Integrand size = 60, antiderivative size = 20 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=\frac {x^3 \log \left (\frac {3 x}{e}\right )}{\frac {1}{e^5}-2 x} \] Output:
x^3*ln(3*x/exp(1))/(-2*x+exp(-5))
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(20)=40\).
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.65 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=\frac {\left (1-2 e^5 x\right ) \log (x)-\left (1-2 e^5 x+8 e^{15} x^3\right ) (-1+\log (3 x))}{8 e^{10} \left (-1+2 e^5 x\right )} \] Input:
Integrate[(E^5*x^2 - 2*E^10*x^3 + (3*E^5*x^2 - 4*E^10*x^3)*Log[(3*x)/E])/( 1 - 4*E^5*x + 4*E^10*x^2),x]
Output:
((1 - 2*E^5*x)*Log[x] - (1 - 2*E^5*x + 8*E^15*x^3)*(-1 + Log[3*x]))/(8*E^1 0*(-1 + 2*E^5*x))
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(20)=40\).
Time = 0.96 (sec) , antiderivative size = 89, normalized size of antiderivative = 4.45, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.117, Rules used = {7277, 27, 7292, 27, 25, 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.
\(\displaystyle \int \frac {-2 e^{10} x^3+e^5 x^2+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{4 e^{10} x^2-4 e^5 x+1} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 16 e^{10} \int \frac {-2 e^{10} x^3+e^5 x^2+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{16 e^{10} \left (1-2 e^5 x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {-2 e^{10} x^3+e^5 x^2+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{\left (1-2 e^5 x\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^5 x^2 \left (2 e^5 x-4 e^5 x \log (3 x)+3 \log (3 x)-2\right )}{\left (1-2 e^5 x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^5 \int -\frac {x^2 \left (4 e^5 \log (3 x) x-2 e^5 x-3 \log (3 x)+2\right )}{\left (1-2 e^5 x\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -e^5 \int \frac {x^2 \left (4 e^5 \log (3 x) x-2 e^5 x-3 \log (3 x)+2\right )}{\left (1-2 e^5 x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -e^5 \int \left (\frac {x^2 \left (4 e^5 x-3\right ) \log (3 x)}{\left (2 e^5 x-1\right )^2}-\frac {2 x^2 \left (e^5 x-1\right )}{\left (2 e^5 x-1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^5 \left (-\frac {x^2}{2 e^5}+\frac {x^2 \log (3 x)}{2 e^5}-\frac {x}{4 e^{10}}+\frac {1}{8 e^{15} \left (1-2 e^5 x\right )}-\frac {x \log (3 x)}{4 e^{10} \left (1-2 e^5 x\right )}+\frac {x \log (3 x)}{4 e^{10}}\right )\) |
Input:
Int[(E^5*x^2 - 2*E^10*x^3 + (3*E^5*x^2 - 4*E^10*x^3)*Log[(3*x)/E])/(1 - 4* E^5*x + 4*E^10*x^2),x]
Output:
-(E^5*(-1/4*x/E^10 - x^2/(2*E^5) + 1/(8*E^15*(1 - 2*E^5*x)) + (x*Log[3*x]) /(4*E^10) + (x^2*Log[3*x])/(2*E^5) - (x*Log[3*x])/(4*E^10*(1 - 2*E^5*x))))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25
method | result | size |
norman | \(-\frac {x^{3} {\mathrm e}^{5} \ln \left (3 x \,{\mathrm e}^{-1}\right )}{2 x \,{\mathrm e}^{5}-1}\) | \(25\) |
parallelrisch | \(-\frac {x^{3} {\mathrm e}^{5} \ln \left (3 x \,{\mathrm e}^{-1}\right )}{2 x \,{\mathrm e}^{5}-1}\) | \(25\) |
risch | \(-\frac {\left (8 \,{\mathrm e}^{15} x^{3}-2 x \,{\mathrm e}^{5}+1\right ) {\mathrm e}^{-10} \ln \left (3 x \,{\mathrm e}^{-1}\right )}{8 \left (2 x \,{\mathrm e}^{5}-1\right )}-\frac {{\mathrm e}^{-10} \ln \left (x \right )}{8}\) | \(41\) |
