\(\int \frac {524288 x^9+(20-10485760 x^8-2621440 x^9) \log (3)}{-x^2+(20 x+5 x^2) \log (3)} \, dx\) [1849]

Optimal result

Integrand size = 42, antiderivative size = 24 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=2-65536 x^8+\log (x)-\log \left (4+x-\frac {x}{5 \log (3)}\right ) \]

[Out]

2-ln(x-1/5*x/ln(3)+4)+ln(x)-65536*x^8

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2011, 1607, 1634} \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-65536 x^8+\log (x)-\log (20 \log (3)-x (1-\log (243))) \]

[In]

Int[(524288*x^9 + (20 - 10485760*x^8 - 2621440*x^9)*Log[3])/(-x^2 + (20*x + 5*x^2)*Log[3]),x]

[Out]

-65536*x^8 + Log[x] - Log[20*Log[3] - x*(1 - Log[243])]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2011

Int[(Pq_)*(u_)^(p_.), x_Symbol] :> Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && PolyQ[Pq, x] && QuadraticQ
[u, x] &&  !QuadraticMatchQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{20 x \log (3)-x^2 (1-\log (243))} \, dx \\ & = \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{x (20 \log (3)+x (-1+\log (243)))} \, dx \\ & = \int \left (\frac {1}{x}-524288 x^7+\frac {1-\log (243)}{20 \log (3)-x (1-\log (243))}\right ) \, dx \\ & = -65536 x^8+\log (x)-\log (20 \log (3)-x (1-\log (243))) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-4 \left (16384 x^8-\frac {\log (x)}{4}+\frac {1}{4} \log (-x+20 \log (3)+x \log (243))\right ) \]

[In]

Integrate[(524288*x^9 + (20 - 10485760*x^8 - 2621440*x^9)*Log[3])/(-x^2 + (20*x + 5*x^2)*Log[3]),x]

[Out]

-4*(16384*x^8 - Log[x]/4 + Log[-x + 20*Log[3] + x*Log[243]]/4)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04

method	result	size
norman	\(-65536 x^{8}-\ln \left (5 x \ln \left (3\right )+20 \ln \left (3\right )-x \right )+\ln \left (x \right )\)	\(25\)
risch	\(-65536 x^{8}+\ln \left (-x \right )-\ln \left (x \left (5 \ln \left (3\right )-1\right )+20 \ln \left (3\right )\right )\)	\(27\)
parallelrisch	\(-65536 x^{8}+\ln \left (x \right )-\ln \left (\frac {5 x \ln \left (3\right )+20 \ln \left (3\right )-x}{5 \ln \left (3\right )-1}\right )\)	\(34\)
default	\(-65536 x^{8}+\ln \left (x \right )+\frac {4 \left (-\frac {5 \ln \left (3\right )}{4}+\frac {1}{4}\right ) \ln \left (5 x \ln \left (3\right )+20 \ln \left (3\right )-x \right )}{5 \ln \left (3\right )-1}\)	\(39\)
meijerg	\(\ln \left (x \right )-2 \ln \left (2\right )-\ln \left (5\right )-\ln \left (\ln \left (3\right )\right )+\ln \left (5 \ln \left (3\right )-1\right )-\ln \left (1+\frac {x \left (5 \ln \left (3\right )-1\right )}{20 \ln \left (3\right )}\right )+\frac {512000000000 \ln \left (3\right )^{8} \left (-131072 \ln \left (3\right )+\frac {131072}{5}\right ) \left (-\frac {x \left (5 \ln \left (3\right )-1\right ) \left (-\frac {63 x^{7} \left (5 \ln \left (3\right )-1\right )^{7}}{256000000 \ln \left (3\right )^{7}}+\frac {9 x^{6} \left (5 \ln \left (3\right )-1\right )^{6}}{1600000 \ln \left (3\right )^{6}}-\frac {21 x^{5} \left (5 \ln \left (3\right )-1\right )^{5}}{160000 \ln \left (3\right )^{5}}+\frac {63 x^{4} \left (5 \ln \left (3\right )-1\right )^{4}}{20000 \ln \left (3\right )^{4}}-\frac {63 x^{3} \left (5 \ln \left (3\right )-1\right )^{3}}{800 \ln \left (3\right )^{3}}+\frac {21 x^{2} \left (5 \ln \left (3\right )-1\right )^{2}}{10 \ln \left (3\right )^{2}}-\frac {63 x \left (5 \ln \left (3\right )-1\right )}{\ln \left (3\right )}+2520\right )}{50400 \ln \left (3\right )}+\ln \left (1+\frac {x \left (5 \ln \left (3\right )-1\right )}{20 \ln \left (3\right )}\right )\right )}{\left (5 \ln \left (3\right )-1\right )^{9}}-\frac {13421772800000000 \ln \left (3\right )^{8} \left (\frac {x \left (5 \ln \left (3\right )-1\right ) \left (\frac {3 x^{6} \left (5 \ln \left (3\right )-1\right )^{6}}{1600000 \ln \left (3\right )^{6}}-\frac {7 x^{5} \left (5 \ln \left (3\right )-1\right )^{5}}{160000 \ln \left (3\right )^{5}}+\frac {21 x^{4} \left (5 \ln \left (3\right )-1\right )^{4}}{20000 \ln \left (3\right )^{4}}-\frac {21 x^{3} \left (5 \ln \left (3\right )-1\right )^{3}}{800 \ln \left (3\right )^{3}}+\frac {7 x^{2} \left (5 \ln \left (3\right )-1\right )^{2}}{10 \ln \left (3\right )^{2}}-\frac {21 x \left (5 \ln \left (3\right )-1\right )}{\ln \left (3\right )}+840\right )}{16800 \ln \left (3\right )}-\ln \left (1+\frac {x \left (5 \ln \left (3\right )-1\right )}{20 \ln \left (3\right )}\right )\right )}{\left (5 \ln \left (3\right )-1\right )^{8}}\)	\(355\)

