3.7.67 \(\int \frac {e^{\frac {2 x}{-8+4 x}} (-4+5 x-x^2)+(60-60 x+15 x^2) \log (4)}{20-20 x+5 x^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac {1}{5} e^{\frac {x}{2 (-2+x)}} x+3 x \log (4) \]

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Rubi [F]  time = 0.34, 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 {2 x}{-8+4 x}} \left (-4+5 x-x^2\right )+\left (60-60 x+15 x^2\right ) \log (4)}{20-20 x+5 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x)/(-8 + 4*x))*(-4 + 5*x - x^2) + (60 - 60*x + 15*x^2)*Log[4])/(20 - 20*x + 5*x^2),x]

[Out]

(-2*E^(1/2 + (-2 + x)^(-1)))/5 - (Sqrt[E]*ExpIntegralEi[(-2 + x)^(-1)])/5 + 3*x*Log[4] - Defer[Int][E^(x/(2*(-
2 + x))), x]/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 x}{-8+4 x}} \left (-4+5 x-x^2\right )+\left (60-60 x+15 x^2\right ) \log (4)}{5 (-2+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {e^{\frac {2 x}{-8+4 x}} \left (-4+5 x-x^2\right )+\left (60-60 x+15 x^2\right ) \log (4)}{(-2+x)^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {e^{\frac {x}{2 (-2+x)}} \left (4-5 x+x^2\right )}{(-2+x)^2}+15 \log (4)\right ) \, dx\\ &=3 x \log (4)-\frac {1}{5} \int \frac {e^{\frac {x}{2 (-2+x)}} \left (4-5 x+x^2\right )}{(-2+x)^2} \, dx\\ &=3 x \log (4)-\frac {1}{5} \int \left (e^{\frac {x}{2 (-2+x)}}+\frac {e^{\frac {x}{2 (-2+x)}}}{2-x}-\frac {2 e^{\frac {x}{2 (-2+x)}}}{(-2+x)^2}\right ) \, dx\\ &=3 x \log (4)-\frac {1}{5} \int e^{\frac {x}{2 (-2+x)}} \, dx-\frac {1}{5} \int \frac {e^{\frac {x}{2 (-2+x)}}}{2-x} \, dx+\frac {2}{5} \int \frac {e^{\frac {x}{2 (-2+x)}}}{(-2+x)^2} \, dx\\ &=3 x \log (4)-\frac {1}{5} \int e^{\frac {x}{2 (-2+x)}} \, dx-\frac {1}{5} \int \frac {e^{\frac {1}{2}+\frac {1}{-2+x}}}{2-x} \, dx+\frac {2}{5} \int \frac {e^{\frac {1}{2}+\frac {1}{-2+x}}}{(-2+x)^2} \, dx\\ &=-\frac {2}{5} e^{\frac {1}{2}+\frac {1}{-2+x}}-\frac {1}{5} \sqrt {e} \text {Ei}\left (\frac {1}{-2+x}\right )+3 x \log (4)-\frac {1}{5} \int e^{\frac {x}{2 (-2+x)}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 1.09 \begin {gather*} -\frac {1}{5} e^{\frac {1}{2} \left (1+\frac {2}{-2+x}\right )} x+x \log (64) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*x)/(-8 + 4*x))*(-4 + 5*x - x^2) + (60 - 60*x + 15*x^2)*Log[4])/(20 - 20*x + 5*x^2),x]

[Out]

-1/5*(E^((1 + 2/(-2 + x))/2)*x) + x*Log[64]

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fricas [A]  time = 0.98, size = 18, normalized size = 0.78 \begin {gather*} -\frac {1}{5} \, x e^{\left (\frac {x}{2 \, {\left (x - 2\right )}}\right )} + 6 \, x \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+5*x-4)*exp(x/(4*x-8))^2+2*(15*x^2-60*x+60)*log(2))/(5*x^2-20*x+20),x, algorithm="fricas")

[Out]

-1/5*x*e^(1/2*x/(x - 2)) + 6*x*log(2)

