3.38.22 \(\int \frac {e^{-\frac {50 e^{3-x}}{x}} (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} (18 x^2+18 x^3)+e^{\frac {25 e^{3-x}}{x}} (-72 x^2+e^{3-x} (-1800-3600 x-1800 x^2)))}{x^2} \, dx\)

Optimal. Leaf size=23 \[ 9 \left (1-4 e^{-\frac {25 e^{3-x}}{x}}+x\right )^2 \]

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Rubi [F]  time = 3.74, 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^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3 - x)*(7200 + 7200*x) + E^((50*E^(3 - x))/x)*(18*x^2 + 18*x^3) + E^((25*E^(3 - x))/x)*(-72*x^2 + E^(3
- x)*(-1800 - 3600*x - 1800*x^2)))/(E^((50*E^(3 - x))/x)*x^2),x]

[Out]

18*x + 9*x^2 - 1800*Defer[Int][E^(3 - (25*E^(3 - x))/x - x), x] - 72*Defer[Int][E^((-25*E^(3 - x))/x), x] + 72
00*Defer[Int][E^(3 - (50*E^(3 - x))/x - x)/x^2, x] - 1800*Defer[Int][E^(3 - (25*E^(3 - x))/x - x)/x^2, x] + 72
00*Defer[Int][E^(3 - (50*E^(3 - x))/x - x)/x, x] - 3600*Defer[Int][E^(3 - (25*E^(3 - x))/x - x)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18 e^{-\frac {50 e^{3-x}}{x}-x} \left (e^{\frac {25 e^{3-x}}{x}+x} x^2-100 e^3 (1+x)\right ) \left (-4+e^{\frac {25 e^{3-x}}{x}} (1+x)\right )}{x^2} \, dx\\ &=18 \int \frac {e^{-\frac {50 e^{3-x}}{x}-x} \left (e^{\frac {25 e^{3-x}}{x}+x} x^2-100 e^3 (1+x)\right ) \left (-4+e^{\frac {25 e^{3-x}}{x}} (1+x)\right )}{x^2} \, dx\\ &=18 \int \left (e^{-\frac {25 e^{3-x}}{x}} \left (-4+e^{\frac {25 e^{3-x}}{x}}+e^{\frac {25 e^{3-x}}{x}} x\right )-\frac {100 e^{3-\frac {50 e^{3-x}}{x}-x} (1+x) \left (-4+e^{\frac {25 e^{3-x}}{x}}+e^{\frac {25 e^{3-x}}{x}} x\right )}{x^2}\right ) \, dx\\ &=18 \int e^{-\frac {25 e^{3-x}}{x}} \left (-4+e^{\frac {25 e^{3-x}}{x}}+e^{\frac {25 e^{3-x}}{x}} x\right ) \, dx-1800 \int \frac {e^{3-\frac {50 e^{3-x}}{x}-x} (1+x) \left (-4+e^{\frac {25 e^{3-x}}{x}}+e^{\frac {25 e^{3-x}}{x}} x\right )}{x^2} \, dx\\ &=18 \int \left (1-4 e^{-\frac {25 e^{3-x}}{x}}+x\right ) \, dx-1800 \int \frac {e^{3-\frac {50 e^{3-x}}{x}-x} (1+x) \left (-4+e^{\frac {25 e^{3-x}}{x}} (1+x)\right )}{x^2} \, dx\\ &=18 x+9 x^2-72 \int e^{-\frac {25 e^{3-x}}{x}} \, dx-1800 \int \left (-\frac {4 e^{3-\frac {50 e^{3-x}}{x}-x} (1+x)}{x^2}+\frac {e^{3-\frac {25 e^{3-x}}{x}-x} (1+x)^2}{x^2}\right ) \, dx\\ &=18 x+9 x^2-72 \int e^{-\frac {25 e^{3-x}}{x}} \, dx-1800 \int \frac {e^{3-\frac {25 e^{3-x}}{x}-x} (1+x)^2}{x^2} \, dx+7200 \int \frac {e^{3-\frac {50 e^{3-x}}{x}-x} (1+x)}{x^2} \, dx\\ &=18 x+9 x^2-72 \int e^{-\frac {25 e^{3-x}}{x}} \, dx-1800 \int \left (e^{3-\frac {25 e^{3-x}}{x}-x}+\frac {e^{3-\frac {25 e^{3-x}}{x}-x}}{x^2}+\frac {2 e^{3-\frac {25 e^{3-x}}{x}-x}}{x}\right ) \, dx+7200 \int \left (\frac {e^{3-\frac {50 e^{3-x}}{x}-x}}{x^2}+\frac {e^{3-\frac {50 e^{3-x}}{x}-x}}{x}\right ) \, dx\\ &=18 x+9 x^2-72 \int e^{-\frac {25 e^{3-x}}{x}} \, dx-1800 \int e^{3-\frac {25 e^{3-x}}{x}-x} \, dx-1800 \int \frac {e^{3-\frac {25 e^{3-x}}{x}-x}}{x^2} \, dx-3600 \int \frac {e^{3-\frac {25 e^{3-x}}{x}-x}}{x} \, dx+7200 \int \frac {e^{3-\frac {50 e^{3-x}}{x}-x}}{x^2} \, dx+7200 \int \frac {e^{3-\frac {50 e^{3-x}}{x}-x}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.00, size = 47, normalized size = 2.04 \begin {gather*} 18 \left (8 e^{-\frac {50 e^{3-x}}{x}}+e^{-\frac {25 e^{3-x}}{x}} (-4-4 x)+x+\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 - x)*(7200 + 7200*x) + E^((50*E^(3 - x))/x)*(18*x^2 + 18*x^3) + E^((25*E^(3 - x))/x)*(-72*x^2
+ E^(3 - x)*(-1800 - 3600*x - 1800*x^2)))/(E^((50*E^(3 - x))/x)*x^2),x]