parts | \(-\frac {x^{2}}{4}-\frac {x}{4 \,{\mathrm e}^{5}}-\frac {{\mathrm e}^{-10} \ln \left (2 x \,{\mathrm e}^{5}-1\right )}{8}-\frac {{\mathrm e}^{5} \left ({\mathrm e}\right )^{3} \left (\frac {\left (\frac {9 x^{2} \ln \left (\frac {3 x}{{\mathrm e}}\right )}{2 \left ({\mathrm e}\right )^{2}}-\frac {9 x^{2}}{4 \left ({\mathrm e}\right )^{2}}\right ) {\mathrm e}^{6}}{{\mathrm e}^{12}}+\frac {3 \left (\frac {3 x \ln \left (\frac {3 x}{{\mathrm e}}\right )}{{\mathrm e}}-\frac {3 x}{{\mathrm e}}\right ) \left (4 \left ({\mathrm e}^{6}\right )^{2}-3 \,{\mathrm e}^{12}\right )}{4 \left ({\mathrm e}^{12}\right )^{2}}+\frac {27 \left (-\frac {\ln \left (\frac {3 x}{{\mathrm e}}\right ) \left (\ln \left (\frac {-\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}+3 \,{\mathrm e}^{6}}{3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )-\ln \left (\frac {\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}-3 \,{\mathrm e}^{6}}{-3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )\right )}{12 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}-\frac {\operatorname {dilog}\left (\frac {-\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}+3 \,{\mathrm e}^{6}}{3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )-\operatorname {dilog}\left (\frac {\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}-3 \,{\mathrm e}^{6}}{-3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )}{12 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right ) {\mathrm e}^{-12}}{4}\right )}{9}\) | \(342\) |
derivativedivides | \(\frac {{\mathrm e} \left (-\left ({\mathrm e}\right )^{2} {\mathrm e}^{5} \left (\frac {\frac {9 x^{2} {\mathrm e}^{6}}{\left ({\mathrm e}\right )^{2}}+\frac {9 x}{{\mathrm e}}}{4 \left ({\mathrm e}^{6}\right )^{2}}+\frac {9 \,{\mathrm e}^{-12} \ln \left (\frac {6 x \,{\mathrm e}^{6}}{{\mathrm e}}-3\right )}{8 \,{\mathrm e}^{6}}\right )-\left ({\mathrm e}\right )^{2} {\mathrm e}^{5} \left (\frac {\left (\frac {9 x^{2} \ln \left (\frac {3 x}{{\mathrm e}}\right )}{2 \left ({\mathrm e}\right )^{2}}-\frac {9 x^{2}}{4 \left ({\mathrm e}\right )^{2}}\right ) {\mathrm e}^{6}}{{\mathrm e}^{12}}+\frac {3 \left (\frac {3 x \ln \left (\frac {3 x}{{\mathrm e}}\right )}{{\mathrm e}}-\frac {3 x}{{\mathrm e}}\right ) \left (4 \left ({\mathrm e}^{6}\right )^{2}-3 \,{\mathrm e}^{12}\right )}{4 \left ({\mathrm e}^{12}\right )^{2}}+\frac {27 \left (-\frac {\ln \left (\frac {3 x}{{\mathrm e}}\right ) \left (\ln \left (\frac {-\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}+3 \,{\mathrm e}^{6}}{3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )-\ln \left (\frac {\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}-3 \,{\mathrm e}^{6}}{-3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )\right )}{12 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}-\frac {\operatorname {dilog}\left (\frac {-\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}+3 \,{\mathrm e}^{6}}{3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )-\operatorname {dilog}\left (\frac {\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}-3 \,{\mathrm e}^{6}}{-3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )}{12 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right ) {\mathrm e}^{-12}}{4}\right )\right )}{9}\) | \(376\) |
