3.89.72 \(\int e^{-78+56 x-13 x^2+x^3+e^x (-16+8 x-x^2)} (-55+82 x-29 x^2+3 x^3+e^3 (56-26 x+3 x^2)+e^x (8-14 x+7 x^2-x^3+e^3 (-8+6 x-x^2))) \, dx\)

Optimal. Leaf size=29 \[ e^{2-(4-x) (-4+x) \left (-5-e^x+x\right )} \left (-1+e^3+x\right ) \]

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Rubi [F]  time = 4.72, 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 \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*(-55 + 82*x - 29*x^2 + 3*x^3 + E^3*(56 - 26*x + 3*x^
2) + E^x*(8 - 14*x + 7*x^2 - x^3 + E^3*(-8 + 6*x - x^2))),x]

[Out]

-55*Defer[Int][E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2)), x] + 56*Defer[Int][E^(-75 + 56*x - 13*x^
2 + x^3 + E^x*(-16 + 8*x - x^2)), x] + 8*(1 - E^3)*Defer[Int][E^(-78 + 57*x - 13*x^2 + x^3 + E^x*(-16 + 8*x -
x^2)), x] + 82*Defer[Int][E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*x, x] - 26*Defer[Int][E^(-75 +
56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*x, x] - 2*(7 - 3*E^3)*Defer[Int][E^(-78 + 57*x - 13*x^2 + x^3 +
E^x*(-16 + 8*x - x^2))*x, x] - 29*Defer[Int][E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*x^2, x] + 3
*Defer[Int][E^(-75 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*x^2, x] + (7 - E^3)*Defer[Int][E^(-78 + 57*x
- 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*x^2, x] + 3*Defer[Int][E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x
- x^2))*x^3, x] - Defer[Int][E^(-78 + 57*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-55 \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right )+82 \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x-29 \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2+3 \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^3-\exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (-1+e^3+x\right ) \left (8-6 x+x^2\right )+\exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (56-26 x+3 x^2\right )\right ) \, dx\\ &=3 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^3 \, dx-29 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2 \, dx-55 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \, dx+82 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x \, dx-\int \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (-1+e^3+x\right ) \left (8-6 x+x^2\right ) \, dx+\int \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (56-26 x+3 x^2\right ) \, dx\\ &=3 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^3 \, dx-29 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2 \, dx-55 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \, dx+82 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x \, dx+\int \left (56 \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right )-26 \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x+3 \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2\right ) \, dx-\int \left (8 \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (-1+e^3\right )-2 \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (-7+3 e^3\right ) x+\exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \left (-7+e^3\right ) x^2+\exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^3\right ) \, dx\\ &=3 \int \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2 \, dx+3 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^3 \, dx-26 \int \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x \, dx-29 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2 \, dx-55 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \, dx+56 \int \exp \left (-75+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \, dx+82 \int \exp \left (-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x \, dx-\left (2 \left (7-3 e^3\right )\right ) \int \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x \, dx+\left (8 \left (1-e^3\right )\right ) \int \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) \, dx-\left (-7+e^3\right ) \int \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^2 \, dx-\int \exp \left (-78+57 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )\right ) x^3 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 32, normalized size = 1.10 \begin {gather*} e^{-78-e^x (-4+x)^2+56 x-13 x^2+x^3} \left (-1+e^3+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*(-55 + 82*x - 29*x^2 + 3*x^3 + E^3*(56 - 26*x
+ 3*x^2) + E^x*(8 - 14*x + 7*x^2 - x^3 + E^3*(-8 + 6*x - x^2))),x]

[Out]

E^(-78 - E^x*(-4 + x)^2 + 56*x - 13*x^2 + x^3)*(-1 + E^3 + x)

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fricas [A]  time = 0.51, size = 32, normalized size = 1.10 \begin {gather*} {\left (x + e^{3} - 1\right )} e^{\left (x^{3} - 13 \, x^{2} - {\left (x^{2} - 8 \, x + 16\right )} e^{x} + 56 \, x - 78\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^
2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x, algorithm="fricas")

[Out]