[In]

int(((-2621440*x^9-10485760*x^8+20)*ln(3)+524288*x^9)/((5*x^2+20*x)*ln(3)-x^2),x,method=_RETURNVERBOSE)

[Out]

-65536*x^8-ln(5*x*ln(3)+20*ln(3)-x)+ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-65536 \, x^{8} - \log \left (5 \, {\left (x + 4\right )} \log \left (3\right ) - x\right ) + \log \left (x\right ) \]

[In]

integrate(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x)*log(3)-x^2),x, algorithm="fricas")

[Out]

-65536*x^8 - log(5*(x + 4)*log(3) - x) + log(x)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=- 65536 x^{8} + \log {\left (x \right )} - \log {\left (x + \frac {20 \log {\left (3 \right )}}{-1 + 5 \log {\left (3 \right )}} \right )} \]

[In]

integrate(((-2621440*x**9-10485760*x**8+20)*ln(3)+524288*x**9)/((5*x**2+20*x)*ln(3)-x**2),x)

[Out]

-65536*x**8 + log(x) - log(x + 20*log(3)/(-1 + 5*log(3)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-65536 \, x^{8} - \log \left (x {\left (5 \, \log \left (3\right ) - 1\right )} + 20 \, \log \left (3\right )\right ) + \log \left (x\right ) \]

[In]

integrate(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x)*log(3)-x^2),x, algorithm="maxima")

[Out]

-65536*x^8 - log(x*(5*log(3) - 1) + 20*log(3)) + log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (22) = 44\).

Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.12 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-\frac {65536 \, {\left (390625 \, x^{8} \log \left (3\right )^{8} - 625000 \, x^{8} \log \left (3\right )^{7} + 437500 \, x^{8} \log \left (3\right )^{6} - 175000 \, x^{8} \log \left (3\right )^{5} + 43750 \, x^{8} \log \left (3\right )^{4} - 7000 \, x^{8} \log \left (3\right )^{3} + 700 \, x^{8} \log \left (3\right )^{2} - 40 \, x^{8} \log \left (3\right ) + x^{8}\right )}}{390625 \, \log \left (3\right )^{8} - 625000 \, \log \left (3\right )^{7} + 437500 \, \log \left (3\right )^{6} - 175000 \, \log \left (3\right )^{5} + 43750 \, \log \left (3\right )^{4} - 7000 \, \log \left (3\right )^{3} + 700 \, \log \left (3\right )^{2} - 40 \, \log \left (3\right ) + 1} - \log \left ({\left | 5 \, x \log \left (3\right ) - x + 20 \, \log \left (3\right ) \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x)*log(3)-x^2),x, algorithm="giac")

