3.23.22 \(\int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} (-4 x^2+x^3)}{x^2}} (-200-175 x-25 x^3+e^{2 x} (-175 x^3+50 x^4))}{25 e^{\frac {2 (4+7 x+2 x^2-x^3+e^{2 x} (-4 x^2+x^3))}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} (-4 x^2+x^3)}{x^2}} x^4+x^5} \, dx\) [2222]

3.23.22.1 Optimal result

Integrand size = 160, antiderivative size = 30 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=\frac {5}{-5 e^{(-4+x) \left (e^{2 x}-\frac {(1+x)^2}{x^2}\right )}+x} \]

output

5/(x-5*exp((exp(2*x)-(1+x)^2/x^2)*(x-4)))
 

3.23.22.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=-\frac {5 e^x}{5 e^{2+e^{2 x} (-4+x)+\frac {4}{x^2}+\frac {7}{x}}-e^x x} \]

input

Integrate[(-5*x^3 + E^((4 + 7*x + 2*x^2 - x^3 + E^(2*x)*(-4*x^2 + x^3))/x^ 
2)*(-200 - 175*x - 25*x^3 + E^(2*x)*(-175*x^3 + 50*x^4)))/(25*E^((2*(4 + 7 
*x + 2*x^2 - x^3 + E^(2*x)*(-4*x^2 + x^3)))/x^2)*x^3 - 10*E^((4 + 7*x + 2* 
x^2 - x^3 + E^(2*x)*(-4*x^2 + x^3))/x^2)*x^4 + x^5),x]
 

output

(-5*E^x)/(5*E^(2 + E^(2*x)*(-4 + x) + 4/x^2 + 7/x) - E^x*x)
 

3.23.22.3 Rubi [F]

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.

\(\displaystyle \int \frac {\left (-25 x^3+e^{2 x} \left (50 x^4-175 x^3\right )-175 x-200\right ) \exp \left (\frac {-x^3+2 x^2+e^{2 x} \left (x^3-4 x^2\right )+7 x+4}{x^2}\right )-5 x^3}{25 x^3 \exp \left (\frac {2 \left (-x^3+2 x^2+e^{2 x} \left (x^3-4 x^2\right )+7 x+4\right )}{x^2}\right )-10 x^4 \exp \left (\frac {-x^3+2 x^2+e^{2 x} \left (x^3-4 x^2\right )+7 x+4}{x^2}\right )+x^5} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{2 x} \left (\left (-25 x^3+e^{2 x} \left (50 x^4-175 x^3\right )-175 x-200\right ) \exp \left (\frac {-x^3+2 x^2+e^{2 x} \left (x^3-4 x^2\right )+7 x+4}{x^2}\right )-5 x^3\right )}{x^3 \left (5 e^{\frac {4}{x^2}+e^{2 x} (x-4)+\frac {7}{x}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {e^{-\frac {4}{x^2}-e^{2 x} (x-4)+x-\frac {7}{x}-2} \left (x^3+7 x+8\right )}{x^3}-\frac {5 e^{2 x} \left (x^3-50 e^{\frac {8}{x^2}+2 e^{2 x} (x-4)+\frac {14}{x}+4} x^2+x^2+175 e^{\frac {8}{x^2}+2 e^{2 x} (x-4)+\frac {14}{x}+4} x+7 x+8\right )}{x^2 \left (e^x x-5 e^{\frac {4}{x^2}+e^{2 x} (x-4)+\frac {7}{x}+2}\right )^2}-\frac {e^{-\frac {4}{x^2}-e^{2 x} (x-4)+2 x-\frac {7}{x}-2} \left (x^3+50 e^{\frac {8}{x^2}+2 e^{2 x} (x-4)+\frac {14}{x}+4} x^2-175 e^{\frac {8}{x^2}+2 e^{2 x} (x-4)+\frac {14}{x}+4} x+7 x+8\right )}{x^2 \left (5 e^{\frac {4}{x^2}+e^{2 x} (x-4)+\frac {7}{x}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {5 e^x \left (-e^x x^3+5 e^{\frac {4}{x^2}+e^{2 x} (x-4)+2 x+\frac {7}{x}+2} (2 x-7) x^3-5 e^{\frac {4}{x^2}+e^{2 x} (x-4)+\frac {7}{x}+2} \left (x^3+7 x+8\right )\right )}{x^3 \left (5 e^{\frac {4}{x^2}+e^{2 x} (x-4)+\frac {7}{x}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 5 \int -\frac {e^x \left (e^x x^3+5 e^{-e^{2 x} (4-x)+2 x+\frac {7}{x}+\frac {4}{x^2}+2} (7-2 x) x^3+5 e^{-e^{2 x} (4-x)+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )\right )}{x^3 \left (5 e^{-e^{2 x} (4-x)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -5 \int \frac {e^x \left (e^x x^3+5 e^{-e^{2 x} (4-x)+2 x+\frac {7}{x}+\frac {4}{x^2}+2} (7-2 x) x^3+5 e^{-e^{2 x} (4-x)+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )\right )}{x^3 \left (5 e^{-e^{2 x} (4-x)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}-\frac {e^x \left (100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}\right )}{x^2 \left (e^x x-5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -5 \int \frac {e^{2 x} x^3-5 e^{e^{2 x} (x-4)+3 x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7) x^3+5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} \left (x^3+7 x+8\right )}{x^3 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (-\frac {5 e^{e^{2 x} (x-4)+x+\frac {7}{x}+\frac {4}{x^2}+2} (2 x-7)}{x^2}-\frac {100 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-x-350 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4}}{x^3}-\frac {25 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} \left (-x^3+50 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-x^2-175 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )^2}+\frac {5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2} \left (-x^3+150 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x^2-2 x^2-525 e^{2 e^{2 x} (x-4)+\frac {14}{x}+\frac {8}{x^2}+4} x-7 x-8\right )}{x^4 \left (5 e^{e^{2 x} (x-4)+\frac {7}{x}+\frac {4}{x^2}+2}-e^x x\right )}\right )dx\)