[Out]

18*(8/E^((50*E^(3 - x))/x) + (-4 - 4*x)/E^((25*E^(3 - x))/x) + x + x^2/2)

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fricas [B]  time = 1.35, size = 53, normalized size = 2.30 \begin {gather*} 9 \, {\left ({\left (x^{2} + 2 \, x\right )} e^{\left (\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )} - 8 \, {\left (x + 1\right )} e^{\left (\frac {25 \, e^{\left (-x + 3\right )}}{x}\right )} + 16\right )} e^{\left (-\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*exp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(
7200*x+7200)*exp(3-x))/x^2/exp(25*exp(3-x)/x)^2,x, algorithm="fricas")

[Out]

9*((x^2 + 2*x)*e^(50*e^(-x + 3)/x) - 8*(x + 1)*e^(25*e^(-x + 3)/x) + 16)*e^(-50*e^(-x + 3)/x)

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giac [B]  time = 1.74, size = 96, normalized size = 4.17 \begin {gather*} 9 \, {\left (x^{2} e^{\left (-x + 3\right )} + 2 \, x e^{\left (-x + 3\right )} - 8 \, x e^{\left (-\frac {x^{2} - 3 \, x + 25 \, e^{\left (-x + 3\right )}}{x}\right )} + 16 \, e^{\left (-\frac {x^{2} - 3 \, x + 50 \, e^{\left (-x + 3\right )}}{x}\right )} - 8 \, e^{\left (-\frac {x^{2} - 3 \, x + 25 \, e^{\left (-x + 3\right )}}{x}\right )}\right )} e^{\left (x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*exp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(
7200*x+7200)*exp(3-x))/x^2/exp(25*exp(3-x)/x)^2,x, algorithm="giac")

[Out]

