Integrand size = 147, antiderivative size = 33 \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=\frac {3}{x-e^x x+\frac {1}{2} \left (-e^x+\frac {x^2}{(49-x)^2}\right )} \]
Time = 4.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=-\frac {6 (-49+x)^2}{e^x (-49+x)^2 (1+2 x)+x \left (-4802+195 x-2 x^2\right )} \]
Integrate[(-69177612 + 5618340*x - 172284*x^2 + 2352*x^3 - 12*x^4 + E^x*(1 03766418 + 60706884*x - 5387844*x^2 + 169344*x^3 - 2334*x^4 + 12*x^5))/(23 059204*x^2 - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + E^x*(-23059204*x - 44240826*x^2 + 3697736*x^3 - 114074*x^4 + 1560*x^5 - 8*x^6) + E^(2*x)*(5 764801 + 22588608*x + 21191226*x^2 - 1824956*x^3 + 56841*x^4 - 780*x^5 + 4 *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.
| \(\displaystyle \int \frac {-12 x^4+2352 x^3-172284 x^2+e^x \left (12 x^5-2334 x^4+169344 x^3-5387844 x^2+60706884 x+103766418\right )+5618340 x-69177612}{4 x^6-780 x^5+57233 x^4-1872780 x^3+23059204 x^2+e^x \left (-8 x^6+1560 x^5-114074 x^4+3697736 x^3-44240826 x^2-23059204 x\right )+e^{2 x} \left (4 x^6-780 x^5+56841 x^4-1824956 x^3+21191226 x^2+22588608 x+5764801\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 (49-x) \left (2 \left (x^3-147 x^2+7154 x-117649\right )-e^x (x-49)^3 (2 x+3)\right )}{\left (x \left (-2 x^2+195 x-4802\right )+e^x (2 x+1) (x-49)^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \int \frac {(49-x) \left (e^x (49-x)^3 (2 x+3)-2 \left (-x^3+147 x^2-7154 x+117649\right )\right )}{\left (e^x (49-x)^2 (2 x+1)-x \left (2 x^2-195 x+4802\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {(2 x+3) (x-49)^2}{(2 x+1) \left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )}+\frac {4 x^6-780 x^5+57037 x^4-1848476 x^3+22086897 x^2+12465992 x-11529602}{(2 x+1) \left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \left (476623 \int \frac {1}{\left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx+11512746 \int \frac {x}{\left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx-938595 \int \frac {x^2}{\left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx+28714 \int \frac {x^3}{\left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx-12006225 \int \frac {1}{(2 x+1) \left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx+\frac {4605}{2} \int \frac {1}{2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x}dx-97 \int \frac {x}{2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x}dx+\int \frac {x^2}{2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x}dx+\frac {9801}{2} \int \frac {1}{(2 x+1) \left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )}dx+2 \int \frac {x^5}{\left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx-391 \int \frac {x^4}{\left (2 e^x x^3-2 x^3-195 e^x x^2+195 x^2+4704 e^x x-4802 x+2401 e^x\right )^2}dx\right )\) |
Int[(-69177612 + 5618340*x - 172284*x^2 + 2352*x^3 - 12*x^4 + E^x*(1037664 18 + 60706884*x - 5387844*x^2 + 169344*x^3 - 2334*x^4 + 12*x^5))/(23059204 *x^2 - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + E^x*(-23059204*x - 4424 0826*x^2 + 3697736*x^3 - 114074*x^4 + 1560*x^5 - 8*x^6) + E^(2*x)*(5764801 + 22588608*x + 21191226*x^2 - 1824956*x^3 + 56841*x^4 - 780*x^5 + 4*x^6)) ,x]
3.13.62.3.1 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 = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(-\frac {6 \left (x -49\right )^{2}}{2 \,{\mathrm e}^{x} x^{3}-195 \,{\mathrm e}^{x} x^{2}-2 x^{3}+4704 \,{\mathrm e}^{x} x +195 x^{2}+2401 \,{\mathrm e}^{x}-4802 x}\) | \(47\) |
| norman | \(\frac {-6 x^{2}+588 x -14406}{2 \,{\mathrm e}^{x} x^{3}-195 \,{\mathrm e}^{x} x^{2}-2 x^{3}+4704 \,{\mathrm e}^{x} x +195 x^{2}+2401 \,{\mathrm e}^{x}-4802 x}\) | \(51\) |
| parallelrisch | \(\frac {-12 x^{2}+1176 x -28812}{4 \,{\mathrm e}^{x} x^{3}-390 \,{\mathrm e}^{x} x^{2}-4 x^{3}+9408 \,{\mathrm e}^{x} x +390 x^{2}+4802 \,{\mathrm e}^{x}-9604 x}\) | \(52\) |