default | \(\frac {{\mathrm e} \left (-\left ({\mathrm e}\right )^{2} {\mathrm e}^{5} \left (\frac {\frac {9 x^{2} {\mathrm e}^{6}}{\left ({\mathrm e}\right )^{2}}+\frac {9 x}{{\mathrm e}}}{4 \left ({\mathrm e}^{6}\right )^{2}}+\frac {9 \,{\mathrm e}^{-12} \ln \left (\frac {6 x \,{\mathrm e}^{6}}{{\mathrm e}}-3\right )}{8 \,{\mathrm e}^{6}}\right )-\left ({\mathrm e}\right )^{2} {\mathrm e}^{5} \left (\frac {\left (\frac {9 x^{2} \ln \left (\frac {3 x}{{\mathrm e}}\right )}{2 \left ({\mathrm e}\right )^{2}}-\frac {9 x^{2}}{4 \left ({\mathrm e}\right )^{2}}\right ) {\mathrm e}^{6}}{{\mathrm e}^{12}}+\frac {3 \left (\frac {3 x \ln \left (\frac {3 x}{{\mathrm e}}\right )}{{\mathrm e}}-\frac {3 x}{{\mathrm e}}\right ) \left (4 \left ({\mathrm e}^{6}\right )^{2}-3 \,{\mathrm e}^{12}\right )}{4 \left ({\mathrm e}^{12}\right )^{2}}+\frac {27 \left (-\frac {\ln \left (\frac {3 x}{{\mathrm e}}\right ) \left (\ln \left (\frac {-\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}+3 \,{\mathrm e}^{6}}{3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )-\ln \left (\frac {\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}-3 \,{\mathrm e}^{6}}{-3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )\right )}{12 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}-\frac {\operatorname {dilog}\left (\frac {-\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}+3 \,{\mathrm e}^{6}}{3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )-\operatorname {dilog}\left (\frac {\frac {6 x \,{\mathrm e}^{12}}{{\mathrm e}}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}-3 \,{\mathrm e}^{6}}{-3 \,{\mathrm e}^{6}+3 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right )}{12 \sqrt {\left ({\mathrm e}^{6}\right )^{2}-{\mathrm e}^{12}}}\right ) {\mathrm e}^{-12}}{4}\right )\right )}{9}\) | \(376\) |
Input:
int(((-4*x^3*exp(5)^2+3*x^2*exp(5))*ln(3*x/exp(1))-2*x^3*exp(5)^2+x^2*exp( 5))/(4*x^2*exp(5)^2-4*x*exp(5)+1),x,method=_RETURNVERBOSE)
Output:
-x^3*exp(5)*ln(3*x/exp(1))/(2*x*exp(5)-1)
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=-\frac {x^{3} e^{5} \log \left (3 \, x e^{\left (-1\right )}\right )}{2 \, x e^{5} - 1} \] Input:
integrate(((-4*x^3*exp(5)^2+3*x^2*exp(5))*log(3*x/exp(1))-2*x^3*exp(5)^2+x ^2*exp(5))/(4*x^2*exp(5)^2-4*x*exp(5)+1),x, algorithm="fricas")
Output:
-x^3*e^5*log(3*x*e^(-1))/(2*x*e^5 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=- \frac {\log {\left (x \right )}}{8 e^{10}} + \frac {\left (- 8 x^{3} e^{15} + 2 x e^{5} - 1\right ) \log {\left (\frac {3 x}{e} \right )}}{16 x e^{15} - 8 e^{10}} \] Input:
integrate(((-4*x**3*exp(5)**2+3*x**2*exp(5))*ln(3*x/exp(1))-2*x**3*exp(5)* *2+x**2*exp(5))/(4*x**2*exp(5)**2-4*x*exp(5)+1),x)
Output:
-exp(-10)*log(x)/8 + (-8*x**3*exp(15) + 2*x*exp(5) - 1)*log(3*x*exp(-1))/( 16*x*exp(15) - 8*exp(10))
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (21) = 42\).
