\(\int \frac {5 x^6+48 x^7+112 x^8+(80-40 x^5-384 x^6-896 x^7) \log (3)+(80 x^4+768 x^5+1792 x^6) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx\) [6170]

Optimal result

Integrand size = 75, antiderivative size = 26 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=-2+x^5 (1+4 x)^2+\frac {20 x}{-x+4 \log (3)} \]

[Out]

(1+4*x)^2*x^5+20/(4*ln(3)-x)*x-2

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {27, 1864} \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=16 x^7+8 x^6+x^5-\frac {80 \log (3)}{x-4 \log (3)} \]

[In]

Int[(5*x^6 + 48*x^7 + 112*x^8 + (80 - 40*x^5 - 384*x^6 - 896*x^7)*Log[3] + (80*x^4 + 768*x^5 + 1792*x^6)*Log[3
]^2)/(x^2 - 8*x*Log[3] + 16*Log[3]^2),x]

[Out]

x^5 + 8*x^6 + 16*x^7 - (80*Log[3])/(x - 4*Log[3])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{(x-4 \log (3))^2} \, dx \\ & = \int \left (5 x^4+48 x^5+112 x^6+\frac {80 \log (3)}{(x-4 \log (3))^2}\right ) \, dx \\ & = x^5+8 x^6+16 x^7-\frac {80 \log (3)}{x-4 \log (3)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(26)=52\).

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=\frac {16 x^8+x^7 (8-64 \log (3))+x^6 (1-32 \log (3))-4 x^5 \log (3)-1024 x \log ^5(3) (1+16 \log (3))^2+16 \log (3) \left (-5+256 \log ^5(3)+8192 \log ^6(3)+65536 \log ^7(3)\right )}{x-4 \log (3)} \]

[In]

Integrate[(5*x^6 + 48*x^7 + 112*x^8 + (80 - 40*x^5 - 384*x^6 - 896*x^7)*Log[3] + (80*x^4 + 768*x^5 + 1792*x^6)
*Log[3]^2)/(x^2 - 8*x*Log[3] + 16*Log[3]^2),x]

[Out]

(16*x^8 + x^7*(8 - 64*Log[3]) + x^6*(1 - 32*Log[3]) - 4*x^5*Log[3] - 1024*x*Log[3]^5*(1 + 16*Log[3])^2 + 16*Lo
g[3]*(-5 + 256*Log[3]^5 + 8192*Log[3]^6 + 65536*Log[3]^7))/(x - 4*Log[3])

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04

method	result	size
default	\(16 x^{7}+8 x^{6}+x^{5}-\frac {80 \ln \left (3\right )}{-4 \ln \left (3\right )+x}\)	\(27\)
risch	\(16 x^{7}+8 x^{6}+x^{5}+\frac {20 \ln \left (3\right )}{\ln \left (3\right )-\frac {x}{4}}\)	\(27\)
norman	\(\frac {-16 x^{8}+\left (64 \ln \left (3\right )-8\right ) x^{7}+\left (32 \ln \left (3\right )-1\right ) x^{6}+4 x^{5} \ln \left (3\right )+80 \ln \left (3\right )}{4 \ln \left (3\right )-x}\)	\(49\)
gosper	\(\frac {64 \ln \left (3\right ) x^{7}-16 x^{8}+32 x^{6} \ln \left (3\right )-8 x^{7}+4 x^{5} \ln \left (3\right )-x^{6}+80 \ln \left (3\right )}{4 \ln \left (3\right )-x}\)	\(53\)
parallelrisch	\(\frac {64 \ln \left (3\right ) x^{7}-16 x^{8}+32 x^{6} \ln \left (3\right )-8 x^{7}+4 x^{5} \ln \left (3\right )-x^{6}+80 \ln \left (3\right )}{4 \ln \left (3\right )-x}\)	\(53\)
meijerg	\(-1835008 \ln \left (3\right )^{7} \left (-\frac {x \left (-\frac {45 x^{7}}{16384 \ln \left (3\right )^{7}}-\frac {15 x^{6}}{1024 \ln \left (3\right )^{6}}-\frac {21 x^{5}}{256 \ln \left (3\right )^{5}}-\frac {63 x^{4}}{128 \ln \left (3\right )^{4}}-\frac {105 x^{3}}{32 \ln \left (3\right )^{3}}-\frac {105 x^{2}}{4 \ln \left (3\right )^{2}}-\frac {315 x}{\ln \left (3\right )}+2520\right )}{1260 \ln \left (3\right ) \left (1-\frac {x}{4 \ln \left (3\right )}\right )}-8 \ln \left (1-\frac {x}{4 \ln \left (3\right )}\right )\right )+65536 \ln \left (3\right )^{6} \left (-56 \ln \left (3\right )+3\right ) \left (\frac {x \left (-\frac {5 x^{6}}{1024 \ln \left (3\right )^{6}}-\frac {7 x^{5}}{256 \ln \left (3\right )^{5}}-\frac {21 x^{4}}{128 \ln \left (3\right )^{4}}-\frac {35 x^{3}}{32 \ln \left (3\right )^{3}}-\frac {35 x^{2}}{4 \ln \left (3\right )^{2}}-\frac {105 x}{\ln \left (3\right )}+840\right )}{480 \ln \left (3\right ) \left (1-\frac {x}{4 \ln \left (3\right )}\right )}+7 \ln \left (1-\frac {x}{4 \ln \left (3\right )}\right )\right )-16384 \ln \left (3\right )^{5} \left (112 \ln \left (3\right )^{2}-24 \ln \left (3\right )+\frac {5}{16}\right ) \left (-\frac {x \left (-\frac {7 x^{5}}{512 \ln \left (3\right )^{5}}-\frac {21 x^{4}}{256 \ln \left (3\right )^{4}}-\frac {35 x^{3}}{64 \ln \left (3\right )^{3}}-\frac {35 x^{2}}{8 \ln \left (3\right )^{2}}-\frac {105 x}{2 \ln \left (3\right )}+420\right )}{280 \ln \left (3\right ) \left (1-\frac {x}{4 \ln \left (3\right )}\right )}-6 \ln \left (1-\frac {x}{4 \ln \left (3\right )}\right )\right )+4096 \ln \left (3\right )^{4} \left (48 \ln \left (3\right )^{2}-\frac {5 \ln \left (3\right )}{2}\right ) \left (\frac {x \left (-\frac {3 x^{4}}{256 \ln \left (3\right )^{4}}-\frac {5 x^{3}}{64 \ln \left (3\right )^{3}}-\frac {5 x^{2}}{8 \ln \left (3\right )^{2}}-\frac {15 x}{2 \ln \left (3\right )}+60\right )}{48 \ln \left (3\right ) \left (1-\frac {x}{4 \ln \left (3\right )}\right )}+5 \ln \left (1-\frac {x}{4 \ln \left (3\right )}\right )\right )-5120 \ln \left (3\right )^{5} \left (-\frac {x \left (-\frac {5 x^{3}}{64 \ln \left (3\right )^{3}}-\frac {5 x^{2}}{8 \ln \left (3\right )^{2}}-\frac {15 x}{2 \ln \left (3\right )}+60\right )}{60 \ln \left (3\right ) \left (1-\frac {x}{4 \ln \left (3\right )}\right )}-4 \ln \left (1-\frac {x}{4 \ln \left (3\right )}\right )\right )+\frac {5 x}{\ln \left (3\right ) \left (1-\frac {x}{4 \ln \left (3\right )}\right )}\)	\(459\)