input

Int[(-5*x^3 + E^((4 + 7*x + 2*x^2 - x^3 + E^(2*x)*(-4*x^2 + x^3))/x^2)*(-2 
00 - 175*x - 25*x^3 + E^(2*x)*(-175*x^3 + 50*x^4)))/(25*E^((2*(4 + 7*x + 2 
*x^2 - x^3 + E^(2*x)*(-4*x^2 + x^3)))/x^2)*x^3 - 10*E^((4 + 7*x + 2*x^2 - 
x^3 + E^(2*x)*(-4*x^2 + x^3))/x^2)*x^4 + x^5),x]
 

output

$Aborted
 

3.23.22.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.23.22.4 Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {5}{x -5 \,{\mathrm e}^{\frac {\left (x -4\right ) \left ({\mathrm e}^{2 x} x^{2}-x^{2}-2 x -1\right )}{x^{2}}}}\)	\(35\)
parallelrisch	\(\frac {5}{x -5 \,{\mathrm e}^{\frac {\left (x^{3}-4 x^{2}\right ) {\mathrm e}^{2 x}-x^{3}+2 x^{2}+7 x +4}{x^{2}}}}\)	\(43\)

input

int((((50*x^4-175*x^3)*exp(2*x)-25*x^3-175*x-200)*exp(((x^3-4*x^2)*exp(2*x 
)-x^3+2*x^2+7*x+4)/x^2)-5*x^3)/(25*x^3*exp(((x^3-4*x^2)*exp(2*x)-x^3+2*x^2 
+7*x+4)/x^2)^2-10*x^4*exp(((x^3-4*x^2)*exp(2*x)-x^3+2*x^2+7*x+4)/x^2)+x^5) 
,x,method=_RETURNVERBOSE)
 

output

5/(x-5*exp((x-4)*(exp(2*x)*x^2-x^2-2*x-1)/x^2))
 

3.23.22.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=\frac {5}{x - 5 \, e^{\left (-\frac {x^{3} - 2 \, x^{2} - {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (2 \, x\right )} - 7 \, x - 4}{x^{2}}\right )}} \]

input

integrate((((50*x^4-175*x^3)*exp(2*x)-25*x^3-175*x-200)*exp(((x^3-4*x^2)*e 
xp(2*x)-x^3+2*x^2+7*x+4)/x^2)-5*x^3)/(25*x^3*exp(((x^3-4*x^2)*exp(2*x)-x^3 
+2*x^2+7*x+4)/x^2)^2-10*x^4*exp(((x^3-4*x^2)*exp(2*x)-x^3+2*x^2+7*x+4)/x^2 
)+x^5),x, algorithm=\
 

output

5/(x - 5*e^(-(x^3 - 2*x^2 - (x^3 - 4*x^2)*e^(2*x) - 7*x - 4)/x^2))
 

3.23.22.6 Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=- \frac {1}{- \frac {x}{5} + e^{\frac {- x^{3} + 2 x^{2} + 7 x + \left (x^{3} - 4 x^{2}\right ) e^{2 x} + 4}{x^{2}}}} \]

input

integrate((((50*x**4-175*x**3)*exp(2*x)-25*x**3-175*x-200)*exp(((x**3-4*x* 
*2)*exp(2*x)-x**3+2*x**2+7*x+4)/x**2)-5*x**3)/(25*x**3*exp(((x**3-4*x**2)* 
exp(2*x)-x**3+2*x**2+7*x+4)/x**2)**2-10*x**4*exp(((x**3-4*x**2)*exp(2*x)-x 
**3+2*x**2+7*x+4)/x**2)+x**5),x)
 

output

-1/(-x/5 + exp((-x**3 + 2*x**2 + 7*x + (x**3 - 4*x**2)*exp(2*x) + 4)/x**2) 
)
 

