Integrand size = 61, antiderivative size = 24 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=3+x+x^2+\log (4)+\frac {x}{2+\frac {1}{3} (x-\log (5))} \] Output:
x+3+2*ln(2)+x^2+x/(-1/3*ln(5)+1/3*x+2)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=(6+x-\log (5))^2+\frac {3 (-6+\log (5))}{6+x-\log (5)}+(6+x-\log (5)) (-11+2 \log (5)) \] Input:
Integrate[(54 + 84*x + 25*x^2 + 2*x^3 + (-15 - 26*x - 4*x^2)*Log[5] + (1 + 2*x)*Log[5]^2)/(36 + 12*x + x^2 + (-12 - 2*x)*Log[5] + Log[5]^2),x]
Output:
(6 + x - Log[5])^2 + (3*(-6 + Log[5]))/(6 + x - Log[5]) + (6 + x - Log[5]) *(-11 + 2*Log[5])
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {2452, 2188, 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.
\(\displaystyle \int \frac {2 x^3+25 x^2+\left (-4 x^2-26 x-15\right ) \log (5)+84 x+(2 x+1) \log ^2(5)+54}{x^2+12 x+(-2 x-12) \log (5)+36+\log ^2(5)} \, dx\) |
\(\Big \downarrow \) 2452 |
\(\displaystyle \int \frac {2 x^3+25 x^2+\left (-4 x^2-26 x-15\right ) \log (5)+84 x+(2 x+1) \log ^2(5)+54}{x^2+2 x (6-\log (5))+(\log (5)-6)^2}dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (-\frac {3 (\log (5)-6)}{x^2+2 x (6-\log (5))+(\log (5)-6)^2}+2 x+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2+x-\frac {3 (6-\log (5))}{x+6-\log (5)}\) |
Input:
Int[(54 + 84*x + 25*x^2 + 2*x^3 + (-15 - 26*x - 4*x^2)*Log[5] + (1 + 2*x)* Log[5]^2)/(36 + 12*x + x^2 + (-12 - 2*x)*Log[5] + Log[5]^2),x]
Output:
x + x^2 - (3*(6 - Log[5]))/(6 + x - Log[5])
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
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]
Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
default | \(x +x^{2}-\frac {-3 \ln \left (5\right )+18}{-\ln \left (5\right )+x +6}\) | \(23\) |
risch | \(x^{2}+x -\frac {3 \ln \left (5\right )}{\ln \left (5\right )-x -6}+\frac {18}{\ln \left (5\right )-x -6}\) | \(30\) |
norman | \(\frac {-x^{3}+\left (-7+\ln \left (5\right )\right ) x^{2}+54+\ln \left (5\right )^{2}-15 \ln \left (5\right )}{\ln \left (5\right )-x -6}\) | \(34\) |
gosper | \(\frac {x^{2} \ln \left (5\right )-x^{3}+\ln \left (5\right )^{2}-7 x^{2}-15 \ln \left (5\right )+54}{\ln \left (5\right )-x -6}\) | \(37\) |
parallelrisch | \(\frac {x^{2} \ln \left (5\right )-x^{3}+\ln \left (5\right )^{2}-7 x^{2}-15 \ln \left (5\right )+54}{\ln \left (5\right )-x -6}\) | \(37\) |
meijerg | \(\left (-4 \ln \left (5\right )+25\right ) \left (-\ln \left (5\right )+6\right ) \left (\frac {x \left (\frac {3 x}{-\ln \left (5\right )+6}+6\right )}{3 \left (-\ln \left (5\right )+6\right ) \left (1+\frac {x}{-\ln \left (5\right )+6}\right )}-2 \ln \left (1+\frac {x}{-\ln \left (5\right )+6}\right )\right )+\left (-2 \ln \left (5\right )+12\right ) \left (-\ln \left (5\right )+6\right ) \left (-\frac {x \left (-\frac {2 x^{2}}{\left (-\ln \left (5\right )+6\right )^{2}}+\frac {6 x}{-\ln \left (5\right )+6}+12\right )}{4 \left (-\ln \left (5\right )+6\right ) \left (1+\frac {x}{-\ln \left (5\right )+6}\right )}+3 \ln \left (1+\frac {x}{-\ln \left (5\right )+6}\right )\right )+\left (2 \ln \left (5\right )^{2}-26 \ln \left (5\right )+84\right ) \left (-\frac {x}{\left (-\ln \left (5\right )+6\right ) \left (1+\frac {x}{-\ln \left (5\right )+6}\right )}+\ln \left (1+\frac {x}{-\ln \left (5\right )+6}\right )\right )+\frac {\ln \left (5\right )^{2} x}{\left (-\ln \left (5\right )+6\right )^{2} \left (1+\frac {x}{-\ln \left (5\right )+6}\right )}-\frac {15 \ln \left (5\right ) x}{\left (-\ln \left (5\right )+6\right )^{2} \left (1+\frac {x}{-\ln \left (5\right )+6}\right )}+\frac {54 x}{\left (-\ln \left (5\right )+6\right )^{2} \left (1+\frac {x}{-\ln \left (5\right )+6}\right )}\) | \(281\) |
