Integrand size = 201, antiderivative size = 33 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {x}{4+2 x+\frac {1}{5} \left (e^{e^5}+e^{\frac {x^2}{-2+3 x}}\right ) x} \] Output:
x/(2*x+1/5*(exp(x^2/(-2+3*x))+exp(exp(5)))*x+4)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {5 x}{20+\left (10+e^{e^5}+e^{\frac {x^2}{-2+3 x}}\right ) x} \] Input:
Integrate[(400 - 1200*x + 900*x^2 + E^(x^2/(-2 + 3*x))*(20*x^3 - 15*x^4))/ (1600 - 3200*x - 800*x^2 + 2400*x^3 + 900*x^4 + E^(2*E^5)*(4*x^2 - 12*x^3 + 9*x^4) + E^((2*x^2)/(-2 + 3*x))*(4*x^2 - 12*x^3 + 9*x^4) + E^(x^2/(-2 + 3*x))*(160*x - 400*x^2 + 120*x^3 + 180*x^4) + E^E^5*(160*x - 400*x^2 + 120 *x^3 + 180*x^4 + E^(x^2/(-2 + 3*x))*(8*x^2 - 24*x^3 + 18*x^4))),x]
Output:
(5*x)/(20 + (10 + E^E^5 + E^(x^2/(-2 + 3*x)))*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.
| \(\displaystyle \int \frac {900 x^2+e^{\frac {x^2}{3 x-2}} \left (20 x^3-15 x^4\right )-1200 x+400}{900 x^4+2400 x^3-800 x^2+e^{\frac {2 x^2}{3 x-2}} \left (9 x^4-12 x^3+4 x^2\right )+e^{2 e^5} \left (9 x^4-12 x^3+4 x^2\right )+e^{\frac {x^2}{3 x-2}} \left (180 x^4+120 x^3-400 x^2+160 x\right )+e^{e^5} \left (180 x^4+120 x^3-400 x^2+e^{\frac {x^2}{3 x-2}} \left (18 x^4-24 x^3+8 x^2\right )+160 x\right )-3200 x+1600} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (180 x^2-3 e^{\frac {x^2}{3 x-2}} x^4+4 e^{\frac {x^2}{3 x-2}} x^3-240 x+80\right )}{(2-3 x)^2 \left (\left (e^{\frac {x^2}{3 x-2}}+10+e^{e^5}\right ) x+20\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int \frac {-3 e^{-\frac {x^2}{2-3 x}} x^4+4 e^{-\frac {x^2}{2-3 x}} x^3+180 x^2-240 x+80}{(2-3 x)^2 \left (\left (10+e^{e^5}+e^{-\frac {x^2}{2-3 x}}\right ) x+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 5 \int \left (\frac {\left (3 \left (10+e^{e^5}\right ) x^4+4 \left (5-e^{e^5}\right ) x^3+100 x^2-240 x+80\right ) x^2}{(2-3 x)^2 \left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2 \left (\left (10+e^{e^5}\right ) x+20\right )^2}+\frac {\left (-3 \left (10+e^{e^5}\right ) x^4-4 \left (5-e^{e^5}\right ) x^3-280 x^2+480 x-160\right ) x}{(2-3 x)^2 \left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right ) \left (\left (10+e^{e^5}\right ) x+20\right )^2}+\frac {20}{\left (\left (10+e^{e^5}\right ) x+20\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 5 \int \frac {e^{\frac {x^2}{2-3 x}} \left ((4-3 x) x^3+20 e^{\frac {x^2}{2-3 x}} (2-3 x)^2\right )}{(2-3 x)^2 \left (e^{\frac {x^2}{2-3 x}+e^5} x+x+10 e^{\frac {x^2}{2-3 x}} (x+2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 5 \int \left (\frac {e^{\frac {x^2}{2-3 x}} x \left (-3 \left (10+e^{e^5}\right ) x^4-4 \left (5-e^{e^5}\right ) x^3-100 x^2+240 x-80\right )}{(2-3 x)^2 \left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2 \left (\left (10+e^{e^5}\right ) x+20\right )}+\frac {20 e^{\frac {x^2}{2-3 x}}}{\left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right ) \left (\left (10+e^{e^5}\right ) x+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 5 \int \frac {e^{\frac {x^2}{2-3 x}} \left ((4-3 x) x^3+20 e^{\frac {x^2}{2-3 x}} (2-3 x)^2\right )}{(2-3 x)^2 \left (e^{\frac {x^2}{2-3 