Integrand size = 153, antiderivative size = 30 \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=x \left (x+\frac {(-4+x) (5+x)}{5 \left (e^x-\frac {25}{x^2}+e^x x\right )}\right ) \]
Time = 7.41 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=\frac {1}{5} x^2 \left (5+\frac {x \left (-20+x+x^2\right )}{-25+e^x x^2 (1+x)}\right ) \]
Integrate[(6250*x + 1500*x^2 - 100*x^3 - 125*x^4 + 10*E^(2*x)*x^5 + 10*E^( 2*x)*x^7 + E^x*x*(-500*x^3 + 21*x^5 + 20*E^x*x^5 + x^6 - x^7) + E^x*(-500* x^3 - 20*x^4 + 22*x^5 + 2*x^6 - x^7))/(3125 - 250*E^x*x^2 + 5*E^(2*x)*x^4 + 5*E^(2*x)*x^6 + E^x*x*(-250*x^2 + 10*E^x*x^4)),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.
| \(\displaystyle \int \frac {10 e^{2 x} x^7+10 e^{2 x} x^5-125 x^4-100 x^3+1500 x^2+e^x \left (-x^7+x^6+20 e^x x^5+21 x^5-500 x^3\right ) x+e^x \left (-x^7+2 x^6+22 x^5-20 x^4-500 x^3\right )+6250 x}{5 e^{2 x} x^6+5 e^{2 x} x^4-250 e^x x^2+e^x \left (10 e^x x^4-250 x^2\right ) x+3125} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {10 e^{2 x} x^7+10 e^{2 x} x^5-125 x^4-100 x^3+1500 x^2+e^x \left (-x^7+x^6+20 e^x x^5+21 x^5-500 x^3\right ) x+e^x \left (-x^7+2 x^6+22 x^5-20 x^4-500 x^3\right )+6250 x}{5 \left (-e^x x^3-e^x x^2+25\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {10 e^{2 x} x^7+10 e^{2 x} x^5-125 x^4-100 x^3+1500 x^2-e^x \left (x^7-x^6-20 e^x x^5-21 x^5+500 x^3\right ) x+6250 x-e^x \left (x^7-2 x^6-22 x^5+20 x^4+500 x^3\right )}{\left (-e^x x^3-e^x x^2+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (-\frac {\left (x^4-23 x^2-22 x+20\right ) x^2}{(x+1) \left (e^x x^3+e^x x^2-25\right )}-\frac {25 \left (x^4+5 x^3-14 x^2-78 x-40\right ) x^2}{(x+1) \left (e^x x^3+e^x x^2-25\right )^2}+10 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (500 \int \frac {1}{\left (e^x x^3+e^x x^2-25\right )^2}dx-500 \int \frac {x}{\left (e^x x^3+e^x x^2-25\right )^2}dx+1500 \int \frac {x^2}{\left (e^x x^3+e^x x^2-25\right )^2}dx+450 \int \frac {x^3}{\left (e^x x^3+e^x x^2-25\right )^2}dx-500 \int \frac {1}{(x+1) \left (e^x x^3+e^x x^2-25\right )^2}dx+20 \int \frac {1}{e^x x^3+e^x x^2-25}dx-20 \int \frac {x}{e^x x^3+e^x x^2-25}dx+22 \int \frac {x^3}{e^x x^3+e^x x^2-25}dx-20 \int \frac {1}{(x+1) \left (e^x x^3+e^x x^2-25\right )}dx-25 \int \frac {x^5}{\left (e^x x^3+e^x x^2-25\right )^2}dx-\int \frac {x^5}{e^x x^3+e^x x^2-25}dx-100 \int \frac {x^4}{\left (e^x x^3+e^x x^2-25\right )^2}dx+\int \frac {x^4}{e^x x^3+e^x x^2-25}dx+5 x^2\right )\) |
Int[(6250*x + 1500*x^2 - 100*x^3 - 125*x^4 + 10*E^(2*x)*x^5 + 10*E^(2*x)*x ^7 + E^x*x*(-500*x^3 + 21*x^5 + 20*E^x*x^5 + x^6 - x^7) + E^x*(-500*x^3 - 20*x^4 + 22*x^5 + 2*x^6 - x^7))/(3125 - 250*E^x*x^2 + 5*E^(2*x)*x^4 + 5*E^ (2*x)*x^6 + E^x*x*(-250*x^2 + 10*E^x*x^4)),x]
3.10.72.3.1 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]]
Time = 4.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(x^{2}+\frac {\left (x^{2}+x -20\right ) x^{3}}{5 \,{\mathrm e}^{x} x^{3}+5 \,{\mathrm e}^{x} x^{2}-125}\) | \(32\) |
| norman | \(\frac {x^{5} {\mathrm e}^{x}+{\mathrm e}^{x} x^{4}-25 x^{2}-4 x^{3}+\frac {x^{4}}{5}+\frac {x^{5}}{5}}{{\mathrm e}^{x} x^{3}+{\mathrm e}^{x} x^{2}-25}\) | \(51\) |
| parallelrisch | \(\frac {25 x^{5}+125 \,{\mathrm e}^{x} x^{4}+125 \,{\mathrm e}^{x +\ln \left (x \right )} x^{4}+25 x^{4}-500 x^{3}-3125 x^{2}}{125 \,{\mathrm e}^{x} x^{2}+125 x^{2} {\mathrm e}^{x +\ln \left (x \right )}-3125}\) | \(60\) |
