\(\int \frac {(810+385 x-10 x^2) \log (5)+(-810-395 x+5 x^2) \log (5) \log (\frac {1}{81} (81 x-x^2))+(81-x) \log (5) \log ^2(\frac {1}{81} (81 x-x^2))}{-4050 x^2-1975 x^3+25 x^4+(1620 x+790 x^2-10 x^3) \log (2+x) \log (\frac {1}{81} (81 x-x^2))+(-162-79 x+x^2) \log ^2(2+x) \log ^2(\frac {1}{81} (81 x-x^2))} \, dx\) [4137]

Optimal result

Integrand size = 148, antiderivative size = 25 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {\log (5)}{\log (2+x)-\frac {5 x}{\log \left (x-\frac {x^2}{81}\right )}} \]

[Out]

ln(5)/(ln(2+x)-5*x/ln(-1/81*x^2+x))

Rubi [F]

\[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx \]

[In]

Int[((810 + 385*x - 10*x^2)*Log[5] + (-810 - 395*x + 5*x^2)*Log[5]*Log[(81*x - x^2)/81] + (81 - x)*Log[5]*Log[
(81*x - x^2)/81]^2)/(-4050*x^2 - 1975*x^3 + 25*x^4 + (1620*x + 790*x^2 - 10*x^3)*Log[2 + x]*Log[(81*x - x^2)/8
1] + (-162 - 79*x + x^2)*Log[2 + x]^2*Log[(81*x - x^2)/81]^2),x]

[Out]

