\(\int e^{\frac {1}{3} (-20+3 x+e^x (-9+9 x))} (1+3 e^x x) \, dx\) [6688]

Optimal result

Integrand size = 29, antiderivative size = 17 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=e^{-\frac {20}{3}-3 e^x (1-x)+x} \]

[Out]

exp(x-3*(1-x)*exp(x)-20/3)

Rubi [F]

\[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=\int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx \]

[In]

Int[E^((-20 + 3*x + E^x*(-9 + 9*x))/3)*(1 + 3*E^x*x),x]

[Out]

Defer[Int][E^((-20 + 3*x + E^x*(-9 + 9*x))/3), x] + 3*Defer[Int][E^((-20 - 9*E^x + 6*x + 9*E^x*x)/3)*x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )}+3 e^{x+\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} x\right ) \, dx \\ & = 3 \int e^{x+\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} x \, dx+\int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \, dx \\ & = 3 \int e^{\frac {1}{3} \left (-20-9 e^x+6 x+9 e^x x\right )} x \, dx+\int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=e^{-\frac {20}{3}+3 e^x (-1+x)+x} \]

[In]

Integrate[E^((-20 + 3*x + E^x*(-9 + 9*x))/3)*(1 + 3*E^x*x),x]

[Out]

E^(-20/3 + 3*E^x*(-1 + x) + x)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\({\mathrm e}^{\frac {\left (9 x -9\right ) {\mathrm e}^{x}}{3}+x -\frac {20}{3}}\)	\(14\)
default	\({\mathrm e}^{\frac {\left (9 x -9\right ) {\mathrm e}^{x}}{3}+x -\frac {20}{3}}\)	\(14\)
norman	\({\mathrm e}^{\frac {\left (9 x -9\right ) {\mathrm e}^{x}}{3}+x -\frac {20}{3}}\)	\(14\)
risch	\({\mathrm e}^{3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}+x -\frac {20}{3}}\)	\(14\)
parallelrisch	\({\mathrm e}^{3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}+x -\frac {20}{3}}\)	\(14\)

[In]

int((3*exp(x)*x+1)*exp(1/3*(9*x-9)*exp(x)+x-20/3),x,method=_RETURNVERBOSE)

[Out]

exp(1/3*(9*x-9)*exp(x)+x-20/3)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=e^{\left (3 \, {\left (x - 1\right )} e^{x} + x - \frac {20}{3}\right )} \]

[In]

integrate((3*exp(x)*x+1)*exp(1/3*(9*x-9)*exp(x)+x-20/3),x, algorithm="fricas")

[Out]

e^(3*(x - 1)*e^x + x - 20/3)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=e^{x + \left (3 x - 3\right ) e^{x} - \frac {20}{3}} \]

[In]

integrate((3*exp(x)*x+1)*exp(1/3*(9*x-9)*exp(x)+x-20/3),x)

[Out]

exp(x + (3*x - 3)*exp(x) - 20/3)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=e^{\left (3 \, x e^{x} + x - 3 \, e^{x} - \frac {20}{3}\right )} \]

[In]

integrate((3*exp(x)*x+1)*exp(1/3*(9*x-9)*exp(x)+x-20/3),x, algorithm="maxima")

[Out]

e^(3*x*e^x + x - 3*e^x - 20/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx=e^{\left (3 \, x e^{x} + x - 3 \, e^{x} - \frac {20}{3}\right )} \]

[In]

integrate((3*exp(x)*x+1)*exp(1/3*(9*x-9)*exp(x)+x-20/3),x, algorithm="giac")

[Out]

e^(3*x*e^x + x - 3*e^x - 20/3)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx={\mathrm {e}}^{3\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-\frac {20}{3}}\,{\mathrm {e}}^{-3\,{\mathrm {e}}^x}\,{\mathrm {e}}^x \]

[In]

int(exp(x + (exp(x)*(9*x - 9))/3 - 20/3)*(3*x*exp(x) + 1),x)

[Out]

exp(3*x*exp(x))*exp(-20/3)*exp(-3*exp(x))*exp(x)