\(\int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{(48-24 x+3 x^2) \log ^3(5 x)} \, dx\) [6629]

Optimal result

Integrand size = 68, antiderivative size = 22 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=5 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} \]

[Out]

exp(ln(5)+1/3*x/(x-4)/ln(5*x)^2-2)

Rubi [F]

\[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=\int \frac {\exp \left (\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}\right ) (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx \]

[In]

Int[(E^((x + (24 - 6*x + (-12 + 3*x)*Log[5])*Log[5*x]^2)/((-12 + 3*x)*Log[5*x]^2))*(8 - 2*x - 4*Log[5*x]))/((4
8 - 24*x + 3*x^2)*Log[5*x]^3),x]

[Out]

(-10*Defer[Int][E^(-2 + x/(3*(-4 + x)*Log[5*x]^2))/((-4 + x)*Log[5*x]^3), x])/3 - (20*Defer[Int][E^(-2 + x/(3*
(-4 + x)*Log[5*x]^2))/((-4 + x)^2*Log[5*x]^2), x])/3

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}\right ) (8-2 x-4 \log (5 x))}{3 (-4+x)^2 \log ^3(5 x)} \, dx \\ & = \frac {1}{3} \int \frac {\exp \left (\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}\right ) (8-2 x-4 \log (5 x))}{(-4+x)^2 \log ^3(5 x)} \, dx \\ & = \frac {1}{3} \int \frac {10 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} (4-x-2 \log (5 x))}{(4-x)^2 \log ^3(5 x)} \, dx \\ & = \frac {10}{3} \int \frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} (4-x-2 \log (5 x))}{(4-x)^2 \log ^3(5 x)} \, dx \\ & = \frac {10}{3} \int \left (-\frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x) \log ^3(5 x)}-\frac {2 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x)^2 \log ^2(5 x)}\right ) \, dx \\ & = -\left (\frac {10}{3} \int \frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x) \log ^3(5 x)} \, dx\right )-\frac {20}{3} \int \frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x)^2 \log ^2(5 x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=5 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} \]

[In]

Integrate[(E^((x + (24 - 6*x + (-12 + 3*x)*Log[5])*Log[5*x]^2)/((-12 + 3*x)*Log[5*x]^2))*(8 - 2*x - 4*Log[5*x]
))/((48 - 24*x + 3*x^2)*Log[5*x]^3),x]

[Out]

5*E^(-2 + x/(3*(-4 + x)*Log[5*x]^2))

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (\left (3 x -12\right ) \ln \left (5\right )-6 x +24\right ) \ln \left (5 x \right )^{2}+x}{\left (3 x -12\right ) \ln \left (5 x \right )^{2}}}\)	\(38\)
risch	\(5^{\frac {x}{x -4}} \left (\frac {1}{625}\right )^{\frac {1}{x -4}} {\mathrm e}^{-\frac {6 x \ln \left (5 x \right )^{2}-24 \ln \left (5 x \right )^{2}-x}{3 \left (x -4\right ) \ln \left (5 x \right )^{2}}}\)	\(53\)

[In]

int((-4*ln(5*x)-2*x+8)*exp((((3*x-12)*ln(5)-6*x+24)*ln(5*x)^2+x)/(3*x-12)/ln(5*x)^2)/(3*x^2-24*x+48)/ln(5*x)^3
,x,method=_RETURNVERBOSE)

[Out]

exp((((3*x-12)*ln(5)-6*x+24)*ln(5*x)^2+x)/(3*x-12)/ln(5*x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=e^{\left (\frac {3 \, {\left ({\left (x - 4\right )} \log \left (5\right ) - 2 \, x + 8\right )} \log \left (5 \, x\right )^{2} + x}{3 \, {\left (x - 4\right )} \log \left (5 \, x\right )^{2}}\right )} \]

[In]

integrate((-4*log(5*x)-2*x+8)*exp((((3*x-12)*log(5)-6*x+24)*log(5*x)^2+x)/(3*x-12)/log(5*x)^2)/(3*x^2-24*x+48)
/log(5*x)^3,x, algorithm="fricas")

[Out]

e^(1/3*(3*((x - 4)*log(5) - 2*x + 8)*log(5*x)^2 + x)/((x - 4)*log(5*x)^2))

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=e^{\frac {x + \left (- 6 x + \left (3 x - 12\right ) \log {\left (5 \right )} + 24\right ) \log {\left (5 x \right )}^{2}}{\left (3 x - 12\right ) \log {\left (5 x \right )}^{2}}} \]

[In]

integrate((-4*ln(5*x)-2*x+8)*exp((((3*x-12)*ln(5)-6*x+24)*ln(5*x)**2+x)/(3*x-12)/ln(5*x)**2)/(3*x**2-24*x+48)/
ln(5*x)**3,x)

[Out]

exp((x + (-6*x + (3*x - 12)*log(5) + 24)*log(5*x)**2)/((3*x - 12)*log(5*x)**2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).