9*(x^2*e^(-x + 3) + 2*x*e^(-x + 3) - 8*x*e^(-(x^2 - 3*x + 25*e^(-x + 3))/x) + 16*e^(-(x^2 - 3*x + 50*e^(-x + 3
))/x) - 8*e^(-(x^2 - 3*x + 25*e^(-x + 3))/x))*e^(x - 3)

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maple [A]  time = 0.06, size = 42, normalized size = 1.83

method	result	size
risch	\(9 x^{2}+18 x +\left (-72 x -72\right ) {\mathrm e}^{-\frac {25 \,{\mathrm e}^{3-x}}{x}}+144 \,{\mathrm e}^{-\frac {50 \,{\mathrm e}^{3-x}}{x}}\)	\(42\)
norman	\(\frac {\left (144 x -72 x \,{\mathrm e}^{\frac {25 \,{\mathrm e}^{3-x}}{x}}+9 x^{3} {\mathrm e}^{\frac {50 \,{\mathrm e}^{3-x}}{x}}-72 \,{\mathrm e}^{\frac {25 \,{\mathrm e}^{3-x}}{x}} x^{2}+18 \,{\mathrm e}^{\frac {50 \,{\mathrm e}^{3-x}}{x}} x^{2}\right ) {\mathrm e}^{-\frac {50 \,{\mathrm e}^{3-x}}{x}}}{x}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*exp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(7200*x
+7200)*exp(3-x))/x^2/exp(25*exp(3-x)/x)^2,x,method=_RETURNVERBOSE)

[Out]

9*x^2+18*x+(-72*x-72)*exp(-25*exp(3-x)/x)+144*exp(-50*exp(3-x)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 9 \, x^{2} + 18 \, x + 144 \, e^{\left (-\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )} - 18 \, \int \frac {4 \, {\left (25 \, x^{2} e^{3} + x^{2} e^{x} + 50 \, x e^{3} + 25 \, e^{3}\right )} e^{\left (-x - \frac {25 \, e^{\left (-x + 3\right )}}{x}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*exp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(
7200*x+7200)*exp(3-x))/x^2/exp(25*exp(3-x)/x)^2,x, algorithm="maxima")

[Out]

9*x^2 + 18*x + 144*e^(-50*e^(-x + 3)/x) - 18*integrate(4*(25*x^2*e^3 + x^2*e^x + 50*x*e^3 + 25*e^3)*e^(-x - 25
*e^(-x + 3)/x)/x^2, x)

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mupad [B]  time = 2.35, size = 42, normalized size = 1.83 \begin {gather*} 18\,x+144\,{\mathrm {e}}^{-\frac {50\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3}{x}}-{\mathrm {e}}^{-\frac {25\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3}{x}}\,\left (72\,x+72\right )+9\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(50*exp(3 - x))/x)*(exp(3 - x)*(7200*x + 7200) + exp((50*exp(3 - x))/x)*(18*x^2 + 18*x^3) - exp((25*
exp(3 - x))/x)*(exp(3 - x)*(3600*x + 1800*x^2 + 1800) + 72*x^2)))/x^2,x)

[Out]

18*x + 144*exp(-(50*exp(-x)*exp(3))/x) - exp(-(25*exp(-x)*exp(3))/x)*(72*x + 72) + 9*x^2

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sympy [B]  time = 5.73, size = 36, normalized size = 1.57 \begin {gather*} 9 x^{2} + 18 x + \left (- 72 x - 72\right ) e^{- \frac {25 e^{3 - x}}{x}} + 144 e^{- \frac {50 e^{3 - x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**3+18*x**2)*exp(25*exp(3-x)/x)**2+((-1800*x**2-3600*x-1800)*exp(3-x)-72*x**2)*exp(25*exp(3-x)
/x)+(7200*x+7200)*exp(3-x))/x**2/exp(25*exp(3-x)/x)**2,x)

[Out]

9*x**2 + 18*x + (-72*x - 72)*exp(-25*exp(3 - x)/x) + 144*exp(-50*exp(3 - x)/x)

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