Integrand size = 69, antiderivative size = 22 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=\frac {16 e^{16} \left (1+\frac {x}{5}\right )^4}{\log \left (\frac {x}{25}\right )} \]
Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=\frac {16 e^{16} (5+x)^4}{625 \log \left (\frac {x}{25}\right )} \]
Integrate[(E^16*(-10000 - 8000*x - 2400*x^2 - 320*x^3 - 16*x^4) + E^16*(80 00*x + 4800*x^2 + 960*x^3 + 64*x^4)*Log[x/25])/(625*x*Log[x/25]^2),x]
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(22)=44\).
Time = 0.75 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {27, 27, 7239, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{16} \left (-16 x^4-320 x^3-2400 x^2-8000 x-10000\right )+e^{16} \left (64 x^4+960 x^3+4800 x^2+8000 x\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{625} \int -\frac {16 \left (e^{16} \left (x^4+20 x^3+150 x^2+500 x+625\right )-4 e^{16} \left (x^4+15 x^3+75 x^2+125 x\right ) \log \left (\frac {x}{25}\right )\right )}{x \log ^2\left (\frac {x}{25}\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {16}{625} \int \frac {e^{16} \left (x^4+20 x^3+150 x^2+500 x+625\right )-4 e^{16} \left (x^4+15 x^3+75 x^2+125 x\right ) \log \left (\frac {x}{25}\right )}{x \log ^2\left (\frac {x}{25}\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -\frac {16}{625} \int \frac {e^{16} (x+5)^3 \left (-4 \log \left (\frac {x}{25}\right ) x+x+5\right )}{x \log ^2\left (\frac {x}{25}\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {16}{625} e^{16} \int \frac {(x+5)^3 \left (-4 \log \left (\frac {x}{25}\right ) x+x+5\right )}{x \log ^2\left (\frac {x}{25}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {16}{625} e^{16} \int \left (\frac {(x+5)^4}{x \log ^2\left (\frac {x}{25}\right )}-\frac {4 (x+5)^3}{\log \left (\frac {x}{25}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {16}{625} e^{16} \left (-\frac {x^4}{\log \left (\frac {x}{25}\right )}-\frac {20 x^3}{\log \left (\frac {x}{25}\right )}-\frac {150 x^2}{\log \left (\frac {x}{25}\right )}-\frac {500 x}{\log \left (\frac {x}{25}\right )}-\frac {625}{\log \left (\frac {x}{25}\right )}\right )\) |
Int[(E^16*(-10000 - 8000*x - 2400*x^2 - 320*x^3 - 16*x^4) + E^16*(8000*x + 4800*x^2 + 960*x^3 + 64*x^4)*Log[x/25])/(625*x*Log[x/25]^2),x]
(-16*E^16*(-625/Log[x/25] - (500*x)/Log[x/25] - (150*x^2)/Log[x/25] - (20* x^3)/Log[x/25] - x^4/Log[x/25]))/625
3.27.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32
method | result | size |
risch | \(\frac {16 \,{\mathrm e}^{16} \left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right )}{625 \ln \left (\frac {x}{25}\right )}\) | \(29\) |
norman | \(\frac {16 \,{\mathrm e}^{16}+\frac {64 x \,{\mathrm e}^{16}}{5}+\frac {96 x^{2} {\mathrm e}^{16}}{25}+\frac {64 x^{3} {\mathrm e}^{16}}{125}+\frac {16 \,{\mathrm e}^{16} x^{4}}{625}}{\ln \left (\frac {x}{25}\right )}\) | \(49\) |
parallelrisch | \(\frac {16 \,{\mathrm e}^{16} x^{4}+320 x^{3} {\mathrm e}^{16}+2400 x^{2} {\mathrm e}^{16}+8000 x \,{\mathrm e}^{16}+10000 \,{\mathrm e}^{16}}{625 \ln \left (\frac {x}{25}\right )}\) | \(50\) |
