Integrand size = 106, antiderivative size = 28 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx=-1-e^5+e^x+e^{x+\frac {e^3}{5+x+2 x^2}} \] Output:
exp(exp(3)/(2*x^2+x+5)+x)+exp(x)-exp(5)-1
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx=e^x+e^{x+\frac {e^3}{5+x+2 x^2}} \] Input:
Integrate[(E^x*(25 + 10*x + 21*x^2 + 4*x^3 + 4*x^4) + E^((E^3 + 5*x + x^2 + 2*x^3)/(5 + x + 2*x^2))*(25 + E^3*(-1 - 4*x) + 10*x + 21*x^2 + 4*x^3 + 4 *x^4))/(25 + 10*x + 21*x^2 + 4*x^3 + 4*x^4),x]
Output:
E^x + E^(x + E^3/(5 + x + 2*x^2))
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.
\(\displaystyle \int \frac {e^x \left (4 x^4+4 x^3+21 x^2+10 x+25\right )+e^{\frac {2 x^3+x^2+5 x+e^3}{2 x^2+x+5}} \left (4 x^4+4 x^3+21 x^2+10 x+e^3 (-4 x-1)+25\right )}{4 x^4+4 x^3+21 x^2+10 x+25} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {16 i \left (e^x \left (4 x^4+4 x^3+21 x^2+10 x+25\right )+e^{\frac {2 x^3+x^2+5 x+e^3}{2 x^2+x+5}} \left (4 x^4+4 x^3+21 x^2+10 x+e^3 (-4 x-1)+25\right )\right )}{39 \sqrt {39} \left (-4 x+i \sqrt {39}-1\right )}+\frac {16 i \left (e^x \left (4 x^4+4 x^3+21 x^2+10 x+25\right )+e^{\frac {2 x^3+x^2+5 x+e^3}{2 x^2+x+5}} \left (4 x^4+4 x^3+21 x^2+10 x+e^3 (-4 x-1)+25\right )\right )}{39 \sqrt {39} \left (4 x+i \sqrt {39}+1\right )}-\frac {16 \left (e^x \left (4 x^4+4 x^3+21 x^2+10 x+25\right )+e^{\frac {2 x^3+x^2+5 x+e^3}{2 x^2+x+5}} \left (4 x^4+4 x^3+21 x^2+10 x+e^3 (-4 x-1)+25\right )\right )}{39 \left (-4 x+i \sqrt {39}-1\right )^2}-\frac {16 \left (e^x \left (4 x^4+4 x^3+21 x^2+10 x+25\right )+e^{\frac {2 x^3+x^2+5 x+e^3}{2 x^2+x+5}} \left (4 x^4+4 x^3+21 x^2+10 x+e^3 (-4 x-1)+25\right )\right )}{39 \left (4 x+i \sqrt {39}+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (e^{\frac {e^3}{2 x^2+x+5}+x}-\frac {e^{\frac {2 x^3+7 x^2+8 x+e^3+15}{2 x^2+x+5}} (4 x+1)}{\left (2 x^2+x+5\right )^2}+e^x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{x+\frac {e^3}{2 x^2+x+5}}dx-\frac {16}{39} \left (1-i \sqrt {39}\right ) \int \frac {e^{\frac {2 x^3+7 x^2+8 x+e^3+15}{2 x^2+x+5}}}{\left (-4 x+i \sqrt {39}-1\right )^2}dx+\frac {16}{39} \int \frac {e^{\frac {2 x^3+7 x^2+8 x+e^3+15}{2 x^2+x+5}}}{\left (-4 x+i \sqrt {39}-1\right )^2}dx-\frac {16}{39} \left (1+i \sqrt {39}\right ) \int \frac {e^{\frac {2 x^3+7 x^2+8 x+e^3+15}{2 x^2+x+5}}}{\left (4 x+i \sqrt {39}+1\right )^2}dx+\frac {16}{39} \int \frac {e^{\frac {2 x^3+7 x^2+8 x+e^3+15}{2 x^2+x+5}}}{\left (4 x+i \sqrt {39}+1\right )^2}dx+e^x\) |
Input:
Int[(E^x*(25 + 10*x + 21*x^2 + 4*x^3 + 4*x^4) + E^((E^3 + 5*x + x^2 + 2*x^ 3)/(5 + x + 2*x^2))*(25 + E^3*(-1 - 4*x) + 10*x + 21*x^2 + 4*x^3 + 4*x^4)) /(25 + 10*x + 21*x^2 + 4*x^3 + 4*x^4),x]
Output:
$Aborted
Time = 2.57 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
risch | \({\mathrm e}^{x}+{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}}\) | \(30\) |
parallelrisch | \({\mathrm e}^{x}+{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}}\) | \(30\) |
parts | \(\frac {{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}} x +2 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}} x^{2}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}}}{2 x^{2}+x +5}+{\mathrm e}^{x}\) | \(103\) |
