Integrand size = 168, antiderivative size = 21 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\left (3+\left (x+e^x x\right ) \left (4+\frac {x}{2+x}\right )\right )^x \] Output:
exp(ln((x/(2+x)+4)*(exp(x)*x+x)+3)*x)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\left (\frac {6+\left (11+8 e^x\right ) x+5 \left (1+e^x\right ) x^2}{2+x}\right )^x \] Input:
Integrate[(((6 + 11*x + 5*x^2 + E^x*(8*x + 5*x^2))/(2 + x))^x*(16*x + 20*x ^2 + 5*x^3 + E^x*(16*x + 36*x^2 + 23*x^3 + 5*x^4) + (12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3))*Log[(6 + 11*x + 5*x^2 + E^x*(8*x + 5* x^2))/(2 + x)]))/(12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3) ),x]
Output:
((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + 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.
| \(\displaystyle \int \frac {\left (\frac {5 x^2+e^x \left (5 x^2+8 x\right )+11 x+6}{x+2}\right )^x \left (5 x^3+20 x^2+\left (5 x^3+21 x^2+e^x \left (5 x^3+18 x^2+16 x\right )+28 x+12\right ) \log \left (\frac {5 x^2+e^x \left (5 x^2+8 x\right )+11 x+6}{x+2}\right )+e^x \left (5 x^4+23 x^3+36 x^2+16 x\right )+16 x\right )}{5 x^3+21 x^2+e^x \left (5 x^3+18 x^2+16 x\right )+28 x+12} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (\frac {5 x^2+e^x \left (5 x^2+8 x\right )+11 x+6}{x+2}\right )^x \left (5 x^3+20 x^2+\left (5 x^3+21 x^2+e^x \left (5 x^3+18 x^2+16 x\right )+28 x+12\right ) \log \left (\frac {5 x^2+e^x \left (5 x^2+8 x\right )+11 x+6}{x+2}\right )+e^x \left (5 x^4+23 x^3+36 x^2+16 x\right )+16 x\right )}{(x+2) \left (5 e^x x^2+5 x^2+8 e^x x+11 x+6\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (\frac {5 x^2+e^x \left (5 x^2+8 x\right )+11 x+6}{x+2}\right )^x \left (5 x^3+23 x^2+5 x^2 \log \left (\frac {5 \left (e^x+1\right ) x^2+\left (8 e^x+11\right ) x+6}{x+2}\right )+18 x \log \left (\frac {5 \left (e^x+1\right ) x^2+\left (8 e^x+11\right ) x+6}{x+2}\right )+16 \log \left (\frac {5 \left (e^x+1\right ) x^2+\left (8 e^x+11\right ) x+6}{x+2}\right )+36 x+16\right )}{(x+2) (5 x+8)}-\frac {\left (25 x^4+95 x^3+133 x^2+108 x+48\right ) \left (\frac {5 x^2+e^x \left (5 x^2+8 x\right )+11 x+6}{x+2}\right )^x}{(5 x+8) \left (5 e^x x^2+5 x^2+8 e^x x+11 x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^x-4 \int e^x \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}dx-5 \int x \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}dx-3 \int e^x x \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}dx-5 \int e^x x^2 \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}dx-8 \int \frac {\left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}}{(x+2)^2}dx-8 \int \frac {e^x \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}}{(x+2)^2}dx+4 \int \frac {\left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}}{x+2}dx+12 \int \frac {e^x \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^{x-1}}{x+2}dx+\int \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^xdx+\int x \left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^xdx+2 \int \frac {\left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^x}{x+2}dx-8 \int \frac {\left (\frac {5 \left (1+e^x\right ) x^2+\left (11+8 e^x\right ) x+6}{x+2}\right )^x}{5 x+8}dx-\frac {36}{5} \int \frac {\left (\frac {5 x^2+11 x+e^x \left (5 x^2+8 x\right )+6}{x+2}\right )^x}{5 e^x x^2+5 x^2+8 e^x x+11 x+6}dx-9 \int \frac {x \left (\frac {5 x^2+11 x+e^x \left (5 x^2+8 x\right )+6}{x+2}\right )^x}{5 e^x x^2+5 x^2+8 e^x x+11 x+6}dx-11 \int \frac {x^2 \left (\frac {5 x^2+11 x+e^x \left (5 x^2+8 x\right )+6}{x+2}\right )^x}{5 e^x x^2+5 x^2+8 e^x x+11 x+6}dx-5 \int \frac {x^3 \left (\frac {5 x^2+11 x+e^x \left (5 x^2+8 x\right )+6}{x+2}\right )^x}{5 e^x x^2+5 x^2+8 e^x x+11 x+6}dx+\frac {48}{5} \int \frac {\left (\frac {5 x^2+11 x+e^x \left (5 x^2+8 x\right )+6}{x+2}\right )^x}{(5 x+8) \left (5 e^x x^2+5 x^2+8 e^x x+11 x+6\right )}dx\) |