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 7.50 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=-\frac {1}{8} \, {\left (2 \, {\left (x^{2} e^{5} + 2 \, x\right )} e^{\left (-15\right )} + 3 \, e^{\left (-20\right )} \log \left (2 \, x e^{5} - 1\right ) - \frac {1}{2 \, x e^{25} - e^{20}}\right )} e^{10} + \frac {1}{8} \, {\left (2 \, x e^{\left (-10\right )} + 2 \, e^{\left (-15\right )} \log \left (2 \, x e^{5} - 1\right ) - \frac {1}{2 \, x e^{20} - e^{15}}\right )} e^{5} + \frac {1}{8} \, e^{\left (-10\right )} \log \left (2 \, x e^{5} - 1\right ) - \frac {4 \, x^{3} {\left (2 \, \log \left (3\right ) - 3\right )} e^{15} + 8 \, x^{3} e^{15} \log \left (x\right ) - 2 \, x^{2} e^{10} - 2 \, x {\left (\log \left (3\right ) - 2\right )} e^{5} + \log \left (3\right ) - 1}{8 \, {\left (2 \, x e^{15} - e^{10}\right )}} \] Input:
integrate(((-4*x^3*exp(5)^2+3*x^2*exp(5))*log(3*x/exp(1))-2*x^3*exp(5)^2+x ^2*exp(5))/(4*x^2*exp(5)^2-4*x*exp(5)+1),x, algorithm="maxima")
Output:
-1/8*(2*(x^2*e^5 + 2*x)*e^(-15) + 3*e^(-20)*log(2*x*e^5 - 1) - 1/(2*x*e^25 - e^20))*e^10 + 1/8*(2*x*e^(-10) + 2*e^(-15)*log(2*x*e^5 - 1) - 1/(2*x*e^ 20 - e^15))*e^5 + 1/8*e^(-10)*log(2*x*e^5 - 1) - 1/8*(4*x^3*(2*log(3) - 3) *e^15 + 8*x^3*e^15*log(x) - 2*x^2*e^10 - 2*x*(log(3) - 2)*e^5 + log(3) - 1 )/(2*x*e^15 - e^10)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.15 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=-\frac {8 \, x^{3} e^{15} \log \left (3 \, x\right ) - 8 \, x^{3} e^{15} - 2 \, x e^{5} \log \left (3 \, x\right ) + 2 \, x e^{5} \log \left (x\right ) + 2 \, x e^{5} + \log \left (3 \, x\right ) - \log \left (x\right ) - 1}{8 \, {\left (2 \, x e^{15} - e^{10}\right )}} \] Input:
integrate(((-4*x^3*exp(5)^2+3*x^2*exp(5))*log(3*x/exp(1))-2*x^3*exp(5)^2+x ^2*exp(5))/(4*x^2*exp(5)^2-4*x*exp(5)+1),x, algorithm="giac")
Output:
-1/8*(8*x^3*e^15*log(3*x) - 8*x^3*e^15 - 2*x*e^5*log(3*x) + 2*x*e^5*log(x) + 2*x*e^5 + log(3*x) - log(x) - 1)/(2*x*e^15 - e^10)
Time = 1.49 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=-\frac {x^3\,{\mathrm {e}}^5\,\left (\ln \left (3\,x\right )-1\right )}{2\,x\,{\mathrm {e}}^5-1} \] Input:
int((x^2*exp(5) - 2*x^3*exp(10) + log(3*x*exp(-1))*(3*x^2*exp(5) - 4*x^3*e xp(10)))/(4*x^2*exp(10) - 4*x*exp(5) + 1),x)
Output:
-(x^3*exp(5)*(log(3*x) - 1))/(2*x*exp(5) - 1)
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {e^5 x^2-2 e^{10} x^3+\left (3 e^5 x^2-4 e^{10} x^3\right ) \log \left (\frac {3 x}{e}\right )}{1-4 e^5 x+4 e^{10} x^2} \, dx=-\frac {\mathrm {log}\left (\frac {3 x}{e}\right ) e^{5} x^{3}}{2 e^{5} x -1} \] Input:
int(((-4*x^3*exp(5)^2+3*x^2*exp(5))*log(3*x/exp(1))-2*x^3*exp(5)^2+x^2*exp (5))/(4*x^2*exp(5)^2-4*x*exp(5)+1),x)
Output:
( - log((3*x)/e)*e**5*x**3)/(2*e**5*x - 1)