3.11.11 \(\int \frac {e^x (-625 x+625 x^2)+e^{\frac {e^{-x} (-4+e^x (2 x+x^2))}{x}} (2500+2500 x+625 e^x x^2)}{e^{x+\frac {2 e^{-x} (-4+e^x (2 x+x^2))}{x}} x^2+e^x (4 x^2+4 x^3+x^4)+e^x (-4 x^2-2 x^3) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} (-4+e^x (2 x+x^2))}{x}} (e^x (4 x^2+2 x^3)-2 e^x x^2 \log (5 x))} \, dx\)

Optimal. Leaf size=30 \[ \frac {625}{-2-e^{2-\frac {4 e^{-x}}{x}+x}-x+\log (5 x)} \]

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Rubi [F]  time = 20.88, 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 (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-625*x + 625*x^2) + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(2500 + 2500*x + 625*E^x*x^2))/(E^(x + (2*(-4
+ E^x*(2*x + x^2)))/(E^x*x))*x^2 + E^x*(4*x^2 + 4*x^3 + x^4) + E^x*(-4*x^2 - 2*x^3)*Log[5*x] + E^x*x^2*Log[5*
x]^2 + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(E^x*(4*x^2 + 2*x^3) - 2*E^x*x^2*Log[5*x])),x]

[Out]

