\(\int \frac {(15 e^5-15 x) \log ^2(x)+((30 e^5-30 x) \log (x)+(-15 e^5+45 x) \log ^2(x)) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx\) [6720]

Optimal result

Integrand size = 79, antiderivative size = 24 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=-5+\frac {15 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2 x} \]

[Out]

15/x*ln(x)^2*ln(2*x)/(exp(5)-x)^2-5

Rubi [F]

\[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx \]

[In]

Int[((15*E^5 - 15*x)*Log[x]^2 + ((30*E^5 - 30*x)*Log[x] + (-15*E^5 + 45*x)*Log[x]^2)*Log[2*x])/(E^15*x^2 - 3*E
^10*x^3 + 3*E^5*x^4 - x^5),x]

[Out]

30/(E^10*x) + (210*Log[E^5 - x])/E^15 + (15*(6 + Log[2])*Log[E^5 - x])/E^15 + (30*Log[x])/(E^10*x) + (15*Log[1
- E^5/x]*Log[x])/E^15 + (15*x*Log[x]^2)/(E^15*(E^5 - x)) - (30*Log[1 - E^5/x]*Log[x]^2)/E^15 - (10*Log[x]^3)/
E^15 - (30*(1 + Log[x]))/(E^10*x) + (15*Log[1 - E^5/x]*Log[2*x])/E^15 + (15*Log[x]*Log[2*x])/(E^5*(E^5 - x)^2)
- (15*x^2*Log[x]*Log[2*x])/(E^15*(E^5 - x)^2) + (30*Log[x]^2*Log[2*x])/E^15 + (15*Log[x]^2*Log[2*x])/(E^10*x)
- (30*PolyLog[2, E^5/x])/E^15 + (60*Log[x]*PolyLog[2, E^5/x])/E^15 - (60*PolyLog[2, 1 - x/E^5])/E^15 + (60*Po
lyLog[3, E^5/x])/E^15 + (60*Defer[Int][(Log[x]*Log[2*x])/(E^5 - x), x])/E^15 + (30*Defer[Int][(Log[x]^2*Log[2*
x])/(E^5 - x)^3, x])/E^5 + (15*Defer[Int][(Log[x]^2*Log[2*x])/(E^5 - x)^2, x])/E^10