x}+e^5} x+x+10 e^{\frac {x^2}{2-3 x}} (x+2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 5 \int \left (\frac {e^{\frac {x^2}{2-3 x}} x \left (-3 \left (10+e^{e^5}\right ) x^4-4 \left (5-e^{e^5}\right ) x^3-100 x^2+240 x-80\right )}{(2-3 x)^2 \left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2 \left (\left (10+e^{e^5}\right ) x+20\right )}+\frac {20 e^{\frac {x^2}{2-3 x}}}{\left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right ) \left (\left (10+e^{e^5}\right ) x+20\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 5 \left (-\frac {4 \left (125-e^{e^5}\right ) \int \frac {e^{\frac {x^2}{2-3 x}}}{\left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2}dx}{27 \left (10+e^{e^5}\right )}+\frac {16}{27} \int \frac {e^{\frac {x^2}{2-3 x}}}{(2-3 x)^2 \left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2}dx-\frac {1}{3} \int \frac {e^{\frac {x^2}{2-3 x}} x^2}{\left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2}dx+\frac {16}{27} \int \frac {e^{\frac {x^2}{2-3 x}}}{(3 x-2) \left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2}dx+\frac {400 \int \frac {e^{\frac {x^2}{2-3 x}}}{\left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right )^2 \left (\left (10+e^{e^5}\right ) x+20\right )}dx}{10+e^{e^5}}+20 \int \frac {e^{\frac {x^2}{2-3 x}}}{\left (10 e^{\frac {x^2}{2-3 x}} \left (1+\frac {e^{e^5}}{10}\right ) x+x+20 e^{\frac {x^2}{2-3 x}}\right ) \left (\left (10+e^{e^5}\right ) x+20\right )}dx\right )\) |
Input:
Int[(400 - 1200*x + 900*x^2 + E^(x^2/(-2 + 3*x))*(20*x^3 - 15*x^4))/(1600 - 3200*x - 800*x^2 + 2400*x^3 + 900*x^4 + E^(2*E^5)*(4*x^2 - 12*x^3 + 9*x^ 4) + E^((2*x^2)/(-2 + 3*x))*(4*x^2 - 12*x^3 + 9*x^4) + E^(x^2/(-2 + 3*x))* (160*x - 400*x^2 + 120*x^3 + 180*x^4) + E^E^5*(160*x - 400*x^2 + 120*x^3 + 180*x^4 + E^(x^2/(-2 + 3*x))*(8*x^2 - 24*x^3 + 18*x^4))),x]
Output:
$Aborted
Time = 0.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {5 x}{x \,{\mathrm e}^{\frac {x^{2}}{-2+3 x}}+{\mathrm e}^{{\mathrm e}^{5}} x +10 x +20}\) | \(30\) |
| parallelrisch | \(\frac {5 x}{x \,{\mathrm e}^{\frac {x^{2}}{-2+3 x}}+{\mathrm e}^{{\mathrm e}^{5}} x +10 x +20}\) | \(30\) |
| norman | \(\frac {15 x^{2}-10 x}{\left (-2+3 x \right ) \left (x \,{\mathrm e}^{\frac {x^{2}}{-2+3 x}}+{\mathrm e}^{{\mathrm e}^{5}} x +10 x +20\right )}\) | \(44\) |
Input:
int(((-15*x^4+20*x^3)*exp(x^2/(-2+3*x))+900*x^2-1200*x+400)/((9*x^4-12*x^3 +4*x^2)*exp(exp(5))^2+((18*x^4-24*x^3+8*x^2)*exp(x^2/(-2+3*x))+180*x^4+120 *x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(-2+3*x))^2+( 180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(-2+3*x))+900*x^4+2400*x^3-800*x^2- 3200*x+1600),x,method=_RETURNVERBOSE)
Output:
5*x/(x*exp(x^2/(-2+3*x))+exp(exp(5))*x+10*x+20)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {5 \, x}{x e^{\left (\frac {x^{2}}{3 \, x - 2}\right )} + x e^{\left (e^{5}\right )} + 10 \, x + 20} \] Input:
integrate(((-15*x^4+20*x^3)*exp(x^2/(-2+3*x))+900*x^2-1200*x+400)/((9*x^4- 12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-24*x^3+8*x^2)*exp(x^2/(-2+3*x))+180*x ^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(-2+3*x ))^2+(180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(-2+3*x))+900*x^4+2400*x^3-80 0*x^2-3200*x+1600),x, algorithm="fricas")
