3.51.30 \(\int (1+e^{2+e^{-3 x+x^2}+x} (e^{10} (4-12 x+4 x^2+4 x^3)+e^{10-3 x+x^2} (-12 x+32 x^2-28 x^3+8 x^4))) \, dx\)

Optimal. Leaf size=22 \[ x+4 e^{12+e^{(-3+x) x}+x} (-1+x)^2 x \]

________________________________________________________________________________________

Rubi [F]  time = 1.59, 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 \left (1+e^{2+e^{-3 x+x^2}+x} \left (e^{10} \left (4-12 x+4 x^2+4 x^3\right )+e^{10-3 x+x^2} \left (-12 x+32 x^2-28 x^3+8 x^4\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1 + E^(2 + E^(-3*x + x^2) + x)*(E^10*(4 - 12*x + 4*x^2 + 4*x^3) + E^(10 - 3*x + x^2)*(-12*x + 32*x^2 - 28*
x^3 + 8*x^4)),x]

[Out]

x + 4*Defer[Int][E^(12 + E^(-3*x + x^2) + x), x] - 12*Defer[Int][E^(12 + E^(-3*x + x^2) + x)*x, x] - 12*Defer[
Int][E^(12 + E^(-3*x + x^2) - 2*x + x^2)*x, x] + 4*Defer[Int][E^(12 + E^(-3*x + x^2) + x)*x^2, x] + 32*Defer[I
nt][E^(12 + E^(-3*x + x^2) - 2*x + x^2)*x^2, x] + 4*Defer[Int][E^(12 + E^(-3*x + x^2) + x)*x^3, x] - 28*Defer[
Int][E^(12 + E^(-3*x + x^2) - 2*x + x^2)*x^3, x] + 8*Defer[Int][E^(12 + E^(-3*x + x^2) - 2*x + x^2)*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+\int e^{2+e^{-3 x+x^2}+x} \left (e^{10} \left (4-12 x+4 x^2+4 x^3\right )+e^{10-3 x+x^2} \left (-12 x+32 x^2-28 x^3+8 x^4\right )\right ) \, dx\\ &=x+\int \left (4 e^{12+e^{-3 x+x^2}-2 x+x^2} (-1+x)^2 x (-3+2 x)+4 e^{12+e^{-3 x+x^2}+x} \left (1-3 x+x^2+x^3\right )\right ) \, dx\\ &=x+4 \int e^{12+e^{-3 x+x^2}-2 x+x^2} (-1+x)^2 x (-3+2 x) \, dx+4 \int e^{12+e^{-3 x+x^2}+x} \left (1-3 x+x^2+x^3\right ) \, dx\\ &=x+4 \int \left (e^{12+e^{-3 x+x^2}+x}-3 e^{12+e^{-3 x+x^2}+x} x+e^{12+e^{-3 x+x^2}+x} x^2+e^{12+e^{-3 x+x^2}+x} x^3\right ) \, dx+4 \int \left (-3 e^{12+e^{-3 x+x^2}-2 x+x^2} x+8 e^{12+e^{-3 x+x^2}-2 x+x^2} x^2-7 e^{12+e^{-3 x+x^2}-2 x+x^2} x^3+2 e^{12+e^{-3 x+x^2}-2 x+x^2} x^4\right ) \, dx\\ &=x+4 \int e^{12+e^{-3 x+x^2}+x} \, dx+4 \int e^{12+e^{-3 x+x^2}+x} x^2 \, dx+4 \int e^{12+e^{-3 x+x^2}+x} x^3 \, dx+8 \int e^{12+e^{-3 x+x^2}-2 x+x^2} x^4 \, dx-12 \int e^{12+e^{-3 x+x^2}+x} x \, dx-12 \int e^{12+e^{-3 x+x^2}-2 x+x^2} x \, dx-28 \int e^{12+e^{-3 x+x^2}-2 x+x^2} x^3 \, dx+32 \int e^{12+e^{-3 x+x^2}-2 x+x^2} x^2 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.64, size = 22, normalized size = 1.00 \begin {gather*} x+4 e^{12+e^{(-3+x) x}+x} (-1+x)^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(2 + E^(-3*x + x^2) + x)*(E^10*(4 - 12*x + 4*x^2 + 4*x^3) + E^(10 - 3*x + x^2)*(-12*x + 32*x^2
- 28*x^3 + 8*x^4)),x]

[Out]

x + 4*E^(12 + E^((-3 + x)*x) + x)*(-1 + x)^2*x

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 36, normalized size = 1.64 \begin {gather*} 4 \, {\left (x^{3} - 2 \, x^{2} + x\right )} e^{\left ({\left ({\left (x + 2\right )} e^{10} + e^{\left (x^{2} - 3 \, x + 10\right )}\right )} e^{\left (-10\right )} + 10\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-28*x^3+32*x^2-12*x)*exp(5)^2*exp(x^2-3*x)+(4*x^3+4*x^2-12*x+4)*exp(5)^2)*exp(exp(x^2-3*x)+2+
x)+1,x, algorithm="fricas")