Rubi steps \begin{align*} \text {integral}& = \int \frac {15 \log (x) \left (-2 \left (-e^5+x\right ) \log (2 x)-\log (x) \left (-e^5+x+\left (e^5-3 x\right ) \log (2 x)\right )\right )}{\left (e^5-x\right )^3 x^2} \, dx \\ & = 15 \int \frac {\log (x) \left (-2 \left (-e^5+x\right ) \log (2 x)-\log (x) \left (-e^5+x+\left (e^5-3 x\right ) \log (2 x)\right )\right )}{\left (e^5-x\right )^3 x^2} \, dx \\ & = 15 \int \left (\frac {\log ^2(x)}{\left (e^5-x\right )^2 x^2}-\frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^3 x^2}\right ) \, dx \\ & = 15 \int \frac {\log ^2(x)}{\left (e^5-x\right )^2 x^2} \, dx-15 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^3 x^2} \, dx \\ & = 15 \int \left (\frac {\log ^2(x)}{e^{10} \left (e^5-x\right )^2}+\frac {\log ^2(x)}{e^{10} x^2}+\frac {2 \log ^2(x)}{e^{10} \left (e^5-x\right ) x}\right ) \, dx-15 \int \left (\frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{10} \left (e^5-x\right )^3}+\frac {2 \log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {3 \log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{20} \left (e^5-x\right )}+\frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{15} x^2}+\frac {3 \log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^{20} x}\right ) \, dx \\ & = -\frac {45 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{e^5-x} \, dx}{e^{20}}-\frac {45 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{x} \, dx}{e^{20}}-\frac {15 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{x^2} \, dx}{e^{15}}-\frac {30 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}+\frac {15 \int \frac {\log ^2(x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {15 \int \frac {\log ^2(x)}{x^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log (x) \left (-2 e^5+2 x+e^5 \log (x)-3 x \log (x)\right ) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^{10}}+\frac {30 \int \frac {\log ^2(x)}{\left (e^5-x\right ) x} \, dx}{e^{10}} \\ & = -\frac {15 \log ^2(x)}{e^{10} x}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}-\frac {45 \int \left (2 \log (x) \log (2 x)-\frac {2 e^5 \log (x) \log (2 x)}{x}-3 \log ^2(x) \log (2 x)+\frac {e^5 \log ^2(x) \log (2 x)}{x}\right ) \, dx}{e^{20}}-\frac {45 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{e^5-x}+\frac {2 x \log (x) \log (2 x)}{e^5-x}+\frac {e^5 \log ^2(x) \log (2 x)}{e^5-x}-\frac {3 x \log ^2(x) \log (2 x)}{e^5-x}\right ) \, dx}{e^{20}}-\frac {15 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{x^2}+\frac {2 \log (x) \log (2 x)}{x}+\frac {e^5 \log ^2(x) \log (2 x)}{x^2}-\frac {3 \log ^2(x) \log (2 x)}{x}\right ) \, dx}{e^{15}}-\frac {30 \int \frac {\log (x)}{e^5-x} \, dx}{e^{15}}-\frac {30 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{\left (e^5-x\right )^2}+\frac {2 x \log (x) \log (2 x)}{\left (e^5-x\right )^2}+\frac {e^5 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2}-\frac {3 x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2}\right ) \, dx}{e^{15}}+\frac {60 \int \frac {\log \left (1-\frac {e^5}{x}\right ) \log (x)}{x} \, dx}{e^{15}}-\frac {15 \int \left (-\frac {2 e^5 \log (x) \log (2 x)}{\left (e^5-x\right )^3}+\frac {2 x \log (x) \log (2 x)}{\left (e^5-x\right )^3}+\frac {e^5 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3}-\frac {3 x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3}\right ) \, dx}{e^{10}}+\frac {30 \int \frac {\log (x)}{x^2} \, dx}{e^{10}} \\ & = -\frac {30}{e^{10} x}+\frac {150 \log \left (e^5-x\right )}{e^{15}}-\frac {30 \log (x)}{e^{10} x}-\frac {15 \log ^2(x)}{e^{10} x}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {90 \int \log (x) \log (2 x) \, dx}{e^{20}}-\frac {90 \int \frac {x \log (x) \log (2 x)}{e^5-x} \, dx}{e^{20}}+\frac {135 \int \log ^2(x) \log (2 x) \, dx}{e^{20}}+\frac {135 \int \frac {x \log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{20}}-\frac {30 \int \frac {\log (x) \log (2 x)}{x} \, dx}{e^{15}}-\frac {30 \int \frac {\log \left (\frac {x}{e^5}\right )}{e^5-x} \, dx}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {60 \int \frac {x \log (x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}-\frac {60 \int \frac {\text {Li}_2\left (\frac {e^5}{x}\right )}{x} \, dx}{e^{15}}+\frac {90 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {90 \int \frac {\log (x) \log (2 x)}{x} \, dx}{e^{15}}+\frac {90 \int \frac {x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{x^2} \, dx}{e^{10}}+\frac {30 \int \frac {\log (x) \log (2 x)}{x^2} \, dx}{e^{10}}-\frac {30 \int \frac {x \log (x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^{10}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {45 \int \frac {x \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^{10}}+\frac {60 \int \frac {\log (x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {30 \int \frac {\log (x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5} \\ & = -\frac {30}{e^{10} x}+\frac {150 \log \left (e^5-x\right )}{e^{15}}-\frac {30 \log (x)}{e^{10} x}-\frac {15 \log ^2(x)}{e^{10} x}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}+\frac {360 x \log (2 x)}{e^{20}}+\frac {15 \log (x) \log (2 x)}{e^5 \left (e^5-x\right )^2}+\frac {60 \log (x) \log (2 x)}{e^{10} \left (e^5-x\right )}-\frac {360 x \log (x) \log (2 x)}{e^{20}}-\frac {15 x^2 \log (x) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {30 \log ^2(x) \log (2 x)}{e^{15}}+\frac {15 \log ^2(x) \log (2 x)}{e^{10} x}+\frac {135 x \log ^2(x) \log (2 x)}{e^{20}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {30 \text {Li}_2\left (1-\frac {x}{e^5}\right )}{e^{15}}+\frac {60 \text {Li}_3\left (\frac {e^5}{x}\right )}{e^{15}}+\frac {90 \int (-1+\log (x)) \, dx}{e^{20}}-\frac {90 \int \left (-\log (x) \log (2 x)+\frac {e^5 \log (x) \log (2 x)}{e^5-x}\right ) \, dx}{e^{20}}-\frac {135 \int \left (2-2 \log (x)+\log ^2(x)\right ) \, dx}{e^{20}}+\frac {135 \int \left (-\log ^2(x) \log (2 x)+\frac {e^5 \log ^2(x) \log (2 x)}{e^5-x}\right ) \, dx}{e^{20}}+\frac {30 \int \frac {\log ^2(x)}{2 x} \, dx}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {60 \int \left (\frac {e^5 \log (x) \log (2 x)}{\left (e^5-x\right )^2}-\frac {\log (x) \log (2 x)}{e^5-x}\right ) \, dx}{e^{15}}-\frac {90 \int \frac {\log ^2(x)}{2 x} \, dx}{e^{15}}+\frac {90 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {90 \int \left (\frac {e^5 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2}-\frac {\log ^2(x) \log (2 x)}{e^5-x}\right ) \, dx}{e^{15}}+\frac {15 \int \frac {-2-2 \log (x)-\log ^2(x)}{x^2} \, dx}{e^{10}}-\frac {30 \int \frac {-1-\log (x)}{x^2} \, dx}{e^{10}}+\frac {30 \int \frac {x \log (x)}{2 e^5 \left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {30 \int \frac {x \log (2 x)}{2 e^5 \left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {45 \int \left (\frac {e^5 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^3}-\frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2}\right ) \, dx}{e^{10}}-\frac {60 \int \frac {\log (x)}{\left (e^5-x\right ) x} \, dx}{e^{10}}-\frac {60 \int \frac {\log (2 x)}{\left (e^5-x\right ) x} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}-\frac {30 \int \frac {\log (x)}{2 \left (e^5-x\right )^2 x} \, dx}{e^5}-\frac {30 \int \frac {\log (2 x)}{2 \left (e^5-x\right )^2 x} \, dx}{e^5} \\ & = -\frac {60}{e^{10} x}-\frac {360 x}{e^{20}}+\frac {150 \log \left (e^5-x\right )}{e^{15}}-\frac {30 \log (x)}{e^{10} x}+\frac {60 \log \left (1-\frac {e^5}{x}\right ) \log (x)}{e^{15}}-\frac {15 \log ^2(x)}{e^{10} x}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}-\frac {30 (1+\log (x))}{e^{10} x}+\frac {360 x \log (2 x)}{e^{20}}+\frac {60 \log \left (1-\frac {e^5}{x}\right ) \log (2 x)}{e^{15}}+\frac {15 \log (x) \log (2 x)}{e^5 \left (e^5-x\right )^2}+\frac {60 \log (x) \log (2 x)}{e^{10} \left (e^5-x\right )}-\frac {360 x \log (x) \log (2 x)}{e^{20}}-\frac {15 x^2 \log (x) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {30 \log ^2(x) \log (2 x)}{e^{15}}+\frac {15 \log ^2(x) \log (2 x)}{e^{10} x}+\frac {135 x \log ^2(x) \log (2 x)}{e^{20}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {30 \text {Li}_2\left (1-\frac {x}{e^5}\right )}{e^{15}}+\frac {60 \text {Li}_3\left (\frac {e^5}{x}\right )}{e^{15}}+\frac {90 \int \log (x) \, dx}{e^{20}}+\frac {90 \int \log (x) \log (2 x) \, dx}{e^{20}}-\frac {135 \int \log ^2(x) \, dx}{e^{20}}-\frac {135 \int \log ^2(x) \log (2 x) \, dx}{e^{20}}+\frac {270 \int \log (x) \, dx}{e^{20}}+\frac {15 \int \frac {x \log (x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}+\frac {15 \int \frac {\log ^2(x)}{x} \, dx}{e^{15}}+\frac {15 \int \frac {x \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{15}}-\frac {45 \int \frac {\log ^2(x)}{x} \, dx}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-2 \frac {60 \int \frac {\log \left (1-\frac {e^5}{x}\right )}{x} \, dx}{e^{15}}+\frac {60 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {90 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {135 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {15 \int \left (-\frac {2}{x^2}-\frac {2 \log (x)}{x^2}-\frac {\log ^2(x)}{x^2}\right ) \, dx}{e^{10}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {60 \int \frac {\log (x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {90 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log (x)}{\left (e^5-x\right )^2 x} \, dx}{e^5}-\frac {15 \int \frac {\log (2 x)}{\left (e^5-x\right )^2 x} \, dx}{e^5}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5} \\ & = -\frac {30}{e^{10} x}-\frac {720 x}{e^{20}}+\frac {150 \log \left (e^5-x\right )}{e^{15}}-\frac {30 \log (x)}{e^{10} x}+\frac {360 x \log (x)}{e^{20}}+\frac {15 x \log (x)}{e^{15} \left (e^5-x\right )}+\frac {60 \log \left (1-\frac {e^5}{x}\right ) \log (x)}{e^{15}}-\frac {15 \log ^2(x)}{e^{10} x}-\frac {135 x \log ^2(x)}{e^{20}}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}-\frac {30 (1+\log (x))}{e^{10} x}+\frac {15 x \log (2 x)}{e^{15} \left (e^5-x\right )}+\frac {60 \log \left (1-\frac {e^5}{x}\right ) \log (2 x)}{e^{15}}+\frac {15 \log (x) \log (2 x)}{e^5 \left (e^5-x\right )^2}-\frac {15 x^2 \log (x) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {30 \log ^2(x) \log (2 x)}{e^{15}}+\frac {15 \log ^2(x) \log (2 x)}{e^{10} x}-\frac {120 \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {30 \text {Li}_2\left (1-\frac {x}{e^5}\right )}{e^{15}}+\frac {60 \text {Li}_3\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {90 \int (-1+\log (x)) \, dx}{e^{20}}+\frac {135 \int \left (2-2 \log (x)+\log ^2(x)\right ) \, dx}{e^{20}}+\frac {270 \int \log (x) \, dx}{e^{20}}-\frac {15 \int \frac {1+\log (x)}{e^5-x} \, dx}{e^{15}}-\frac {15 \int \frac {1+\log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {15 \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right )}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {45 \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right )}{e^{15}}+\frac {60 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {90 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {135 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {15 \int \frac {\log (x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log (x)}{\left (e^5-x\right ) x} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x)}{x^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log (2 x)}{\left (e^5-x\right ) x} \, dx}{e^{10}}-\frac {30 \int \frac {\log (x)}{x^2} \, dx}{e^{10}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {60 \int \frac {\log (x)}{\left (e^5-x\right ) x} \, dx}{e^{10}}+\frac {60 \int \frac {\log (2 x)}{\left (e^5-x\right ) x} \, dx}{e^{10}}+\frac {90 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5} \\ & = -\frac {630 x}{e^{20}}+\frac {240 \log \left (e^5-x\right )}{e^{15}}+\frac {15 (6+\log (2)) \log \left (e^5-x\right )}{e^{15}}+\frac {630 x \log (x)}{e^{20}}+\frac {15 \log \left (1-\frac {e^5}{x}\right ) \log (x)}{e^{15}}-\frac {135 x \log ^2(x)}{e^{20}}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}-\frac {10 \log ^3(x)}{e^{15}}-\frac {30 (1+\log (x))}{e^{10} x}+\frac {15 \log \left (1-\frac {e^5}{x}\right ) \log (2 x)}{e^{15}}+\frac {15 \log (x) \log (2 x)}{e^5 \left (e^5-x\right )^2}-\frac {15 x^2 \log (x) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {30 \log ^2(x) \log (2 x)}{e^{15}}+\frac {15 \log ^2(x) \log (2 x)}{e^{10} x}-\frac {120 \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {30 \text {Li}_2\left (1-\frac {x}{e^5}\right )}{e^{15}}+\frac {60 \text {Li}_3\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {90 \int \log (x) \, dx}{e^{20}}+\frac {135 \int \log ^2(x) \, dx}{e^{20}}-\frac {270 \int \log (x) \, dx}{e^{20}}+2 \frac {15 \int \frac {1}{e^5-x} \, dx}{e^{15}}-2 \frac {15 \int \frac {\log \left (1-\frac {e^5}{x}\right )}{x} \, dx}{e^{15}}-2 \frac {15 \int \frac {\log \left (\frac {x}{e^5}\right )}{e^5-x} \, dx}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+2 \frac {60 \int \frac {\log \left (1-\frac {e^5}{x}\right )}{x} \, dx}{e^{15}}+\frac {60 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {90 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {135 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {30 \int \frac {\log (x)}{x^2} \, dx}{e^{10}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {90 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5} \\ & = \frac {30}{e^{10} x}-\frac {270 x}{e^{20}}+\frac {210 \log \left (e^5-x\right )}{e^{15}}+\frac {15 (6+\log (2)) \log \left (e^5-x\right )}{e^{15}}+\frac {30 \log (x)}{e^{10} x}+\frac {270 x \log (x)}{e^{20}}+\frac {15 \log \left (1-\frac {e^5}{x}\right ) \log (x)}{e^{15}}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}-\frac {10 \log ^3(x)}{e^{15}}-\frac {30 (1+\log (x))}{e^{10} x}+\frac {15 \log \left (1-\frac {e^5}{x}\right ) \log (2 x)}{e^{15}}+\frac {15 \log (x) \log (2 x)}{e^5 \left (e^5-x\right )^2}-\frac {15 x^2 \log (x) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {30 \log ^2(x) \log (2 x)}{e^{15}}+\frac {15 \log ^2(x) \log (2 x)}{e^{10} x}-\frac {30 \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {60 \text {Li}_2\left (1-\frac {x}{e^5}\right )}{e^{15}}+\frac {60 \text {Li}_3\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {270 \int \log (x) \, dx}{e^{20}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {60 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {90 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {135 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {90 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5} \\ & = \frac {30}{e^{10} x}+\frac {210 \log \left (e^5-x\right )}{e^{15}}+\frac {15 (6+\log (2)) \log \left (e^5-x\right )}{e^{15}}+\frac {30 \log (x)}{e^{10} x}+\frac {15 \log \left (1-\frac {e^5}{x}\right ) \log (x)}{e^{15}}+\frac {15 x \log ^2(x)}{e^{15} \left (e^5-x\right )}-\frac {30 \log \left (1-\frac {e^5}{x}\right ) \log ^2(x)}{e^{15}}-\frac {10 \log ^3(x)}{e^{15}}-\frac {30 (1+\log (x))}{e^{10} x}+\frac {15 \log \left (1-\frac {e^5}{x}\right ) \log (2 x)}{e^{15}}+\frac {15 \log (x) \log (2 x)}{e^5 \left (e^5-x\right )^2}-\frac {15 x^2 \log (x) \log (2 x)}{e^{15} \left (e^5-x\right )^2}+\frac {30 \log ^2(x) \log (2 x)}{e^{15}}+\frac {15 \log ^2(x) \log (2 x)}{e^{10} x}-\frac {30 \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}+\frac {60 \log (x) \text {Li}_2\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {60 \text {Li}_2\left (1-\frac {x}{e^5}\right )}{e^{15}}+\frac {60 \text {Li}_3\left (\frac {e^5}{x}\right )}{e^{15}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {60 \int \frac {\log (x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {90 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}+\frac {135 \int \frac {\log ^2(x) \log (2 x)}{e^5-x} \, dx}{e^{15}}-\frac {30 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}+\frac {90 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^2} \, dx}{e^{10}}-\frac {15 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5}+\frac {45 \int \frac {\log ^2(x) \log (2 x)}{\left (e^5-x\right )^3} \, dx}{e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\frac {15 \log ^2(x) \log (2 x)}{\left (e^5-x\right )^2 x} \]

