Integrand size = 82, antiderivative size = 34 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=\left (1+x^2\right ) \left (x-\frac {(4+x)^2}{2+\frac {4}{x}-x}\right )-\log (5) \log (x) \] Output:
(x-(4+x)^2/(4/x+2-x))*(x^2+1)-ln(5)*ln(x)
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=42 x+10 x^2+2 x^3+\frac {40 (13+11 x)}{-4-2 x+x^2}-\log (5) \log (x) \] Input:
Integrate[(-48*x - 48*x^2 - 192*x^3 - 152*x^4 - 62*x^5 - 4*x^6 + 6*x^7 + ( -16 - 16*x + 4*x^2 + 4*x^3 - x^4)*Log[5])/(16*x + 16*x^2 - 4*x^3 - 4*x^4 + x^5),x]
Output:
42*x + 10*x^2 + 2*x^3 + (40*(13 + 11*x))/(-4 - 2*x + x^2) - Log[5]*Log[x]
Time = 0.41 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2026, 2462, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {6 x^7-4 x^6-62 x^5-152 x^4-192 x^3-48 x^2+\left (-x^4+4 x^3+4 x^2-16 x-16\right ) \log (5)-48 x}{x^5-4 x^4-4 x^3+16 x^2+16 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {6 x^7-4 x^6-62 x^5-152 x^4-192 x^3-48 x^2+\left (-x^4+4 x^3+4 x^2-16 x-16\right ) \log (5)-48 x}{x \left (x^4-4 x^3-4 x^2+16 x+16\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (6 x^2-\frac {440}{x^2-2 x-4}-\frac {80 (24 x+31)}{\left (x^2-2 x-4\right )^2}+20 x-\frac {\log (5)}{x}+42\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 x^3+10 x^2-\frac {40 (11 x+13)}{-x^2+2 x+4}+42 x-\log (5) \log (x)\) |
Input:
Int[(-48*x - 48*x^2 - 192*x^3 - 152*x^4 - 62*x^5 - 4*x^6 + 6*x^7 + (-16 - 16*x + 4*x^2 + 4*x^3 - x^4)*Log[5])/(16*x + 16*x^2 - 4*x^3 - 4*x^4 + x^5), x]
Output:
42*x + 10*x^2 + 2*x^3 - (40*(13 + 11*x))/(4 + 2*x - x^2) - Log[5]*Log[x]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09
method | result | size |
risch | \(2 x^{3}+10 x^{2}+42 x +\frac {440 x +520}{x^{2}-2 x -4}-\ln \left (5\right ) \ln \left (x \right )\) | \(37\) |
default | \(2 x^{3}+10 x^{2}+42 x -\ln \left (5\right ) \ln \left (x \right )-\frac {40 \left (-11 x -13\right )}{x^{2}-2 x -4}\) | \(38\) |
norman | \(\frac {2 x^{5}+6 x^{4}+14 x^{3}+24 x +24}{x^{2}-2 x -4}-\ln \left (5\right ) \ln \left (x \right )\) | \(39\) |
parallelrisch | \(-\frac {-2 x^{5}+x^{2} \ln \left (5\right ) \ln \left (x \right )-6 x^{4}-2 x \ln \left (5\right ) \ln \left (x \right )-14 x^{3}-24-4 \ln \left (5\right ) \ln \left (x \right )-24 x}{x^{2}-2 x -4}\) | \(54\) |
Input:
int(((-x^4+4*x^3+4*x^2-16*x-16)*ln(5)+6*x^7-4*x^6-62*x^5-152*x^4-192*x^3-4 8*x^2-48*x)/(x^5-4*x^4-4*x^3+16*x^2+16*x),x,method=_RETURNVERBOSE)
Output:
2*x^3+10*x^2+42*x+(440*x+520)/(x^2-2*x-4)-ln(5)*ln(x)
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=\frac {2 \, x^{5} + 6 \, x^{4} + 14 \, x^{3} - {\left (x^{2} - 2 \, x - 4\right )} \log \left (5\right ) \log \left (x\right ) - 124 \, x^{2} + 272 \, x + 520}{x^{2} - 2 \, x - 4} \] Input:
integrate(((-x^4+4*x^3+4*x^2-16*x-16)*log(5)+6*x^7-4*x^6-62*x^5-152*x^4-19 2*x^3-48*x^2-48*x)/(x^5-4*x^4-4*x^3+16*x^2+16*x),x, algorithm="fricas")