Log[5]/Log[2 + x] - 10*Log[5]*Defer[Int][(5*x - Log[2 + x]*Log[x - x^2/81])^(-2), x] - 405*Log[5]*Defer[Int][1
/((-81 + x)*(5*x - Log[2 + x]*Log[x - x^2/81])^2), x] - 100*Log[5]*Defer[Int][1/((2 + x)*Log[2 + x]^2*(5*x - L
og[2 + x]*Log[x - x^2/81])^2), x] - 20*Log[5]*Defer[Int][1/((2 + x)*Log[2 + x]^2*(5*x - Log[2 + x]*Log[x - x^2
/81])), x] + 50*Log[5]*Defer[Int][1/(Log[2 + x]^2*(-5*x + Log[2 + x]*Log[x - x^2/81])^2), x] - 25*Log[5]*Defer
[Int][x/(Log[2 + x]^2*(-5*x + Log[2 + x]*Log[x - x^2/81])^2), x] + 25*Log[5]*Defer[Int][x/(Log[2 + x]*(-5*x +
Log[2 + x]*Log[x - x^2/81])^2), x] - 10*Log[5]*Defer[Int][1/(Log[2 + x]^2*(-5*x + Log[2 + x]*Log[x - x^2/81]))
, x] + 5*Log[5]*Defer[Int][1/(Log[2 + x]*(-5*x + Log[2 + x]*Log[x - x^2/81])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (5) \left (-810-385 x+10 x^2-5 \left (-162-79 x+x^2\right ) \log \left (x-\frac {x^2}{81}\right )+(-81+x) \log ^2\left (x-\frac {x^2}{81}\right )\right )}{\left (162+79 x-x^2\right ) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx \\ & = \log (5) \int \frac {-810-385 x+10 x^2-5 \left (-162-79 x+x^2\right ) \log \left (x-\frac {x^2}{81}\right )+(-81+x) \log ^2\left (x-\frac {x^2}{81}\right )}{\left (162+79 x-x^2\right ) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx \\ & = \log (5) \int \left (-\frac {1}{(2+x) \log ^2(2+x)}+\frac {5 \left (405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)\right )}{(-81+x) (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {5 (-2 x+2 \log (2+x)+x \log (2+x))}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx \\ & = -\left (\log (5) \int \frac {1}{(2+x) \log ^2(2+x)} \, dx\right )+(5 \log (5)) \int \frac {405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)}{(-81+x) (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(5 \log (5)) \int \frac {-2 x+2 \log (2+x)+x \log (2+x)}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx \\ & = -\left (\log (5) \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,2+x\right )\right )+(5 \log (5)) \int \frac {5 (-81+x) x^2-5 x \left (-162-79 x+x^2\right ) \log (2+x)-\left (162+77 x-2 x^2\right ) \log ^2(2+x)}{(81-x) (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(5 \log (5)) \int \left (-\frac {2 x}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}+\frac {2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}+\frac {x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx \\ & = -\left (\log (5) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (2+x)\right )\right )-(5 \log (5)) \int \frac {x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx+(5 \log (5)) \int \left (\frac {405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)}{83 (-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {-405 x^2+5 x^3+810 x \log (2+x)+395 x^2 \log (2+x)-5 x^3 \log (2+x)-162 \log ^2(2+x)-77 x \log ^2(2+x)+2 x^2 \log ^2(2+x)}{83 (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right ) \, dx+(10 \log (5)) \int \frac {x}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(10 \log (5)) \int \frac {1}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx \\ & = \frac {\log (5)}{\log (2+x)}+\frac {1}{83} (5 \log (5)) \int \frac {405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (5 \log (5)) \int \frac {-405 x^2+5 x^3+810 x \log (2+x)+395 x^2 \log (2+x)-5 x^3 \log (2+x)-162 \log ^2(2+x)-77 x \log ^2(2+x)+2 x^2 \log ^2(2+x)}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(5 \log (5)) \int \left (-\frac {2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}-\frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx-(10 \log (5)) \int \frac {1}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx+(10 \log (5)) \int \left (-\frac {2}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}-\frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx \\ & = \frac {\log (5)}{\log (2+x)}+\frac {1}{83} (5 \log (5)) \int \frac {5 (-81+x) x^2-5 x \left (-162-79 x+x^2\right ) \log (2+x)-\left (162+77 x-2 x^2\right ) \log ^2(2+x)}{(81-x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (5 \log (5)) \int \frac {5 (-81+x) x^2-5 x \left (-162-79 x+x^2\right ) \log (2+x)+\left (-162-77 x+2 x^2\right ) \log ^2(2+x)}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+(5 \log (5)) \int \frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(10 \log (5)) \int \frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(20 \log (5)) \int \frac {1}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx \\ & = \frac {\log (5)}{\log (2+x)}+\frac {1}{83} (5 \log (5)) \int \left (\frac {162}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {77 x}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {2 x^2}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {405 x^2}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {5 x^3}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {810 x}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {395 x^2}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {5 x^3}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right ) \, dx+\frac {1}{83} (5 \log (5)) \int \left (-\frac {162}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {77 x}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {2 x^2}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {405 x^2}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {5 x^3}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {810 x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {395 x^2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {5 x^3}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right ) \, dx+(5 \log (5)) \int \frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(10 \log (5)) \int \frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(20 \log (5)) \int \frac {1}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx \\ & = \frac {\log (5)}{\log (2+x)}-\frac {1}{83} (10 \log (5)) \int \frac {x^2}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (10 \log (5)) \int \frac {x^2}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-\frac {1}{83} (25 \log (5)) \int \frac {x^3}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (25 \log (5)) \int \frac {x^3}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (25 \log (5)) \int \frac {x^3}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-\frac {1}{83} (25 \log (5)) \int \frac {x^3}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (385 \log (5)) \int \frac {x}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-\frac {1}{83} (385 \log (5)) \int \frac {x}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+(5 \log (5)) \int \frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx+\frac {1}{83} (810 \log (5)) \int \frac {1}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-\frac {1}{83} (810 \log (5)) \int \frac {1}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(10 \log (5)) \int \frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(20 \log (5)) \int \frac {1}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-\frac {1}{83} (1975 \log (5)) \int \frac {x^2}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (1975 \log (5)) \int \frac {x^2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (2025 \log (5)) \int \frac {x^2}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-\frac {1}{83} (2025 \log (5)) \int \frac {x^2}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-\frac {1}{83} (4050 \log (5)) \int \frac {x}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (4050 \log (5)) \int \frac {x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {\log (5) \log \left (x-\frac {x^2}{81}\right )}{-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )} \]

