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

Optimal. Leaf size=25 \[ \frac {\log (5)}{\log (2+x)-\frac {5 x}{\log \left (x-\frac {x^2}{81}\right )}} \]

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Rubi [F]  time = 12.67, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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) \operatorname {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) \operatorname {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\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.18, size = 34, normalized size = 1.36 \begin {gather*} \frac {\log (5) \log \left (x-\frac {x^2}{81}\right )}{-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.57, size = 30, normalized size = 1.20 \begin {gather*} \frac {\log \relax (5) \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.44, size = 53, normalized size = 2.12 \begin {gather*} -\frac {5 \, x \log \relax (5)}{4 \, \log \relax (3) \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 \relax (5)}{\log \left (x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.19, size = 174, normalized size = 6.96

method	result	size
risch	\(\frac {\ln \relax (5)}{\ln \left (2+x \right )}-\frac {10 \ln \relax (5) x}{\ln \left (2+x \right ) \left (i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -81\right )\right ) \mathrm {csgn}\left (i x \left (x -81\right )\right ) \ln \left (2+x \right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )+2 i \pi \mathrm {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \,\mathrm {csgn}\left (i \left (x -81\right )\right ) \mathrm {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \mathrm {csgn}\left (i x \left (x -81\right )\right )^{3} \ln \left (2+x \right )-2 i \pi \ln \left (2+x \right )+8 \ln \left (2+x \right ) \ln \relax (3)-2 \ln \left (2+x \right ) \ln \relax (x )-2 \ln \left (2+x \right ) \ln \left (x -81\right )+10 x \right )}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[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(2+x)-10*ln(5)*x/ln(2+x)/(I*Pi*csgn(I*x)*csgn(I*(x-81))*csgn(I*x*(x-81))*ln(2+x)-I*Pi*csgn(I*x)*csgn(I
*x*(x-81))^2*ln(2+x)+2*I*Pi*csgn(I*x*(x-81))^2*ln(2+x)-I*Pi*csgn(I*(x-81))*csgn(I*x*(x-81))^2*ln(2+x)-I*Pi*csg
n(I*x*(x-81))^3*ln(2+x)-2*I*Pi*ln(2+x)+8*ln(2+x)*ln(3)-2*ln(2+x)*ln(x)-2*ln(2+x)*ln(x-81)+10*x)

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maxima [B]  time = 0.51, size = 56, normalized size = 2.24 \begin {gather*} \frac {4 \, \log \relax (5) \log \relax (3) - \log \relax (5) \log \relax (x) - \log \relax (5) \log \left (-x + 81\right )}{{\left (4 \, \log \relax (3) - \log \relax (x)\right )} \log \left (x + 2\right ) - \log \left (x + 2\right ) \log \left (-x + 81\right ) + 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.95, size = 32, normalized size = 1.28 \begin {gather*} -\frac {\ln \relax (5)\,\ln \left (x-\frac {x^2}{81}\right )}{5\,x-\ln \left (x+2\right )\,\ln \left (x-\frac {x^2}{81}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.42, size = 37, normalized size = 1.48 \begin {gather*} \frac {5 x \log {\relax (5 )}}{- 5 x \log {\left (x + 2 \right )} + \log {\left (x + 2 \right )}^{2} \log {\left (- \frac {x^{2}}{81} + x \right )}} + \frac {\log {\relax (5 )}}{\log {\left (x + 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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