Integrand size = 126, antiderivative size = 32 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 \left (2+e^x-x-x^2\right )}{-x+e^{x/5} (3+x)} \]
Time = 7.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 \left (2+e^x-x-x^2\right )}{-x+e^{x/5} (3+x)} \]
Integrate[(90 + 30*x + 45*x^2 + 15*x^3 + E^x*(45 - 30*x - 15*x^2) + E^((x + 5*Log[3 + x])/5)*(-93 - 87*x - 3*x^2 + 3*x^3 + E^x*(21 + 12*x)))/(15*x^2 + 5*x^3 + E^((2*(x + 5*Log[3 + x]))/5)*(15 + 5*x) + E^((x + 5*Log[3 + x]) /5)*(-30*x - 10*x^2)),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.
| \(\displaystyle \int \frac {15 x^3+45 x^2+e^x \left (-15 x^2-30 x+45\right )+\left (3 x^3-3 x^2-87 x+e^x (12 x+21)-93\right ) e^{\frac {1}{5} (x+5 \log (x+3))}+30 x+90}{5 x^3+15 x^2+\left (-10 x^2-30 x\right ) e^{\frac {1}{5} (x+5 \log (x+3))}+(5 x+15) e^{\frac {2}{5} (x+5 \log (x+3))}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 \left (5 \left (x^2+2\right )+e^{x/5} \left (x^3-x^2-29 x-31\right )-5 e^x (x-1)+e^{6 x/5} (4 x+7)\right )}{5 \left (x-e^{x/5} (x+3)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^x (1-x)+e^{6 x/5} (4 x+7)+5 \left (x^2+2\right )-e^{x/5} \left (-x^3+x^2+29 x+31\right )}{\left (x-e^{x/5} (x+3)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (-\frac {5 (x-12) x^3}{(x+3)^6}+\frac {e^{x/5} \left (x^2-2 x+45\right ) x^2}{(x+3)^5}+\frac {e^{2 x/5} \left (2 x^2+x+30\right ) x}{(x+3)^4}+\frac {e^{4 x/5} (4 x+7)}{(x+3)^2}+\frac {e^{3 x/5} \left (3 x^2+4 x+15\right )}{(x+3)^3}+\frac {x^8+14 x^7+45 x^6-294 x^5-2865 x^4-10782 x^3-20358 x^2-19602 x-7533}{(x+3)^6 \left (e^{x/5} x-x+3 e^{x/5}\right )}+\frac {x^9+19 x^8+135 x^7+396 x^6-45 x^5-3357 x^4-7668 x^3-3807 x^2+7047 x+7290}{(x+3)^6 \left (e^{x/5} x-x+3 e^{x/5}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{5} \left (\int \frac {x^3}{\left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx+\int \frac {x^2}{\left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx+\int \frac {x^2}{e^{x/5} x-x+3 e^{x/5}}dx+45 \int \frac {1}{\left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx-18 \int \frac {x}{\left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx-3645 \int \frac {1}{(x+3)^6 \left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx+5346 \int \frac {1}{(x+3)^5 \left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx-2592 \int \frac {1}{(x+3)^4 \left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx+135 \int \frac {1}{(x+3)^3 \left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx+315 \int \frac {1}{(x+3)^2 \left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx-180 \int \frac {1}{(x+3) \left (e^{x/5} x-x+3 e^{x/5}\right )^2}dx-18 \int \frac {1}{e^{x/5} x-x+3 e^{x/5}}dx-4 \int \frac {x}{e^{x/5} x-x+3 e^{x/5}}dx+7290 \int \frac {1}{(x+3)^6 \left (e^{x/5} x-x+3 e^{x/5}\right )}dx-9882 \int \frac {1}{(x+3)^5 \left (e^{x/5} x-x+3 e^{x/5}\right )}dx+4995 \int \frac {1}{(x+3)^4 \left (e^{x/5} x-x+3 e^{x/5}\right )}dx-1080 \int \frac {1}{(x+3)^3 \left (e^{x/5} x-x+3 e^{x/5}\right )}dx+60 \int \frac {1}{(x+3)^2 \left (e^{x/5} x-x+3 e^{x/5}\right )}dx+30 \int \frac {1}{(x+3) \left (e^{x/5} x-x+3 e^{x/5}\right )}dx+\frac {5 x^4}{(x+3)^5}+\frac {5 e^{x/5}}{x+3}+\frac {5 e^{2 x/5}}{x+3}+\frac {5 e^{3 x/5}}{x+3}+\frac {5 e^{4 x/5}}{x+3}-\frac {45 e^{x/5}}{(x+3)^2}-\frac {30 e^{2 x/5}}{(x+3)^2}-\frac {15 e^{3 x/5}}{(x+3)^2}+\frac {135 e^{x/5}}{(x+3)^3}+\frac {45 e^{2 x/5}}{(x+3)^3}-\frac {135 e^{x/5}}{(x+3)^4}\right )\) |