parts | \(\frac {64 \,{\mathrm e}^{16} \left (-390625 \,\operatorname {Ei}_{1}\left (-4 \ln \left (\frac {x}{25}\right )\right )-234375 \,\operatorname {Ei}_{1}\left (-3 \ln \left (\frac {x}{25}\right )\right )-46875 \,\operatorname {Ei}_{1}\left (-2 \ln \left (\frac {x}{25}\right )\right )-3125 \,\operatorname {Ei}_{1}\left (-\ln \left (\frac {x}{25}\right )\right )\right )}{625}-\frac {16 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{\ln \left (\frac {x}{25}\right )}-1562500 \,\operatorname {Ei}_{1}\left (-4 \ln \left (\frac {x}{25}\right )\right )-\frac {20 x^{3}}{\ln \left (\frac {x}{25}\right )}-937500 \,\operatorname {Ei}_{1}\left (-3 \ln \left (\frac {x}{25}\right )\right )-\frac {150 x^{2}}{\ln \left (\frac {x}{25}\right )}-187500 \,\operatorname {Ei}_{1}\left (-2 \ln \left (\frac {x}{25}\right )\right )-\frac {500 x}{\ln \left (\frac {x}{25}\right )}-12500 \,\operatorname {Ei}_{1}\left (-\ln \left (\frac {x}{25}\right )\right )-\frac {625}{\ln \left (\frac {x}{25}\right )}\right )}{625}\) | \(146\) |
derivativedivides | \(-40000 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-4 \ln \left (\frac {x}{25}\right )\right )-24000 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-3 \ln \left (\frac {x}{25}\right )\right )-10000 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{390625 \ln \left (\frac {x}{25}\right )}-4 \,\operatorname {Ei}_{1}\left (-4 \ln \left (\frac {x}{25}\right )\right )\right )-4800 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-2 \ln \left (\frac {x}{25}\right )\right )-8000 \,{\mathrm e}^{16} \left (-\frac {x^{3}}{15625 \ln \left (\frac {x}{25}\right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-\ln \left (\frac {x}{25}\right )\right )-2400 \,{\mathrm e}^{16} \left (-\frac {x^{2}}{625 \ln \left (\frac {x}{25}\right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \left (-\frac {x}{25 \ln \left (\frac {x}{25}\right )}-\operatorname {Ei}_{1}\left (-\ln \left (\frac {x}{25}\right )\right )\right )+\frac {16 \,{\mathrm e}^{16}}{\ln \left (\frac {x}{25}\right )}\) | \(180\) |
default | \(-40000 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-4 \ln \left (\frac {x}{25}\right )\right )-24000 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-3 \ln \left (\frac {x}{25}\right )\right )-10000 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{390625 \ln \left (\frac {x}{25}\right )}-4 \,\operatorname {Ei}_{1}\left (-4 \ln \left (\frac {x}{25}\right )\right )\right )-4800 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-2 \ln \left (\frac {x}{25}\right )\right )-8000 \,{\mathrm e}^{16} \left (-\frac {x^{3}}{15625 \ln \left (\frac {x}{25}\right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \operatorname {Ei}_{1}\left (-\ln \left (\frac {x}{25}\right )\right )-2400 \,{\mathrm e}^{16} \left (-\frac {x^{2}}{625 \ln \left (\frac {x}{25}\right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \left (-\frac {x}{25 \ln \left (\frac {x}{25}\right )}-\operatorname {Ei}_{1}\left (-\ln \left (\frac {x}{25}\right )\right )\right )+\frac {16 \,{\mathrm e}^{16}}{\ln \left (\frac {x}{25}\right )}\) | \(180\) |
