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

Optimal. Leaf size=24 \[ 2-65536 x^8+\log (x)-\log \left (4+x-\frac {x}{5 \log (3)}\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1986, 1593, 1620} \begin {gather*} -65536 x^8+\log (x)-\log (20 \log (3)-x (1-\log (243))) \end {gather*}

Antiderivative was successfully verified.

[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 1593

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 1620

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 1986

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 1.29 \begin {gather*} -4 \left (16384 x^8-\frac {\log (x)}{4}+\frac {1}{4} \log (-x+20 \log (3)+x \log (243))\right ) \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.75, size = 22, normalized size = 0.92 \begin {gather*} -65536 \, x^{8} - \log \left (5 \, {\left (x + 4\right )} \log \relax (3) - x\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.20, size = 147, normalized size = 6.12 \begin {gather*} -\frac {65536 \, {\left (390625 \, x^{8} \log \relax (3)^{8} - 625000 \, x^{8} \log \relax (3)^{7} + 437500 \, x^{8} \log \relax (3)^{6} - 175000 \, x^{8} \log \relax (3)^{5} + 43750 \, x^{8} \log \relax (3)^{4} - 7000 \, x^{8} \log \relax (3)^{3} + 700 \, x^{8} \log \relax (3)^{2} - 40 \, x^{8} \log \relax (3) + x^{8}\right )}}{390625 \, \log \relax (3)^{8} - 625000 \, \log \relax (3)^{7} + 437500 \, \log \relax (3)^{6} - 175000 \, \log \relax (3)^{5} + 43750 \, \log \relax (3)^{4} - 7000 \, \log \relax (3)^{3} + 700 \, \log \relax (3)^{2} - 40 \, \log \relax (3) + 1} - \log \left ({\left | 5 \, x \log \relax (3) - x + 20 \, \log \relax (3) \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.34, size = 25, normalized size = 1.04

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.38, size = 24, normalized size = 1.00 \begin {gather*} -65536 \, x^{8} - \log \left (x {\left (5 \, \log \relax (3) - 1\right )} + 20 \, \log \relax (3)\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.18, size = 325, normalized size = 13.54 \begin {gather*} \frac {10\,x^6\,\ln \relax (3)\,\left (\frac {10485760\,\ln \relax (3)}{\ln \left (243\right )-1}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-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 \relax (3)-524288\right )}{8\,\left (\ln \left (243\right )-1\right )}-x^7\,\left (\frac {10485760\,\ln \relax (3)}{7\,\left (\ln \left (243\right )-1\right )}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-524288\right )}{7\,{\left (\ln \left (243\right )-1\right )}^2}\right )-\frac {64000000\,x\,{\ln \relax (3)}^6\,\left (\frac {10485760\,\ln \relax (3)}{\ln \left (243\right )-1}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^6}-\frac {80\,x^5\,{\ln \relax (3)}^2\,\left (\frac {10485760\,\ln \relax (3)}{\ln \left (243\right )-1}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^2}+\frac {2000\,x^4\,{\ln \relax (3)}^3\,\left (\frac {10485760\,\ln \relax (3)}{\ln \left (243\right )-1}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-524288\right )}{{\left (\ln \left (243\right )-1\right )}^2}\right )}{{\left (\ln \left (243\right )-1\right )}^3}-\frac {160000\,x^3\,{\ln \relax (3)}^4\,\left (\frac {10485760\,\ln \relax (3)}{\ln \left (243\right )-1}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-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 \relax (3)}^5\,\left (\frac {10485760\,\ln \relax (3)}{\ln \left (243\right )-1}-\frac {20\,\ln \relax (3)\,\left (2621440\,\ln \relax (3)-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 \relax (3)}+1{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.51, size = 22, normalized size = 0.92 \begin {gather*} - 65536 x^{8} + \log {\relax (x )} - \log {\left (x + \frac {20 \log {\relax (3 )}}{-1 + 5 \log {\relax (3 )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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