3.57.25 \(\int \frac {e^4 x+e^4 \log (2)+625^{\frac {x}{e^4}} (5 e^4+5 \log (2) \log (625))}{5^{1+\frac {4 x}{e^4}} e^4+e^4 x} \, dx\)

Optimal. Leaf size=19 \[ 5+x+\log (2) \log \left (5^{1+\frac {4 x}{e^4}}+x\right ) \]

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Rubi [F]  time = 0.48, 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^4 x+e^4 \log (2)+625^{\frac {x}{e^4}} \left (5 e^4+5 \log (2) \log (625)\right )}{5^{1+\frac {4 x}{e^4}} e^4+e^4 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^4*x + E^4*Log[2] + 625^(x/E^4)*(5*E^4 + 5*Log[2]*Log[625]))/(5^(1 + (4*x)/E^4)*E^4 + E^4*x),x]

[Out]

x*(1 + (Log[2]*Log[625])/E^4) + Log[2]*Defer[Int][(5^(1 + (4*x)/E^4) + x)^(-1), x] - (Log[2]*Log[625]*Defer[In
t][x/(5^(1 + (4*x)/E^4) + x), x])/E^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^4 x+e^4 \log (2)+625^{\frac {x}{e^4}} \left (5 e^4+5 \log (2) \log (625)\right )}{e^4 \left (5^{1+\frac {4 x}{e^4}}+x\right )} \, dx\\ &=\frac {\int \frac {e^4 x+e^4 \log (2)+625^{\frac {x}{e^4}} \left (5 e^4+5 \log (2) \log (625)\right )}{5^{1+\frac {4 x}{e^4}}+x} \, dx}{e^4}\\ &=\frac {\int \left (-\frac {\log (2) \left (-e^4+x \log (625)\right )}{5^{1+\frac {4 x}{e^4}}+x}+e^4 \left (1+\frac {\log (2) \log (625)}{e^4}\right )\right ) \, dx}{e^4}\\ &=x \left (1+\frac {\log (2) \log (625)}{e^4}\right )-\frac {\log (2) \int \frac {-e^4+x \log (625)}{5^{1+\frac {4 x}{e^4}}+x} \, dx}{e^4}\\ &=x \left (1+\frac {\log (2) \log (625)}{e^4}\right )-\frac {\log (2) \int \left (-\frac {e^4}{5^{1+\frac {4 x}{e^4}}+x}+\frac {x \log (625)}{5^{1+\frac {4 x}{e^4}}+x}\right ) \, dx}{e^4}\\ &=x \left (1+\frac {\log (2) \log (625)}{e^4}\right )+\log (2) \int \frac {1}{5^{1+\frac {4 x}{e^4}}+x} \, dx-\frac {(\log (2) \log (625)) \int \frac {x}{5^{1+\frac {4 x}{e^4}}+x} \, dx}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^4 x+e^4 \log (2)+625^{\frac {x}{e^4}} \left (5 e^4+5 \log (2) \log (625)\right )}{5^{1+\frac {4 x}{e^4}} e^4+e^4 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^4*x + E^4*Log[2] + 625^(x/E^4)*(5*E^4 + 5*Log[2]*Log[625]))/(5^(1 + (4*x)/E^4)*E^4 + E^4*x),x]

[Out]

Integrate[(E^4*x + E^4*Log[2] + 625^(x/E^4)*(5*E^4 + 5*Log[2]*Log[625]))/(5^(1 + (4*x)/E^4)*E^4 + E^4*x), x]

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fricas [A]  time = 0.78, size = 17, normalized size = 0.89 \begin {gather*} \log \relax (2) \log \left (5 \cdot 5^{4 \, x e^{\left (-4\right )}} + x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*log(2)*log(5)+5*exp(4))*exp(4*x*log(5)/exp(4))+exp(4)*log(2)+x*exp(4))/(5*exp(4)*exp(4*x*log(5)
/exp(4))+x*exp(4)),x, algorithm="fricas")

[Out]

log(2)*log(5*5^(4*x*e^(-4)) + x) + x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*log(2)*log(5)+5*exp(4))*exp(4*x*log(5)/exp(4))+exp(4)*log(2)+x*exp(4))/(5*exp(4)*exp(4*x*log(5)
/exp(4))+x*exp(4)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.30, size = 21, normalized size = 1.11

method	result	size
norman	\(x +\ln \relax (2) \ln \left (x +5 \,{\mathrm e}^{4 x \ln \relax (5) {\mathrm e}^{-4}}\right )\)	\(21\)
risch	\({\mathrm e}^{4} {\mathrm e}^{-4} x +\ln \relax (2) \ln \left (\frac {x}{5}+625^{x \,{\mathrm e}^{-4}}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*ln(2)*ln(5)+5*exp(4))*exp(4*x*ln(5)/exp(4))+exp(4)*ln(2)+x*exp(4))/(5*exp(4)*exp(4*x*ln(5)/exp(4))+x*
exp(4)),x,method=_RETURNVERBOSE)

[Out]

x+ln(2)*ln(x+5*exp(4*x*ln(5)/exp(4)))

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maxima [A]  time = 0.57, size = 17, normalized size = 0.89 \begin {gather*} \log \relax (2) \log \left (5^{4 \, x e^{\left (-4\right )}} + \frac {1}{5} \, x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*log(2)*log(5)+5*exp(4))*exp(4*x*log(5)/exp(4))+exp(4)*log(2)+x*exp(4))/(5*exp(4)*exp(4*x*log(5)
/exp(4))+x*exp(4)),x, algorithm="maxima")

[Out]

log(2)*log(5^(4*x*e^(-4)) + 1/5*x) + x

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mupad [B]  time = 3.56, size = 17, normalized size = 0.89 \begin {gather*} x+\ln \left (x+5\,5^{4\,x\,{\mathrm {e}}^{-4}}\right )\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4)*log(2) + x*exp(4) + exp(4*x*exp(-4)*log(5))*(5*exp(4) + 20*log(2)*log(5)))/(x*exp(4) + 5*exp(4*x*e
xp(-4)*log(5))*exp(4)),x)

[Out]

x + log(x + 5*5^(4*x*exp(-4)))*log(2)

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sympy [A]  time = 0.16, size = 20, normalized size = 1.05 \begin {gather*} x + \log {\relax (2 )} \log {\left (\frac {x}{5} + e^{\frac {4 x \log {\relax (5 )}}{e^{4}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*ln(2)*ln(5)+5*exp(4))*exp(4*x*ln(5)/exp(4))+exp(4)*ln(2)+x*exp(4))/(5*exp(4)*exp(4*x*ln(5)/exp(
4))+x*exp(4)),x)

[Out]

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

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