-625*Defer[Int][E^(8/(E^x*x))/(2*E^(4/(E^x*x)) + E^(2 + x) + E^(4/(E^x*x))*x - E^(4/(E^x*x))*Log[5*x])^2, x] +
2500*Defer[Int][E^(2 + 4/(E^x*x))/(x^2*(2*E^(4/(E^x*x)) + E^(2 + x) + E^(4/(E^x*x))*x - E^(4/(E^x*x))*Log[5*x
])^2), x] + 2500*Defer[Int][E^(2 + 4/(E^x*x))/(x*(2*E^(4/(E^x*x)) + E^(2 + x) + E^(4/(E^x*x))*x - E^(4/(E^x*x)
)*Log[5*x])^2), x] - 625*Defer[Int][E^(8/(E^x*x))/(x*(2*E^(4/(E^x*x)) + E^(2 + x) + E^(4/(E^x*x))*x - E^(4/(E^
x*x))*Log[5*x])^2), x] - 625*Defer[Int][(E^(8/(E^x*x))*x)/(2*E^(4/(E^x*x)) + E^(2 + x) + E^(4/(E^x*x))*x - E^(
4/(E^x*x))*Log[5*x])^2, x] + 625*Defer[Int][E^(4/(E^x*x))/(2*E^(4/(E^x*x)) + E^(2 + x) + E^(4/(E^x*x))*x - E^(
4/(E^x*x))*Log[5*x]), x] + 625*Defer[Int][(E^(8/(E^x*x))*Log[5*x])/(-2*E^(4/(E^x*x)) - E^(2 + x) - E^(4/(E^x*x
))*x + E^(4/(E^x*x))*Log[5*x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {625 e^{\frac {4 e^{-x}}{x}} \left (e^{\frac {4 e^{-x}}{x}} (-1+x) x+e^{2+x} x^2+4 e^2 (1+x)\right )}{x^2 \left (e^{2+x}+e^{\frac {4 e^{-x}}{x}} (2+x)-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx\\ &=625 \int \frac {e^{\frac {4 e^{-x}}{x}} \left (e^{\frac {4 e^{-x}}{x}} (-1+x) x+e^{2+x} x^2+4 e^2 (1+x)\right )}{x^2 \left (e^{2+x}+e^{\frac {4 e^{-x}}{x}} (2+x)-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx\\ &=625 \int \left (\frac {e^{\frac {4 e^{-x}}{x}}}{2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)}-\frac {e^{\frac {4 e^{-x}}{x}} \left (-4 e^2-4 e^2 x+e^{\frac {4 e^{-x}}{x}} x+e^{\frac {4 e^{-x}}{x}} x^2+e^{\frac {4 e^{-x}}{x}} x^3-e^{\frac {4 e^{-x}}{x}} x^2 \log (5 x)\right )}{x^2 \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}\right ) \, dx\\ &=625 \int \frac {e^{\frac {4 e^{-x}}{x}}}{2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)} \, dx-625 \int \frac {e^{\frac {4 e^{-x}}{x}} \left (-4 e^2-4 e^2 x+e^{\frac {4 e^{-x}}{x}} x+e^{\frac {4 e^{-x}}{x}} x^2+e^{\frac {4 e^{-x}}{x}} x^3-e^{\frac {4 e^{-x}}{x}} x^2 \log (5 x)\right )}{x^2 \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx\\ &=625 \int \frac {e^{\frac {4 e^{-x}}{x}}}{2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)} \, dx-625 \int \frac {-4 e^{2+\frac {4 e^{-x}}{x}} (1+x)+e^{\frac {8 e^{-x}}{x}} x \left (1+x+x^2\right )-e^{\frac {8 e^{-x}}{x}} x^2 \log (5 x)}{x^2 \left (e^{2+x}+e^{\frac {4 e^{-x}}{x}} (2+x)-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx\\ &=625 \int \frac {e^{\frac {4 e^{-x}}{x}}}{2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)} \, dx-625 \int \left (\frac {e^{\frac {8 e^{-x}}{x}}}{\left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}-\frac {4 e^{2+\frac {4 e^{-x}}{x}}}{x^2 \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}-\frac {4 e^{2+\frac {4 e^{-x}}{x}}}{x \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}+\frac {e^{\frac {8 e^{-x}}{x}}}{x \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}+\frac {e^{\frac {8 e^{-x}}{x}} x}{\left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}-\frac {e^{\frac {8 e^{-x}}{x}} \log (5 x)}{\left (-2 e^{\frac {4 e^{-x}}{x}}-e^{2+x}-e^{\frac {4 e^{-x}}{x}} x+e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2}\right ) \, dx\\ &=-\left (625 \int \frac {e^{\frac {8 e^{-x}}{x}}}{\left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx\right )-625 \int \frac {e^{\frac {8 e^{-x}}{x}}}{x \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx-625 \int \frac {e^{\frac {8 e^{-x}}{x}} x}{\left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx+625 \int \frac {e^{\frac {4 e^{-x}}{x}}}{2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)} \, dx+625 \int \frac {e^{\frac {8 e^{-x}}{x}} \log (5 x)}{\left (-2 e^{\frac {4 e^{-x}}{x}}-e^{2+x}-e^{\frac {4 e^{-x}}{x}} x+e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx+2500 \int \frac {e^{2+\frac {4 e^{-x}}{x}}}{x^2 \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx+2500 \int \frac {e^{2+\frac {4 e^{-x}}{x}}}{x \left (2 e^{\frac {4 e^{-x}}{x}}+e^{2+x}+e^{\frac {4 e^{-x}}{x}} x-e^{\frac {4 e^{-x}}{x}} \log (5 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 56, normalized size = 1.87 \begin {gather*} -\frac {625 e^{\frac {4 e^{-x}}{x}}}{e^{2+x}+e^{\frac {4 e^{-x}}{x}} (2+x)-e^{\frac {4 e^{-x}}{x}} \log (5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-625*x + 625*x^2) + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(2500 + 2500*x + 625*E^x*x^2))/(E^(x +
(2*(-4 + E^x*(2*x + x^2)))/(E^x*x))*x^2 + E^x*(4*x^2 + 4*x^3 + x^4) + E^x*(-4*x^2 - 2*x^3)*Log[5*x] + E^x*x^2*
Log[5*x]^2 + E^((-4 + E^x*(2*x + x^2))/(E^x*x))*(E^x*(4*x^2 + 2*x^3) - 2*E^x*x^2*Log[5*x])),x]

[Out]

(-625*E^(4/(E^x*x)))/(E^(2 + x) + E^(4/(E^x*x))*(2 + x) - E^(4/(E^x*x))*Log[5*x])

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fricas [A]  time = 0.62, size = 34, normalized size = 1.13 \begin {gather*} -\frac {625}{x + e^{\left (\frac {{\left ({\left (x^{2} + 2 \, x\right )} e^{x} - 4\right )} e^{\left (-x\right )}}{x}\right )} - \log \left (5 \, x\right ) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625*x^2-625*x)*exp(x))/(x^2*exp(x)
*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/e
xp(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x^2)*exp(x)),x, algorithm="fricas")

