3.87.88 \(\int \frac {e^{-e^x+10 x+3 x^2} (12 x^2+38 x^3+14 x^4-6 x^5+e^x (-4 x^3+x^4))}{32-16 x+2 x^2} \, dx\)

Optimal. Leaf size=34 \[ \frac {e^{-e^x+x+3 (-x+x (4+x))} x^3}{2 (4-x)} \]

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Rubi [F]  time = 2.90, 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 \frac {e^{-e^x+10 x+3 x^2} \left (12 x^2+38 x^3+14 x^4-6 x^5+e^x \left (-4 x^3+x^4\right )\right )}{32-16 x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-E^x + 10*x + 3*x^2)*(12*x^2 + 38*x^3 + 14*x^4 - 6*x^5 + E^x*(-4*x^3 + x^4)))/(32 - 16*x + 2*x^2),x]

[Out]

-274*Defer[Int][E^(-E^x + 10*x + 3*x^2), x] + 8*Defer[Int][E^(-E^x + 11*x + 3*x^2), x] + 32*Defer[Int][E^(-E^x
+ 10*x + 3*x^2)/(-4 + x)^2, x] - 1088*Defer[Int][E^(-E^x + 10*x + 3*x^2)/(-4 + x), x] + 32*Defer[Int][E^(-E^x
+ 11*x + 3*x^2)/(-4 + x), x] - 69*Defer[Int][E^(-E^x + 10*x + 3*x^2)*x, x] + 2*Defer[Int][E^(-E^x + 11*x + 3*
x^2)*x, x] - 17*Defer[Int][E^(-E^x + 10*x + 3*x^2)*x^2, x] + Defer[Int][E^(-E^x + 11*x + 3*x^2)*x^2, x]/2 - 3*
Defer[Int][E^(-E^x + 10*x + 3*x^2)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-e^x+10 x+3 x^2} \left (12 x^2+38 x^3+14 x^4-6 x^5+e^x \left (-4 x^3+x^4\right )\right )}{2 (-4+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{-e^x+10 x+3 x^2} \left (12 x^2+38 x^3+14 x^4-6 x^5+e^x \left (-4 x^3+x^4\right )\right )}{(-4+x)^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {12 e^{-e^x+10 x+3 x^2} x^2}{(-4+x)^2}+\frac {38 e^{-e^x+10 x+3 x^2} x^3}{(-4+x)^2}+\frac {e^{-e^x+11 x+3 x^2} x^3}{-4+x}+\frac {14 e^{-e^x+10 x+3 x^2} x^4}{(-4+x)^2}-\frac {6 e^{-e^x+10 x+3 x^2} x^5}{(-4+x)^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{-e^x+11 x+3 x^2} x^3}{-4+x} \, dx-3 \int \frac {e^{-e^x+10 x+3 x^2} x^5}{(-4+x)^2} \, dx+6 \int \frac {e^{-e^x+10 x+3 x^2} x^2}{(-4+x)^2} \, dx+7 \int \frac {e^{-e^x+10 x+3 x^2} x^4}{(-4+x)^2} \, dx+19 \int \frac {e^{-e^x+10 x+3 x^2} x^3}{(-4+x)^2} \, dx\\ &=\frac {1}{2} \int \left (16 e^{-e^x+11 x+3 x^2}+\frac {64 e^{-e^x+11 x+3 x^2}}{-4+x}+4 e^{-e^x+11 x+3 x^2} x+e^{-e^x+11 x+3 x^2} x^2\right ) \, dx-3 \int \left (256 e^{-e^x+10 x+3 x^2}+\frac {1024 e^{-e^x+10 x+3 x^2}}{(-4+x)^2}+\frac {1280 e^{-e^x+10 x+3 x^2}}{-4+x}+48 e^{-e^x+10 x+3 x^2} x+8 e^{-e^x+10 x+3 x^2} x^2+e^{-e^x+10 x+3 x^2} x^3\right ) \, dx+6 \int \left (e^{-e^x+10 x+3 x^2}+\frac {16 e^{-e^x+10 x+3 x^2}}{(-4+x)^2}+\frac {8 e^{-e^x+10 x+3 x^2}}{-4+x}\right ) \, dx+7 \int \left (48 e^{-e^x+10 x+3 x^2}+\frac {256 e^{-e^x+10 x+3 x^2}}{(-4+x)^2}+\frac {256 e^{-e^x+10 x+3 x^2}}{-4+x}+8 e^{-e^x+10 x+3 x^2} x+e^{-e^x+10 x+3 x^2} x^2\right ) \, dx+19 \int \left (8 e^{-e^x+10 x+3 x^2}+\frac {64 e^{-e^x+10 x+3 x^2}}{(-4+x)^2}+\frac {48 e^{-e^x+10 x+3 x^2}}{-4+x}+e^{-e^x+10 x+3 x^2} x\right ) \, dx\\ &=\frac {1}{2} \int e^{-e^x+11 x+3 x^2} x^2 \, dx+2 \int e^{-e^x+11 x+3 x^2} x \, dx-3 \int e^{-e^x+10 x+3 x^2} x^3 \, dx+6 \int e^{-e^x+10 x+3 x^2} \, dx+7 \int e^{-e^x+10 x+3 x^2} x^2 \, dx+8 \int e^{-e^x+11 x+3 x^2} \, dx+19 \int e^{-e^x+10 x+3 x^2} x \, dx-24 \int e^{-e^x+10 x+3 x^2} x^2 \, dx+32 \int \frac {e^{-e^x+11 x+3 x^2}}{-4+x} \, dx+48 \int \frac {e^{-e^x+10 x+3 x^2}}{-4+x} \, dx+56 \int e^{-e^x+10 x+3 x^2} x \, dx+96 \int \frac {e^{-e^x+10 x+3 x^2}}{(-4+x)^2} \, dx-144 \int e^{-e^x+10 x+3 x^2} x \, dx+152 \int e^{-e^x+10 x+3 x^2} \, dx+336 \int e^{-e^x+10 x+3 x^2} \, dx-768 \int e^{-e^x+10 x+3 x^2} \, dx+912 \int \frac {e^{-e^x+10 x+3 x^2}}{-4+x} \, dx+1216 \int \frac {e^{-e^x+10 x+3 x^2}}{(-4+x)^2} \, dx+1792 \int \frac {e^{-e^x+10 x+3 x^2}}{(-4+x)^2} \, dx+1792 \int \frac {e^{-e^x+10 x+3 x^2}}{-4+x} \, dx-3072 \int \frac {e^{-e^x+10 x+3 x^2}}{(-4+x)^2} \, dx-3840 \int \frac {e^{-e^x+10 x+3 x^2}}{-4+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.73, size = 27, normalized size = 0.79 \begin {gather*} -\frac {e^{-e^x+x (10+3 x)} x^3}{2 (-4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^x + 10*x + 3*x^2)*(12*x^2 + 38*x^3 + 14*x^4 - 6*x^5 + E^x*(-4*x^3 + x^4)))/(32 - 16*x + 2*x^2
),x]

