Integrand size = 57, antiderivative size = 34 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=3+\left (3-e^{\frac {e^x}{3}+2 x}+x (3+x)\right ) \left (x+3 \left (-3+x^2\right )\right ) \] Output:
(3*x^2+x-9)*((3+x)*x-exp(1/3*exp(x)+2*x)+3)+3
Time = 0.72 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=-24 x+3 x^2+10 x^3+3 x^4-e^{\frac {e^x}{3}+2 x} \left (-9+x+3 x^2\right ) \] Input:
Integrate[(-72 + 18*x + 90*x^2 + 36*x^3 + E^((E^x + 6*x)/3)*(51 - 24*x - 1 8*x^2 + E^x*(9 - x - 3*x^2)))/3,x]
Output:
-24*x + 3*x^2 + 10*x^3 + 3*x^4 - E^(E^x/3 + 2*x)*(-9 + x + 3*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.
\(\displaystyle \int \frac {1}{3} \left (36 x^3+90 x^2+e^{\frac {1}{3} \left (6 x+e^x\right )} \left (-18 x^2+e^x \left (-3 x^2-x+9\right )-24 x+51\right )+18 x-72\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \left (36 x^3+90 x^2+18 x+e^{\frac {1}{3} \left (6 x+e^x\right )} \left (-18 x^2-24 x+e^x \left (-3 x^2-x+9\right )+51\right )-72\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-18 \int e^{\frac {1}{3} \left (6 x+e^x\right )} x^2dx-3 \int e^{\frac {1}{3} \left (9 x+e^x\right )} x^2dx+51 \int e^{\frac {1}{3} \left (6 x+e^x\right )}dx+9 \int e^{\frac {1}{3} \left (9 x+e^x\right )}dx-24 \int e^{\frac {1}{3} \left (6 x+e^x\right )} xdx-\int e^{\frac {1}{3} \left (9 x+e^x\right )} xdx+9 x^4+30 x^3+9 x^2-72 x\right )\) |
Input:
Int[(-72 + 18*x + 90*x^2 + 36*x^3 + E^((E^x + 6*x)/3)*(51 - 24*x - 18*x^2 + E^x*(9 - x - 3*x^2)))/3,x]
Output:
$Aborted
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {\left (-9 x^{2}-3 x +27\right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}}{3}+3 x^{4}+10 x^{3}+3 x^{2}-24 x\) | \(41\) |
default | \(-24 x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x -3 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x^{2}+9 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}+3 x^{2}+10 x^{3}+3 x^{4}\) | \(57\) |
norman | \(-24 x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x -3 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x^{2}+9 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}+3 x^{2}+10 x^{3}+3 x^{4}\) | \(57\) |
parallelrisch | \(-24 x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x -3 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x} x^{2}+9 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{3}+2 x}+3 x^{2}+10 x^{3}+3 x^{4}\) | \(57\) |
Input:
int(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x^3+30 *x^2+6*x-24,x,method=_RETURNVERBOSE)
Output:
1/3*(-9*x^2-3*x+27)*exp(1/3*exp(x)+2*x)+3*x^4+10*x^3+3*x^2-24*x
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=3 \, x^{4} + 10 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} + x - 9\right )} e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} - 24 \, x \] Input:
integrate(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12* x^3+30*x^2+6*x-24,x, algorithm="fricas")
Output:
3*x^4 + 10*x^3 + 3*x^2 - (3*x^2 + x - 9)*e^(2*x + 1/3*e^x) - 24*x
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=3 x^{4} + 10 x^{3} + 3 x^{2} - 24 x + \left (- 3 x^{2} - x + 9\right ) e^{2 x + \frac {e^{x}}{3}} \] Input:
integrate(1/3*((-3*x**2-x+9)*exp(x)-18*x**2-24*x+51)*exp(1/3*exp(x)+2*x)+1 2*x**3+30*x**2+6*x-24,x)
Output:
3*x**4 + 10*x**3 + 3*x**2 - 24*x + (-3*x**2 - x + 9)*exp(2*x + exp(x)/3)
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=3 \, x^{4} + 10 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} + x - 9\right )} e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} - 24 \, x \] Input:
integrate(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12* x^3+30*x^2+6*x-24,x, algorithm="maxima")
Output:
3*x^4 + 10*x^3 + 3*x^2 - (3*x^2 + x - 9)*e^(2*x + 1/3*e^x) - 24*x
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=3 \, x^{4} + 10 \, x^{3} - 3 \, x^{2} e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} + 3 \, x^{2} - x e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} - 24 \, x + 9 \, e^{\left (2 \, x + \frac {1}{3} \, e^{x}\right )} \] Input:
integrate(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12* x^3+30*x^2+6*x-24,x, algorithm="giac")
Output:
3*x^4 + 10*x^3 - 3*x^2*e^(2*x + 1/3*e^x) + 3*x^2 - x*e^(2*x + 1/3*e^x) - 2 4*x + 9*e^(2*x + 1/3*e^x)
Time = 3.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=9\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{3}}-24\,x-3\,x^2\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{3}}-x\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{3}}+3\,x^2+10\,x^3+3\,x^4 \] Input:
int(6*x + 30*x^2 + 12*x^3 - (exp(2*x + exp(x)/3)*(24*x + exp(x)*(x + 3*x^2 - 9) + 18*x^2 - 51))/3 - 24,x)
Output:
9*exp(2*x + exp(x)/3) - 24*x - 3*x^2*exp(2*x + exp(x)/3) - x*exp(2*x + exp (x)/3) + 3*x^2 + 10*x^3 + 3*x^4
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.82 \[ \int \frac {1}{3} \left (-72+18 x+90 x^2+36 x^3+e^{\frac {1}{3} \left (e^x+6 x\right )} \left (51-24 x-18 x^2+e^x \left (9-x-3 x^2\right )\right )\right ) \, dx=-3 e^{\frac {e^{x}}{3}+2 x} x^{2}-e^{\frac {e^{x}}{3}+2 x} x +9 e^{\frac {e^{x}}{3}+2 x}+3 x^{4}+10 x^{3}+3 x^{2}-24 x \] Input:
int(1/3*((-3*x^2-x+9)*exp(x)-18*x^2-24*x+51)*exp(1/3*exp(x)+2*x)+12*x^3+30 *x^2+6*x-24,x)
Output:
- 3*e**((e**x + 6*x)/3)*x**2 - e**((e**x + 6*x)/3)*x + 9*e**((e**x + 6*x) /3) + 3*x**4 + 10*x**3 + 3*x**2 - 24*x