3.31.49 \(\int \frac {e^{2 x-\frac {4}{x^2 \log (5)}} (8-16 x+8 x^2+(-2 x^4+2 x^5) \log (5)+e^{\frac {4}{x^2 \log (5)}} (2 x^4+2 x^5) \log (5)+e^{\frac {2}{x^2 \log (5)}} (-8 x+8 x^2+(-2 x^3+4 x^5) \log (5)))}{x^3 \log (5)} \, dx\)

Optimal. Leaf size=29 \[ e^{2 x} \left (e^{-\frac {2}{x^2 \log (5)}} (1-x)-x\right )^2 \]

________________________________________________________________________________________

Rubi [F]  time = 4.69, 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^{2 x-\frac {4}{x^2 \log (5)}} \left (8-16 x+8 x^2+\left (-2 x^4+2 x^5\right ) \log (5)+e^{\frac {4}{x^2 \log (5)}} \left (2 x^4+2 x^5\right ) \log (5)+e^{\frac {2}{x^2 \log (5)}} \left (-8 x+8 x^2+\left (-2 x^3+4 x^5\right ) \log (5)\right )\right )}{x^3 \log (5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x - 4/(x^2*Log[5]))*(8 - 16*x + 8*x^2 + (-2*x^4 + 2*x^5)*Log[5] + E^(4/(x^2*Log[5]))*(2*x^4 + 2*x^5)
*Log[5] + E^(2/(x^2*Log[5]))*(-8*x + 8*x^2 + (-2*x^3 + 4*x^5)*Log[5])))/(x^3*Log[5]),x]

[Out]

E^(2*x)*x^2 - 2*Defer[Int][E^(2*x - 2/(x^2*Log[5])), x] + (8*Defer[Int][E^(2*x - 4/(x^2*Log[5]))/x^3, x])/Log[
5] - (16*Defer[Int][E^(2*x - 4/(x^2*Log[5]))/x^2, x])/Log[5] - (8*Defer[Int][E^(2*x - 2/(x^2*Log[5]))/x^2, x])
/Log[5] + (8*Defer[Int][E^(2*x - 4/(x^2*Log[5]))/x, x])/Log[5] + (8*Defer[Int][E^(2*x - 2/(x^2*Log[5]))/x, x])
/Log[5] - 2*Defer[Int][E^(2*x - 4/(x^2*Log[5]))*x, x] + 2*Defer[Int][E^(2*x - 4/(x^2*Log[5]))*x^2, x] + (2*Log
[25]*Defer[Int][E^(2*x - 2/(x^2*Log[5]))*x^2, x])/Log[5]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{2 x-\frac {4}{x^2 \log (5)}} \left (8-16 x+8 x^2+\left (-2 x^4+2 x^5\right ) \log (5)+e^{\frac {4}{x^2 \log (5)}} \left (2 x^4+2 x^5\right ) \log (5)+e^{\frac {2}{x^2 \log (5)}} \left (-8 x+8 x^2+\left (-2 x^3+4 x^5\right ) \log (5)\right )\right )}{x^3} \, dx}{\log (5)}\\ &=\frac {\int \frac {2 e^{2 x-\frac {4}{x^2 \log (5)}} \left (1-x-e^{\frac {2}{x^2 \log (5)}} x\right ) \left (4-4 x-e^{\frac {2}{x^2 \log (5)}} x^3 \log (5)-x^4 \log (5)-e^{\frac {2}{x^2 \log (5)}} x^4 \log (5)\right )}{x^3} \, dx}{\log (5)}\\ &=\frac {2 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}} \left (1-x-e^{\frac {2}{x^2 \log (5)}} x\right ) \left (4-4 x-e^{\frac {2}{x^2 \log (5)}} x^3 \log (5)-x^4 \log (5)-e^{\frac {2}{x^2 \log (5)}} x^4 \log (5)\right )}{x^3} \, dx}{\log (5)}\\ &=\frac {2 \int \left (e^{2 x} x (1+x) \log (5)+\frac {e^{2 x-\frac {4}{x^2 \log (5)}} (-1+x) \left (-4+4 x+x^4 \log (5)\right )}{x^3}+\frac {e^{2 x-\frac {2}{x^2 \log (5)}} \left (-4+4 x-x^2 \log (5)+x^4 \log (25)\right )}{x^2}\right ) \, dx}{\log (5)}\\ &=2 \int e^{2 x} x (1+x) \, dx+\frac {2 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}} (-1+x) \left (-4+4 x+x^4 \log (5)\right )}{x^3} \, dx}{\log (5)}+\frac {2 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}} \left (-4+4 x-x^2 \log (5)+x^4 \log (25)\right )}{x^2} \, dx}{\log (5)}\\ &=2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx+\frac {2 \int \left (\frac {4 e^{2 x-\frac {4}{x^2 \log (5)}}}{x^3}-\frac {8 e^{2 x-\frac {4}{x^2 \log (5)}}}{x^2}+\frac {4 e^{2 x-\frac {4}{x^2 \log (5)}}}{x}-e^{2 x-\frac {4}{x^2 \log (5)}} x \log (5)+e^{2 x-\frac {4}{x^2 \log (5)}} x^2 \log (5)\right ) \, dx}{\log (5)}+\frac {2 \int \left (-\frac {4 e^{2 x-\frac {2}{x^2 \log (5)}}}{x^2}+\frac {4 e^{2 x-\frac {2}{x^2 \log (5)}}}{x}-e^{2 x-\frac {2}{x^2 \log (5)}} \log (5)+e^{2 x-\frac {2}{x^2 \log (5)}} x^2 \log (25)\right ) \, dx}{\log (5)}\\ &=-\left (2 \int e^{2 x-\frac {2}{x^2 \log (5)}} \, dx\right )+2 \int e^{2 x} x \, dx-2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x \, dx+2 \int e^{2 x} x^2 \, dx+2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x^2 \, dx+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^3} \, dx}{\log (5)}-\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x} \, dx}{\log (5)}-\frac {16 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {(2 \log (25)) \int e^{2 x-\frac {2}{x^2 \log (5)}} x^2 \, dx}{\log (5)}\\ &=e^{2 x} x+e^{2 x} x^2-2 \int e^{2 x-\frac {2}{x^2 \log (5)}} \, dx-2 \int e^{2 x} x \, dx-2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x \, dx+2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x^2 \, dx+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^3} \, dx}{\log (5)}-\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x} \, dx}{\log (5)}-\frac {16 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {(2 \log (25)) \int e^{2 x-\frac {2}{x^2 \log (5)}} x^2 \, dx}{\log (5)}-\int e^{2 x} \, dx\\ &=-\frac {e^{2 x}}{2}+e^{2 x} x^2-2 \int e^{2 x-\frac {2}{x^2 \log (5)}} \, dx-2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x \, dx+2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x^2 \, dx+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^3} \, dx}{\log (5)}-\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x} \, dx}{\log (5)}-\frac {16 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {(2 \log (25)) \int e^{2 x-\frac {2}{x^2 \log (5)}} x^2 \, dx}{\log (5)}+\int e^{2 x} \, dx\\ &=e^{2 x} x^2-2 \int e^{2 x-\frac {2}{x^2 \log (5)}} \, dx-2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x \, dx+2 \int e^{2 x-\frac {4}{x^2 \log (5)}} x^2 \, dx+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^3} \, dx}{\log (5)}-\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x} \, dx}{\log (5)}+\frac {8 \int \frac {e^{2 x-\frac {2}{x^2 \log (5)}}}{x} \, dx}{\log (5)}-\frac {16 \int \frac {e^{2 x-\frac {4}{x^2 \log (5)}}}{x^2} \, dx}{\log (5)}+\frac {(2 \log (25)) \int e^{2 x-\frac {2}{x^2 \log (5)}} x^2 \, dx}{\log (5)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.96, size = 34, normalized size = 1.17 \begin {gather*} e^{2 x-\frac {4}{x^2 \log (5)}} \left (-1+x+e^{\frac {2}{x^2 \log (5)}} x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x - 4/(x^2*Log[5]))*(8 - 16*x + 8*x^2 + (-2*x^4 + 2*x^5)*Log[5] + E^(4/(x^2*Log[5]))*(2*x^4 +
2*x^5)*Log[5] + E^(2/(x^2*Log[5]))*(-8*x + 8*x^2 + (-2*x^3 + 4*x^5)*Log[5])))/(x^3*Log[5]),x]