Int[(90 + 30*x + 45*x^2 + 15*x^3 + E^x*(45 - 30*x - 15*x^2) + E^((x + 5*Lo g[3 + x])/5)*(-93 - 87*x - 3*x^2 + 3*x^3 + E^x*(21 + 12*x)))/(15*x^2 + 5*x ^3 + E^((2*(x + 5*Log[3 + x]))/5)*(15 + 5*x) + E^((x + 5*Log[3 + x])/5)*(- 30*x - 10*x^2)),x]
3.16.92.3.1 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 = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(\frac {15 x^{2}-30+15 x -15 \,{\mathrm e}^{x}}{5 x -5 \,{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}}\) | \(32\) |
| default | \(\frac {-6+3 \,{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}+3 x^{2}}{x -{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}}+\frac {3 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{\frac {x}{5}}-x +3 \,{\mathrm e}^{\frac {x}{5}}}\) | \(62\) |
| parts | \(\frac {-6+3 \,{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}+3 x^{2}}{x -{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}}+\frac {3 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{\frac {x}{5}}-x +3 \,{\mathrm e}^{\frac {x}{5}}}\) | \(62\) |
| risch | \(\frac {3 \,{\mathrm e}^{x} x^{3}-3 x^{5}+27 \,{\mathrm e}^{x} x^{2}-30 x^{4}+81 \,{\mathrm e}^{x} x -102 x^{3}+81 \,{\mathrm e}^{x}-108 x^{2}+81 x +162}{\left (x \,{\mathrm e}^{\frac {x}{5}}-x +3 \,{\mathrm e}^{\frac {x}{5}}\right ) \left (3+x \right )^{3}}\) | \(73\) |
int((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(ln(3+x)+1/5*x)+(-15*x^2-30 *x+45)*exp(x)+15*x^3+45*x^2+30*x+90)/((5*x+15)*exp(ln(3+x)+1/5*x)^2+(-10*x ^2-30*x)*exp(ln(3+x)+1/5*x)+5*x^3+15*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.41 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 \, {\left (x^{7} + 16 \, x^{6} + 103 \, x^{5} + 330 \, x^{4} + 495 \, x^{3} + 108 \, x^{2} - 567 \, x - e^{\left (x + 5 \, \log \left (x + 3\right )\right )} - 486\right )}}{x^{6} + 15 \, x^{5} + 90 \, x^{4} + 270 \, x^{3} + 405 \, x^{2} - {\left (x^{5} + 15 \, x^{4} + 90 \, x^{3} + 270 \, x^{2} + 405 \, x + 243\right )} e^{\left (\frac {1}{5} \, x + \log \left (x + 3\right )\right )} + 243 \, x} \]
integrate((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(log(3+x)+1/5*x)+(-15 *x^2-30*x+45)*exp(x)+15*x^3+45*x^2+30*x+90)/((5*x+15)*exp(log(3+x)+1/5*x)^ 2+(-10*x^2-30*x)*exp(log(3+x)+1/5*x)+5*x^3+15*x^2),x, algorithm=\
3*(x^7 + 16*x^6 + 103*x^5 + 330*x^4 + 495*x^3 + 108*x^2 - 567*x - e^(x + 5 *log(x + 3)) - 486)/(x^6 + 15*x^5 + 90*x^4 + 270*x^3 + 405*x^2 - (x^5 + 15 *x^4 + 90*x^3 + 270*x^2 + 405*x + 243)*e^(1/5*x + log(x + 3)) + 243*x)
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (22) = 44\).