Input:
int(((1+2*x)*ln(5)^2+(-4*x^2-26*x-15)*ln(5)+2*x^3+25*x^2+84*x+54)/(ln(5)^2 +(-2*x-12)*ln(5)+x^2+12*x+36),x,method=_RETURNVERBOSE)
Output:
x+x^2-(-3*ln(5)+18)/(-ln(5)+x+6)
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=\frac {x^{3} + 7 \, x^{2} - {\left (x^{2} + x - 3\right )} \log \left (5\right ) + 6 \, x - 18}{x - \log \left (5\right ) + 6} \] Input:
integrate(((1+2*x)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/ (log(5)^2+(-2*x-12)*log(5)+x^2+12*x+36),x, algorithm="fricas")
Output:
(x^3 + 7*x^2 - (x^2 + x - 3)*log(5) + 6*x - 18)/(x - log(5) + 6)
Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=x^{2} + x + \frac {-18 + 3 \log {\left (5 \right )}}{x - \log {\left (5 \right )} + 6} \] Input:
integrate(((1+2*x)*ln(5)**2+(-4*x**2-26*x-15)*ln(5)+2*x**3+25*x**2+84*x+54 )/(ln(5)**2+(-2*x-12)*ln(5)+x**2+12*x+36),x)
Output:
x**2 + x + (-18 + 3*log(5))/(x - log(5) + 6)
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=x^{2} + x + \frac {3 \, {\left (\log \left (5\right ) - 6\right )}}{x - \log \left (5\right ) + 6} \] Input:
integrate(((1+2*x)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/ (log(5)^2+(-2*x-12)*log(5)+x^2+12*x+36),x, algorithm="maxima")
Output:
x^2 + x + 3*(log(5) - 6)/(x - log(5) + 6)
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=x^{2} + x + \frac {3 \, {\left (\log \left (5\right ) - 6\right )}}{x - \log \left (5\right ) + 6} \] Input:
integrate(((1+2*x)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/ (log(5)^2+(-2*x-12)*log(5)+x^2+12*x+36),x, algorithm="giac")
Output:
x^2 + x + 3*(log(5) - 6)/(x - log(5) + 6)
Time = 1.57 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=x\,\left (2\,\ln \left (25\right )-\ln \left (625\right )+1\right )+x^2-\frac {\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}-\ln \left (25\right )\,1{}\mathrm {i}+12{}\mathrm {i}}{\sqrt {\ln \left (25\right )-2\,\ln \left (5\right )}\,\sqrt {2\,\ln \left (5\right )+\ln \left (25\right )-24}}\right )\,\left (\ln \left (125\right )-18\right )\,2{}\mathrm {i}}{\sqrt {\ln \left (25\right )-2\,\ln \left (5\right )}\,\sqrt {2\,\ln \left (5\right )+\ln \left (25\right )-24}} \] Input:
int((84*x - log(5)*(26*x + 4*x^2 + 15) + log(5)^2*(2*x + 1) + 25*x^2 + 2*x ^3 + 54)/(12*x - log(5)*(2*x + 12) + log(5)^2 + x^2 + 36),x)
Output:
x*(2*log(25) - log(625) + 1) + x^2 - (atan((x*2i - log(25)*1i + 12i)/((log (25) - 2*log(5))^(1/2)*(2*log(5) + log(25) - 24)^(1/2)))*(log(125) - 18)*2 i)/((log(25) - 2*log(5))^(1/2)*(2*log(5) + log(25) - 24)^(1/2))
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {54+84 x+25 x^2+2 x^3+\left (-15-26 x-4 x^2\right ) \log (5)+(1+2 x) \log ^2(5)}{36+12 x+x^2+(-12-2 x) \log (5)+\log ^2(5)} \, dx=\frac {x \left (\mathrm {log}\left (5\right ) x +\mathrm {log}\left (5\right )-x^{2}-7 x -9\right )}{\mathrm {log}\left (5\right )-x -6} \] Input:
int(((1+2*x)*log(5)^2+(-4*x^2-26*x-15)*log(5)+2*x^3+25*x^2+84*x+54)/(log(5 )^2+(-2*x-12)*log(5)+x^2+12*x+36),x)
Output:
(x*(log(5)*x + log(5) - x**2 - 7*x - 9))/(log(5) - x - 6)