[Out]

-1/2*(E^(-E^x + x*(10 + 3*x))*x^3)/(-4 + x)

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fricas [A]  time = 0.76, size = 24, normalized size = 0.71 \begin {gather*} -\frac {x^{3} e^{\left (3 \, x^{2} + 10 \, x - e^{x}\right )}}{2 \, {\left (x - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-4*x^3)*exp(x)-6*x^5+14*x^4+38*x^3+12*x^2)*exp(3*x^2+9*x)*exp(x-exp(x))/(2*x^2-16*x+32),x, algo
rithm="fricas")

[Out]

-1/2*x^3*e^(3*x^2 + 10*x - e^x)/(x - 4)

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giac [A]  time = 0.31, size = 24, normalized size = 0.71 \begin {gather*} -\frac {x^{3} e^{\left (3 \, x^{2} + 10 \, x - e^{x}\right )}}{2 \, {\left (x - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-4*x^3)*exp(x)-6*x^5+14*x^4+38*x^3+12*x^2)*exp(3*x^2+9*x)*exp(x-exp(x))/(2*x^2-16*x+32),x, algo
rithm="giac")

[Out]

-1/2*x^3*e^(3*x^2 + 10*x - e^x)/(x - 4)

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maple [A]  time = 0.47, size = 25, normalized size = 0.74

method	result	size
risch	\(-\frac {x^{3} {\mathrm e}^{3 x^{2}-{\mathrm e}^{x}+10 x}}{2 \left (x -4\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4-4*x^3)*exp(x)-6*x^5+14*x^4+38*x^3+12*x^2)*exp(3*x^2+9*x)*exp(x-exp(x))/(2*x^2-16*x+32),x,method=_RET
URNVERBOSE)

[Out]

-1/2*x^3/(x-4)*exp(3*x^2-exp(x)+10*x)

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maxima [A]  time = 0.45, size = 24, normalized size = 0.71 \begin {gather*} -\frac {x^{3} e^{\left (3 \, x^{2} + 10 \, x - e^{x}\right )}}{2 \, {\left (x - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-4*x^3)*exp(x)-6*x^5+14*x^4+38*x^3+12*x^2)*exp(3*x^2+9*x)*exp(x-exp(x))/(2*x^2-16*x+32),x, algo
rithm="maxima")

[Out]

-1/2*x^3*e^(3*x^2 + 10*x - e^x)/(x - 4)

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mupad [B]  time = 5.64, size = 27, normalized size = 0.79 \begin {gather*} -\frac {x^3\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{3\,x^2}\,{\mathrm {e}}^{-{\mathrm {e}}^x}}{2\,\left (x-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - exp(x))*exp(9*x + 3*x^2)*(12*x^2 - exp(x)*(4*x^3 - x^4) + 38*x^3 + 14*x^4 - 6*x^5))/(2*x^2 - 16*x
+ 32),x)

[Out]

-(x^3*exp(10*x)*exp(3*x^2)*exp(-exp(x)))/(2*(x - 4))

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sympy [A]  time = 0.25, size = 26, normalized size = 0.76 \begin {gather*} - \frac {x^{3} e^{x - e^{x}} e^{3 x^{2} + 9 x}}{2 x - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4-4*x**3)*exp(x)-6*x**5+14*x**4+38*x**3+12*x**2)*exp(3*x**2+9*x)*exp(x-exp(x))/(2*x**2-16*x+32)
,x)

[Out]

-x**3*exp(x - exp(x))*exp(3*x**2 + 9*x)/(2*x - 8)

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