3.47.25 \(\int \frac {1}{36} e^{\frac {1}{9} (-45-9 e^x+4 x)} (-56-40 x+16 x^2+e^x (-441+252 x-36 x^2)) \, dx\)

Optimal. Leaf size=28 \[ e^{-e^x-5 \left (1+\frac {x}{9}\right )+x} \left (\frac {7}{2}-x\right )^2 \]

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Rubi [F]  time = 1.00, 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 {1}{36} e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} \left (-56-40 x+16 x^2+e^x \left (-441+252 x-36 x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-45 - 9*E^x + 4*x)/9)*(-56 - 40*x + 16*x^2 + E^x*(-441 + 252*x - 36*x^2)))/36,x]

[Out]

(-14*Defer[Int][E^((-45 - 9*E^x + 4*x)/9), x])/9 - (49*Defer[Int][E^((-45 - 9*E^x + 13*x)/9), x])/4 - (10*Defe
r[Int][E^((-45 - 9*E^x + 4*x)/9)*x, x])/9 + 7*Defer[Int][E^((-45 - 9*E^x + 13*x)/9)*x, x] + (4*Defer[Int][E^((
-45 - 9*E^x + 4*x)/9)*x^2, x])/9 - Defer[Int][E^((-45 - 9*E^x + 13*x)/9)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{36} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} \left (-56-40 x+16 x^2+e^x \left (-441+252 x-36 x^2\right )\right ) \, dx\\ &=\frac {1}{36} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} (7-2 x) \left (-8-63 e^x-8 x+18 e^x x\right ) \, dx\\ &=\frac {1}{36} \int \left (-9 e^{x+\frac {1}{9} \left (-45-9 e^x+4 x\right )} (-7+2 x)^2+8 e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} \left (-7-5 x+2 x^2\right )\right ) \, dx\\ &=\frac {2}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} \left (-7-5 x+2 x^2\right ) \, dx-\frac {1}{4} \int e^{x+\frac {1}{9} \left (-45-9 e^x+4 x\right )} (-7+2 x)^2 \, dx\\ &=\frac {2}{9} \int \left (-7 e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )}-5 e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} x+2 e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} x^2\right ) \, dx-\frac {1}{4} \int e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )} (7-2 x)^2 \, dx\\ &=-\left (\frac {1}{4} \int \left (49 e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )}-28 e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )} x+4 e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )} x^2\right ) \, dx\right )+\frac {4}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} x^2 \, dx-\frac {10}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} x \, dx-\frac {14}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} \, dx\\ &=\frac {4}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} x^2 \, dx-\frac {10}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} x \, dx-\frac {14}{9} \int e^{\frac {1}{9} \left (-45-9 e^x+4 x\right )} \, dx+7 \int e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )} x \, dx-\frac {49}{4} \int e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )} \, dx-\int e^{\frac {1}{9} \left (-45-9 e^x+13 x\right )} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 25, normalized size = 0.89 \begin {gather*} \frac {1}{4} e^{-5-e^x+\frac {4 x}{9}} (-7+2 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-45 - 9*E^x + 4*x)/9)*(-56 - 40*x + 16*x^2 + E^x*(-441 + 252*x - 36*x^2)))/36,x]

[Out]

(E^(-5 - E^x + (4*x)/9)*(-7 + 2*x)^2)/4

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fricas [A]  time = 0.76, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, {\left (4 \, x^{2} - 28 \, x + 49\right )} e^{\left (\frac {4}{9} \, x - e^{x} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/36*((-36*x^2+252*x-441)*exp(x)+16*x^2-40*x-56)/exp(exp(x)-4/9*x+5),x, algorithm="fricas")

[Out]

1/4*(4*x^2 - 28*x + 49)*e^(4/9*x - e^x - 5)

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giac [B]  time = 0.17, size = 42, normalized size = 1.50 \begin {gather*} \frac {1}{4} \, {\left (4 \, x^{2} e^{\left (\frac {4}{9} \, x - e^{x}\right )} - 28 \, x e^{\left (\frac {4}{9} \, x - e^{x}\right )} + 49 \, e^{\left (\frac {4}{9} \, x - e^{x}\right )}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/36*((-36*x^2+252*x-441)*exp(x)+16*x^2-40*x-56)/exp(exp(x)-4/9*x+5),x, algorithm="giac")

[Out]

1/4*(4*x^2*e^(4/9*x - e^x) - 28*x*e^(4/9*x - e^x) + 49*e^(4/9*x - e^x))*e^(-5)

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maple [A]  time = 0.06, size = 20, normalized size = 0.71

method	result	size
norman	\(\left (x^{2}-7 x +\frac {49}{4}\right ) {\mathrm e}^{-{\mathrm e}^{x}+\frac {4 x}{9}-5}\)	\(20\)
risch	\(\frac {\left (36 x^{2}-252 x +441\right ) {\mathrm e}^{-{\mathrm e}^{x}+\frac {4 x}{9}-5}}{36}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/36*((-36*x^2+252*x-441)*exp(x)+16*x^2-40*x-56)/exp(exp(x)-4/9*x+5),x,method=_RETURNVERBOSE)

[Out]

(x^2-7*x+49/4)/exp(exp(x)-4/9*x+5)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{36} \, \int {\left (16 \, x^{2} - 9 \, {\left (4 \, x^{2} - 28 \, x + 49\right )} e^{x} - 40 \, x - 56\right )} e^{\left (\frac {4}{9} \, x - e^{x} - 5\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/36*((-36*x^2+252*x-441)*exp(x)+16*x^2-40*x-56)/exp(exp(x)-4/9*x+5),x, algorithm="maxima")

[Out]

1/36*integrate((16*x^2 - 9*(4*x^2 - 28*x + 49)*e^x - 40*x - 56)*e^(4/9*x - e^x - 5), x)

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mupad [B]  time = 3.34, size = 20, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {e}}^{\frac {4\,x}{9}}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\left (2\,x-7\right )}^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp((4*x)/9 - exp(x) - 5)*((10*x)/9 + (exp(x)*(36*x^2 - 252*x + 441))/36 - (4*x^2)/9 + 14/9),x)

[Out]

(exp((4*x)/9)*exp(-5)*exp(-exp(x))*(2*x - 7)^2)/4

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sympy [A]  time = 0.22, size = 22, normalized size = 0.79 \begin {gather*} \frac {\left (4 x^{2} - 28 x + 49\right ) e^{\frac {4 x}{9} - e^{x} - 5}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/36*((-36*x**2+252*x-441)*exp(x)+16*x**2-40*x-56)/exp(exp(x)-4/9*x+5),x)

[Out]

(4*x**2 - 28*x + 49)*exp(4*x/9 - exp(x) - 5)/4

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