[Out]

-65536*(390625*x^8*log(3)^8 - 625000*x^8*log(3)^7 + 437500*x^8*log(3)^6 - 175000*x^8*log(3)^5 + 43750*x^8*log(
3)^4 - 7000*x^8*log(3)^3 + 700*x^8*log(3)^2 - 40*x^8*log(3) + x^8)/(390625*log(3)^8 - 625000*log(3)^7 + 437500
*log(3)^6 - 175000*log(3)^5 + 43750*log(3)^4 - 7000*log(3)^3 + 700*log(3)^2 - 40*log(3) + 1) - log(abs(5*x*log
(3) - x + 20*log(3))) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 325, normalized size of antiderivative = 13.54 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=\frac {10\,x^6\,\ln \left (3\right )\,\left (\frac {10485760\,\ln \left (3\right )}{\ln \left (243\right )-1}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{3\,\left (\ln \left (243\right )-1\right )}-\frac {x^8\,\left (2621440\,\ln \left (3\right )-524288\right )}{8\,\left (\ln \left (243\right )-1\right )}-x^7\,\left (\frac {10485760\,\ln \left (3\right )}{7\,\left (\ln \left (243\right )-1\right )}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{7\,{\left (\ln \left (243\right )-1\right )}^2}\right )-\frac {64000000\,x\,{\ln \left (3\right )}^6\,\left (\frac {10485760\,\ln \left (3\right )}{\ln \left (243\right )-1}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^6}-\frac {80\,x^5\,{\ln \left (3\right )}^2\,\left (\frac {10485760\,\ln \left (3\right )}{\ln \left (243\right )-1}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^2}+\frac {2000\,x^4\,{\ln \left (3\right )}^3\,\left (\frac {10485760\,\ln \left (3\right )}{\ln \left (243\right )-1}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^3}-\frac {160000\,x^3\,{\ln \left (3\right )}^4\,\left (\frac {10485760\,\ln \left (3\right )}{\ln \left (243\right )-1}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{3\,{\left (\ln \left (243\right )-1\right )}^4}+\frac {1600000\,x^2\,{\ln \left (3\right )}^5\,\left (\frac {10485760\,\ln \left (3\right )}{\ln \left (243\right )-1}-\frac {20\,\ln \left (3\right )\,\left (2621440\,\ln \left (3\right )-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^5}+\mathrm {atan}\left (\frac {x\,\left (2\,\ln \left (243\right )-2\right )\,1{}\mathrm {i}}{20\,\ln \left (3\right )}+1{}\mathrm {i}\right )\,2{}\mathrm {i} \]

[In]

int(-(log(3)*(10485760*x^8 + 2621440*x^9 - 20) - 524288*x^9)/(log(3)*(20*x + 5*x^2) - x^2),x)

[Out]

atan((x*(2*log(243) - 2)*1i)/(20*log(3)) + 1i)*2i - x^7*((10485760*log(3))/(7*(log(243) - 1)) - (20*log(3)*(26
21440*log(3) - 524288))/(7*(log(243) - 1)^2)) - (x^8*(2621440*log(3) - 524288))/(8*(log(243) - 1)) + (10*x^6*l
og(3)*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(3*(log(243
) - 1)) - (64000000*x*log(3)^6*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 524288))/(log(
243) - 1)^2))/(log(243) - 1)^6 - (80*x^5*log(3)^2*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(
3) - 524288))/(log(243) - 1)^2))/(log(243) - 1)^2 + (2000*x^4*log(3)^3*((10485760*log(3))/(log(243) - 1) - (20
*log(3)*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(log(243) - 1)^3 - (160000*x^3*log(3)^4*((10485760*log(3
))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(3*(log(243) - 1)^4) + (1600000*x
^2*log(3)^5*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(log(
243) - 1)^5