Integrand size = 115, antiderivative size = 30 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {x}{\left (-\frac {1}{5}-\frac {2}{x}-x^2\right ) \log \left (4+\frac {5}{8+x}\right )} \]
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 x^2}{\left (10+x+5 x^3\right ) \log \left (\frac {37+4 x}{8+x}\right )} \]
Integrate[(-250*x^2 - 25*x^3 - 125*x^5 + (-29600*x - 8380*x^2 - 745*x^3 + 7380*x^4 + 1725*x^5 + 100*x^6)*Log[(37 + 4*x)/(8 + x)])/((29600 + 12820*x + 2076*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725*x^7 + 100*x ^8)*Log[(37 + 4*x)/(8 + x)]^2),x]
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.
\(\displaystyle \int \frac {-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )}{\left (100 x^8+1725 x^7+7440 x^6+1090 x^5+9864 x^4+29749 x^3+2076 x^2+12820 x+29600\right ) \log ^2\left (\frac {4 x+37}{x+8}\right )} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )}{32716820 (x+8) \log ^2\left (\frac {4 x+37}{x+8}\right )}-\frac {16384 \left (-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )\right )}{320594245445 (4 x+37) \log ^2\left (\frac {4 x+37}{x+8}\right )}+\frac {\left (-37317230145 x^2+265035817480 x-1817856266829\right ) \left (-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )\right )}{419552968850395396 \left (5 x^3+x+10\right ) \log ^2\left (\frac {4 x+37}{x+8}\right )}+\frac {\left (89505 x^2-511400 x+2216181\right ) \left (-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )\right )}{647729086 \left (5 x^3+x+10\right )^2 \log ^2\left (\frac {4 x+37}{x+8}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {16384 \int \frac {x (x+8) \left (5 x^3-x-20\right )}{\log \left (\frac {4 x+37}{x+8}\right )}dx}{64118849089}+\frac {\int \frac {x (4 x+37) \left (5 x^3-x-20\right )}{\log \left (\frac {4 x+37}{x+8}\right )}dx}{6543364}+\frac {25 \int \frac {x^2 \left (37317230145 x^2-265035817480 x+1817856266829\right )}{\log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{419552968850395396}-\frac {25 \int \frac {x^2 \left (89505 x^2-511400 x+2216181\right )}{\left (5 x^3+x+10\right ) \log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{647729086}-\frac {5 \int \frac {x^2 \left (5 x^3+x+10\right )}{(x+8) \log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{6543364}+\frac {81920 \int \frac {x^2 \left (5 x^3+x+10\right )}{(4 x+37) \log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{64118849089}+\frac {5 \int \frac {x (x+8) (4 x+37) \left (89505 x^2-511400 x+2216181\right ) \left (5 x^3-x-20\right )}{\left (5 x^3+x+10\right )^2 \log \left (\frac {4 x+37}{x+8}\right )}dx}{647729086}-\frac {5 \int \frac {x (x+8) (4 x+37) \left (37317230145 x^2-265035817480 x+1817856266829\right ) \left (5 x^3-x-20\right )}{\left (5 x^3+x+10\right ) \log \left (\frac {4 x+37}{x+8}\right )}dx}{419552968850395396}\) |
Int[(-250*x^2 - 25*x^3 - 125*x^5 + (-29600*x - 8380*x^2 - 745*x^3 + 7380*x ^4 + 1725*x^5 + 100*x^6)*Log[(37 + 4*x)/(8 + x)])/((29600 + 12820*x + 2076 *x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725*x^7 + 100*x^8)*Lo g[(37 + 4*x)/(8 + x)]^2),x]
3.21.54.3.1 Defintions of rubi rules used
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] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 13.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\) | \(30\) |
parallelrisch | \(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\) | \(30\) |
derivativedivides | \(\frac {160}{1279 \ln \left (4+\frac {5}{x +8}\right )}+\frac {\frac {257200 \left (4+\frac {5}{x +8}\right )^{2}}{1279}-\frac {4015200}{1279}-\frac {11888625}{1279 \left (x +8\right )}}{\left (2558 \left (4+\frac {5}{x +8}\right )^{3}-35501 \left (4+\frac {5}{x +8}\right )^{2}+403679+\frac {821120}{x +8}\right ) \ln \left (4+\frac {5}{x +8}\right )}\) | \(89\) |
default | \(\frac {160}{1279 \ln \left (4+\frac {5}{x +8}\right )}+\frac {\frac {257200 \left (4+\frac {5}{x +8}\right )^{2}}{1279}-\frac {4015200}{1279}-\frac {11888625}{1279 \left (x +8\right )}}{\left (2558 \left (4+\frac {5}{x +8}\right )^{3}-35501 \left (4+\frac {5}{x +8}\right )^{2}+403679+\frac {821120}{x +8}\right ) \ln \left (4+\frac {5}{x +8}\right )}\) | \(89\) |
int(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*ln((4*x+37)/(x+8 ))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864*x^4+29 749*x^3+2076*x^2+12820*x+29600)/ln((4*x+37)/(x+8))^2,x,method=_RETURNVERBO SE)
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )} \]
integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+3 7)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864 *x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorithm= \
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=- \frac {5 x^{2}}{\left (5 x^{3} + x + 10\right ) \log {\left (\frac {4 x + 37}{x + 8} \right )}} \]
integrate(((100*x**6+1725*x**5+7380*x**4-745*x**3-8380*x**2-29600*x)*ln((4 *x+37)/(x+8))-125*x**5-25*x**3-250*x**2)/(100*x**8+1725*x**7+7440*x**6+109 0*x**5+9864*x**4+29749*x**3+2076*x**2+12820*x+29600)/ln((4*x+37)/(x+8))**2 ,x)
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (4 \, x + 37\right ) - {\left (5 \, x^{3} + x + 10\right )} \log \left (x + 8\right )} \]
integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+3 7)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864 *x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorithm= \
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (29) = 58\).
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.57 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {5 \, {\left (\frac {64 \, {\left (4 \, x + 37\right )}^{3}}{{\left (x + 8\right )}^{3}} - \frac {848 \, {\left (4 \, x + 37\right )}^{2}}{{\left (x + 8\right )}^{2}} + \frac {3737 \, {\left (4 \, x + 37\right )}}{x + 8} - 5476\right )}}{\frac {2558 \, {\left (4 \, x + 37\right )}^{3} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{3}} - \frac {35501 \, {\left (4 \, x + 37\right )}^{2} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{2}} + \frac {164224 \, {\left (4 \, x + 37\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{x + 8} - 253217 \, \log \left (\frac {4 \, x + 37}{x + 8}\right )} \]
integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+3 7)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864 *x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorithm= \
5*(64*(4*x + 37)^3/(x + 8)^3 - 848*(4*x + 37)^2/(x + 8)^2 + 3737*(4*x + 37 )/(x + 8) - 5476)/(2558*(4*x + 37)^3*log((4*x + 37)/(x + 8))/(x + 8)^3 - 3 5501*(4*x + 37)^2*log((4*x + 37)/(x + 8))/(x + 8)^2 + 164224*(4*x + 37)*lo g((4*x + 37)/(x + 8))/(x + 8) - 253217*log((4*x + 37)/(x + 8)))
Time = 14.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {4\,x^3+\frac {4\,x}{5}+8}{5\,x^3+x+10}-\frac {5\,x^2}{\ln \left (\frac {4\,x+37}{x+8}\right )\,\left (5\,x^3+x+10\right )} \]