3.2.50 \(\int \frac {e^{-\frac {2 x^2}{36+4 e^2+e (24-8 x)-24 x+4 x^2}} (1350+50 e^3+e^2 (450-150 x)-1350 x+300 x^2-50 x^3+e (1350-900 x+100 x^2))}{27+e^3+e^2 (9-3 x)-27 x+9 x^2-x^3+e (27-18 x+3 x^2)} \, dx\)

Optimal. Leaf size=20 \[ 50 e^{-\frac {x^2}{2 (3+e-x)^2}} x \]

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Rubi [F]  time = 1.44, 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 {\exp \left (-\frac {2 x^2}{36+4 e^2+e (24-8 x)-24 x+4 x^2}\right ) \left (1350+50 e^3+e^2 (450-150 x)-1350 x+300 x^2-50 x^3+e \left (1350-900 x+100 x^2\right )\right )}{27+e^3+e^2 (9-3 x)-27 x+9 x^2-x^3+e \left (27-18 x+3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1350 + 50*E^3 + E^2*(450 - 150*x) - 1350*x + 300*x^2 - 50*x^3 + E*(1350 - 900*x + 100*x^2))/(E^((2*x^2)/(
36 + 4*E^2 + E*(24 - 8*x) - 24*x + 4*x^2))*(27 + E^3 + E^2*(9 - 3*x) - 27*x + 9*x^2 - x^3 + E*(27 - 18*x + 3*x
^2))),x]

[Out]

50*Defer[Int][E^(-1/2*x^2/(3 + E - x)^2), x] - 50*(3 + E)^3*Defer[Int][1/(E^(x^2/(2*(3 + E - x)^2))*(3 + E - x
)^3), x] + 100*(3 + E)^2*Defer[Int][1/(E^(x^2/(2*(3 + E - x)^2))*(3 + E - x)^2), x] - 50*(3 + E)*Defer[Int][1/
(E^(x^2/(2*(3 + E - x)^2))*(3 + E - x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 e^{-\frac {x^2}{2 (3+e-x)^2}} \left ((3+e)^3-3 (3+e)^2 x+2 (3+e) x^2-x^3\right )}{(3+e-x)^3} \, dx\\ &=50 \int \frac {e^{-\frac {x^2}{2 (3+e-x)^2}} \left ((3+e)^3-3 (3+e)^2 x+2 (3+e) x^2-x^3\right )}{(3+e-x)^3} \, dx\\ &=50 \int \left (e^{-\frac {x^2}{2 (3+e-x)^2}}-\frac {e^{-\frac {x^2}{2 (3+e-x)^2}} (3+e)^3}{(3+e-x)^3}+\frac {2 e^{-\frac {x^2}{2 (3+e-x)^2}} (3+e)^2}{(3+e-x)^2}+\frac {(-3-e) e^{-\frac {x^2}{2 (3+e-x)^2}}}{3+e-x}\right ) \, dx\\ &=50 \int e^{-\frac {x^2}{2 (3+e-x)^2}} \, dx-(50 (3+e)) \int \frac {e^{-\frac {x^2}{2 (3+e-x)^2}}}{3+e-x} \, dx+\left (100 (3+e)^2\right ) \int \frac {e^{-\frac {x^2}{2 (3+e-x)^2}}}{(3+e-x)^2} \, dx-\left (50 (3+e)^3\right ) \int \frac {e^{-\frac {x^2}{2 (3+e-x)^2}}}{(3+e-x)^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.84, size = 20, normalized size = 1.00 \begin {gather*} 50 e^{-\frac {x^2}{2 (3+e-x)^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1350 + 50*E^3 + E^2*(450 - 150*x) - 1350*x + 300*x^2 - 50*x^3 + E*(1350 - 900*x + 100*x^2))/(E^((2*
x^2)/(36 + 4*E^2 + E*(24 - 8*x) - 24*x + 4*x^2))*(27 + E^3 + E^2*(9 - 3*x) - 27*x + 9*x^2 - x^3 + E*(27 - 18*x
+ 3*x^2))),x]

[Out]

(50*x)/E^(x^2/(2*(3 + E - x)^2))

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fricas [A]  time = 0.63, size = 28, normalized size = 1.40 \begin {gather*} 50 \, x e^{\left (-\frac {x^{2}}{2 \, {\left (x^{2} - 2 \, {\left (x - 3\right )} e - 6 \, x + e^{2} + 9\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*exp(1)^3+(-150*x+450)*exp(1)^2+(100*x^2-900*x+1350)*exp(1)-50*x^3+300*x^2-1350*x+1350)/(exp(1)^3
+(-3*x+9)*exp(1)^2+(3*x^2-18*x+27)*exp(1)-x^3+9*x^2-27*x+27)/exp(x^2/(4*exp(1)^2+(-8*x+24)*exp(1)+4*x^2-24*x+3
6))^2,x, algorithm="fricas")

[Out]

50*x*e^(-1/2*x^2/(x^2 - 2*(x - 3)*e - 6*x + e^2 + 9))

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giac [A]  time = 1.77, size = 30, normalized size = 1.50 \begin {gather*} 50 \, x e^{\left (-\frac {x^{2}}{2 \, {\left (x^{2} - 2 \, x e - 6 \, x + e^{2} + 6 \, e + 9\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*exp(1)^3+(-150*x+450)*exp(1)^2+(100*x^2-900*x+1350)*exp(1)-50*x^3+300*x^2-1350*x+1350)/(exp(1)^3
+(-3*x+9)*exp(1)^2+(3*x^2-18*x+27)*exp(1)-x^3+9*x^2-27*x+27)/exp(x^2/(4*exp(1)^2+(-8*x+24)*exp(1)+4*x^2-24*x+3
6))^2,x, algorithm="giac")