[Out]

4*(x^3 - 2*x^2 + x)*e^(((x + 2)*e^10 + e^(x^2 - 3*x + 10))*e^(-10) + 10) + x

________________________________________________________________________________________

giac [B]  time = 0.18, size = 51, normalized size = 2.32 \begin {gather*} 4 \, x^{3} e^{\left (x + e^{\left (x^{2} - 3 \, x\right )} + 12\right )} - 8 \, x^{2} e^{\left (x + e^{\left (x^{2} - 3 \, x\right )} + 12\right )} + 4 \, x e^{\left (x + e^{\left (x^{2} - 3 \, x\right )} + 12\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-28*x^3+32*x^2-12*x)*exp(5)^2*exp(x^2-3*x)+(4*x^3+4*x^2-12*x+4)*exp(5)^2)*exp(exp(x^2-3*x)+2+
x)+1,x, algorithm="giac")

[Out]

4*x^3*e^(x + e^(x^2 - 3*x) + 12) - 8*x^2*e^(x + e^(x^2 - 3*x) + 12) + 4*x*e^(x + e^(x^2 - 3*x) + 12) + x

________________________________________________________________________________________

maple [A]  time = 0.15, size = 24, normalized size = 1.09

method	result	size
risch	\(4 x \left (x^{2}-2 x +1\right ) {\mathrm e}^{12+x +{\mathrm e}^{x \left (x -3\right )}}+x\)	\(24\)
default	\(x +4 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{x^{2}-3 x}+2+x} x -8 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{x^{2}-3 x}+2+x} x^{2}+4 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{x^{2}-3 x}+2+x} x^{3}\)	\(64\)
norman	\(x +4 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{x^{2}-3 x}+2+x} x -8 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{x^{2}-3 x}+2+x} x^{2}+4 \,{\mathrm e}^{10} {\mathrm e}^{{\mathrm e}^{x^{2}-3 x}+2+x} x^{3}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^4-28*x^3+32*x^2-12*x)*exp(5)^2*exp(x^2-3*x)+(4*x^3+4*x^2-12*x+4)*exp(5)^2)*exp(exp(x^2-3*x)+2+x)+1,x
,method=_RETURNVERBOSE)

[Out]

4*x*(x^2-2*x+1)*exp(12+x+exp(x*(x-3)))+x

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 33, normalized size = 1.50 \begin {gather*} 4 \, {\left (x^{3} e^{12} - 2 \, x^{2} e^{12} + x e^{12}\right )} e^{\left (x + e^{\left (x^{2} - 3 \, x\right )}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-28*x^3+32*x^2-12*x)*exp(5)^2*exp(x^2-3*x)+(4*x^3+4*x^2-12*x+4)*exp(5)^2)*exp(exp(x^2-3*x)+2+
x)+1,x, algorithm="maxima")

[Out]

4*(x^3*e^12 - 2*x^2*e^12 + x*e^12)*e^(x + e^(x^2 - 3*x)) + x

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 54, normalized size = 2.45 \begin {gather*} x-8\,x^2\,{\mathrm {e}}^{x+{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{x^2}+12}+4\,x^3\,{\mathrm {e}}^{x+{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{x^2}+12}+4\,x\,{\mathrm {e}}^{x+{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{x^2}+12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + exp(x^2 - 3*x) + 2)*(exp(10)*(4*x^2 - 12*x + 4*x^3 + 4) - exp(10)*exp(x^2 - 3*x)*(12*x - 32*x^2 +
28*x^3 - 8*x^4)) + 1,x)

[Out]

x - 8*x^2*exp(x + exp(-3*x)*exp(x^2) + 12) + 4*x^3*exp(x + exp(-3*x)*exp(x^2) + 12) + 4*x*exp(x + exp(-3*x)*ex
p(x^2) + 12)

________________________________________________________________________________________

sympy [A]  time = 84.44, size = 37, normalized size = 1.68 \begin {gather*} x + \left (4 x^{3} e^{10} - 8 x^{2} e^{10} + 4 x e^{10}\right ) e^{x + e^{x^{2} - 3 x} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**4-28*x**3+32*x**2-12*x)*exp(5)**2*exp(x**2-3*x)+(4*x**3+4*x**2-12*x+4)*exp(5)**2)*exp(exp(x**
2-3*x)+2+x)+1,x)

[Out]

x + (4*x**3*exp(10) - 8*x**2*exp(10) + 4*x*exp(10))*exp(x + exp(x**2 - 3*x) + 2)

________________________________________________________________________________________