[In]

Integrate[((15*E^5 - 15*x)*Log[x]^2 + ((30*E^5 - 30*x)*Log[x] + (-15*E^5 + 45*x)*Log[x]^2)*Log[2*x])/(E^15*x^2
- 3*E^10*x^3 + 3*E^5*x^4 - x^5),x]

[Out]

(15*Log[x]^2*Log[2*x])/((E^5 - x)^2*x)

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21

method	result	size
parallelrisch	\(\frac {15 \ln \left (x \right )^{2} \ln \left (2 x \right )}{x \left ({\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}\right )}\)	\(29\)
risch	\(\frac {15 \ln \left (x \right )^{3}}{x \left ({\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}\right )}+\frac {15 \ln \left (2\right ) \ln \left (x \right )^{2}}{x \left ({\mathrm e}^{10}-2 x \,{\mathrm e}^{5}+x^{2}\right )}\)	\(48\)

[In]

int((((-15*exp(5)+45*x)*ln(x)^2+(30*exp(5)-30*x)*ln(x))*ln(2*x)+(15*exp(5)-15*x)*ln(x)^2)/(x^2*exp(5)^3-3*x^3*
exp(5)^2+3*x^4*exp(5)-x^5),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.64 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\frac {15 \, {\left (\log \left (2\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}}{x^{3} - 2 \, x^{2} e^{5} + x e^{10}} \]

[In]

integrate((((-15*exp(5)+45*x)*log(x)^2+(30*exp(5)-30*x)*log(x))*log(2*x)+(15*exp(5)-15*x)*log(x)^2)/(x^2*exp(5
)^3-3*x^3*exp(5)^2+3*x^4*exp(5)-x^5),x, algorithm="fricas")

[Out]

15*(log(2)*log(x)^2 + log(x)^3)/(x^3 - 2*x^2*e^5 + x*e^10)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).

Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\frac {15 \log {\left (x \right )}^{3}}{x^{3} - 2 x^{2} e^{5} + x e^{10}} + \frac {15 \log {\left (2 \right )} \log {\left (x \right )}^{2}}{x^{3} - 2 x^{2} e^{5} + x e^{10}} \]

[In]

integrate((((-15*exp(5)+45*x)*ln(x)**2+(30*exp(5)-30*x)*ln(x))*ln(2*x)+(15*exp(5)-15*x)*ln(x)**2)/(x**2*exp(5)
**3-3*x**3*exp(5)**2+3*x**4*exp(5)-x**5),x)

[Out]

15*log(x)**3/(x**3 - 2*x**2*exp(5) + x*exp(10)) + 15*log(2)*log(x)**2/(x**3 - 2*x**2*exp(5) + x*exp(10))

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\frac {15 \, {\left (\log \left (2\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}}{x^{3} - 2 \, x^{2} e^{5} + x e^{10}} \]

[In]

integrate((((-15*exp(5)+45*x)*log(x)^2+(30*exp(5)-30*x)*log(x))*log(2*x)+(15*exp(5)-15*x)*log(x)^2)/(x^2*exp(5
)^3-3*x^3*exp(5)^2+3*x^4*exp(5)-x^5),x, algorithm="maxima")

[Out]

15*(log(2)*log(x)^2 + log(x)^3)/(x^3 - 2*x^2*e^5 + x*e^10)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\frac {15 \, {\left (\log \left (2\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}}{x^{3} - 2 \, x^{2} e^{5} + x e^{10}} \]

[In]

integrate((((-15*exp(5)+45*x)*log(x)^2+(30*exp(5)-30*x)*log(x))*log(2*x)+(15*exp(5)-15*x)*log(x)^2)/(x^2*exp(5
)^3-3*x^3*exp(5)^2+3*x^4*exp(5)-x^5),x, algorithm="giac")

[Out]

15*(log(2)*log(x)^2 + log(x)^3)/(x^3 - 2*x^2*e^5 + x*e^10)

Mupad [B] (verification not implemented)

Time = 12.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (15 e^5-15 x\right ) \log ^2(x)+\left (\left (30 e^5-30 x\right ) \log (x)+\left (-15 e^5+45 x\right ) \log ^2(x)\right ) \log (2 x)}{e^{15} x^2-3 e^{10} x^3+3 e^5 x^4-x^5} \, dx=\frac {15\,{\ln \left (x\right )}^2\,\left (\ln \left (2\right )+\ln \left (x\right )\right )}{x\,{\left (x-{\mathrm {e}}^5\right )}^2} \]

[In]

int(-(log(x)^2*(15*x - 15*exp(5)) + log(2*x)*(log(x)*(30*x - 30*exp(5)) - log(x)^2*(45*x - 15*exp(5))))/(3*x^4
*exp(5) - 3*x^3*exp(10) + x^2*exp(15) - x^5),x)

[Out]

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