+\int \frac {x^2}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx\right )\) |
Int[(24*x + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8 + E^x*(64* x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8))/(9 - 24*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + E^(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + E^x*(-96*x^3 - 96*x^4 + 128*x^7 + 256*x^8 + 128*x^9)),x]
3.9.57.3.1 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]]
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34
method | result | size |
risch | \(-\frac {4 x^{2} \left (1+x \right )^{2}}{16 \,{\mathrm e}^{x} x^{4}+4 x^{5}+16 \,{\mathrm e}^{x} x^{3}+4 x^{4}-3}\) | \(39\) |
norman | \(\frac {-4 x^{4}-8 x^{3}-4 x^{2}}{16 \,{\mathrm e}^{x} x^{4}+4 x^{5}+16 \,{\mathrm e}^{x} x^{3}+4 x^{4}-3}\) | \(46\) |
parallelrisch | \(\frac {-64 x^{4}-128 x^{3}-64 x^{2}}{256 \,{\mathrm e}^{x} x^{4}+64 x^{5}+256 \,{\mathrm e}^{x} x^{3}+64 x^{4}-48}\) | \(47\) |
int(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+3 2*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x^9+256 *x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x^4+9), x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 4 \, x^{4} + 16 \, {\left (x^{4} + x^{3}\right )} e^{x} - 3} \end {dmath*}
integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80 *x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x ^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x ^4+9),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\frac {- 4 x^{4} - 8 x^{3} - 4 x^{2}}{4 x^{5} + 4 x^{4} + \left (16 x^{4} + 16 x^{3}\right ) e^{x} - 3} \end {dmath*}
integrate(((64*x**8+192*x**7+256*x**6+192*x**5+64*x**4)*exp(x)+16*x**8+64* x**7+80*x**6+32*x**5+48*x**3+72*x**2+24*x)/((256*x**8+512*x**7+256*x**6)*e xp(x)**2+(128*x**9+256*x**8+128*x**7-96*x**4-96*x**3)*exp(x)+16*x**10+32*x **9+16*x**8-24*x**5-24*x**4+9),x)
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 4 \, x^{4} + 16 \, {\left (x^{4} + x^{3}\right )} e^{x} - 3} \end {dmath*}
integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80 *x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x ^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x ^4+9),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 16 \, x^{4} e^{x} + 4 \, x^{4} + 16 \, x^{3} e^{x} - 3} \end {dmath*}
integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80 *x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x ^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x ^4+9),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\int \frac {24\,x+{\mathrm {e}}^x\,\left (64\,x^8+192\,x^7+256\,x^6+192\,x^5+64\,x^4\right )+72\,x^2+48\,x^3+32\,x^5+80\,x^6+64\,x^7+16\,x^8}{{\mathrm {e}}^{2\,x}\,\left (256\,x^8+512\,x^7+256\,x^6\right )+{\mathrm {e}}^x\,\left (128\,x^9+256\,x^8+128\,x^7-96\,x^4-96\,x^3\right )-24\,x^4-24\,x^5+16\,x^8+32\,x^9+16\,x^{10}+9} \,d x \end {dmath*}
int((24*x + exp(x)*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8) + 72*x^ 2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8)/(exp(2*x)*(256*x^6 + 512*x ^7 + 256*x^8) + exp(x)*(128*x^7 - 96*x^4 - 96*x^3 + 256*x^8 + 128*x^9) - 2 4*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + 9),x)