[Out]

E^(2*x - 4/(x^2*Log[5]))*(-1 + x + E^(2/(x^2*Log[5]))*x)^2

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 60, normalized size = 2.07 \begin {gather*} {\left (x^{2} e^{\left (\frac {4}{x^{2} \log \relax (5)}\right )} + x^{2} + 2 \, {\left (x^{2} - x\right )} e^{\left (\frac {2}{x^{2} \log \relax (5)}\right )} - 2 \, x + 1\right )} e^{\left (\frac {2 \, {\left (x^{3} \log \relax (5) - 2\right )}}{x^{2} \log \relax (5)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+2*x^4)*log(5)*exp(2/x^2/log(5))^2+((4*x^5-2*x^3)*log(5)+8*x^2-8*x)*exp(2/x^2/log(5))+(2*x^5-
2*x^4)*log(5)+8*x^2-16*x+8)*exp(x)^2/x^3/log(5)/exp(2/x^2/log(5))^2,x, algorithm="fricas")

[Out]

(x^2*e^(4/(x^2*log(5))) + x^2 + 2*(x^2 - x)*e^(2/(x^2*log(5))) - 2*x + 1)*e^(2*(x^3*log(5) - 2)/(x^2*log(5)))

________________________________________________________________________________________

giac [B]  time = 0.27, size = 132, normalized size = 4.55 \begin {gather*} \frac {x^{2} e^{\left (2 \, x\right )} \log \relax (5) + 2 \, x^{2} e^{\left (\frac {2 \, {\left (x^{3} \log \relax (5) - 1\right )}}{x^{2} \log \relax (5)}\right )} \log \relax (5) + x^{2} e^{\left (\frac {2 \, {\left (x^{3} \log \relax (5) - 2\right )}}{x^{2} \log \relax (5)}\right )} \log \relax (5) - 2 \, x e^{\left (\frac {2 \, {\left (x^{3} \log \relax (5) - 1\right )}}{x^{2} \log \relax (5)}\right )} \log \relax (5) - 2 \, x e^{\left (\frac {2 \, {\left (x^{3} \log \relax (5) - 2\right )}}{x^{2} \log \relax (5)}\right )} \log \relax (5) + e^{\left (\frac {2 \, {\left (x^{3} \log \relax (5) - 2\right )}}{x^{2} \log \relax (5)}\right )} \log \relax (5)}{\log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+2*x^4)*log(5)*exp(2/x^2/log(5))^2+((4*x^5-2*x^3)*log(5)+8*x^2-8*x)*exp(2/x^2/log(5))+(2*x^5-
2*x^4)*log(5)+8*x^2-16*x+8)*exp(x)^2/x^3/log(5)/exp(2/x^2/log(5))^2,x, algorithm="giac")

[Out]

(x^2*e^(2*x)*log(5) + 2*x^2*e^(2*(x^3*log(5) - 1)/(x^2*log(5)))*log(5) + x^2*e^(2*(x^3*log(5) - 2)/(x^2*log(5)
))*log(5) - 2*x*e^(2*(x^3*log(5) - 1)/(x^2*log(5)))*log(5) - 2*x*e^(2*(x^3*log(5) - 2)/(x^2*log(5)))*log(5) +
e^(2*(x^3*log(5) - 2)/(x^2*log(5)))*log(5))/log(5)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 61, normalized size = 2.10

method	result	size
risch	\({\mathrm e}^{2 x} x^{2}+2 x \left (x -1\right ) {\mathrm e}^{\frac {2 x^{3} \ln \relax (5)-2}{x^{2} \ln \relax (5)}}+\left (x^{2}-2 x +1\right ) {\mathrm e}^{\frac {2 x^{3} \ln \relax (5)-4}{x^{2} \ln \relax (5)}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5+2*x^4)*ln(5)*exp(2/x^2/ln(5))^2+((4*x^5-2*x^3)*ln(5)+8*x^2-8*x)*exp(2/x^2/ln(5))+(2*x^5-2*x^4)*ln(
5)+8*x^2-16*x+8)*exp(x)^2/x^3/ln(5)/exp(2/x^2/ln(5))^2,x,method=_RETURNVERBOSE)

[Out]

exp(2*x)*x^2+2*x*(x-1)*exp(2*(x^3*ln(5)-1)/x^2/ln(5))+(x^2-2*x+1)*exp(2*(x^3*ln(5)-2)/x^2/ln(5))

________________________________________________________________________________________

maxima [B]  time = 0.75, size = 94, normalized size = 3.24 \begin {gather*} \frac {{\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} \log \relax (5) + {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \log \relax (5) + 4 \, {\left (x^{2} \log \relax (5) - x \log \relax (5)\right )} e^{\left (2 \, x - \frac {2}{x^{2} \log \relax (5)}\right )} + 2 \, {\left (x^{2} \log \relax (5) - 2 \, x \log \relax (5) + \log \relax (5)\right )} e^{\left (2 \, x - \frac {4}{x^{2} \log \relax (5)}\right )}}{2 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+2*x^4)*log(5)*exp(2/x^2/log(5))^2+((4*x^5-2*x^3)*log(5)+8*x^2-8*x)*exp(2/x^2/log(5))+(2*x^5-
2*x^4)*log(5)+8*x^2-16*x+8)*exp(x)^2/x^3/log(5)/exp(2/x^2/log(5))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.21, size = 32, normalized size = 1.10 \begin {gather*} {\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-\frac {4}{x^2\,\ln \relax (5)}}\,{\left (x+x\,{\mathrm {e}}^{\frac {2}{x^2\,\ln \relax (5)}}-1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*exp(-4/(x^2*log(5)))*(16*x + log(5)*(2*x^4 - 2*x^5) + exp(2/(x^2*log(5)))*(8*x + log(5)*(2*x^3
- 4*x^5) - 8*x^2) - 8*x^2 - exp(4/(x^2*log(5)))*log(5)*(2*x^4 + 2*x^5) - 8))/(x^3*log(5)),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 2.33, size = 68, normalized size = 2.34 \begin {gather*} x^{2} e^{2 x} + \left (2 x^{2} e^{2 x} - 2 x e^{2 x}\right ) e^{- \frac {2}{x^{2} \log {\relax (5 )}}} + \left (x^{2} e^{2 x} - 2 x e^{2 x} + e^{2 x}\right ) e^{- \frac {4}{x^{2} \log {\relax (5 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5+2*x**4)*ln(5)*exp(2/x**2/ln(5))**2+((4*x**5-2*x**3)*ln(5)+8*x**2-8*x)*exp(2/x**2/ln(5))+(2*
x**5-2*x**4)*ln(5)+8*x**2-16*x+8)*exp(x)**2/x**3/ln(5)/exp(2/x**2/ln(5))**2,x)

[Out]

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

________________________________________________________________________________________