[In]

Integrate[((810 + 385*x - 10*x^2)*Log[5] + (-810 - 395*x + 5*x^2)*Log[5]*Log[(81*x - x^2)/81] + (81 - x)*Log[5
]*Log[(81*x - x^2)/81]^2)/(-4050*x^2 - 1975*x^3 + 25*x^4 + (1620*x + 790*x^2 - 10*x^3)*Log[2 + x]*Log[(81*x -
x^2)/81] + (-162 - 79*x + x^2)*Log[2 + x]^2*Log[(81*x - x^2)/81]^2),x]

[Out]

(Log[5]*Log[x - x^2/81])/(-5*x + Log[2 + x]*Log[x - x^2/81])

Maple [A] (verified)

Time = 12.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
parallelrisch	\(-\frac {\ln \left (5\right ) \ln \left (-\frac {1}{81} x^{2}+x \right )}{-\ln \left (-\frac {1}{81} x^{2}+x \right ) \ln \left (2+x \right )+5 x}\)	\(33\)
default	\(-\ln \left (5\right ) \left (-\frac {1}{\ln \left (2+x \right )}+\frac {5 x}{\ln \left (2+x \right ) \left (4 \ln \left (2+x \right ) \ln \left (3\right )-\ln \left (2+x \right ) \ln \left (-\left (2+x \right )^{2}+4+85 x \right )+5 x \right )}\right )\)	\(56\)
risch	\(\frac {\ln \left (5\right )}{\ln \left (2+x \right )}-\frac {10 x \ln \left (5\right )}{\ln \left (2+x \right ) \left (i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x -81\right )\right ) \operatorname {csgn}\left (i x \left (x -81\right )\right ) \ln \left (2+x \right )-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )+2 i \pi \operatorname {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \,\operatorname {csgn}\left (i \left (x -81\right )\right ) \operatorname {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \operatorname {csgn}\left (i x \left (x -81\right )\right )^{3} \ln \left (2+x \right )-2 i \pi \ln \left (2+x \right )+8 \ln \left (2+x \right ) \ln \left (3\right )-2 \ln \left (x \right ) \ln \left (2+x \right )-2 \ln \left (2+x \right ) \ln \left (x -81\right )+10 x \right )}\)	\(174\)

[In]

int(((81-x)*ln(5)*ln(-1/81*x^2+x)^2+(5*x^2-395*x-810)*ln(5)*ln(-1/81*x^2+x)+(-10*x^2+385*x+810)*ln(5))/((x^2-7
9*x-162)*ln(-1/81*x^2+x)^2*ln(2+x)^2+(-10*x^3+790*x^2+1620*x)*ln(-1/81*x^2+x)*ln(2+x)+25*x^4-1975*x^3-4050*x^2
),x,method=_RETURNVERBOSE)

[Out]

-ln(5)*ln(-1/81*x^2+x)/(-ln(-1/81*x^2+x)*ln(2+x)+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {\log \left (5\right ) \log \left (-\frac {1}{81} \, x^{2} + x\right )}{\log \left (-\frac {1}{81} \, x^{2} + x\right ) \log \left (x + 2\right ) - 5 \, x} \]

[In]

integrate(((81-x)*log(5)*log(-1/81*x^2+x)^2+(5*x^2-395*x-810)*log(5)*log(-1/81*x^2+x)+(-10*x^2+385*x+810)*log(
5))/((x^2-79*x-162)*log(-1/81*x^2+x)^2*log(2+x)^2+(-10*x^3+790*x^2+1620*x)*log(-1/81*x^2+x)*log(2+x)+25*x^4-19
75*x^3-4050*x^2),x, algorithm="fricas")