int((10*x^5*exp(x+ln(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*exp(x+ln (x))+10*x^5*exp(x)^2+(-x^7+2*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125*x^4-100 *x^3+1500*x^2+6250*x)/(5*x^4*exp(x+ln(x))^2+(10*exp(x)*x^4-250*x^2)*exp(x+ ln(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=\frac {x^{5} + x^{4} - 20 \, x^{3} - 125 \, x^{2} + 5 \, {\left (x^{4} + x^{3}\right )} e^{\left (x + \log \left (x\right )\right )}}{5 \, {\left ({\left (x^{2} + x\right )} e^{\left (x + \log \left (x\right )\right )} - 25\right )}} \]
integrate((10*x^5*exp(x+log(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*e xp(x+log(x))+10*x^5*exp(x)^2+(-x^7+2*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125 *x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+log(x))^2+(10*exp(x)*x^4-250*x^ 2)*exp(x+log(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x, algorithm=\
1/5*(x^5 + x^4 - 20*x^3 - 125*x^2 + 5*(x^4 + x^3)*e^(x + log(x)))/((x^2 + x)*e^(x + log(x)) - 25)
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=x^{2} + \frac {x^{5} + x^{4} - 20 x^{3}}{\left (5 x^{3} + 5 x^{2}\right ) e^{x} - 125} \]
integrate((10*x**5*exp(x+ln(x))**2+(20*x**5*exp(x)-x**7+x**6+21*x**5-500*x **3)*exp(x+ln(x))+10*x**5*exp(x)**2+(-x**7+2*x**6+22*x**5-20*x**4-500*x**3 )*exp(x)-125*x**4-100*x**3+1500*x**2+6250*x)/(5*x**4*exp(x+ln(x))**2+(10*e xp(x)*x**4-250*x**2)*exp(x+ln(x))+5*exp(x)**2*x**4-250*exp(x)*x**2+3125),x )
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=\frac {x^{5} + x^{4} - 20 \, x^{3} - 125 \, x^{2} + 5 \, {\left (x^{5} + x^{4}\right )} e^{x}}{5 \, {\left ({\left (x^{3} + x^{2}\right )} e^{x} - 25\right )}} \]
integrate((10*x^5*exp(x+log(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*e xp(x+log(x))+10*x^5*exp(x)^2+(-x^7+2*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125 *x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+log(x))^2+(10*exp(x)*x^4-250*x^ 2)*exp(x+log(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=\frac {5 \, x^{5} e^{x} + x^{5} + 5 \, x^{4} e^{x} + x^{4} - 20 \, x^{3} - 125 \, x^{2}}{5 \, {\left (x^{3} e^{x} + x^{2} e^{x} - 25\right )}} \]
integrate((10*x^5*exp(x+log(x))^2+(20*x^5*exp(x)-x^7+x^6+21*x^5-500*x^3)*e xp(x+log(x))+10*x^5*exp(x)^2+(-x^7+2*x^6+22*x^5-20*x^4-500*x^3)*exp(x)-125 *x^4-100*x^3+1500*x^2+6250*x)/(5*x^4*exp(x+log(x))^2+(10*exp(x)*x^4-250*x^ 2)*exp(x+log(x))+5*exp(x)^2*x^4-250*exp(x)*x^2+3125),x, algorithm=\
Timed out. \[ \int \frac {6250 x+1500 x^2-100 x^3-125 x^4+10 e^{2 x} x^5+10 e^{2 x} x^7+e^x x \left (-500 x^3+21 x^5+20 e^x x^5+x^6-x^7\right )+e^x \left (-500 x^3-20 x^4+22 x^5+2 x^6-x^7\right )}{3125-250 e^x x^2+5 e^{2 x} x^4+5 e^{2 x} x^6+e^x x \left (-250 x^2+10 e^x x^4\right )} \, dx=\int \frac {6250\,x+10\,x^5\,{\mathrm {e}}^{2\,x+2\,\ln \left (x\right )}-{\mathrm {e}}^x\,\left (x^7-2\,x^6-22\,x^5+20\,x^4+500\,x^3\right )+{\mathrm {e}}^{x+\ln \left (x\right )}\,\left (20\,x^5\,{\mathrm {e}}^x-500\,x^3+21\,x^5+x^6-x^7\right )+10\,x^5\,{\mathrm {e}}^{2\,x}+1500\,x^2-100\,x^3-125\,x^4}{5\,x^4\,{\mathrm {e}}^{2\,x+2\,\ln \left (x\right )}-250\,x^2\,{\mathrm {e}}^x+{\mathrm {e}}^{x+\ln \left (x\right )}\,\left (10\,x^4\,{\mathrm {e}}^x-250\,x^2\right )+5\,x^4\,{\mathrm {e}}^{2\,x}+3125} \,d x \]
int((6250*x + 10*x^5*exp(2*x + 2*log(x)) - exp(x)*(500*x^3 + 20*x^4 - 22*x ^5 - 2*x^6 + x^7) + exp(x + log(x))*(20*x^5*exp(x) - 500*x^3 + 21*x^5 + x^ 6 - x^7) + 10*x^5*exp(2*x) + 1500*x^2 - 100*x^3 - 125*x^4)/(5*x^4*exp(2*x + 2*log(x)) - 250*x^2*exp(x) + exp(x + log(x))*(10*x^4*exp(x) - 250*x^2) + 5*x^4*exp(2*x) + 3125),x)
int((6250*x + 10*x^5*exp(2*x + 2*log(x)) - exp(x)*(500*x^3 + 20*x^4 - 22*x ^5 - 2*x^6 + x^7) + exp(x + log(x))*(20*x^5*exp(x) - 500*x^3 + 21*x^5 + x^ 6 - x^7) + 10*x^5*exp(2*x) + 1500*x^2 - 100*x^3 - 125*x^4)/(5*x^4*exp(2*x + 2*log(x)) - 250*x^2*exp(x) + exp(x + log(x))*(10*x^4*exp(x) - 250*x^2) + 5*x^4*exp(2*x) + 3125), x)