Integrand size = 71, antiderivative size = 25 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=e^{3+x-\frac {4 \left (-2+x^2\right )}{(3+x)^2}}-x+5 \log (5) \] Output:
5*ln(5)+exp(3+x-2*(x^2-2)/(3+x)/(3/2+1/2*x))-x
Time = 0.74 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=e^{-1+x-\frac {28}{(3+x)^2}+\frac {24}{3+x}}-x \] Input:
Integrate[(-27 - 27*x - 9*x^2 - x^3 + E^((35 + 27*x + 5*x^2 + x^3)/(9 + 6* x + x^2))*(11 + 3*x + 9*x^2 + x^3))/(27 + 27*x + 9*x^2 + x^3),x]
Output:
E^(-1 + x - 28/(3 + x)^2 + 24/(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.
\(\displaystyle \int \frac {-x^3-9 x^2+e^{\frac {x^3+5 x^2+27 x+35}{x^2+6 x+9}} \left (x^3+9 x^2+3 x+11\right )-27 x-27}{x^3+9 x^2+27 x+27} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-x^3-9 x^2+e^{\frac {x^3+5 x^2+27 x+35}{x^2+6 x+9}} \left (x^3+9 x^2+3 x+11\right )-27 x-27}{(x+3)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {x^3}{(x+3)^3}-\frac {9 x^2}{(x+3)^3}+\frac {e^{\frac {x^3+5 x^2+27 x+35}{(x+3)^2}} \left (x^3+9 x^2+3 x+11\right )}{(x+3)^3}-\frac {27 x}{(x+3)^3}-\frac {27}{(x+3)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{\frac {x^3+5 x^2+27 x+35}{(x+3)^2}}dx+56 \int \frac {e^{\frac {x^3+5 x^2+27 x+35}{(x+3)^2}}}{(x+3)^3}dx-24 \int \frac {e^{\frac {x^3+5 x^2+27 x+35}{(x+3)^2}}}{(x+3)^2}dx-\frac {9 x^2}{2 (x+3)^2}-x-\frac {27}{x+3}+\frac {81}{2 (x+3)^2}\) |
Input:
Int[(-27 - 27*x - 9*x^2 - x^3 + E^((35 + 27*x + 5*x^2 + x^3)/(9 + 6*x + x^ 2))*(11 + 3*x + 9*x^2 + x^3))/(27 + 27*x + 9*x^2 + x^3),x]
Output:
$Aborted
Time = 0.63 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{\left (3+x \right )^{2}}}\) | \(25\) |
parallelrisch | \(-x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}+18\) | \(31\) |
parts | \(-x +\frac {{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x^{2}+6 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x +9 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}}{\left (3+x \right )^{2}}\) | \(96\) |
norman | \(\frac {27 x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x^{2}-x^{3}+6 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x +9 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}+54}{\left (3+x \right )^{2}}\) | \(101\) |
orering | \(\frac {\left (-8+x \right ) \left (\left (x^{3}+9 x^{2}+3 x +11\right ) {\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}-x^{3}-9 x^{2}-27 x -27\right )}{x^{3}+9 x^{2}+27 x +27}-\frac {\left (x^{4}-78 x^{2}-40 x -115\right ) \left (3+x \right )^{3} \left (\frac {\left (3 x^{2}+18 x +3\right ) {\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}+\left (x^{3}+9 x^{2}+3 x +11\right ) \left (\frac {3 x^{2}+10 x +27}{x^{2}+6 x +9}-\frac {\left (x^{3}+5 x^{2}+27 x +35\right ) \left (2 x +6\right )}{\left (x^{2}+6 x +9\right )^{2}}\right ) {\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}-3 x^{2}-18 x -27}{x^{3}+9 x^{2}+27 x +27}-\frac {\left (\left (x^{3}+9 x^{2}+3 x +11\right ) {\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}-x^{3}-9 x^{2}-27 x -27\right ) \left (3 x^{2}+18 x +27\right )}{\left (x^{3}+9 x^{2}+27 x +27\right )^{2}}\right )}{x^{6}+18 x^{5}+87 x^{4}+124 x^{3}+471 x^{2}+354 x -95}\) | \(360\) |
Input:
int(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x^2-27* x-27)/(x^3+9*x^2+27*x+27),x,method=_RETURNVERBOSE)
Output:
-x+exp((x^3+5*x^2+27*x+35)/(3+x)^2)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=-x + e^{\left (\frac {x^{3} + 5 \, x^{2} + 27 \, x + 35}{x^{2} + 6 \, x + 9}\right )} \] Input:
integrate(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x ^2-27*x-27)/(x^3+9*x^2+27*x+27),x, algorithm="fricas")
Output:
-x + e^((x^3 + 5*x^2 + 27*x + 35)/(x^2 + 6*x + 9))
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=- x + e^{\frac {x^{3} + 5 x^{2} + 27 x + 35}{x^{2} + 6 x + 9}} \] Input:
integrate(((x**3+9*x**2+3*x+11)*exp((x**3+5*x**2+27*x+35)/(x**2+6*x+9))-x* *3-9*x**2-27*x-27)/(x**3+9*x**2+27*x+27),x)
Output:
-x + exp((x**3 + 5*x**2 + 27*x + 35)/(x**2 + 6*x + 9))
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=-x - \frac {27 \, {\left (4 \, x + 9\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27 \, {\left (2 \, x + 5\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27 \, {\left (2 \, x + 3\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + e^{\left (x - \frac {28}{x^{2} + 6 \, x + 9} + \frac {24}{x + 3} - 1\right )} \] Input:
integrate(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x ^2-27*x-27)/(x^3+9*x^2+27*x+27),x, algorithm="maxima")
Output:
-x - 27/2*(4*x + 9)/(x^2 + 6*x + 9) + 27/2*(2*x + 5)/(x^2 + 6*x + 9) + 27/ 2*(2*x + 3)/(x^2 + 6*x + 9) + 27/2/(x^2 + 6*x + 9) + e^(x - 28/(x^2 + 6*x + 9) + 24/(x + 3) - 1)
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=-x + e^{\left (\frac {9 \, x^{3} + 10 \, x^{2} + 33 \, x}{9 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {35}{9}\right )} \] Input:
integrate(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x ^2-27*x-27)/(x^3+9*x^2+27*x+27),x, algorithm="giac")
Output:
-x + e^(1/9*(9*x^3 + 10*x^2 + 33*x)/(x^2 + 6*x + 9) + 35/9)
Time = 0.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx={\mathrm {e}}^{\frac {x^3}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {5\,x^2}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {35}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {27\,x}{x^2+6\,x+9}}-x \] Input:
int(-(27*x - exp((27*x + 5*x^2 + x^3 + 35)/(6*x + x^2 + 9))*(3*x + 9*x^2 + x^3 + 11) + 9*x^2 + x^3 + 27)/(27*x + 9*x^2 + x^3 + 27),x)
Output:
exp(x^3/(6*x + x^2 + 9))*exp((5*x^2)/(6*x + x^2 + 9))*exp(35/(6*x + x^2 + 9))*exp((27*x)/(6*x + x^2 + 9)) - x
Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx=\frac {e^{\frac {x^{3}+6 x^{2}+33 x +44}{x^{2}+6 x +9}}-e x}{e} \] Input:
int(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x^2-27* x-27)/(x^3+9*x^2+27*x+27),x)
Output:
(e**((x**3 + 6*x**2 + 33*x + 44)/(x**2 + 6*x + 9)) - e*x)/e