3.23.22.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=\frac {5 \, e^{\left (x + 4 \, e^{\left (2 \, x\right )}\right )}}{x e^{\left (x + 4 \, e^{\left (2 \, x\right )}\right )} - 5 \, e^{\left (x e^{\left (2 \, x\right )} + \frac {7}{x} + \frac {4}{x^{2}} + 2\right )}} \]

input

integrate((((50*x^4-175*x^3)*exp(2*x)-25*x^3-175*x-200)*exp(((x^3-4*x^2)*e 
xp(2*x)-x^3+2*x^2+7*x+4)/x^2)-5*x^3)/(25*x^3*exp(((x^3-4*x^2)*exp(2*x)-x^3 
+2*x^2+7*x+4)/x^2)^2-10*x^4*exp(((x^3-4*x^2)*exp(2*x)-x^3+2*x^2+7*x+4)/x^2 
)+x^5),x, algorithm=\
 

output

5*e^(x + 4*e^(2*x))/(x*e^(x + 4*e^(2*x)) - 5*e^(x*e^(2*x) + 7/x + 4/x^2 + 
2))
 

3.23.22.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1342 vs. \(2 (30) = 60\).

Time = 0.54 (sec) , antiderivative size = 1342, normalized size of antiderivative = 44.73 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=\text {Too large to display} \]

input

integrate((((50*x^4-175*x^3)*exp(2*x)-25*x^3-175*x-200)*exp(((x^3-4*x^2)*e 
xp(2*x)-x^3+2*x^2+7*x+4)/x^2)-5*x^3)/(25*x^3*exp(((x^3-4*x^2)*exp(2*x)-x^3 
+2*x^2+7*x+4)/x^2)^2-10*x^4*exp(((x^3-4*x^2)*exp(2*x)-x^3+2*x^2+7*x+4)/x^2 
)+x^5),x, algorithm=\
 

output

5*(2*x^5*e^(4*x) - 7*x^4*e^(4*x) - x^4*e^(2*x) - 10*x^4*e^(4*x + (x^3*e^(2 
*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) - x^3*e^(2*x) + 35*x^3*e 
^(4*x + (x^3*e^(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) + 5*x^3 
*e^((x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) - 7*x^2*e^( 
2*x) + 5*x^2*e^((x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) 
 - 8*x*e^(2*x) + 35*x*e^((x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x 
+ 4)/x^2) + 40*e^((x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^ 
2))/(2*x^6*e^(4*x) - 7*x^5*e^(4*x) - x^5*e^(2*x) - 10*x^5*e^(4*x + (x^3*e^ 
(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) - 10*x^5*e^(2*x + (x^3 
*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) - x^4*e^(2*x) + 35* 
x^4*e^(4*x + (x^3*e^(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) + 
50*x^4*e^(2*x + (x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2 
+ (x^3*e^(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) + 35*x^4*e^(2 
*x + (x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) + 5*x^4*e^ 
(2*x + (x^3*e^(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) + 5*x^4* 
e^((x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) - 7*x^3*e^(2 
*x) - 175*x^3*e^(2*x + (x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 
4)/x^2 + (x^3*e^(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) + 5*x^ 
3*e^(2*x + (x^3*e^(2*x) - x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2) - 25 
*x^3*e^((x^3*e^(2*x) + x^3 - 4*x^2*e^(2*x) + 2*x^2 + 7*x + 4)/x^2 + (x^...
 

3.23.22.9 Mupad [B] (verification not implemented)

Time = 12.89 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-5 x^3+e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} \left (-200-175 x-25 x^3+e^{2 x} \left (-175 x^3+50 x^4\right )\right )}{25 e^{\frac {2 \left (4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )\right )}{x^2}} x^3-10 e^{\frac {4+7 x+2 x^2-x^3+e^{2 x} \left (-4 x^2+x^3\right )}{x^2}} x^4+x^5} \, dx=\frac {5}{x-5\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{\frac {4}{x^2}}\,{\mathrm {e}}^{7/x}} \]

input

int(-(5*x^3 + exp((7*x - exp(2*x)*(4*x^2 - x^3) + 2*x^2 - x^3 + 4)/x^2)*(1 
75*x + exp(2*x)*(175*x^3 - 50*x^4) + 25*x^3 + 200))/(x^5 - 10*x^4*exp((7*x 
 - exp(2*x)*(4*x^2 - x^3) + 2*x^2 - x^3 + 4)/x^2) + 25*x^3*exp((2*(7*x - e 
xp(2*x)*(4*x^2 - x^3) + 2*x^2 - x^3 + 4))/x^2)),x)
 

output

5/(x - 5*exp(-4*exp(2*x))*exp(-x)*exp(2)*exp(x*exp(2*x))*exp(4/x^2)*exp(7/ 
x))