(x + e^3 - 1)*e^(x^3 - 13*x^2 - (x^2 - 8*x + 16)*e^x + 56*x - 78)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (3 \, x^{3} - 29 \, x^{2} + {\left (3 \, x^{2} - 26 \, x + 56\right )} e^{3} - {\left (x^{3} - 7 \, x^{2} + {\left (x^{2} - 6 \, x + 8\right )} e^{3} + 14 \, x - 8\right )} e^{x} + 82 \, x - 55\right )} e^{\left (x^{3} - 13 \, x^{2} - {\left (x^{2} - 8 \, x + 16\right )} e^{x} + 56 \, x - 78\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^
2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x, algorithm="giac")

[Out]

integrate((3*x^3 - 29*x^2 + (3*x^2 - 26*x + 56)*e^3 - (x^3 - 7*x^2 + (x^2 - 6*x + 8)*e^3 + 14*x - 8)*e^x + 82*
x - 55)*e^(x^3 - 13*x^2 - (x^2 - 8*x + 16)*e^x + 56*x - 78), x)

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maple [A]  time = 0.10, size = 37, normalized size = 1.28

method	result	size
risch	\(\left ({\mathrm e}^{3}+x -1\right ) {\mathrm e}^{-{\mathrm e}^{x} x^{2}+x^{3}+8 \,{\mathrm e}^{x} x -13 x^{2}-16 \,{\mathrm e}^{x}+56 x -78}\)	\(37\)
norman	\(x \,{\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}+\left ({\mathrm e}^{3}-1\right ) {\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-
16)*exp(x)+x^3-13*x^2+56*x-78),x,method=_RETURNVERBOSE)

[Out]

(exp(3)+x-1)*exp(-exp(x)*x^2+x^3+8*exp(x)*x-13*x^2-16*exp(x)+56*x-78)

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maxima [A]  time = 33.82, size = 36, normalized size = 1.24 \begin {gather*} {\left (x + e^{3} - 1\right )} e^{\left (x^{3} - x^{2} e^{x} - 13 \, x^{2} + 8 \, x e^{x} + 56 \, x - 16 \, e^{x} - 78\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^
2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x, algorithm="maxima")

[Out]

(x + e^3 - 1)*e^(x^3 - x^2*e^x - 13*x^2 + 8*x*e^x + 56*x - 16*e^x - 78)

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mupad [B]  time = 0.15, size = 111, normalized size = 3.83 \begin {gather*} {\mathrm {e}}^{8\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{56\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-75}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-13\,x^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^x}-{\mathrm {e}}^{8\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{56\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-78}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-13\,x^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^x}+x\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{56\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-78}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-13\,x^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(56*x - exp(x)*(x^2 - 8*x + 16) - 13*x^2 + x^3 - 78)*(82*x + exp(3)*(3*x^2 - 26*x + 56) - exp(x)*(14*x
+ exp(3)*(x^2 - 6*x + 8) - 7*x^2 + x^3 - 8) - 29*x^2 + 3*x^3 - 55),x)

[Out]

exp(8*x*exp(x))*exp(56*x)*exp(x^3)*exp(-75)*exp(-x^2*exp(x))*exp(-13*x^2)*exp(-16*exp(x)) - exp(8*x*exp(x))*ex
p(56*x)*exp(x^3)*exp(-78)*exp(-x^2*exp(x))*exp(-13*x^2)*exp(-16*exp(x)) + x*exp(8*x*exp(x))*exp(56*x)*exp(x^3)
*exp(-78)*exp(-x^2*exp(x))*exp(-13*x^2)*exp(-16*exp(x))

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sympy [A]  time = 0.33, size = 32, normalized size = 1.10 \begin {gather*} \left (x - 1 + e^{3}\right ) e^{x^{3} - 13 x^{2} + 56 x + \left (- x^{2} + 8 x - 16\right ) e^{x} - 78} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+6*x-8)*exp(3)-x**3+7*x**2-14*x+8)*exp(x)+(3*x**2-26*x+56)*exp(3)+3*x**3-29*x**2+82*x-55)*ex
p((-x**2+8*x-16)*exp(x)+x**3-13*x**2+56*x-78),x)

[Out]

(x - 1 + exp(3))*exp(x**3 - 13*x**2 + 56*x + (-x**2 + 8*x - 16)*exp(x) - 78)

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