∫ e−x(e−2+2x(5−5x)+5x2+e−2+x(−1+2x))_____________________________

Optimal. Leaf size=31 \[ 4-\frac {-x+\frac {e^{-x} x}{5}}{\frac {e^{-2+x}}{5}+x} \]

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Rubi [F] time = 0.88, 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^{-x} \left (e^{-2+2 x} (5-5 x)+5 x^2+e^{-2+x} (-1+2 x)\right )}{e^{-4+2 x}+10 e^{-2+x} x+25 x^2} \, dx \end {gather*}

Int[(E^(-2 + 2*x)*(5 - 5*x) + 5*x^2 + E^(-2 + x)*(-1 + 2*x))/(E^x*(E^(-4 + 2*x) + 10*E^(-2 + x)*x + 25*x^2)),x

-1/5*1/E^x - E^2*Defer[Int][(E^x + 5*E^2*x)^(-2), x] + E^2*(1 - 25*E^2)*Defer[Int][x/(E^x + 5*E^2*x)^2, x] + 2

5*E^4*Defer[Int][x^2/(E^x + 5*E^2*x)^2, x] - ((1 - 25*E^2)*Defer[Int][(E^x + 5*E^2*x)^(-1), x])/5 - 5*E^2*Defe

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 \left (-1-5 e^x (-1+x)+2 x+5 e^{2-x} x^2\right )}{\left (e^x+5 e^2 x\right )^2} \, dx\\ &=e^2 \int \frac {-1-5 e^x (-1+x)+2 x+5 e^{2-x} x^2}{\left (e^x+5 e^2 x\right )^2} \, dx\\ &=e^2 \int \left (\frac {e^{-2-x}}{5}+\frac {(-1+x) \left (1+25 e^2 x\right )}{\left (e^x+5 e^2 x\right )^2}-\frac {1-25 e^2+25 e^2 x}{5 e^2 \left (e^x+5 e^2 x\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {1-25 e^2+25 e^2 x}{e^x+5 e^2 x} \, dx\right )+\frac {1}{5} e^2 \int e^{-2-x} \, dx+e^2 \int \frac {(-1+x) \left (1+25 e^2 x\right )}{\left (e^x+5 e^2 x\right )^2} \, dx\\ &=-\frac {e^{-x}}{5}-\frac {1}{5} \int \left (\frac {1-25 e^2}{e^x+5 e^2 x}+\frac {25 e^2 x}{e^x+5 e^2 x}\right ) \, dx+e^2 \int \left (-\frac {1}{\left (e^x+5 e^2 x\right )^2}-\frac {\left (-1+25 e^2\right ) x}{\left (e^x+5 e^2 x\right )^2}+\frac {25 e^2 x^2}{\left (e^x+5 e^2 x\right )^2}\right ) \, dx\\ &=-\frac {e^{-x}}{5}-e^2 \int \frac {1}{\left (e^x+5 e^2 x\right )^2} \, dx-\left (5 e^2\right ) \int \frac {x}{e^x+5 e^2 x} \, dx+\left (25 e^4\right ) \int \frac {x^2}{\left (e^x+5 e^2 x\right )^2} \, dx-\frac {1}{5} \left (1-25 e^2\right ) \int \frac {1}{e^x+5 e^2 x} \, dx+\left (e^2 \left (1-25 e^2\right )\right ) \int \frac {x}{\left (e^x+5 e^2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.78, size = 30, normalized size = 0.97 \begin {gather*} -\frac {e^2 \left (e^{-2+x}+e^{-x} x\right )}{e^x+5 e^2 x} \end {gather*}

Integrate[(E^(-2 + 2*x)*(5 - 5*x) + 5*x^2 + E^(-2 + x)*(-1 + 2*x))/(E^x*(E^(-4 + 2*x) + 10*E^(-2 + x)*x + 25*x

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fricas [A] time = 0.65, size = 24, normalized size = 0.77 \begin {gather*} \frac {5 \, x e^{x} - x}{5 \, x e^{x} + e^{\left (2 \, x - 2\right )}} \end {gather*}

integrate(((-5*x+5)*exp(x-2)*exp(x)+(2*x-1)*exp(x-2)+5*x^2)/(exp(x-2)^2+10*x*exp(x-2)+25*x^2)/exp(x),x, algori

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giac [A] time = 0.39, size = 28, normalized size = 0.90 \begin {gather*} -\frac {x e^{2} - 5 \, x e^{\left (x + 2\right )}}{5 \, x e^{\left (x + 2\right )} + e^{\left (2 \, x\right )}} \end {gather*}

integrate(((-5*x+5)*exp(x-2)*exp(x)+(2*x-1)*exp(x-2)+5*x^2)/(exp(x-2)^2+10*x*exp(x-2)+25*x^2)/exp(x),x, algori

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method	result	size

risch	\(-\frac {{\mathrm e}^{-x}}{5}+\frac {25 x +{\mathrm e}^{-2}}{5 \,{\mathrm e}^{x -2}+25 x}\)	\(26\)
norman	\(\frac {\left (5 x \,{\mathrm e}^{2} {\mathrm e}^{x}-{\mathrm e}^{2} x \right ) {\mathrm e}^{-x}}{5 \,{\mathrm e}^{2} x +{\mathrm e}^{x}}\)	\(29\)

int(((-5*x+5)*exp(x-2)*exp(x)+(2*x-1)*exp(x-2)+5*x^2)/(exp(x-2)^2+10*x*exp(x-2)+25*x^2)/exp(x),x,method=_RETUR

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maxima [A] time = 0.57, size = 28, normalized size = 0.90 \begin {gather*} -\frac {x e^{2} - 5 \, x e^{\left (x + 2\right )}}{5 \, x e^{\left (x + 2\right )} + e^{\left (2 \, x\right )}} \end {gather*}

integrate(((-5*x+5)*exp(x-2)*exp(x)+(2*x-1)*exp(x-2)+5*x^2)/(exp(x-2)^2+10*x*exp(x-2)+25*x^2)/exp(x),x, algori

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mupad [B] time = 0.18, size = 24, normalized size = 0.77 \begin {gather*} \frac {x\,{\mathrm {e}}^{2-x}\,\left (5\,{\mathrm {e}}^x-1\right )}{{\mathrm {e}}^x+5\,x\,{\mathrm {e}}^2} \end {gather*}

int((exp(-x)*(exp(x - 2)*(2*x - 1) + 5*x^2 - exp(x - 2)*exp(x)*(5*x - 5)))/(exp(2*x - 4) + 10*x*exp(x - 2) + 2

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sympy [B] time = 0.19, size = 42, normalized size = 1.35 \begin {gather*} \frac {- 25 x e^{2} - 1}{125 x^{2} e^{4} e^{- x} + 25 x e^{2}} - \frac {e^{- x}}{5} + \frac {1}{25 x e^{2}} \end {gather*}

integrate(((-5*x+5)*exp(x-2)*exp(x)+(2*x-1)*exp(x-2)+5*x**2)/(exp(x-2)**2+10*x*exp(x-2)+25*x**2)/exp(x),x)

(-25*x*exp(2) - 1)/(125*x**2*exp(4)*exp(-x) + 25*x*exp(2)) - exp(-x)/5 + exp(-2)/(25*x)

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3.7.79 \(\int \frac {e^{-x} (e^{-2+2 x} (5-5 x)+5 x^2+e^{-2+x} (-1+2 x))}{e^{-4+2 x}+10 e^{-2+x} x+25 x^2} \, dx\)