Input:
Int[(((6 + 11*x + 5*x^2 + E^x*(8*x + 5*x^2))/(2 + x))^x*(16*x + 20*x^2 + 5 *x^3 + E^x*(16*x + 36*x^2 + 23*x^3 + 5*x^4) + (12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3))*Log[(6 + 11*x + 5*x^2 + E^x*(8*x + 5*x^2))/ (2 + x)]))/(12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3)),x]
Output:
$Aborted
Time = 21.99 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57
| method | result | size |
| parallelrisch | \({\mathrm e}^{x \ln \left (\frac {\left (5 x^{2}+8 x \right ) {\mathrm e}^{x}+5 x^{2}+11 x +6}{2+x}\right )}\) | \(33\) |
| risch | \(5^{x} \left (2+x \right )^{-x} {\left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}^{x} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}{2+x}\right ) x \left (-\operatorname {csgn}\left (\frac {i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}{2+x}\right )+\operatorname {csgn}\left (i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}{2+x}\right )+\operatorname {csgn}\left (\frac {i}{2+x}\right )\right )}{2}}\) | \(156\) |
Input:
int((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*ln(((5*x^2+8*x)*exp (x)+5*x^2+11*x+6)/(2+x))+(5*x^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20*x^2+16 *x)*exp(x*ln(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+18*x^2+16*x )*exp(x)+5*x^3+21*x^2+28*x+12),x,method=_RETURNVERBOSE)
Output:
exp(x*ln(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\left (\frac {5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6}{x + 2}\right )^{x} \] Input:
integrate((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8 *x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20 *x^2+16*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+18* x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12),x, algorithm="fricas")
Output:
((5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6)/(x + 2))^x
Timed out. \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((((5*x**3+18*x**2+16*x)*exp(x)+5*x**3+21*x**2+28*x+12)*ln(((5*x* *2+8*x)*exp(x)+5*x**2+11*x+6)/(2+x))+(5*x**4+23*x**3+36*x**2+16*x)*exp(x)+ 5*x**3+20*x**2+16*x)*exp(x*ln(((5*x**2+8*x)*exp(x)+5*x**2+11*x+6)/(2+x)))/ ((5*x**3+18*x**2+16*x)*exp(x)+5*x**3+21*x**2+28*x+12),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=e^{\left (x \log \left (5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6\right ) - x \log \left (x + 2\right )\right )} \] Input:
integrate((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8 *x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20 *x^2+16*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+18* x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12),x, algorithm="maxima")
Output:
e^(x*log(5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6) - x*log(x + 2))
