\(\int e^{-7+x^2-25^{\frac {4 x}{e^2}} (x^2)^{\frac {4 x}{e^2}}} (e^2 (1+2 x^2)+25^{\frac {4 x}{e^2}} (x^2)^{\frac {4 x}{e^2}} (-8 x-4 x \log (25 x^2))) \, dx\) [6680]

Optimal result

Integrand size = 72, antiderivative size = 29 \[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \]

[Out]

x*exp(-exp(2*x*ln(25*x^2)/exp(2))^2+x^2-5)

Rubi [F]

\[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=\int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx \]

[In]

Int[E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*(E^2*(1 + 2*x^2) + 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)*(-8*x
- 4*x*Log[25*x^2])),x]

[Out]

Defer[Int][E^(-5 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)), x] + 2*Defer[Int][E^(-5 + x^2 - 25^((4*x)/E^2)*(x^
2)^((4*x)/E^2))*x^2, x] - 8*Defer[Int][25^((4*x)/E^2)*E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x*(x^2)^
((4*x)/E^2), x] - 4*Log[25*x^2]*Defer[Int][25^((4*x)/E^2)*E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x*(x
^2)^((4*x)/E^2), x] + 8*Defer[Int][Defer[Int][25^((4*x)/E^2)*E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*x
*(x^2)^((4*x)/E^2), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (1+2 x^2\right )-4\ 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \left (2+\log \left (25 x^2\right )\right )\right ) \, dx \\ & = -\left (4 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \left (2+\log \left (25 x^2\right )\right ) \, dx\right )+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (1+2 x^2\right ) \, dx \\ & = -\left (4 \int \left (2\ 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}}+25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \log \left (25 x^2\right )\right ) \, dx\right )+\int \left (e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}}+2 e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2\right ) \, dx \\ & = 2 \int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2 \, dx-4 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \log \left (25 x^2\right ) \, dx-8 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \, dx \\ & = 2 \int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2 \, dx+4 \int \frac {2 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx}{x} \, dx-8 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx-\left (4 \log \left (25 x^2\right )\right ) \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \, dx \\ & = 2 \int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x^2 \, dx-8 \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+8 \int \frac {\int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx}{x} \, dx-\left (4 \log \left (25 x^2\right )\right ) \int 25^{\frac {4 x}{e^2}} e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \left (x^2\right )^{\frac {4 x}{e^2}} \, dx+\int e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=e^{-5+x^2-390625^{\frac {x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \]

[In]

Integrate[E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*(E^2*(1 + 2*x^2) + 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)*
(-8*x - 4*x*Log[25*x^2])),x]

[Out]

E^(-5 + x^2 - 390625^(x/E^2)*(x^2)^((4*x)/E^2))*x

Maple [A] (verified)

Time = 4.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
risch	\(x \,{\mathrm e}^{-\left (25 x^{2}\right )^{4 \,{\mathrm e}^{-2} x}+x^{2}-5}\)	\(24\)
parallelrisch	\(x \,{\mathrm e}^{-{\mathrm e}^{4 x \ln \left (25 x^{2}\right ) {\mathrm e}^{-2}}+x^{2}-5}\)	\(27\)

[In]

int(((-4*x*ln(25*x^2)-8*x)*exp(2*x*ln(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*ln(25*x^2)/exp(2))^2+x^
2-5)/exp(2),x,method=_RETURNVERBOSE)

[Out]

x*exp(-((25*x^2)^(2*exp(-2)*x))^2+x^2-5)

Fricas [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=x e^{\left (x^{2} - \left (25 \, x^{2}\right )^{4 \, x e^{\left (-2\right )}} - 5\right )} \]

[In]

integrate(((-4*x*log(25*x^2)-8*x)*exp(2*x*log(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*log(25*x^2)/exp
(2))^2+x^2-5)/exp(2),x, algorithm="fricas")

[Out]

x*e^(x^2 - (25*x^2)^(4*x*e^(-2)) - 5)

Sympy [A] (verification not implemented)

Time = 23.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=x e^{x^{2} - e^{\frac {4 x \log {\left (25 x^{2} \right )}}{e^{2}}} - 5} \]

[In]

integrate(((-4*x*ln(25*x**2)-8*x)*exp(2*x*ln(25*x**2)/exp(2))**2+(2*x**2+1)*exp(2))*exp(-exp(2*x*ln(25*x**2)/e
xp(2))**2+x**2-5)/exp(2),x)

[Out]

x*exp(x**2 - exp(4*x*exp(-2)*log(25*x**2)) - 5)

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=x e^{\left (x^{2} - e^{\left (8 \, x e^{\left (-2\right )} \log \left (5\right ) + 8 \, x e^{\left (-2\right )} \log \left (x\right )\right )} - 5\right )} \]

[In]

integrate(((-4*x*log(25*x^2)-8*x)*exp(2*x*log(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*log(25*x^2)/exp
(2))^2+x^2-5)/exp(2),x, algorithm="maxima")

[Out]

x*e^(x^2 - e^(8*x*e^(-2)*log(5) + 8*x*e^(-2)*log(x)) - 5)

Giac [F]

\[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=\int { -{\left (4 \, {\left (x \log \left (25 \, x^{2}\right ) + 2 \, x\right )} \left (25 \, x^{2}\right )^{4 \, x e^{\left (-2\right )}} - {\left (2 \, x^{2} + 1\right )} e^{2}\right )} e^{\left (x^{2} - \left (25 \, x^{2}\right )^{4 \, x e^{\left (-2\right )}} - 7\right )} \,d x } \]

[In]

integrate(((-4*x*log(25*x^2)-8*x)*exp(2*x*log(25*x^2)/exp(2))^2+(2*x^2+1)*exp(2))*exp(-exp(2*x*log(25*x^2)/exp
(2))^2+x^2-5)/exp(2),x, algorithm="giac")

[Out]

integrate(-(4*(x*log(25*x^2) + 2*x)*(25*x^2)^(4*x*e^(-2)) - (2*x^2 + 1)*e^2)*e^(x^2 - (25*x^2)^(4*x*e^(-2)) -
7), x)

Mupad [B] (verification not implemented)

Time = 12.89 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx=x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{-{\left (390625\,x^8\right )}^{x\,{\mathrm {e}}^{-2}}} \]

[In]

int(-exp(x^2 - exp(4*x*exp(-2)*log(25*x^2)) - 5)*exp(-2)*(exp(4*x*exp(-2)*log(25*x^2))*(8*x + 4*x*log(25*x^2))
- exp(2)*(2*x^2 + 1)),x)

[Out]

x*exp(x^2)*exp(-5)*exp(-(390625*x^8)^(x*exp(-2)))