int(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*ln(1/25*x)+(-16*x^4-3 20*x^3-2400*x^2-8000*x-10000)*exp(4)^4)/x/ln(1/25*x)^2,x,method=_RETURNVER BOSE)
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=\frac {16 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} e^{16}}{625 \, \log \left (\frac {1}{25} \, x\right )} \]
integrate(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*log(1/25*x)+(-1 6*x^4-320*x^3-2400*x^2-8000*x-10000)*exp(4)^4)/x/log(1/25*x)^2,x, algorith m=\
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=\frac {16 x^{4} e^{16} + 320 x^{3} e^{16} + 2400 x^{2} e^{16} + 8000 x e^{16} + 10000 e^{16}}{625 \log {\left (\frac {x}{25} \right )}} \]
integrate(1/625*((64*x**4+960*x**3+4800*x**2+8000*x)*exp(4)**4*ln(1/25*x)+ (-16*x**4-320*x**3-2400*x**2-8000*x-10000)*exp(4)**4)/x/ln(1/25*x)**2,x)
(16*x**4*exp(16) + 320*x**3*exp(16) + 2400*x**2*exp(16) + 8000*x*exp(16) + 10000*exp(16))/(625*log(x/25))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.59 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=40000 \, {\rm Ei}\left (4 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 24000 \, {\rm Ei}\left (3 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 4800 \, {\rm Ei}\left (2 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 320 \, {\rm Ei}\left (\log \left (\frac {1}{25} \, x\right )\right ) e^{16} - 320 \, e^{16} \Gamma \left (-1, -\log \left (\frac {1}{25} \, x\right )\right ) - 4800 \, e^{16} \Gamma \left (-1, -2 \, \log \left (\frac {1}{25} \, x\right )\right ) - 24000 \, e^{16} \Gamma \left (-1, -3 \, \log \left (\frac {1}{25} \, x\right )\right ) - 40000 \, e^{16} \Gamma \left (-1, -4 \, \log \left (\frac {1}{25} \, x\right )\right ) + \frac {16 \, e^{16}}{\log \left (\frac {1}{25} \, x\right )} \]
integrate(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*log(1/25*x)+(-1 6*x^4-320*x^3-2400*x^2-8000*x-10000)*exp(4)^4)/x/log(1/25*x)^2,x, algorith m=\
40000*Ei(4*log(1/25*x))*e^16 + 24000*Ei(3*log(1/25*x))*e^16 + 4800*Ei(2*lo g(1/25*x))*e^16 + 320*Ei(log(1/25*x))*e^16 - 320*e^16*gamma(-1, -log(1/25* x)) - 4800*e^16*gamma(-1, -2*log(1/25*x)) - 24000*e^16*gamma(-1, -3*log(1/ 25*x)) - 40000*e^16*gamma(-1, -4*log(1/25*x)) + 16*e^16/log(1/25*x)
Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=\frac {16 \, {\left (x^{4} e^{16} + 20 \, x^{3} e^{16} + 150 \, x^{2} e^{16} + 500 \, x e^{16} + 625 \, e^{16}\right )}}{625 \, \log \left (\frac {1}{25} \, x\right )} \]
integrate(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*log(1/25*x)+(-1 6*x^4-320*x^3-2400*x^2-8000*x-10000)*exp(4)^4)/x/log(1/25*x)^2,x, algorith m=\
Time = 13.61 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{625 x \log ^2\left (\frac {x}{25}\right )} \, dx=\frac {16\,{\mathrm {e}}^{16}\,x^6+320\,{\mathrm {e}}^{16}\,x^5+2400\,{\mathrm {e}}^{16}\,x^4+8000\,{\mathrm {e}}^{16}\,x^3+10000\,{\mathrm {e}}^{16}\,x^2}{625\,x^2\,\ln \left (\frac {x}{25}\right )} \]
int(-((exp(16)*(8000*x + 2400*x^2 + 320*x^3 + 16*x^4 + 10000))/625 - (log( x/25)*exp(16)*(8000*x + 4800*x^2 + 960*x^3 + 64*x^4))/625)/(x*log(x/25)^2) ,x)