\(\int \frac {1}{3} (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)) \, dx\) [3997]

Optimal result

Integrand size = 30, antiderivative size = 27 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=e^{5+e^x x}+e^3 x+\frac {1}{3} \left (6+x+2 x^2\right ) \]

[Out]

x*exp(3)+exp(exp(x)*x+5)+2/3*x^2+1/3*x+2

Rubi [F]

\[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=\int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx \]

[In]

Int[(1 + 3*E^3 + 4*x + E^(5 + x + E^x*x)*(3 + 3*x))/3,x]

[Out]

((1 + 3*E^3)*x)/3 + (2*x^2)/3 + Defer[Int][E^(5 + x + E^x*x), x] + Defer[Int][E^(5 + x + E^x*x)*x, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx \\ & = \frac {1}{3} \left (1+3 e^3\right ) x+\frac {2 x^2}{3}+\frac {1}{3} \int e^{5+x+e^x x} (3+3 x) \, dx \\ & = \frac {1}{3} \left (1+3 e^3\right ) x+\frac {2 x^2}{3}+\frac {1}{3} \int \left (3 e^{5+x+e^x x}+3 e^{5+x+e^x x} x\right ) \, dx \\ & = \frac {1}{3} \left (1+3 e^3\right ) x+\frac {2 x^2}{3}+\int e^{5+x+e^x x} \, dx+\int e^{5+x+e^x x} x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=e^{5+e^x x}+\frac {x}{3}+e^3 x+\frac {2 x^2}{3} \]

[In]

Integrate[(1 + 3*E^3 + 4*x + E^(5 + x + E^x*x)*(3 + 3*x))/3,x]

[Out]

E^(5 + E^x*x) + x/3 + E^3*x + (2*x^2)/3

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74

method	result	size
norman	\(\left ({\mathrm e}^{3}+\frac {1}{3}\right ) x +\frac {2 x^{2}}{3}+{\mathrm e}^{{\mathrm e}^{x} x +5}\)	\(20\)
parallelrisch	\(\left ({\mathrm e}^{3}+\frac {1}{3}\right ) x +\frac {2 x^{2}}{3}+{\mathrm e}^{{\mathrm e}^{x} x +5}\)	\(20\)
default	\(\frac {x}{3}+{\mathrm e}^{{\mathrm e}^{x} x +5}+\frac {2 x^{2}}{3}+x \,{\mathrm e}^{3}\)	\(21\)
risch	\(\frac {x}{3}+{\mathrm e}^{{\mathrm e}^{x} x +5}+\frac {2 x^{2}}{3}+x \,{\mathrm e}^{3}\)	\(21\)
parts	\(\frac {x}{3}+{\mathrm e}^{{\mathrm e}^{x} x +5}+\frac {2 x^{2}}{3}+x \,{\mathrm e}^{3}\)	\(21\)

[In]

int(1/3*(3*x+3)*exp(x)*exp(exp(x)*x+5)+exp(3)+4/3*x+1/3,x,method=_RETURNVERBOSE)

[Out]

(exp(3)+1/3)*x+2/3*x^2+exp(exp(x)*x+5)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=\frac {1}{3} \, {\left ({\left (2 \, x^{2} + 3 \, x e^{3} + x\right )} e^{x} + 3 \, e^{\left (x e^{x} + x + 5\right )}\right )} e^{\left (-x\right )} \]

[In]

integrate(1/3*(3*x+3)*exp(x)*exp(exp(x)*x+5)+exp(3)+4/3*x+1/3,x, algorithm="fricas")

[Out]

1/3*((2*x^2 + 3*x*e^3 + x)*e^x + 3*e^(x*e^x + x + 5))*e^(-x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=\frac {2 x^{2}}{3} + x \left (\frac {1}{3} + e^{3}\right ) + e^{x e^{x} + 5} \]

[In]

integrate(1/3*(3*x+3)*exp(x)*exp(exp(x)*x+5)+exp(3)+4/3*x+1/3,x)

[Out]

2*x**2/3 + x*(1/3 + exp(3)) + exp(x*exp(x) + 5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=\frac {2}{3} \, x^{2} + x e^{3} + \frac {1}{3} \, x + e^{\left (x e^{x} + 5\right )} \]

[In]

integrate(1/3*(3*x+3)*exp(x)*exp(exp(x)*x+5)+exp(3)+4/3*x+1/3,x, algorithm="maxima")

[Out]

2/3*x^2 + x*e^3 + 1/3*x + e^(x*e^x + 5)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx=\frac {2}{3} \, x^{2} + x e^{3} + \frac {1}{3} \, x + e^{\left (x e^{x} + 5\right )} \]

[In]

integrate(1/3*(3*x+3)*exp(x)*exp(exp(x)*x+5)+exp(3)+4/3*x+1/3,x, algorithm="giac")

[Out]

2/3*x^2 + x*e^3 + 1/3*x + e^(x*e^x + 5)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {1}{3} \left (1+3 e^3+4 x+e^{5+x+e^x x} (3+3 x)\right ) \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^x+5}+x\,\left ({\mathrm {e}}^3+\frac {1}{3}\right )+\frac {2\,x^2}{3} \]

[In]

int((4*x)/3 + exp(3) + (exp(x*exp(x) + 5)*exp(x)*(3*x + 3))/3 + 1/3,x)

[Out]

exp(x*exp(x) + 5) + x*(exp(3) + 1/3) + (2*x^2)/3