[In]

int(((1792*x^6+768*x^5+80*x^4)*ln(3)^2+(-896*x^7-384*x^6-40*x^5+80)*ln(3)+112*x^8+48*x^7+5*x^6)/(16*ln(3)^2-8*
x*ln(3)+x^2),x,method=_RETURNVERBOSE)

[Out]

16*x^7+8*x^6+x^5-80*ln(3)/(-4*ln(3)+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=\frac {16 \, x^{8} + 8 \, x^{7} + x^{6} - 4 \, {\left (16 \, x^{7} + 8 \, x^{6} + x^{5} + 20\right )} \log \left (3\right )}{x - 4 \, \log \left (3\right )} \]

[In]

integrate(((1792*x^6+768*x^5+80*x^4)*log(3)^2+(-896*x^7-384*x^6-40*x^5+80)*log(3)+112*x^8+48*x^7+5*x^6)/(16*lo
g(3)^2-8*x*log(3)+x^2),x, algorithm="fricas")

[Out]

(16*x^8 + 8*x^7 + x^6 - 4*(16*x^7 + 8*x^6 + x^5 + 20)*log(3))/(x - 4*log(3))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=16 x^{7} + 8 x^{6} + x^{5} - \frac {80 \log {\left (3 \right )}}{x - 4 \log {\left (3 \right )}} \]

[In]

integrate(((1792*x**6+768*x**5+80*x**4)*ln(3)**2+(-896*x**7-384*x**6-40*x**5+80)*ln(3)+112*x**8+48*x**7+5*x**6
)/(16*ln(3)**2-8*x*ln(3)+x**2),x)

[Out]

16*x**7 + 8*x**6 + x**5 - 80*log(3)/(x - 4*log(3))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=16 \, x^{7} + 8 \, x^{6} + x^{5} - \frac {80 \, \log \left (3\right )}{x - 4 \, \log \left (3\right )} \]

[In]

integrate(((1792*x^6+768*x^5+80*x^4)*log(3)^2+(-896*x^7-384*x^6-40*x^5+80)*log(3)+112*x^8+48*x^7+5*x^6)/(16*lo
g(3)^2-8*x*log(3)+x^2),x, algorithm="maxima")

[Out]

16*x^7 + 8*x^6 + x^5 - 80*log(3)/(x - 4*log(3))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=16 \, x^{7} + 8 \, x^{6} + x^{5} - \frac {80 \, \log \left (3\right )}{x - 4 \, \log \left (3\right )} \]

[In]

integrate(((1792*x^6+768*x^5+80*x^4)*log(3)^2+(-896*x^7-384*x^6-40*x^5+80)*log(3)+112*x^8+48*x^7+5*x^6)/(16*lo
g(3)^2-8*x*log(3)+x^2),x, algorithm="giac")

[Out]

16*x^7 + 8*x^6 + x^5 - 80*log(3)/(x - 4*log(3))

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {5 x^6+48 x^7+112 x^8+\left (80-40 x^5-384 x^6-896 x^7\right ) \log (3)+\left (80 x^4+768 x^5+1792 x^6\right ) \log ^2(3)}{x^2-8 x \log (3)+16 \log ^2(3)} \, dx=x^5-\frac {80\,\ln \left (3\right )}{x-4\,\ln \left (3\right )}+8\,x^6+16\,x^7 \]

[In]

int((log(3)^2*(80*x^4 + 768*x^5 + 1792*x^6) - log(3)*(40*x^5 + 384*x^6 + 896*x^7 - 80) + 5*x^6 + 48*x^7 + 112*
x^8)/(16*log(3)^2 - 8*x*log(3) + x^2),x)

[Out]

x^5 - (80*log(3))/(x - 4*log(3)) + 8*x^6 + 16*x^7