Output:
(2*x^5 + 6*x^4 + 14*x^3 - (x^2 - 2*x - 4)*log(5)*log(x) - 124*x^2 + 272*x + 520)/(x^2 - 2*x - 4)
Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=2 x^{3} + 10 x^{2} + 42 x + \frac {440 x + 520}{x^{2} - 2 x - 4} - \log {\left (5 \right )} \log {\left (x \right )} \] Input:
integrate(((-x**4+4*x**3+4*x**2-16*x-16)*ln(5)+6*x**7-4*x**6-62*x**5-152*x **4-192*x**3-48*x**2-48*x)/(x**5-4*x**4-4*x**3+16*x**2+16*x),x)
Output:
2*x**3 + 10*x**2 + 42*x + (440*x + 520)/(x**2 - 2*x - 4) - log(5)*log(x)
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=2 \, x^{3} + 10 \, x^{2} - \log \left (5\right ) \log \left (x\right ) + 42 \, x + \frac {40 \, {\left (11 \, x + 13\right )}}{x^{2} - 2 \, x - 4} \] Input:
integrate(((-x^4+4*x^3+4*x^2-16*x-16)*log(5)+6*x^7-4*x^6-62*x^5-152*x^4-19 2*x^3-48*x^2-48*x)/(x^5-4*x^4-4*x^3+16*x^2+16*x),x, algorithm="maxima")
Output:
2*x^3 + 10*x^2 - log(5)*log(x) + 42*x + 40*(11*x + 13)/(x^2 - 2*x - 4)
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=2 \, x^{3} + 10 \, x^{2} - \log \left (5\right ) \log \left ({\left | x \right |}\right ) + 42 \, x + \frac {40 \, {\left (11 \, x + 13\right )}}{x^{2} - 2 \, x - 4} \] Input:
integrate(((-x^4+4*x^3+4*x^2-16*x-16)*log(5)+6*x^7-4*x^6-62*x^5-152*x^4-19 2*x^3-48*x^2-48*x)/(x^5-4*x^4-4*x^3+16*x^2+16*x),x, algorithm="giac")
Output:
2*x^3 + 10*x^2 - log(5)*log(abs(x)) + 42*x + 40*(11*x + 13)/(x^2 - 2*x - 4 )
Time = 4.13 (sec) , antiderivative size = 325, normalized size of antiderivative = 9.56 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=42\,x-\ln \left (5\right )\,\ln \left (x\right )+10\,x^2+2\,x^3-\frac {\frac {16\,\ln \left (5\right )}{5}-\frac {4\,\ln \left (625\right )}{5}+x\,\left (\frac {14\,\ln \left (5\right )}{5}-\frac {7\,\ln \left (625\right )}{10}+440\right )+520}{-x^2+2\,x+4}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {\frac {\sqrt {5}\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )\,\left (\frac {64\,\ln \left (5\right )}{5}-\frac {6\,\ln \left (625\right )}{5}+x\,\left (\frac {72\,\ln \left (5\right )}{5}-\frac {3\,\ln \left (625\right )}{5}\right )-\frac {3\,\sqrt {5}\,\left (32\,x+8\right )\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )}{100}\right )\,3{}\mathrm {i}}{100}+\frac {\sqrt {5}\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )\,\left (\frac {64\,\ln \left (5\right )}{5}-\frac {6\,\ln \left (625\right )}{5}+x\,\left (\frac {72\,\ln \left (5\right )}{5}-\frac {3\,\ln \left (625\right )}{5}\right )+\frac {3\,\sqrt {5}\,\left (32\,x+8\right )\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )}{100}\right )\,3{}\mathrm {i}}{100}}{2\,x\,\left (\frac {3\,\ln \left (5\right )\,\ln \left (625\right )}{25}+\frac {24\,{\ln \left (5\right )}^2}{25}-\frac {9\,{\ln \left (625\right )}^2}{100}\right )-\frac {12\,\ln \left (5\right )\,\ln \left (625\right )}{5}+\frac {48\,{\ln \left (5\right )}^2}{5}-\frac {3\,\sqrt {5}\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )\,\left (\frac {64\,\ln \left (5\right )}{5}-\frac {6\,\ln \left (625\right )}{5}+x\,\left (\frac {72\,\ln \left (5\right )}{5}-\frac {3\,\ln \left (625\right )}{5}\right )-\frac {3\,\sqrt {5}\,\left (32\,x+8\right )\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )}{100}\right )}{100}+\frac {3\,\sqrt {5}\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )\,\left (\frac {64\,\ln \left (5\right )}{5}-\frac {6\,\ln \left (625\right )}{5}+x\,\left (\frac {72\,\ln \left (5\right )}{5}-\frac {3\,\ln \left (625\right )}{5}\right )+\frac {3\,\sqrt {5}\,\left (32\,x+8\right )\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )}{100}\right )}{100}}\right )\,\left (4\,\ln \left (5\right )-\ln \left (625\right )\right )\,3{}\mathrm {i}}{50} \] Input:
int(-(48*x + log(5)*(16*x - 4*x^2 - 4*x^3 + x^4 + 16) + 48*x^2 + 192*x^3 + 152*x^4 + 62*x^5 + 4*x^6 - 6*x^7)/(16*x + 16*x^2 - 4*x^3 - 4*x^4 + x^5),x )
Output:
42*x - log(5)*log(x) + 10*x^2 + 2*x^3 - ((16*log(5))/5 - (4*log(625))/5 + x*((14*log(5))/5 - (7*log(625))/10 + 440) + 520)/(2*x - x^2 + 4) - (5^(1/2 )*atan(((5^(1/2)*(4*log(5) - log(625))*((64*log(5))/5 - (6*log(625))/5 + x *((72*log(5))/5 - (3*log(625))/5) - (3*5^(1/2)*(32*x + 8)*(4*log(5) - log( 625)))/100)*3i)/100 + (5^(1/2)*(4*log(5) - log(625))*((64*log(5))/5 - (6*l og(625))/5 + x*((72*log(5))/5 - (3*log(625))/5) + (3*5^(1/2)*(32*x + 8)*(4 *log(5) - log(625)))/100)*3i)/100)/(2*x*((3*log(5)*log(625))/25 + (24*log( 5)^2)/25 - (9*log(625)^2)/100) - (12*log(5)*log(625))/5 + (48*log(5)^2)/5 - (3*5^(1/2)*(4*log(5) - log(625))*((64*log(5))/5 - (6*log(625))/5 + x*((7 2*log(5))/5 - (3*log(625))/5) - (3*5^(1/2)*(32*x + 8)*(4*log(5) - log(625) ))/100))/100 + (3*5^(1/2)*(4*log(5) - log(625))*((64*log(5))/5 - (6*log(62 5))/5 + x*((72*log(5))/5 - (3*log(625))/5) + (3*5^(1/2)*(32*x + 8)*(4*log( 5) - log(625)))/100))/100))*(4*log(5) - log(625))*3i)/50
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.62 \[ \int \frac {-48 x-48 x^2-192 x^3-152 x^4-62 x^5-4 x^6+6 x^7+\left (-16-16 x+4 x^2+4 x^3-x^4\right ) \log (5)}{16 x+16 x^2-4 x^3-4 x^4+x^5} \, dx=\frac {-\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{2}+2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x +4 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right )+2 x^{5}+6 x^{4}+14 x^{3}+12 x^{2}-24}{x^{2}-2 x -4} \] Input:
int(((-x^4+4*x^3+4*x^2-16*x-16)*log(5)+6*x^7-4*x^6-62*x^5-152*x^4-192*x^3- 48*x^2-48*x)/(x^5-4*x^4-4*x^3+16*x^2+16*x),x)
Output:
( - log(x)*log(5)*x**2 + 2*log(x)*log(5)*x + 4*log(x)*log(5) + 2*x**5 + 6* x**4 + 14*x**3 + 12*x**2 - 24)/(x**2 - 2*x - 4)