norman | \(\frac {{\mathrm e}^{x} x +{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}} x +2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}} x^{2}+5 \,{\mathrm e}^{x}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+2 x^{3}+x^{2}+5 x}{2 x^{2}+x +5}}}{2 x^{2}+x +5}\) | \(115\) |
Input:
int((((-4*x-1)*exp(3)+4*x^4+4*x^3+21*x^2+10*x+25)*exp((exp(3)+2*x^3+x^2+5* x)/(2*x^2+x+5))+(4*x^4+4*x^3+21*x^2+10*x+25)*exp(x))/(4*x^4+4*x^3+21*x^2+1 0*x+25),x,method=_RETURNVERBOSE)
Output:
exp(x)+exp((exp(3)+2*x^3+x^2+5*x)/(2*x^2+x+5))
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx=e^{x} + e^{\left (\frac {2 \, x^{3} + x^{2} + 5 \, x + e^{3}}{2 \, x^{2} + x + 5}\right )} \] Input:
integrate((((-4*x-1)*exp(3)+4*x^4+4*x^3+21*x^2+10*x+25)*exp((exp(3)+2*x^3+ x^2+5*x)/(2*x^2+x+5))+(4*x^4+4*x^3+21*x^2+10*x+25)*exp(x))/(4*x^4+4*x^3+21 *x^2+10*x+25),x, algorithm="fricas")
Output:
e^x + e^((2*x^3 + x^2 + 5*x + e^3)/(2*x^2 + x + 5))
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx=e^{x} + e^{\frac {2 x^{3} + x^{2} + 5 x + e^{3}}{2 x^{2} + x + 5}} \] Input:
integrate((((-4*x-1)*exp(3)+4*x**4+4*x**3+21*x**2+10*x+25)*exp((exp(3)+2*x **3+x**2+5*x)/(2*x**2+x+5))+(4*x**4+4*x**3+21*x**2+10*x+25)*exp(x))/(4*x** 4+4*x**3+21*x**2+10*x+25),x)
Output:
exp(x) + exp((2*x**3 + x**2 + 5*x + exp(3))/(2*x**2 + x + 5))
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx=e^{\left (x + \frac {e^{3}}{2 \, x^{2} + x + 5}\right )} + e^{x} \] Input:
integrate((((-4*x-1)*exp(3)+4*x^4+4*x^3+21*x^2+10*x+25)*exp((exp(3)+2*x^3+ x^2+5*x)/(2*x^2+x+5))+(4*x^4+4*x^3+21*x^2+10*x+25)*exp(x))/(4*x^4+4*x^3+21 *x^2+10*x+25),x, algorithm="maxima")
Output:
e^(x + e^3/(2*x^2 + x + 5)) + e^x
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx={\left (e^{\left (x + 3\right )} + e^{\left (\frac {10 \, x^{3} - 2 \, x^{2} e^{3} + 5 \, x^{2} - x e^{3} + 25 \, x}{5 \, {\left (2 \, x^{2} + x + 5\right )}} + \frac {1}{5} \, e^{3} + 3\right )}\right )} e^{\left (-3\right )} \] Input:
integrate((((-4*x-1)*exp(3)+4*x^4+4*x^3+21*x^2+10*x+25)*exp((exp(3)+2*x^3+ x^2+5*x)/(2*x^2+x+5))+(4*x^4+4*x^3+21*x^2+10*x+25)*exp(x))/(4*x^4+4*x^3+21 *x^2+10*x+25),x, algorithm="giac")
Output:
(e^(x + 3) + e^(1/5*(10*x^3 - 2*x^2*e^3 + 5*x^2 - x*e^3 + 25*x)/(2*x^2 + x + 5) + 1/5*e^3 + 3))*e^(-3)
Time = 2.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx={\mathrm {e}}^x+{\mathrm {e}}^{\frac {x^2}{2\,x^2+x+5}}\,{\mathrm {e}}^{\frac {2\,x^3}{2\,x^2+x+5}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{2\,x^2+x+5}}\,{\mathrm {e}}^{\frac {5\,x}{2\,x^2+x+5}} \] Input:
int((exp((5*x + exp(3) + x^2 + 2*x^3)/(x + 2*x^2 + 5))*(10*x + 21*x^2 + 4* x^3 + 4*x^4 - exp(3)*(4*x + 1) + 25) + exp(x)*(10*x + 21*x^2 + 4*x^3 + 4*x ^4 + 25))/(10*x + 21*x^2 + 4*x^3 + 4*x^4 + 25),x)
Output:
exp(x) + exp(x^2/(x + 2*x^2 + 5))*exp((2*x^3)/(x + 2*x^2 + 5))*exp(exp(3)/ (x + 2*x^2 + 5))*exp((5*x)/(x + 2*x^2 + 5))
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (25+10 x+21 x^2+4 x^3+4 x^4\right )+e^{\frac {e^3+5 x+x^2+2 x^3}{5+x+2 x^2}} \left (25+e^3 (-1-4 x)+10 x+21 x^2+4 x^3+4 x^4\right )}{25+10 x+21 x^2+4 x^3+4 x^4} \, dx=e^{x} \left (e^{\frac {e^{3}}{2 x^{2}+x +5}}+1\right ) \] Input:
int((((-4*x-1)*exp(3)+4*x^4+4*x^3+21*x^2+10*x+25)*exp((exp(3)+2*x^3+x^2+5* x)/(2*x^2+x+5))+(4*x^4+4*x^3+21*x^2+10*x+25)*exp(x))/(4*x^4+4*x^3+21*x^2+1 0*x+25),x)
Output:
e**x*(e**(e**3/(2*x**2 + x + 5)) + 1)