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giac [A]  time = 0.32, size = 34, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (\frac {x e^{\left (\frac {x}{2 \, {\left (x - 2\right )}}\right )}}{x - 2} - 30 \, \log \relax (2)\right )}}{5 \, {\left (\frac {x}{x - 2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+5*x-4)*exp(x/(4*x-8))^2+2*(15*x^2-60*x+60)*log(2))/(5*x^2-20*x+20),x, algorithm="giac")

[Out]

-2/5*(x*e^(1/2*x/(x - 2))/(x - 2) - 30*log(2))/(x/(x - 2) - 1)

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maple [A]  time = 0.20, size = 19, normalized size = 0.83

method	result	size
risch	\(6 x \ln \relax (2)-\frac {x \,{\mathrm e}^{\frac {x}{2 x -4}}}{5}\)	\(19\)
derivativedivides	\(-\frac {{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}} \left (x -2\right )}{5}+6 \left (x -2\right ) \ln \relax (2)-\frac {2 \,{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}}}{5}\)	\(40\)
default	\(-\frac {{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}} \left (x -2\right )}{5}+6 \left (x -2\right ) \ln \relax (2)-\frac {2 \,{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}}}{5}\)	\(40\)
norman	\(\frac {\frac {2 x \,{\mathrm e}^{\frac {2 x}{4 x -8}}}{5}-\frac {x^{2} {\mathrm e}^{\frac {2 x}{4 x -8}}}{5}+6 x^{2} \ln \relax (2)-24 \ln \relax (2)}{x -2}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+5*x-4)*exp(x/(4*x-8))^2+2*(15*x^2-60*x+60)*ln(2))/(5*x^2-20*x+20),x,method=_RETURNVERBOSE)

[Out]

6*x*ln(2)-1/5*x*exp(1/2*x/(x-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{5} \, x e^{\left (\frac {1}{x - 2} + \frac {1}{2}\right )} + 6 \, {\left (x - \frac {4}{x - 2} + 4 \, \log \left (x - 2\right )\right )} \log \relax (2) + 24 \, {\left (\frac {2}{x - 2} - \log \left (x - 2\right )\right )} \log \relax (2) - \frac {24 \, \log \relax (2)}{x - 2} + \frac {4}{5} \, e^{\left (\frac {1}{x - 2} + \frac {1}{2}\right )} + \frac {4}{5} \, \int \frac {e^{\left (\frac {1}{x - 2} + \frac {1}{2}\right )}}{x^{2} - 4 \, x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+5*x-4)*exp(x/(4*x-8))^2+2*(15*x^2-60*x+60)*log(2))/(5*x^2-20*x+20),x, algorithm="maxima")

[Out]

-1/5*x*e^(1/(x - 2) + 1/2) + 6*(x - 4/(x - 2) + 4*log(x - 2))*log(2) + 24*(2/(x - 2) - log(x - 2))*log(2) - 24
*log(2)/(x - 2) + 4/5*e^(1/(x - 2) + 1/2) + 4/5*integrate(e^(1/(x - 2) + 1/2)/(x^2 - 4*x + 4), x)

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mupad [B]  time = 0.12, size = 18, normalized size = 0.78 \begin {gather*} x\,\left (\ln \left (64\right )-\frac {{\mathrm {e}}^{\frac {2\,x}{4\,x-8}}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(15*x^2 - 60*x + 60) - exp((2*x)/(4*x - 8))*(x^2 - 5*x + 4))/(5*x^2 - 20*x + 20),x)

[Out]

x*(log(64) - exp((2*x)/(4*x - 8))/5)

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sympy [A]  time = 0.17, size = 19, normalized size = 0.83 \begin {gather*} - \frac {x e^{\frac {2 x}{4 x - 8}}}{5} + 6 x \log {\relax (2 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+5*x-4)*exp(x/(4*x-8))**2+2*(15*x**2-60*x+60)*ln(2))/(5*x**2-20*x+20),x)

[Out]

-x*exp(2*x/(4*x - 8))/5 + 6*x*log(2)

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