Time = 0.46 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=5 \, e^{\left (\frac {4}{3 \, {\left (x \log \left (5\right )^{2} + {\left (x - 4\right )} \log \left (x\right )^{2} - 4 \, \log \left (5\right )^{2} + 2 \, {\left (x \log \left (5\right ) - 4 \, \log \left (5\right )\right )} \log \left (x\right )\right )}} + \frac {1}{3 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )}} - 2\right )} \]

[In]

integrate((-4*log(5*x)-2*x+8)*exp((((3*x-12)*log(5)-6*x+24)*log(5*x)^2+x)/(3*x-12)/log(5*x)^2)/(3*x^2-24*x+48)
/log(5*x)^3,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (19) = 38\).

Time = 1.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.23 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=e^{\left (\frac {x \log \left (5\right ) \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} - \frac {2 \, x \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} - \frac {4 \, \log \left (5\right ) \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} + \frac {8 \, \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} + \frac {x}{3 \, {\left (x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}\right )}}\right )} \]

[In]

integrate((-4*log(5*x)-2*x+8)*exp((((3*x-12)*log(5)-6*x+24)*log(5*x)^2+x)/(3*x-12)/log(5*x)^2)/(3*x^2-24*x+48)
/log(5*x)^3,x, algorithm="giac")

[Out]

e^(x*log(5)*log(5*x)^2/(x*log(5*x)^2 - 4*log(5*x)^2) - 2*x*log(5*x)^2/(x*log(5*x)^2 - 4*log(5*x)^2) - 4*log(5)
*log(5*x)^2/(x*log(5*x)^2 - 4*log(5*x)^2) + 8*log(5*x)^2/(x*log(5*x)^2 - 4*log(5*x)^2) + 1/3*x/(x*log(5*x)^2 -
4*log(5*x)^2))

Mupad [B] (verification not implemented)

Time = 12.01 (sec) , antiderivative size = 401, normalized size of antiderivative = 18.23 \[ \int \frac {e^{\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}} (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 x)} \, dx=\frac {5^{\frac {{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+2\,\ln \left (5\right )\,\ln \left (x\right )+{\ln \left (5\right )}^2}}\,{\mathrm {e}}^{\frac {3\,x\,{\ln \left (5\right )}^3}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {6\,x\,{\ln \left (5\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {24\,{\ln \left (x\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {x}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {6\,x\,{\ln \left (x\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {12\,{\ln \left (5\right )}^3}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {24\,{\ln \left (5\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}}{x^{\frac {2\,\left (2\,\ln \left (5\right )-{\ln \left (5\right )}^2\right )}{{\ln \left (x\right )}^2+2\,\ln \left (5\right )\,\ln \left (x\right )+{\ln \left (5\right )}^2}}} \]

[In]

int(-(exp((x + log(5*x)^2*(log(5)*(3*x - 12) - 6*x + 24))/(log(5*x)^2*(3*x - 12)))*(2*x + 4*log(5*x) - 8))/(lo
g(5*x)^3*(3*x^2 - 24*x + 48)),x)

[Out]

(5^(log(x)^2/(log(x)^2 + 2*log(5)*log(x) + log(5)^2))*exp((3*x*log(5)^3)/(3*x*log(x)^2 - 12*log(x)^2 + 3*x*log
(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x)))*exp(-(6*x*log(5)^2)/(3*x*log(x)^2 - 12*log(x)^2 +
3*x*log(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x)))*exp((24*log(x)^2)/(3*x*log(x)^2 - 12*log(
x)^2 + 3*x*log(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x)))*exp(x/(3*x*log(x)^2 - 12*log(x)^2 +
3*x*log(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x)))*exp(-(6*x*log(x)^2)/(3*x*log(x)^2 - 12*lo
g(x)^2 + 3*x*log(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x)))*exp(-(12*log(5)^3)/(3*x*log(x)^2
- 12*log(x)^2 + 3*x*log(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x)))*exp((24*log(5)^2)/(3*x*log
(x)^2 - 12*log(x)^2 + 3*x*log(5)^2 - 24*log(5)*log(x) - 12*log(5)^2 + 6*x*log(5)*log(x))))/x^((2*(2*log(5) - l
og(5)^2))/(log(x)^2 + 2*log(5)*log(x) + log(5)^2))