\[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\int { \frac {{\left (5 \, x^{3} + 20 \, x^{2} + {\left (5 \, x^{4} + 23 \, x^{3} + 36 \, x^{2} + 16 \, x\right )} e^{x} + {\left (5 \, x^{3} + 21 \, x^{2} + {\left (5 \, x^{3} + 18 \, x^{2} + 16 \, x\right )} e^{x} + 28 \, x + 12\right )} \log \left (\frac {5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6}{x + 2}\right ) + 16 \, x\right )} \left (\frac {5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6}{x + 2}\right )^{x}}{5 \, x^{3} + 21 \, x^{2} + {\left (5 \, x^{3} + 18 \, x^{2} + 16 \, x\right )} e^{x} + 28 \, x + 12} \,d x } \] Input:
integrate((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8 *x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20 *x^2+16*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+18* x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12),x, algorithm="giac")
Output:
integrate((5*x^3 + 20*x^2 + (5*x^4 + 23*x^3 + 36*x^2 + 16*x)*e^x + (5*x^3 + 21*x^2 + (5*x^3 + 18*x^2 + 16*x)*e^x + 28*x + 12)*log((5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6)/(x + 2)) + 16*x)*((5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6)/(x + 2))^x/(5*x^3 + 21*x^2 + (5*x^3 + 18*x^2 + 16*x)*e^x + 28*x + 12), x)
Time = 3.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx={\left (\frac {11\,x+5\,x^2\,{\mathrm {e}}^x+8\,x\,{\mathrm {e}}^x+5\,x^2+6}{x+2}\right )}^x \] Input:
int((exp(x*log((11*x + exp(x)*(8*x + 5*x^2) + 5*x^2 + 6)/(x + 2)))*(16*x + log((11*x + exp(x)*(8*x + 5*x^2) + 5*x^2 + 6)/(x + 2))*(28*x + 21*x^2 + 5 *x^3 + exp(x)*(16*x + 18*x^2 + 5*x^3) + 12) + exp(x)*(16*x + 36*x^2 + 23*x ^3 + 5*x^4) + 20*x^2 + 5*x^3))/(28*x + 21*x^2 + 5*x^3 + exp(x)*(16*x + 18* x^2 + 5*x^3) + 12),x)
Output:
((11*x + 5*x^2*exp(x) + 8*x*exp(x) + 5*x^2 + 6)/(x + 2))^x
\[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\text {too large to display} \] Input:
int((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8*x)*ex p(x)+5*x^2+11*x+6)/(2+x))+(5*x^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20*x^2+1 6*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+18*x^2+16 *x)*exp(x)+5*x^3+21*x^2+28*x+12),x)
Output:
5*int(((5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)**x*x**3)/(5*e**x*(x + 2)**x*x**3 + 18*e**x*(x + 2)**x*x**2 + 16*e**x*(x + 2)**x*x + 5*(x + 2)**x *x**3 + 21*(x + 2)**x*x**2 + 28*(x + 2)**x*x + 12*(x + 2)**x),x) + 20*int( ((5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)**x*x**2)/(5*e**x*(x + 2)**x* x**3 + 18*e**x*(x + 2)**x*x**2 + 16*e**x*(x + 2)**x*x + 5*(x + 2)**x*x**3 + 21*(x + 2)**x*x**2 + 28*(x + 2)**x*x + 12*(x + 2)**x),x) + 5*int(((5*e** x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)**x*log((5*e**x*x**2 + 8*e**x*x + 5* x**2 + 11*x + 6)/(x + 2))*x**3)/(5*e**x*(x + 2)**x*x**3 + 18*e**x*(x + 2)* *x*x**2 + 16*e**x*(x + 2)**x*x + 5*(x + 2)**x*x**3 + 21*(x + 2)**x*x**2 + 28*(x + 2)**x*x + 12*(x + 2)**x),x) + 21*int(((5*e**x*x**2 + 8*e**x*x + 5* x**2 + 11*x + 6)**x*log((5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)/(x + 2))*x**2)/(5*e**x*(x + 2)**x*x**3 + 18*e**x*(x + 2)**x*x**2 + 16*e**x*(x + 2)**x*x + 5*(x + 2)**x*x**3 + 21*(x + 2)**x*x**2 + 28*(x + 2)**x*x + 12*( x + 2)**x),x) + 28*int(((5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)**x*lo g((5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)/(x + 2))*x)/(5*e**x*(x + 2) **x*x**3 + 18*e**x*(x + 2)**x*x**2 + 16*e**x*(x + 2)**x*x + 5*(x + 2)**x*x **3 + 21*(x + 2)**x*x**2 + 28*(x + 2)**x*x + 12*(x + 2)**x),x) + 12*int((( 5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)**x*log((5*e**x*x**2 + 8*e**x*x + 5*x**2 + 11*x + 6)/(x + 2)))/(5*e**x*(x + 2)**x*x**3 + 18*e**x*(x + 2)* *x*x**2 + 16*e**x*(x + 2)**x*x + 5*(x + 2)**x*x**3 + 21*(x + 2)**x*x**2...