Integrand size = 87, antiderivative size = 23 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {3}{25+x+\frac {4 x}{\log ^2(5)}}-\log (3-x) \] Output:
3/(4*x/ln(5)^2+x+25)-ln(3-x)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {3 \log ^2(5)}{25 \log ^2(5)+x \left (4+\log ^2(5)\right )}-\log (-3+x) \] Input:
Integrate[(-16*x^2 + (36 - 212*x - 8*x^2)*Log[5]^2 + (-616 - 53*x - x^2)*L og[5]^4)/(-48*x^2 + 16*x^3 + (-600*x + 176*x^2 + 8*x^3)*Log[5]^2 + (-1875 + 475*x + 47*x^2 + x^3)*Log[5]^4),x]
Output:
(3*Log[5]^2)/(25*Log[5]^2 + x*(4 + Log[5]^2)) - Log[-3 + x]
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {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.
\(\displaystyle \int \frac {-16 x^2+\left (-x^2-53 x-616\right ) \log ^4(5)+\left (-8 x^2-212 x+36\right ) \log ^2(5)}{16 x^3-48 x^2+\left (x^3+47 x^2+475 x-1875\right ) \log ^4(5)+\left (8 x^3+176 x^2-600 x\right ) \log ^2(5)} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {1}{3-x}+\frac {3 \log ^2(5) \left (-4-\log ^2(5)\right )}{\left (x \left (4+\log ^2(5)\right )+25 \log ^2(5)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \log ^2(5)}{x \left (4+\log ^2(5)\right )+25 \log ^2(5)}-\log (3-x)\) |
Input:
Int[(-16*x^2 + (36 - 212*x - 8*x^2)*Log[5]^2 + (-616 - 53*x - x^2)*Log[5]^ 4)/(-48*x^2 + 16*x^3 + (-600*x + 176*x^2 + 8*x^3)*Log[5]^2 + (-1875 + 475* x + 47*x^2 + x^3)*Log[5]^4),x]
Output:
(3*Log[5]^2)/(25*Log[5]^2 + x*(4 + Log[5]^2)) - Log[3 - 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 = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39
method | result | size |
default | \(\frac {3 \ln \left (5\right )^{2}}{x \ln \left (5\right )^{2}+25 \ln \left (5\right )^{2}+4 x}-\ln \left (-3+x \right )\) | \(32\) |
risch | \(\frac {3 \ln \left (5\right )^{2}}{x \ln \left (5\right )^{2}+25 \ln \left (5\right )^{2}+4 x}-\ln \left (-3+x \right )\) | \(32\) |
norman | \(-\frac {\left (3 \ln \left (5\right )^{4}+12 \ln \left (5\right )^{2}\right ) x}{25 \ln \left (5\right )^{2} \left (x \ln \left (5\right )^{2}+25 \ln \left (5\right )^{2}+4 x \right )}-\ln \left (-3+x \right )\) | \(46\) |
parallelrisch | \(-\frac {25 \ln \left (-3+x \right ) \ln \left (5\right )^{4} x +625 \ln \left (-3+x \right ) \ln \left (5\right )^{4}+3 \ln \left (5\right )^{4} x +100 \ln \left (-3+x \right ) \ln \left (5\right )^{2} x +12 x \ln \left (5\right )^{2}}{25 \ln \left (5\right )^{2} \left (x \ln \left (5\right )^{2}+25 \ln \left (5\right )^{2}+4 x \right )}\) | \(72\) |
Input:
int(((-x^2-53*x-616)*ln(5)^4+(-8*x^2-212*x+36)*ln(5)^2-16*x^2)/((x^3+47*x^ 2+475*x-1875)*ln(5)^4+(8*x^3+176*x^2-600*x)*ln(5)^2+16*x^3-48*x^2),x,metho d=_RETURNVERBOSE)
Output:
3*ln(5)^2/(x*ln(5)^2+25*ln(5)^2+4*x)-ln(-3+x)
Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {3 \, \log \left (5\right )^{2} - {\left ({\left (x + 25\right )} \log \left (5\right )^{2} + 4 \, x\right )} \log \left (x - 3\right )}{{\left (x + 25\right )} \log \left (5\right )^{2} + 4 \, x} \] Input:
integrate(((-x^2-53*x-616)*log(5)^4+(-8*x^2-212*x+36)*log(5)^2-16*x^2)/((x ^3+47*x^2+475*x-1875)*log(5)^4+(8*x^3+176*x^2-600*x)*log(5)^2+16*x^3-48*x^ 2),x, algorithm="fricas")
Output:
(3*log(5)^2 - ((x + 25)*log(5)^2 + 4*x)*log(x - 3))/((x + 25)*log(5)^2 + 4 *x)
Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=- \log {\left (x - 3 \right )} + \frac {3 \log {\left (5 \right )}^{2}}{x \left (\log {\left (5 \right )}^{2} + 4\right ) + 25 \log {\left (5 \right )}^{2}} \] Input:
integrate(((-x**2-53*x-616)*ln(5)**4+(-8*x**2-212*x+36)*ln(5)**2-16*x**2)/ ((x**3+47*x**2+475*x-1875)*ln(5)**4+(8*x**3+176*x**2-600*x)*ln(5)**2+16*x* *3-48*x**2),x)
Output:
-log(x - 3) + 3*log(5)**2/(x*(log(5)**2 + 4) + 25*log(5)**2)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {3 \, \log \left (5\right )^{2}}{{\left (\log \left (5\right )^{2} + 4\right )} x + 25 \, \log \left (5\right )^{2}} - \log \left (x - 3\right ) \] Input:
integrate(((-x^2-53*x-616)*log(5)^4+(-8*x^2-212*x+36)*log(5)^2-16*x^2)/((x ^3+47*x^2+475*x-1875)*log(5)^4+(8*x^3+176*x^2-600*x)*log(5)^2+16*x^3-48*x^ 2),x, algorithm="maxima")
Output:
3*log(5)^2/((log(5)^2 + 4)*x + 25*log(5)^2) - log(x - 3)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {3 \, \log \left (5\right )^{2}}{x \log \left (5\right )^{2} + 25 \, \log \left (5\right )^{2} + 4 \, x} - \log \left ({\left | x - 3 \right |}\right ) \] Input:
integrate(((-x^2-53*x-616)*log(5)^4+(-8*x^2-212*x+36)*log(5)^2-16*x^2)/((x ^3+47*x^2+475*x-1875)*log(5)^4+(8*x^3+176*x^2-600*x)*log(5)^2+16*x^3-48*x^ 2),x, algorithm="giac")
Output:
3*log(5)^2/(x*log(5)^2 + 25*log(5)^2 + 4*x) - log(abs(x - 3))
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {3\,{\ln \left (5\right )}^2}{4\,x+x\,{\ln \left (5\right )}^2+25\,{\ln \left (5\right )}^2}-\ln \left (x-3\right ) \] Input:
int(-(log(5)^4*(53*x + x^2 + 616) + log(5)^2*(212*x + 8*x^2 - 36) + 16*x^2 )/(log(5)^4*(475*x + 47*x^2 + x^3 - 1875) + log(5)^2*(176*x^2 - 600*x + 8* x^3) - 48*x^2 + 16*x^3),x)
Output:
(3*log(5)^2)/(4*x + x*log(5)^2 + 25*log(5)^2) - log(x - 3)
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {-16 x^2+\left (36-212 x-8 x^2\right ) \log ^2(5)+\left (-616-53 x-x^2\right ) \log ^4(5)}{-48 x^2+16 x^3+\left (-600 x+176 x^2+8 x^3\right ) \log ^2(5)+\left (-1875+475 x+47 x^2+x^3\right ) \log ^4(5)} \, dx=\frac {-25 \,\mathrm {log}\left (x -3\right ) \mathrm {log}\left (5\right )^{2} x -625 \,\mathrm {log}\left (x -3\right ) \mathrm {log}\left (5\right )^{2}-100 \,\mathrm {log}\left (x -3\right ) x -3 \mathrm {log}\left (5\right )^{2} x -12 x}{25 \mathrm {log}\left (5\right )^{2} x +625 \mathrm {log}\left (5\right )^{2}+100 x} \] Input:
int(((-x^2-53*x-616)*log(5)^4+(-8*x^2-212*x+36)*log(5)^2-16*x^2)/((x^3+47* x^2+475*x-1875)*log(5)^4+(8*x^3+176*x^2-600*x)*log(5)^2+16*x^3-48*x^2),x)
Output:
( - 25*log(x - 3)*log(5)**2*x - 625*log(x - 3)*log(5)**2 - 100*log(x - 3)* x - 3*log(5)**2*x - 12*x)/(25*(log(5)**2*x + 25*log(5)**2 + 4*x))