3.42.61 \(\int \frac {e^{-\frac {5 x^2}{5 e^{2 x}-6 e^x x}} (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x (-56 x^2+200 x^3))}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx\)

Optimal. Leaf size=27 \[ 4 e^{\frac {e^{-x} x^2}{-e^x+\frac {6 x}{5}}} x \]

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Rubi [F]  time = 6.59, 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 {5 x^2}{5 e^{2 x}-6 e^x x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{25 e^{3 x}-60 e^{2 x} x+36 e^x x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100*E^(3*x) - 240*E^(2*x)*x + 120*x^3 - 120*x^4 + E^x*(-56*x^2 + 200*x^3))/(E^((5*x^2)/(5*E^(2*x) - 6*E^x
*x))*(25*E^(3*x) - 60*E^(2*x)*x + 36*E^x*x^2)),x]

[Out]

4*Defer[Int][E^((-5*x^2)/(E^x*(5*E^x - 6*x))), x] - 40*Defer[Int][(E^(-x - (5*x^2)/(E^x*(5*E^x - 6*x)))*x^2)/(
5*E^x - 6*x), x] - 120*Defer[Int][(E^(-x - (5*x^2)/(E^x*(5*E^x - 6*x)))*x^3)/(5*E^x - 6*x)^2, x] + 40*Defer[In
t][(E^(-x - (5*x^2)/(E^x*(5*E^x - 6*x)))*x^3)/(5*E^x - 6*x), x] + 120*Defer[Int][(E^(-x - (5*x^2)/(E^x*(5*E^x
- 6*x)))*x^4)/(5*E^x - 6*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \left (100 e^{3 x}-240 e^{2 x} x+120 x^3-120 x^4+e^x \left (-56 x^2+200 x^3\right )\right )}{\left (5 e^x-6 x\right )^2} \, dx\\ &=\int \left (4 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}}+\frac {40 e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^2}{5 e^x-6 x}+\frac {120 e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^3}{\left (5 e^x-6 x\right )^2}\right ) \, dx\\ &=4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \, dx+40 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^2}{5 e^x-6 x} \, dx+120 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} (-1+x) x^3}{\left (5 e^x-6 x\right )^2} \, dx\\ &=4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \, dx+40 \int \left (-\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^2}{5 e^x-6 x}+\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{5 e^x-6 x}\right ) \, dx+120 \int \left (-\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{\left (5 e^x-6 x\right )^2}+\frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^4}{\left (5 e^x-6 x\right )^2}\right ) \, dx\\ &=4 \int e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} \, dx-40 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^2}{5 e^x-6 x} \, dx+40 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{5 e^x-6 x} \, dx-120 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^3}{\left (5 e^x-6 x\right )^2} \, dx+120 \int \frac {e^{-x-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x^4}{\left (5 e^x-6 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.72, size = 26, normalized size = 0.96 \begin {gather*} 4 e^{-\frac {5 e^{-x} x^2}{5 e^x-6 x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100*E^(3*x) - 240*E^(2*x)*x + 120*x^3 - 120*x^4 + E^x*(-56*x^2 + 200*x^3))/(E^((5*x^2)/(5*E^(2*x) -
6*E^x*x))*(25*E^(3*x) - 60*E^(2*x)*x + 36*E^x*x^2)),x]

[Out]

(4*x)/E^((5*x^2)/(E^x*(5*E^x - 6*x)))

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fricas [A]  time = 0.71, size = 23, normalized size = 0.85 \begin {gather*} 4 \, x e^{\left (\frac {5 \, x^{2}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120*x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x
)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*exp(x)*x^2),x, algorithm="fricas")

[Out]

4*x*e^(5*x^2/(6*x*e^x - 5*e^(2*x)))

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giac [A]  time = 0.19, size = 43, normalized size = 1.59 \begin {gather*} 4 \, x e^{\left (-x + \frac {6 \, x^{2} e^{x} + 5 \, x^{2} - 5 \, x e^{\left (2 \, x\right )}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120*x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x
)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*exp(x)*x^2),x, algorithm="giac")

[Out]

4*x*e^(-x + (6*x^2*e^x + 5*x^2 - 5*x*e^(2*x))/(6*x*e^x - 5*e^(2*x)))

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maple [A]  time = 0.06, size = 24, normalized size = 0.89

method	result	size
risch	\(4 x \,{\mathrm e}^{\frac {5 x^{2}}{-5 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x} x}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120*x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x)*x))/
(25*exp(x)^3-60*x*exp(x)^2+36*exp(x)*x^2),x,method=_RETURNVERBOSE)

[Out]

4*x*exp(5*x^2/(-5*exp(2*x)+6*exp(x)*x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -4 \, \int \frac {{\left (30 \, x^{4} - 30 \, x^{3} + 60 \, x e^{\left (2 \, x\right )} - 2 \, {\left (25 \, x^{3} - 7 \, x^{2}\right )} e^{x} - 25 \, e^{\left (3 \, x\right )}\right )} e^{\left (\frac {5 \, x^{2}}{6 \, x e^{x} - 5 \, e^{\left (2 \, x\right )}}\right )}}{36 \, x^{2} e^{x} - 60 \, x e^{\left (2 \, x\right )} + 25 \, e^{\left (3 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*exp(x)^3-240*x*exp(x)^2+(200*x^3-56*x^2)*exp(x)-120*x^4+120*x^3)*exp(-5*x^2/(5*exp(x)^2-6*exp(x
)*x))/(25*exp(x)^3-60*x*exp(x)^2+36*exp(x)*x^2),x, algorithm="maxima")

[Out]

-4*integrate((30*x^4 - 30*x^3 + 60*x*e^(2*x) - 2*(25*x^3 - 7*x^2)*e^x - 25*e^(3*x))*e^(5*x^2/(6*x*e^x - 5*e^(2
*x)))/(36*x^2*e^x - 60*x*e^(2*x) + 25*e^(3*x)), x)

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mupad [B]  time = 3.11, size = 23, normalized size = 0.85 \begin {gather*} 4\,x\,{\mathrm {e}}^{-\frac {5\,x^2}{5\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(5*x^2)/(5*exp(2*x) - 6*x*exp(x)))*(exp(x)*(56*x^2 - 200*x^3) - 100*exp(3*x) + 240*x*exp(2*x) - 120
*x^3 + 120*x^4))/(25*exp(3*x) - 60*x*exp(2*x) + 36*x^2*exp(x)),x)

[Out]

4*x*exp(-(5*x^2)/(5*exp(2*x) - 6*x*exp(x)))

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sympy [A]  time = 4.80, size = 22, normalized size = 0.81 \begin {gather*} 4 x e^{- \frac {5 x^{2}}{- 6 x e^{x} + 5 e^{2 x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((100*exp(x)**3-240*x*exp(x)**2+(200*x**3-56*x**2)*exp(x)-120*x**4+120*x**3)*exp(-5*x**2/(5*exp(x)**2
-6*exp(x)*x))/(25*exp(x)**3-60*x*exp(x)**2+36*exp(x)*x**2),x)

[Out]

4*x*exp(-5*x**2/(-6*x*exp(x) + 5*exp(2*x)))

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