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

Optimal. Leaf size=17 \[ e^{-\frac {20}{3}-3 e^x (1-x)+x} \]

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Rubi [F]  time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{\frac {1}{3} \left (-20+3 x+e^x (-9+9 x)\right )} \left (1+3 e^x x\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 15, normalized size = 0.88 \begin {gather*} e^{-\frac {20}{3}+3 e^x (-1+x)+x} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.62, size = 11, normalized size = 0.65 \begin {gather*} e^{\left (3 \, {\left (x - 1\right )} e^{x} + x - \frac {20}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.17, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (3 \, x e^{x} + x - 3 \, e^{x} - \frac {20}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.04, size = 14, normalized size = 0.82

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.44, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (3 \, x e^{x} + x - 3 \, e^{x} - \frac {20}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.06, size = 16, normalized size = 0.94 \begin {gather*} {\mathrm {e}}^{3\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-\frac {20}{3}}\,{\mathrm {e}}^{-3\,{\mathrm {e}}^x}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.82 \begin {gather*} e^{x + \left (3 x - 3\right ) e^{x} - \frac {20}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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