[Out]

50*x*e^(-1/2*x^2/(x^2 - 2*x*e - 6*x + e^2 + 6*e + 9))

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maple [A]  time = 0.30, size = 35, normalized size = 1.75

method	result	size
gosper	\(50 x \,{\mathrm e}^{-\frac {x^{2}}{2 \left (-2 x \,{\mathrm e}+x^{2}+{\mathrm e}^{2}+6 \,{\mathrm e}-6 x +9\right )}}\)	\(35\)
risch	\(50 x \,{\mathrm e}^{\frac {x^{2}}{4 x \,{\mathrm e}-2 x^{2}-2 \,{\mathrm e}^{2}-12 \,{\mathrm e}+12 x -18}}\)	\(35\)
norman	\(\frac {\left (\left (-100 \,{\mathrm e}-300\right ) x^{2}+\left (50 \,{\mathrm e}^{2}+300 \,{\mathrm e}+450\right ) x +50 x^{3}\right ) {\mathrm e}^{-\frac {2 x^{2}}{4 \,{\mathrm e}^{2}+\left (-8 x +24\right ) {\mathrm e}+4 x^{2}-24 x +36}}}{\left (3-x +{\mathrm e}\right )^{2}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*exp(1)^3+(-150*x+450)*exp(1)^2+(100*x^2-900*x+1350)*exp(1)-50*x^3+300*x^2-1350*x+1350)/(exp(1)^3+(-3*x
+9)*exp(1)^2+(3*x^2-18*x+27)*exp(1)-x^3+9*x^2-27*x+27)/exp(x^2/(4*exp(1)^2+(-8*x+24)*exp(1)+4*x^2-24*x+36))^2,
x,method=_RETURNVERBOSE)

[Out]

50*x/exp(1/4*x^2/(exp(1)^2-2*x*exp(1)+x^2+6*exp(1)-6*x+9))^2

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maxima [B]  time = 0.62, size = 100, normalized size = 5.00 \begin {gather*} 50 \, x e^{\left (-\frac {e^{2}}{2 \, {\left (x^{2} - 2 \, x {\left (e + 3\right )} + e^{2} + 6 \, e + 9\right )}} - \frac {3 \, e}{x^{2} - 2 \, x {\left (e + 3\right )} + e^{2} + 6 \, e + 9} - \frac {e}{x - e - 3} - \frac {9}{2 \, {\left (x^{2} - 2 \, x {\left (e + 3\right )} + e^{2} + 6 \, e + 9\right )}} - \frac {3}{x - e - 3} - \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*exp(1)^3+(-150*x+450)*exp(1)^2+(100*x^2-900*x+1350)*exp(1)-50*x^3+300*x^2-1350*x+1350)/(exp(1)^3
+(-3*x+9)*exp(1)^2+(3*x^2-18*x+27)*exp(1)-x^3+9*x^2-27*x+27)/exp(x^2/(4*exp(1)^2+(-8*x+24)*exp(1)+4*x^2-24*x+3
6))^2,x, algorithm="maxima")

[Out]

50*x*e^(-1/2*e^2/(x^2 - 2*x*(e + 3) + e^2 + 6*e + 9) - 3*e/(x^2 - 2*x*(e + 3) + e^2 + 6*e + 9) - e/(x - e - 3)
- 9/2/(x^2 - 2*x*(e + 3) + e^2 + 6*e + 9) - 3/(x - e - 3) - 1/2)

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mupad [B]  time = 1.27, size = 34, normalized size = 1.70 \begin {gather*} 50\,x\,{\mathrm {e}}^{-\frac {x^2}{12\,\mathrm {e}-12\,x+2\,{\mathrm {e}}^2-4\,x\,\mathrm {e}+2\,x^2+18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*x^2)/(4*exp(2) - 24*x + 4*x^2 - exp(1)*(8*x - 24) + 36))*(50*exp(3) - 1350*x + exp(1)*(100*x^2 -
900*x + 1350) + 300*x^2 - 50*x^3 - exp(2)*(150*x - 450) + 1350))/(exp(3) - 27*x + exp(1)*(3*x^2 - 18*x + 27) +
9*x^2 - x^3 - exp(2)*(3*x - 9) + 27),x)

[Out]

50*x*exp(-x^2/(12*exp(1) - 12*x + 2*exp(2) - 4*x*exp(1) + 2*x^2 + 18))

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sympy [A]  time = 0.78, size = 32, normalized size = 1.60 \begin {gather*} 50 x e^{- \frac {2 x^{2}}{4 x^{2} - 24 x + e \left (24 - 8 x\right ) + 4 e^{2} + 36}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*exp(1)**3+(-150*x+450)*exp(1)**2+(100*x**2-900*x+1350)*exp(1)-50*x**3+300*x**2-1350*x+1350)/(exp
(1)**3+(-3*x+9)*exp(1)**2+(3*x**2-18*x+27)*exp(1)-x**3+9*x**2-27*x+27)/exp(x**2/(4*exp(1)**2+(-8*x+24)*exp(1)+
4*x**2-24*x+36))**2,x)

[Out]

50*x*exp(-2*x**2/(4*x**2 - 24*x + E*(24 - 8*x) + 4*exp(2) + 36))

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