[Out]

-625/(x + e^(((x^2 + 2*x)*e^x - 4)*e^(-x)/x) - log(5*x) + 2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625*x^2-625*x)*exp(x))/(x^2*exp(x)
*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/e
xp(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x^2)*exp(x)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.09, size = 36, normalized size = 1.20

method	result	size
risch	\(-\frac {625}{x -\ln \left (5 x \right )+{\mathrm e}^{\frac {\left ({\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x -4\right ) {\mathrm e}^{-x}}{x}}+2}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625*x^2-625*x)*exp(x))/(x^2*exp(x)*exp((
(x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^2*exp(x)*ln(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x
)+x^2*exp(x)*ln(5*x)^2+(-2*x^3-4*x^2)*exp(x)*ln(5*x)+(x^4+4*x^3+4*x^2)*exp(x)),x,method=_RETURNVERBOSE)

[Out]

-625/(x-ln(5*x)+exp((exp(x)*x^2+2*exp(x)*x-4)*exp(-x)/x)+2)

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maxima [A]  time = 0.95, size = 41, normalized size = 1.37 \begin {gather*} -\frac {625 \, e^{\left (\frac {4 \, e^{\left (-x\right )}}{x}\right )}}{{\left (x - \log \relax (5) - \log \relax (x) + 2\right )} e^{\left (\frac {4 \, e^{\left (-x\right )}}{x}\right )} + e^{\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625*x^2-625*x)*exp(x))/(x^2*exp(x)
*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/e
xp(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x^2)*exp(x)),x, algorithm="maxima")

[Out]

-625*e^(4*e^(-x)/x)/((x - log(5) - log(x) + 2)*e^(4*e^(-x)/x) + e^(x + 2))

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mupad [B]  time = 0.95, size = 30, normalized size = 1.00 \begin {gather*} -\frac {625}{x-\ln \relax (5)-\ln \relax (x)+{\mathrm {e}}^2\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{-x}}{x}}\,{\mathrm {e}}^x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(-x)*(exp(x)*(2*x + x^2) - 4))/x)*(2500*x + 625*x^2*exp(x) + 2500) - exp(x)*(625*x - 625*x^2))/(e
xp((exp(-x)*(exp(x)*(2*x + x^2) - 4))/x)*(exp(x)*(4*x^2 + 2*x^3) - 2*x^2*log(5*x)*exp(x)) + exp(x)*(4*x^2 + 4*
x^3 + x^4) - log(5*x)*exp(x)*(4*x^2 + 2*x^3) + x^2*exp((2*exp(-x)*(exp(x)*(2*x + x^2) - 4))/x)*exp(x) + x^2*lo
g(5*x)^2*exp(x)),x)

[Out]

-625/(x - log(5) - log(x) + exp(2)*exp(-(4*exp(-x))/x)*exp(x) + 2)

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sympy [A]  time = 0.49, size = 29, normalized size = 0.97 \begin {gather*} - \frac {625}{x + e^{\frac {\left (\left (x^{2} + 2 x\right ) e^{x} - 4\right ) e^{- x}}{x}} - \log {\left (5 x \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*exp(x)*x**2+2500*x+2500)*exp(((x**2+2*x)*exp(x)-4)/exp(x)/x)+(625*x**2-625*x)*exp(x))/(x**2*ex
p(x)*exp(((x**2+2*x)*exp(x)-4)/exp(x)/x)**2+(-2*x**2*exp(x)*ln(5*x)+(2*x**3+4*x**2)*exp(x))*exp(((x**2+2*x)*ex
p(x)-4)/exp(x)/x)+x**2*exp(x)*ln(5*x)**2+(-2*x**3-4*x**2)*exp(x)*ln(5*x)+(x**4+4*x**3+4*x**2)*exp(x)),x)

[Out]

-625/(x + exp(((x**2 + 2*x)*exp(x) - 4)*exp(-x)/x) - log(5*x) + 2)

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