int(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*exp(x)- 12*x^4+2352*x^3-172284*x^2+5618340*x-69177612)/((4*x^6-780*x^5+56841*x^4-1 824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*x^5-1140 74*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+57233*x^4 -1872780*x^3+23059204*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=\frac {6 \, {\left (x^{2} - 98 \, x + 2401\right )}}{2 \, x^{3} - 195 \, x^{2} - {\left (2 \, x^{3} - 195 \, x^{2} + 4704 \, x + 2401\right )} e^{x} + 4802 \, x} \]
integrate(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*e xp(x)-12*x^4+2352*x^3-172284*x^2+5618340*x-69177612)/((4*x^6-780*x^5+56841 *x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*x^ 5-114074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+572 33*x^4-1872780*x^3+23059204*x^2),x, algorithm=\
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=\frac {- 6 x^{2} + 588 x - 14406}{- 2 x^{3} + 195 x^{2} - 4802 x + \left (2 x^{3} - 195 x^{2} + 4704 x + 2401\right ) e^{x}} \]
integrate(((12*x**5-2334*x**4+169344*x**3-5387844*x**2+60706884*x+10376641 8)*exp(x)-12*x**4+2352*x**3-172284*x**2+5618340*x-69177612)/((4*x**6-780*x **5+56841*x**4-1824956*x**3+21191226*x**2+22588608*x+5764801)*exp(x)**2+(- 8*x**6+1560*x**5-114074*x**4+3697736*x**3-44240826*x**2-23059204*x)*exp(x) +4*x**6-780*x**5+57233*x**4-1872780*x**3+23059204*x**2),x)
(-6*x**2 + 588*x - 14406)/(-2*x**3 + 195*x**2 - 4802*x + (2*x**3 - 195*x** 2 + 4704*x + 2401)*exp(x))
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=\frac {6 \, {\left (x^{2} - 98 \, x + 2401\right )}}{2 \, x^{3} - 195 \, x^{2} - {\left (2 \, x^{3} - 195 \, x^{2} + 4704 \, x + 2401\right )} e^{x} + 4802 \, x} \]
integrate(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*e xp(x)-12*x^4+2352*x^3-172284*x^2+5618340*x-69177612)/((4*x^6-780*x^5+56841 *x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*x^ 5-114074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+572 33*x^4-1872780*x^3+23059204*x^2),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=-\frac {6 \, {\left (x^{2} - 98 \, x + 2401\right )}}{2 \, x^{3} e^{x} - 2 \, x^{3} - 195 \, x^{2} e^{x} + 195 \, x^{2} + 4704 \, x e^{x} - 4802 \, x + 2401 \, e^{x}} \]
integrate(((12*x^5-2334*x^4+169344*x^3-5387844*x^2+60706884*x+103766418)*e xp(x)-12*x^4+2352*x^3-172284*x^2+5618340*x-69177612)/((4*x^6-780*x^5+56841 *x^4-1824956*x^3+21191226*x^2+22588608*x+5764801)*exp(x)^2+(-8*x^6+1560*x^ 5-114074*x^4+3697736*x^3-44240826*x^2-23059204*x)*exp(x)+4*x^6-780*x^5+572 33*x^4-1872780*x^3+23059204*x^2),x, algorithm=\
-6*(x^2 - 98*x + 2401)/(2*x^3*e^x - 2*x^3 - 195*x^2*e^x + 195*x^2 + 4704*x *e^x - 4802*x + 2401*e^x)
Timed out. \[ \int \frac {-69177612+5618340 x-172284 x^2+2352 x^3-12 x^4+e^x \left (103766418+60706884 x-5387844 x^2+169344 x^3-2334 x^4+12 x^5\right )}{23059204 x^2-1872780 x^3+57233 x^4-780 x^5+4 x^6+e^x \left (-23059204 x-44240826 x^2+3697736 x^3-114074 x^4+1560 x^5-8 x^6\right )+e^{2 x} \left (5764801+22588608 x+21191226 x^2-1824956 x^3+56841 x^4-780 x^5+4 x^6\right )} \, dx=\int \frac {5618340\,x+{\mathrm {e}}^x\,\left (12\,x^5-2334\,x^4+169344\,x^3-5387844\,x^2+60706884\,x+103766418\right )-172284\,x^2+2352\,x^3-12\,x^4-69177612}{23059204\,x^2-{\mathrm {e}}^x\,\left (8\,x^6-1560\,x^5+114074\,x^4-3697736\,x^3+44240826\,x^2+23059204\,x\right )-1872780\,x^3+57233\,x^4-780\,x^5+4\,x^6+{\mathrm {e}}^{2\,x}\,\left (4\,x^6-780\,x^5+56841\,x^4-1824956\,x^3+21191226\,x^2+22588608\,x+5764801\right )} \,d x \]
int((5618340*x + exp(x)*(60706884*x - 5387844*x^2 + 169344*x^3 - 2334*x^4 + 12*x^5 + 103766418) - 172284*x^2 + 2352*x^3 - 12*x^4 - 69177612)/(230592 04*x^2 - exp(x)*(23059204*x + 44240826*x^2 - 3697736*x^3 + 114074*x^4 - 15 60*x^5 + 8*x^6) - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + exp(2*x)*(22 588608*x + 21191226*x^2 - 1824956*x^3 + 56841*x^4 - 780*x^5 + 4*x^6 + 5764 801)),x)
int((5618340*x + exp(x)*(60706884*x - 5387844*x^2 + 169344*x^3 - 2334*x^4 + 12*x^5 + 103766418) - 172284*x^2 + 2352*x^3 - 12*x^4 - 69177612)/(230592 04*x^2 - exp(x)*(23059204*x + 44240826*x^2 - 3697736*x^3 + 114074*x^4 - 15 60*x^5 + 8*x^6) - 1872780*x^3 + 57233*x^4 - 780*x^5 + 4*x^6 + exp(2*x)*(22 588608*x + 21191226*x^2 - 1824956*x^3 + 56841*x^4 - 780*x^5 + 4*x^6 + 5764 801)), x)