3.46.52 \(\int \frac {1}{9} e^{\frac {1}{9} (-4 x-4 e^{4+3 x} x)} (90-9 e^{\frac {1}{9} (4 x+4 e^{4+3 x} x)}-40 x+e^{4+3 x} (-40 x-120 x^2)) \, dx\)

Optimal. Leaf size=23 \[ -x+10 e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x \]

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Rubi [F]  time = 1.83, 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}{9} e^{\frac {1}{9} \left (-4 x-4 e^{4+3 x} x\right )} \left (90-9 e^{\frac {1}{9} \left (4 x+4 e^{4+3 x} x\right )}-40 x+e^{4+3 x} \left (-40 x-120 x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-4*x - 4*E^(4 + 3*x)*x)/9)*(90 - 9*E^((4*x + 4*E^(4 + 3*x)*x)/9) - 40*x + E^(4 + 3*x)*(-40*x - 120*x^
2)))/9,x]

[Out]

-x + 10*Defer[Int][E^((-4*(1 + E^(4 + 3*x))*x)/9), x] - (40*Defer[Int][x/E^((4*(1 + E^(4 + 3*x))*x)/9), x])/9
- (40*Defer[Int][E^((36 + 23*x - 4*E^(4 + 3*x)*x)/9)*x, x])/9 - (40*Defer[Int][E^((36 + 23*x - 4*E^(4 + 3*x)*x
)/9)*x^2, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int e^{\frac {1}{9} \left (-4 x-4 e^{4+3 x} x\right )} \left (90-9 e^{\frac {1}{9} \left (4 x+4 e^{4+3 x} x\right )}-40 x+e^{4+3 x} \left (-40 x-120 x^2\right )\right ) \, dx\\ &=\frac {1}{9} \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} \left (90-9 e^{\frac {1}{9} \left (4 x+4 e^{4+3 x} x\right )}-40 x+e^{4+3 x} \left (-40 x-120 x^2\right )\right ) \, dx\\ &=\frac {1}{9} \int \left (-9+90 e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x}-40 e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x-40 e^{4+3 x-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x (1+3 x)\right ) \, dx\\ &=-x-\frac {40}{9} \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x \, dx-\frac {40}{9} \int e^{4+3 x-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x (1+3 x) \, dx+10 \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} \, dx\\ &=-x-\frac {40}{9} \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x \, dx-\frac {40}{9} \int e^{\frac {1}{9} \left (36+23 x-4 e^{4+3 x} x\right )} x (1+3 x) \, dx+10 \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} \, dx\\ &=-x-\frac {40}{9} \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x \, dx-\frac {40}{9} \int \left (e^{\frac {1}{9} \left (36+23 x-4 e^{4+3 x} x\right )} x+3 e^{\frac {1}{9} \left (36+23 x-4 e^{4+3 x} x\right )} x^2\right ) \, dx+10 \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} \, dx\\ &=-x-\frac {40}{9} \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} x \, dx-\frac {40}{9} \int e^{\frac {1}{9} \left (36+23 x-4 e^{4+3 x} x\right )} x \, dx+10 \int e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x} \, dx-\frac {40}{3} \int e^{\frac {1}{9} \left (36+23 x-4 e^{4+3 x} x\right )} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.77, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{9} \left (-9+90 e^{-\frac {4}{9} \left (1+e^{4+3 x}\right ) x}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-4*x - 4*E^(4 + 3*x)*x)/9)*(90 - 9*E^((4*x + 4*E^(4 + 3*x)*x)/9) - 40*x + E^(4 + 3*x)*(-40*x -
120*x^2)))/9,x]

[Out]

((-9 + 90/E^((4*(1 + E^(4 + 3*x))*x)/9))*x)/9

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fricas [A]  time = 0.64, size = 36, normalized size = 1.57 \begin {gather*} -{\left (x e^{\left (\frac {4}{9} \, x e^{\left (3 \, x + 4\right )} + \frac {4}{9} \, x\right )} - 10 \, x\right )} e^{\left (-\frac {4}{9} \, x e^{\left (3 \, x + 4\right )} - \frac {4}{9} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-9*exp(4/9*x*exp(4+3*x)+4/9*x)+(-120*x^2-40*x)*exp(4+3*x)-40*x+90)/exp(4/9*x*exp(4+3*x)+4/9*x),
x, algorithm="fricas")