Output:
5*x/(x*e^(x^2/(3*x - 2)) + x*e^(e^5) + 10*x + 20)
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {5 x}{x e^{\frac {x^{2}}{3 x - 2}} + 10 x + x e^{e^{5}} + 20} \] Input:
integrate(((-15*x**4+20*x**3)*exp(x**2/(-2+3*x))+900*x**2-1200*x+400)/((9* x**4-12*x**3+4*x**2)*exp(exp(5))**2+((18*x**4-24*x**3+8*x**2)*exp(x**2/(-2 +3*x))+180*x**4+120*x**3-400*x**2+160*x)*exp(exp(5))+(9*x**4-12*x**3+4*x** 2)*exp(x**2/(-2+3*x))**2+(180*x**4+120*x**3-400*x**2+160*x)*exp(x**2/(-2+3 *x))+900*x**4+2400*x**3-800*x**2-3200*x+1600),x)
Output:
5*x/(x*exp(x**2/(3*x - 2)) + 10*x + x*exp(exp(5)) + 20)
Exception generated. \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((-15*x^4+20*x^3)*exp(x^2/(-2+3*x))+900*x^2-1200*x+400)/((9*x^4- 12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-24*x^3+8*x^2)*exp(x^2/(-2+3*x))+180*x ^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(-2+3*x ))^2+(180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(-2+3*x))+900*x^4+2400*x^3-80 0*x^2-3200*x+1600),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {5 \, x}{x e^{\left (\frac {x^{2}}{3 \, x - 2}\right )} + x e^{\left (e^{5}\right )} + 10 \, x + 20} \] Input:
integrate(((-15*x^4+20*x^3)*exp(x^2/(-2+3*x))+900*x^2-1200*x+400)/((9*x^4- 12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-24*x^3+8*x^2)*exp(x^2/(-2+3*x))+180*x ^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(-2+3*x ))^2+(180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(-2+3*x))+900*x^4+2400*x^3-80 0*x^2-3200*x+1600),x, algorithm="giac")
Output:
5*x/(x*e^(x^2/(3*x - 2)) + x*e^(e^5) + 10*x + 20)
Time = 2.94 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {5}{{\mathrm {e}}^{\frac {x^2}{3\,x-2}}+\frac {10\,x+x\,{\mathrm {e}}^{{\mathrm {e}}^5}+20}{x}} \] Input:
int((exp(x^2/(3*x - 2))*(20*x^3 - 15*x^4) - 1200*x + 900*x^2 + 400)/(exp(x ^2/(3*x - 2))*(160*x - 400*x^2 + 120*x^3 + 180*x^4) - 3200*x + exp(2*exp(5 ))*(4*x^2 - 12*x^3 + 9*x^4) + exp((2*x^2)/(3*x - 2))*(4*x^2 - 12*x^3 + 9*x ^4) + exp(exp(5))*(160*x + exp(x^2/(3*x - 2))*(8*x^2 - 24*x^3 + 18*x^4) - 400*x^2 + 120*x^3 + 180*x^4) - 800*x^2 + 2400*x^3 + 900*x^4 + 1600),x)
Output:
5/(exp(x^2/(3*x - 2)) + (10*x + x*exp(exp(5)) + 20)/x)
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.85 \[ \int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx=\frac {-5 e^{\frac {x^{2}}{3 x -2}} x -100}{e^{2 e^{5}} x +e^{\frac {3 e^{5} x -2 e^{5}+x^{2}}{3 x -2}} x +20 e^{e^{5}} x +20 e^{e^{5}}+10 e^{\frac {x^{2}}{3 x -2}} x +100 x +200} \] Input:
int(((-15*x^4+20*x^3)*exp(x^2/(-2+3*x))+900*x^2-1200*x+400)/((9*x^4-12*x^3 +4*x^2)*exp(exp(5))^2+((18*x^4-24*x^3+8*x^2)*exp(x^2/(-2+3*x))+180*x^4+120 *x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(-2+3*x))^2+( 180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(-2+3*x))+900*x^4+2400*x^3-800*x^2- 3200*x+1600),x)
Output:
(5*( - e**(x**2/(3*x - 2))*x - 20))/(e**(2*e**5)*x + e**((3*e**5*x - 2*e** 5 + x**2)/(3*x - 2))*x + 20*e**(e**5)*x + 20*e**(e**5) + 10*e**(x**2/(3*x - 2))*x + 100*x + 200)