Time = 0.32 (sec) , antiderivative size = 366, normalized size of antiderivative = 11.44 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 x^{4}}{x^{5} + 15 x^{4} + 90 x^{3} + 270 x^{2} + 405 x + 243} + \frac {\left (3 x^{9} + 54 x^{8} + 405 x^{7} + 1620 x^{6} + 3645 x^{5} + 4374 x^{4} + 2187 x^{3}\right ) e^{\frac {x}{5}} + \left (3 x^{9} + 63 x^{8} + 567 x^{7} + 2835 x^{6} + 8505 x^{5} + 15309 x^{4} + 15309 x^{3} + 6561 x^{2}\right ) e^{\frac {2 x}{5}} + \left (3 x^{9} + 72 x^{8} + 756 x^{7} + 4536 x^{6} + 17010 x^{5} + 40824 x^{4} + 61236 x^{3} + 52488 x^{2} + 19683 x\right ) e^{\frac {3 x}{5}} + \left (3 x^{9} + 81 x^{8} + 972 x^{7} + 6804 x^{6} + 30618 x^{5} + 91854 x^{4} + 183708 x^{3} + 236196 x^{2} + 177147 x + 59049\right ) e^{\frac {4 x}{5}}}{x^{10} + 30 x^{9} + 405 x^{8} + 3240 x^{7} + 17010 x^{6} + 61236 x^{5} + 153090 x^{4} + 262440 x^{3} + 295245 x^{2} + 196830 x + 59049} + \frac {- 3 x^{7} - 48 x^{6} - 306 x^{5} - 990 x^{4} - 1485 x^{3} - 324 x^{2} + 1701 x + 1458}{- x^{6} - 15 x^{5} - 90 x^{4} - 270 x^{3} - 405 x^{2} - 243 x + \left (x^{6} + 18 x^{5} + 135 x^{4} + 540 x^{3} + 1215 x^{2} + 1458 x + 729\right ) e^{\frac {x}{5}}} \]
integrate((((12*x+21)*exp(x)+3*x**3-3*x**2-87*x-93)*exp(ln(3+x)+1/5*x)+(-1 5*x**2-30*x+45)*exp(x)+15*x**3+45*x**2+30*x+90)/((5*x+15)*exp(ln(3+x)+1/5* x)**2+(-10*x**2-30*x)*exp(ln(3+x)+1/5*x)+5*x**3+15*x**2),x)
3*x**4/(x**5 + 15*x**4 + 90*x**3 + 270*x**2 + 405*x + 243) + ((3*x**9 + 54 *x**8 + 405*x**7 + 1620*x**6 + 3645*x**5 + 4374*x**4 + 2187*x**3)*exp(x/5) + (3*x**9 + 63*x**8 + 567*x**7 + 2835*x**6 + 8505*x**5 + 15309*x**4 + 153 09*x**3 + 6561*x**2)*exp(2*x/5) + (3*x**9 + 72*x**8 + 756*x**7 + 4536*x**6 + 17010*x**5 + 40824*x**4 + 61236*x**3 + 52488*x**2 + 19683*x)*exp(3*x/5) + (3*x**9 + 81*x**8 + 972*x**7 + 6804*x**6 + 30618*x**5 + 91854*x**4 + 18 3708*x**3 + 236196*x**2 + 177147*x + 59049)*exp(4*x/5))/(x**10 + 30*x**9 + 405*x**8 + 3240*x**7 + 17010*x**6 + 61236*x**5 + 153090*x**4 + 262440*x** 3 + 295245*x**2 + 196830*x + 59049) + (-3*x**7 - 48*x**6 - 306*x**5 - 990* x**4 - 1485*x**3 - 324*x**2 + 1701*x + 1458)/(-x**6 - 15*x**5 - 90*x**4 - 270*x**3 - 405*x**2 - 243*x + (x**6 + 18*x**5 + 135*x**4 + 540*x**3 + 1215 *x**2 + 1458*x + 729)*exp(x/5))
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=-\frac {3 \, {\left (x^{2} + x - e^{x} - 2\right )}}{{\left (x + 3\right )} e^{\left (\frac {1}{5} \, x\right )} - x} \]
integrate((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(log(3+x)+1/5*x)+(-15 *x^2-30*x+45)*exp(x)+15*x^3+45*x^2+30*x+90)/((5*x+15)*exp(log(3+x)+1/5*x)^ 2+(-10*x^2-30*x)*exp(log(3+x)+1/5*x)+5*x^3+15*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.38 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=-\frac {3 \, {\left (x^{5} + 10 \, x^{4} - x^{3} e^{x} + 34 \, x^{3} - 9 \, x^{2} e^{x} + 36 \, x^{2} - 27 \, x e^{x} - 27 \, x - 27 \, e^{x} - 54\right )}}{x^{4} e^{\left (\frac {1}{5} \, x\right )} - x^{4} + 12 \, x^{3} e^{\left (\frac {1}{5} \, x\right )} - 9 \, x^{3} + 54 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} - 27 \, x^{2} + 108 \, x e^{\left (\frac {1}{5} \, x\right )} - 27 \, x + 81 \, e^{\left (\frac {1}{5} \, x\right )}} \]
integrate((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(log(3+x)+1/5*x)+(-15 *x^2-30*x+45)*exp(x)+15*x^3+45*x^2+30*x+90)/((5*x+15)*exp(log(3+x)+1/5*x)^ 2+(-10*x^2-30*x)*exp(log(3+x)+1/5*x)+5*x^3+15*x^2),x, algorithm=\
-3*(x^5 + 10*x^4 - x^3*e^x + 34*x^3 - 9*x^2*e^x + 36*x^2 - 27*x*e^x - 27*x - 27*e^x - 54)/(x^4*e^(1/5*x) - x^4 + 12*x^3*e^(1/5*x) - 9*x^3 + 54*x^2*e ^(1/5*x) - 27*x^2 + 108*x*e^(1/5*x) - 27*x + 81*e^(1/5*x))
Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3\,x-3\,{\mathrm {e}}^x+3\,x^2-6}{x-{\mathrm {e}}^{x/5}\,\left (x+3\right )} \]