[Out]

-(x*e^(4/9*x*e^(3*x + 4) + 4/9*x) - 10*x)*e^(-4/9*x*e^(3*x + 4) - 4/9*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {1}{9} \, {\left (40 \, {\left (3 \, x^{2} + x\right )} e^{\left (3 \, x + 4\right )} + 40 \, x + 9 \, e^{\left (\frac {4}{9} \, x e^{\left (3 \, x + 4\right )} + \frac {4}{9} \, x\right )} - 90\right )} e^{\left (-\frac {4}{9} \, x e^{\left (3 \, x + 4\right )} - \frac {4}{9} \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-9*exp(4/9*x*exp(4+3*x)+4/9*x)+(-120*x^2-40*x)*exp(4+3*x)-40*x+90)/exp(4/9*x*exp(4+3*x)+4/9*x),
x, algorithm="giac")

[Out]

integrate(-1/9*(40*(3*x^2 + x)*e^(3*x + 4) + 40*x + 9*e^(4/9*x*e^(3*x + 4) + 4/9*x) - 90)*e^(-4/9*x*e^(3*x + 4
) - 4/9*x), x)

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maple [A]  time = 0.08, size = 20, normalized size = 0.87

method	result	size
risch	\(-x +10 x \,{\mathrm e}^{-\frac {4 x \left ({\mathrm e}^{4+3 x}+1\right )}{9}}\)	\(20\)
norman	\(\left (10 x -x \,{\mathrm e}^{\frac {4 x \,{\mathrm e}^{4+3 x}}{9}+\frac {4 x}{9}}\right ) {\mathrm e}^{-\frac {4 x \,{\mathrm e}^{4+3 x}}{9}-\frac {4 x}{9}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(-9*exp(4/9*x*exp(4+3*x)+4/9*x)+(-120*x^2-40*x)*exp(4+3*x)-40*x+90)/exp(4/9*x*exp(4+3*x)+4/9*x),x,meth
od=_RETURNVERBOSE)

[Out]

-x+10*x*exp(-4/9*x*(exp(4+3*x)+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x - \frac {1}{9} \, \int 10 \, {\left (4 \, {\left (3 \, x^{2} e^{4} + x e^{4}\right )} e^{\left (3 \, x\right )} + 4 \, x - 9\right )} e^{\left (-\frac {4}{9} \, x e^{\left (3 \, x + 4\right )} - \frac {4}{9} \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-9*exp(4/9*x*exp(4+3*x)+4/9*x)+(-120*x^2-40*x)*exp(4+3*x)-40*x+90)/exp(4/9*x*exp(4+3*x)+4/9*x),
x, algorithm="maxima")

[Out]

-x - 1/9*integrate(10*(4*(3*x^2*e^4 + x*e^4)*e^(3*x) + 4*x - 9)*e^(-4/9*x*e^(3*x + 4) - 4/9*x), x)

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mupad [B]  time = 3.12, size = 21, normalized size = 0.91 \begin {gather*} 10\,x\,{\mathrm {e}}^{-\frac {4\,x}{9}-\frac {4\,x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4}{9}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- (4*x)/9 - (4*x*exp(3*x + 4))/9)*((40*x)/9 + exp((4*x)/9 + (4*x*exp(3*x + 4))/9) + (exp(3*x + 4)*(40
*x + 120*x^2))/9 - 10),x)

[Out]

10*x*exp(- (4*x)/9 - (4*x*exp(3*x)*exp(4))/9) - x

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sympy [A]  time = 0.23, size = 24, normalized size = 1.04 \begin {gather*} 10 x e^{- \frac {4 x e^{3 x + 4}}{9} - \frac {4 x}{9}} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-9*exp(4/9*x*exp(4+3*x)+4/9*x)+(-120*x**2-40*x)*exp(4+3*x)-40*x+90)/exp(4/9*x*exp(4+3*x)+4/9*x)
,x)

[Out]

10*x*exp(-4*x*exp(3*x + 4)/9 - 4*x/9) - x

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