[Out]

log(5)*log(-1/81*x^2 + x)/(log(-1/81*x^2 + x)*log(x + 2) - 5*x)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {5 x \log {\left (5 \right )}}{- 5 x \log {\left (x + 2 \right )} + \log {\left (x + 2 \right )}^{2} \log {\left (- \frac {x^{2}}{81} + x \right )}} + \frac {\log {\left (5 \right )}}{\log {\left (x + 2 \right )}} \]

[In]

integrate(((81-x)*ln(5)*ln(-1/81*x**2+x)**2+(5*x**2-395*x-810)*ln(5)*ln(-1/81*x**2+x)+(-10*x**2+385*x+810)*ln(
5))/((x**2-79*x-162)*ln(-1/81*x**2+x)**2*ln(2+x)**2+(-10*x**3+790*x**2+1620*x)*ln(-1/81*x**2+x)*ln(2+x)+25*x**
4-1975*x**3-4050*x**2),x)

[Out]

5*x*log(5)/(-5*x*log(x + 2) + log(x + 2)**2*log(-x**2/81 + x)) + log(5)/log(x + 2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {4 \, \log \left (5\right ) \log \left (3\right ) - \log \left (5\right ) \log \left (x\right ) - \log \left (5\right ) \log \left (-x + 81\right )}{{\left (4 \, \log \left (3\right ) - \log \left (x\right )\right )} \log \left (x + 2\right ) - \log \left (x + 2\right ) \log \left (-x + 81\right ) + 5 \, x} \]

[In]

integrate(((81-x)*log(5)*log(-1/81*x^2+x)^2+(5*x^2-395*x-810)*log(5)*log(-1/81*x^2+x)+(-10*x^2+385*x+810)*log(
5))/((x^2-79*x-162)*log(-1/81*x^2+x)^2*log(2+x)^2+(-10*x^3+790*x^2+1620*x)*log(-1/81*x^2+x)*log(2+x)+25*x^4-19
75*x^3-4050*x^2),x, algorithm="maxima")

[Out]

(4*log(5)*log(3) - log(5)*log(x) - log(5)*log(-x + 81))/((4*log(3) - log(x))*log(x + 2) - log(x + 2)*log(-x +
81) + 5*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).

Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=-\frac {5 \, x \log \left (5\right )}{4 \, \log \left (3\right ) \log \left (x + 2\right )^{2} - \log \left (-x^{2} + 81 \, x\right ) \log \left (x + 2\right )^{2} + 5 \, x \log \left (x + 2\right )} + \frac {\log \left (5\right )}{\log \left (x + 2\right )} \]

[In]

integrate(((81-x)*log(5)*log(-1/81*x^2+x)^2+(5*x^2-395*x-810)*log(5)*log(-1/81*x^2+x)+(-10*x^2+385*x+810)*log(
5))/((x^2-79*x-162)*log(-1/81*x^2+x)^2*log(2+x)^2+(-10*x^3+790*x^2+1620*x)*log(-1/81*x^2+x)*log(2+x)+25*x^4-19
75*x^3-4050*x^2),x, algorithm="giac")

[Out]

-5*x*log(5)/(4*log(3)*log(x + 2)^2 - log(-x^2 + 81*x)*log(x + 2)^2 + 5*x*log(x + 2)) + log(5)/log(x + 2)

Mupad [B] (verification not implemented)

Time = 10.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=-\frac {\ln \left (5\right )\,\ln \left (x-\frac {x^2}{81}\right )}{5\,x-\ln \left (x+2\right )\,\ln \left (x-\frac {x^2}{81}\right )} \]

[In]

int((log(5)*log(x - x^2/81)^2*(x - 81) - log(5)*(385*x - 10*x^2 + 810) + log(5)*log(x - x^2/81)*(395*x - 5*x^2
+ 810))/(4050*x^2 + 1975*x^3 - 25*x^4 + log(x + 2)^2*log(x - x^2/81)^2*(79*x - x^2 + 162) - log(x + 2)*log(x
- x^2/81)*(1620*x + 790*x^2 - 10*x^3)),x)

[Out]

-(log(5)*log(x - x^2/81))/(5*x - log(x + 2)*log(x - x^2/81))